**Wired/Wireless Photonic Communication Systems Using Optical Heterodyning**

Alejandro García Juárez, Ignacio Enrique Zaldívar Huerta, Antonio Baylón Fuentes, María del Rocío Gómez Colín, Luis Arturo García Delgado, Ana Lilia Leal Cruz and Alicia Vera Marquina

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59081

## **1. Introduction**

[25] Selesnik I. W. Hilbert transform pairs of wavelet bases, IEEE Signal Processing Let‐

[26] Ma Y., Yang Q., Tang Y., Chen S., and Shieh W. 1-Tb/s single-channel coherent opti‐ cal OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth ac‐

[27] Ip, Ezra, Lau, A.P.T., Barros, D.J.F., Kahn, J.M. Coherent detection in optical fiber sys‐

ters, 2001; 8(6) 170-173.

168 Advances in Optical Communication

cess. Optics Express 2009; 17(11) 9421-9427.

tems. Optics Express, January 2008; 16(2) 753-791.

Over the past few years, there has been an increasing effort in researching new design of indoor wireless communications systems, due to connectivity that show in a room or in a building. Currently, several companies of telecommunications use purely omnidirectional antennas in their wireless routers to transmit data to laptops in close vicinity [1]. The properties of microstrip patch antennas and arrays with their planar configuration exhibit an attractive option for indoor communications where the gain is considerably enhanced. On the other hand, the generation of microwave and millimetre-wave (mm-wave) signals by using photonic techniques are being used in radio-over-fiber (RoF) systems, distribution antenna systems, broadband wireless access networks, and radar systems etc. In all these applications the microwave signals are generated at a remote central station and distributed transparently to several simplified antenna stations via optical fiber [1]. The main goal of these systems is to reduce infrastructure cost and to overcome the capacity bottleneck in wireless access networks, allowing, at the same time, flexible merging with conventional optical access networks. Thus, in order to design a reliable RoF-based access network infrastructure, RoF techniques must be capable of generating the microwave signals and allow a reliable microwave signals trans‐ mission over the optical link. For broadband wireless systems and distribution antenna systems operating at microwave and millimeter-wave carriers, several photonic techniques for generating microwave signals have been proposed. Among the most common used techniques are: optical heterodyning [2], optical injection locking [3], optical frequency/phase locked loops (OFLL/OPLL) [4], microwave generation using external modulation [5]. Optical injection locking and optical phase-locked loops (OPLL) are expensive in practice. The use of

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external intensity modulation generates frequency doubling or quadrupling of the driven radiofrequency (RF) sinusoid signal [6]. This method requires an external modulator which increases both loss and cost, and is more susceptible to bias drifting of the modulators, which can affect the output spectrum. The key advantage for generating microwave or millimeterwave signals by optical means is that very high-frequency signals with very low phase noise and high purity can be generated. By using optical heterodyne technique it is very easy to tune frequencies with a spectral linewidth of a few ten MHz and over a wide range by simply tuning the wavelength of the two optical input signals; the obtained frequencies are limited only by the photodetector bandwidth [2]. Besides, the generated signals by using this technique can be generally used as both information carriers, and as a local oscillator for transmitting and receiving both analog and digital information signals by using not only RF schemes but also through an optical fiber. On the other hand, microwave photonics, which brings together radiofrequency engineering and optoelectronics, has attracted great interest in the field of telecommunications since it is an excellent alternative for the transmission of services such as high quality audio and video, e-mail, and Internet among others [7]. Furthermore, there has been an increase effort in researching new microwave photonics techniques for different interesting application that attracts interest in research is the filtering of microwave signals by using photonic techniques. The main feature of a photonic filter is that microwave signals are directly processed in the optical domain exploiting advantages inherent to photonics such as low loss, high bandwidth, immunity to electromagnetic interference, and tunability [8]. On the other hand, network architectures such as FTTx, where x can stand for home (H), building (B), neighborhood (N), or curb (C), are a communication architecture in which the final connection to the subscribers is optical fiber. Another important application of photonic telecommunica‐ tions systems, which is very closely related to the FTTx systems, is the distribution of signals by integrating optical and wireless networks and passive optical networks (PONs). This particular type of scheme is referred as fiber-radio system [9]. Along with wavelength division multiplexing (WDM) technique, it would be more advantageous if RoF is integrated with a conventional PON where a base station (BS) plays the role of an optical network unit (ONU) to support both wired and wireless services. This integrated optical access system is capable not only to reduce the cost of multifunction BSs and the whole system but also meet the demands for bandwidth, mobility and connection options of users [10]. In this sense, the purpose of this chapter is to describe two an alternative optical communications systems. The first proposed system uses a couple microstrip antennas for distributing point to point analog TV with coherent demodulation based on optical heterodyne. In the proposed experimental setup, two optical waves at different wavelengths are mixed and applied to a photodetector. Then a beat signal with a frequency equivalent to the spacing of the two wavelengths is obtained at the output of the photodetector. This signal corresponds to a microwave signal located at 2.8 GHz, which it is used as a microwave carrier in the transmitter and as a local oscillator in the receiver of our optical communication system. The feasibility of this technique is to demonstrate the transmission of a TV signal located at 66-72 MHz. The second system deals with the experimental transmission of analog TV signal in a fiber-radio scheme using a microwave photonic filter (MPF). For that purpose, filtering of a microwave band-pass window located at 2.8 GHz is obtained by the interaction of an externally modulated multi‐ mode laser diode emitting at 1.5 μm associated to the chromatic dispersion parameter of an optical fiber. Transmission of TV signal coded on the microwave band-pass window is achieved over an optical link of 20.70 km. Demodulated signal is transmitted via radiofre‐ quency using printed antennas.

#### **2. Optical heterodyne technique**

The basic principle for generating microwave carriers is based on optical heterodyne techni‐ que, it represents a physical process called optical beating or frequency beating, where two phase-locked optical sources with angular frequencies *ω*1 and *ω*<sup>2</sup> are superimposed and injected into a high frequency photodetector that permits to obtain a photocurrent at a frequency *ω*<sup>2</sup> −*ω*1 . To explain this in more detail, let us consider the relation between the generated electrical output signal and the two superimposed optical input waves from a more physical point of view. For simplicity, we assume that the two optical input waves are linearly polarized monochromatic plane waves in the infrared which propagate in the +z direction. Let

$$\mathbf{E}\_1 = \hat{\mathbf{E}}\_1 \exp\left[i\left(o\_1t - k\_1z + \varphi\_1\right)\right] \mathbf{e}\_{1\prime} \tag{1}$$

and

external intensity modulation generates frequency doubling or quadrupling of the driven radiofrequency (RF) sinusoid signal [6]. This method requires an external modulator which increases both loss and cost, and is more susceptible to bias drifting of the modulators, which can affect the output spectrum. The key advantage for generating microwave or millimeterwave signals by optical means is that very high-frequency signals with very low phase noise and high purity can be generated. By using optical heterodyne technique it is very easy to tune frequencies with a spectral linewidth of a few ten MHz and over a wide range by simply tuning the wavelength of the two optical input signals; the obtained frequencies are limited only by the photodetector bandwidth [2]. Besides, the generated signals by using this technique can be generally used as both information carriers, and as a local oscillator for transmitting and receiving both analog and digital information signals by using not only RF schemes but also through an optical fiber. On the other hand, microwave photonics, which brings together radiofrequency engineering and optoelectronics, has attracted great interest in the field of telecommunications since it is an excellent alternative for the transmission of services such as high quality audio and video, e-mail, and Internet among others [7]. Furthermore, there has been an increase effort in researching new microwave photonics techniques for different interesting application that attracts interest in research is the filtering of microwave signals by using photonic techniques. The main feature of a photonic filter is that microwave signals are directly processed in the optical domain exploiting advantages inherent to photonics such as low loss, high bandwidth, immunity to electromagnetic interference, and tunability [8]. On the other hand, network architectures such as FTTx, where x can stand for home (H), building (B), neighborhood (N), or curb (C), are a communication architecture in which the final connection to the subscribers is optical fiber. Another important application of photonic telecommunica‐ tions systems, which is very closely related to the FTTx systems, is the distribution of signals by integrating optical and wireless networks and passive optical networks (PONs). This particular type of scheme is referred as fiber-radio system [9]. Along with wavelength division multiplexing (WDM) technique, it would be more advantageous if RoF is integrated with a conventional PON where a base station (BS) plays the role of an optical network unit (ONU) to support both wired and wireless services. This integrated optical access system is capable not only to reduce the cost of multifunction BSs and the whole system but also meet the demands for bandwidth, mobility and connection options of users [10]. In this sense, the purpose of this chapter is to describe two an alternative optical communications systems. The first proposed system uses a couple microstrip antennas for distributing point to point analog TV with coherent demodulation based on optical heterodyne. In the proposed experimental setup, two optical waves at different wavelengths are mixed and applied to a photodetector. Then a beat signal with a frequency equivalent to the spacing of the two wavelengths is obtained at the output of the photodetector. This signal corresponds to a microwave signal located at 2.8 GHz, which it is used as a microwave carrier in the transmitter and as a local oscillator in the receiver of our optical communication system. The feasibility of this technique is to demonstrate the transmission of a TV signal located at 66-72 MHz. The second system deals with the experimental transmission of analog TV signal in a fiber-radio scheme using a microwave photonic filter (MPF). For that purpose, filtering of a microwave band-pass window located at 2.8 GHz is obtained by the interaction of an externally modulated multi‐

170 Advances in Optical Communication

$$\mathbf{E}\_2 = \hat{\mathbf{E}}\_2 \exp\left[i\left(a\_2 t - k\_2 z + \varphi\_2\right)\right] \mathbf{e}\_{2'} \tag{2}$$

be the complex electrical field vectors of the two optical waves, with field amplitudes E ^ 1 and E ^ <sup>2</sup> , angular frequencies *ω*1 and *ω*2 and wave numbers *k*1 and *k*1 . The phase of each optical input wave is considered by *φ*1 and *φ*1 and e1 and e2 are the unit vectors determining the orientation of the electrical field vector of the linearly polarized optical input waves. The intensities of the constituent waves are given by the magnitude of their Poynting vectors and are therefore given by [11]

$$I\_1 = \frac{1}{2} \left(\frac{\mathcal{E}\_\mathbf{r} \mathcal{E}\_o}{\mu\_o}\right)^{1/2} \left|E\_1\right|^2. \tag{3}$$

$$I\_2 = \frac{1}{2} \left(\frac{\varepsilon\_\mathrm{r} \varepsilon\_o}{\mu\_o}\right)^{1/2} \left|E\_2\right|^2. \tag{4}$$

If the two incident optical waves are perfect plane waves and have precisely the same polarization ( e1 = e2 ), the resulting electrical field E*o* of the optical interference signal is the sum of the two constituent input fields and hence we can write E*<sup>o</sup>* =E1 + E2 . Taking the squared absolute value of the optical interference signal we obtain

$$\begin{aligned} \left| \boldsymbol{E}\_o \right|^2 = \left| \boldsymbol{E}\_1 + \boldsymbol{E}\_2 \right|^2 = \left| \boldsymbol{E}\_1 \right|^2 + \left| \boldsymbol{E}\_2 \right|^2 + \boldsymbol{E}\_1 \boldsymbol{E}\_2^\* + \boldsymbol{E}\_1^\* \boldsymbol{E}\_2 \\ = \left| \boldsymbol{E}\_1 \right|^2 + \left| \boldsymbol{E}\_2 \right|^2 + 2 \left| \boldsymbol{E}\_1 \right| \left| \boldsymbol{E}\_2 \right| \cos \left( (\phi\_2 - \phi\_1)t - (\phi\_2 - \phi\_1) \right). \end{aligned} \tag{5}$$

From equation (5) and by using equations (3) and (4), it follows that the intensity of the interference signal *Io* is given by [11]

$$I\_o = I\_1 + I\_2 + 2\left(I\_1 I\_2\right)^{1/2} \cos\left( (\phi\_2 - \phi\_1)t - (\phi\_2 - \phi\_1) \right). \tag{6}$$

By launching this optical interference signal into a photodetector, a photocurrent *i* is generated which can be expressed as [11]

$$\dot{q} = \frac{\eta\_o q}{h f\_1} P\_1 + \frac{\eta\_o q}{h f\_2} P\_2 + 2 \frac{\eta\_f q}{h} \left(\frac{P\_1 P\_2}{f\_1 f\_2}\right)^{1/2} \cos\left((\alpha\_2 - \alpha\_1)t - (\varphi\_2 - \varphi\_1)\right),\tag{7}$$

where *q* is the electron charge and *P*1 and *P*2 denote the optical power levels of the two constituent optical input waves. The photodetector's DC and high-frequency quantum efficiencies are represented by *ηo* and *η <sup>f</sup> <sup>c</sup>* . It is of course important to consider that the detector's quantum efficiency is not independent of the frequency. Several intrinsic and extrinsic effects such as transit time limitations or microwave losses will eventually limit the high-frequency performance of the detector and thus the detector's DC responsivity *η<sup>o</sup>* is typically much larger than its high-frequency responsivity *η <sup>f</sup> <sup>c</sup>* . In our case, we can further simplify the photocurrent equation (Eq. (7)) by considering the fact that the two optical input waves are close in frequency ( *f* <sup>1</sup> ≈ *f* 2 ) whereas the difference frequency *f <sup>c</sup>* is by far smaller ( *f <sup>c</sup>* = | *f* <sup>2</sup> − *f* <sup>1</sup> | < < *f* 1, *f* <sup>2</sup> ). If we further assume for simplicity that the power levels of the two optical input waves are equal ( *Popt* ≈*P*<sup>1</sup> ≈*P*2 ), Eq. (7) becomes [11]

$$\dot{\mathbf{u}} = \mathbf{2}s\_o P\_{opt} + \mathbf{2}s\_{f\_c} P\_{opt} \cos \left( 2\pi f\_c t + \Delta \varphi \right). \tag{8}$$

Where *Δφ* =*φ*<sup>2</sup> −*φ*1 . Here *so* = *ηoq hf* and *sf <sup>c</sup>* = *η f c q hf* are the photodetector's DC and high frequency responsivities given in A/W. Eq. (8) is the fundamental equation describing optical hetero‐ dyning in a photodetector. The first term is the DC photocurrent generated by the constituent optical input waves and the second term is the desired high-frequency signal oscillating at the difference frequency *f <sup>c</sup>* (down-converter) or intermediate frequency (IF) [1]. In our case it represents the microwave signal that we will use as both information carriers, and as a local oscillator for transmitting and receiving TV signals in a wireless communication system.

#### **3. Experimental scheme for generating microwave signals**

If the two incident optical waves are perfect plane waves and have precisely the same polarization ( e1 = e2 ), the resulting electrical field E*o* of the optical interference signal is the sum of the two constituent input fields and hence we can write E*<sup>o</sup>* =E1 + E2 . Taking the squared

( )

 jj*t*


(5)

(7)

1 2 12 2 1 2 1 2 cos ( ) ( ) .

=+ + - --

( ) ( ) 1 2

ww

2 cos ( ) ( ) , *<sup>c</sup>*

ww

From equation (5) and by using equations (3) and (4), it follows that the intensity of the

By launching this optical interference signal into a photodetector, a photocurrent *i* is generated

1 2 1 2 1 2 21 21

where *q* is the electron charge and *P*1 and *P*2 denote the optical power levels of the two constituent optical input waves. The photodetector's DC and high-frequency quantum

quantum efficiency is not independent of the frequency. Several intrinsic and extrinsic effects such as transit time limitations or microwave losses will eventually limit the high-frequency performance of the detector and thus the detector's DC responsivity *η<sup>o</sup>* is typically much larger

equation (Eq. (7)) by considering the fact that the two optical input waves are close in frequency ( *f* <sup>1</sup> ≈ *f* 2 ) whereas the difference frequency *f <sup>c</sup>* is by far smaller ( *f <sup>c</sup>* = | *f* <sup>2</sup> − *f* <sup>1</sup> | < < *f* 1, *f* <sup>2</sup> ). If we further assume for simplicity that the power levels of the two optical input waves are equal

p

 j

2 2 cos 2 . ( ) *<sup>c</sup> o opt f opt <sup>c</sup> i sP s P ft* = +

responsivities given in A/W. Eq. (8) is the fundamental equation describing optical hetero‐

= *η f c q*

1 2 12 21 21 2 cos ( ) ( ) . *oI I I II* =++

ww

 jj

( )

 jj

. It is of course important to consider that the detector's

. In our case, we can further simplify the photocurrent

+ D (8)

*hf* are the photodetector's DC and high frequency

absolute value of the optical interference signal we obtain

interference signal *Io* is given by [11]

172 Advances in Optical Communication

which can be expressed as [11]

h

efficiencies are represented by *ηo* and *η <sup>f</sup> <sup>c</sup>*

than its high-frequency responsivity *η <sup>f</sup> <sup>c</sup>*

( *Popt* ≈*P*<sup>1</sup> ≈*P*2 ), Eq. (7) becomes [11]

Where *Δφ* =*φ*<sup>2</sup> −*φ*1 . Here *so* =

2 222 \* \* 1 2 1 2 12 12 2 2

*E E EE*

*<sup>o</sup> E E E E E EE EE*

1 2 1 2

*ηoq*

*hf* and *sf <sup>c</sup>*

*iP P t*

æ ö =++ ç ÷ - - è ø

h

*<sup>f</sup> o o q q <sup>q</sup> P P*

*hf hf h f f*

 h

=+ = + + +

The heterodyne technique for generating microwave signals has been done using the experi‐ mental setup shown in Figure 1. In this experiment, two laser diodes emitting at different wavelengths are used. One of them is a tunable laser (New Focus, model TLB-3902) which can be tuned over the C band with a channel spacing of 25 GHz, and the other one is a fiber coupled distributed feedback (DFB) laser source (Thorlabs, model S3FC1550) with a central wavelength at 1550 nm. For the generation of the microwave signals, the outputs of both lasers are coupled to optical isolators to avoid a feedback into the lasers and consequently instabilities to the system. A pair of polarization controllers is used to minimize the angle between the polariza‐ tion directions of both optical sources. Thus, the polarization of the light issued from each optical source is matched and therefore, there is not degradation of the power levels in the microwave signals generated from the photodetector. The output of each controller is launched to a 3 dB coupler to combine both optical spectrums. After that, an optical output signal is received by a fast photodetector (MITEQ model SCMR-50K6G-10-20-10) with a typical gain of 25 dB, and –3 dB bandwidth of 6 GHz, The resulting photocurrent from the photodetector corresponds to the microwave beat signal which is analyzed with an Electrical Spectrum Analyzer (ESA), (Agilent model E4407B). The other optical output resulting from optical coupler is applied to an Optical Spectrum Analyzer (OSA) (Anritsu model MS9710C), for monitoring the wavelength of the two beams.

**Figure 1.** Experimental setup for generating microwave signals by using optical heterodyne technique.

DFB laser can be used to control not only the output power of the fiber coupled laser diode, but also the precise control of the temperature at which the laser is operating. Both controls can be used to tune the fiber coupled laser diode to an optimum operating point, providing a stable output. In this way, it is possible to observe that the wavelength of the DFB laser is shifting, by varying its temperature with a scale of 1 °C. Consequently, the beat signal frequency is continuously over the band of photodetector. On the other hand, the frequency difference from both lasers can be expressed by [1]

$$
\Delta f = \frac{c}{\lambda\_1} - \frac{c}{\lambda\_2} = \frac{c\left(\lambda\_2 - \lambda\_1\right)}{\lambda\_1 \lambda\_2} \approx \frac{c}{\lambda^2} \left| \Delta \lambda \right| \tag{9}
$$

where *λ*1 and *λ*2 are the wavelengths of the two beams, respectively, and *Δλ* is the difference between the two wavelengths. To obtain a microwave signal, in a first step, the tunable laser is biased and its optical spectrum is displayed on the OSA screen. In a second step, the DFB laser is also biased, fixing an optical power of 2.2 mW and its central wavelength is settled as near as possible to the central wavelength of the tunable laser. As can be seen from Figure 2, the value of Δλ = 0.023739 nm is the wavelength difference between both lasers and it corre‐ sponds to the beat signal frequency of 2.8 GHz.

**Figure 2.** Optical spectrum corresponding to the mixed optical sources. The peaks located at λ1=1550.3197 nm and λ2=1550.3435 nm corresponds to the tunable and DFB lasers, respectively.

A precise control of the difference, between the two central wavelengths and by consequence over the frequency difference, is obtained by tuning the DFB. The wavelength variation of the laser source is obtained by changing the junction temperature between 22.8 °C, 23.2 °C and 23.7 °C corresponding to the frequency range of 0 to 5.0 GHz. Figure 3 illustrates the electrical spectrums of four generated microwave signals by using optical heterodyne. These signals are located at *f* <sup>1</sup> =1.0 , *f* <sup>2</sup> =2.0 , *f* <sup>3</sup> =2.8 and *f* <sup>4</sup> =4.0 GHz respectively. It can be seen, that the microwave signals are in good agreement with theoretical value given by Eq. (9). Therefore, when one laser source is operating at a fixed wavelength and the other is being continuously tuned, the beat frequency will shift correspondingly. In particular, the frequency of the microwave drive signal is set at 2.8 GHz.

**Figure 3.** Spectrum for the microwave signal generated by using optical heterodyne.

#### **4. Modulation and demodulation**

DFB laser can be used to control not only the output power of the fiber coupled laser diode, but also the precise control of the temperature at which the laser is operating. Both controls can be used to tune the fiber coupled laser diode to an optimum operating point, providing a stable output. In this way, it is possible to observe that the wavelength of the DFB laser is shifting, by varying its temperature with a scale of 1 °C. Consequently, the beat signal frequency is continuously over the band of photodetector. On the other hand, the frequency

( ) 2 1

where *λ*1 and *λ*2 are the wavelengths of the two beams, respectively, and *Δλ* is the difference between the two wavelengths. To obtain a microwave signal, in a first step, the tunable laser is biased and its optical spectrum is displayed on the OSA screen. In a second step, the DFB laser is also biased, fixing an optical power of 2.2 mW and its central wavelength is settled as near as possible to the central wavelength of the tunable laser. As can be seen from Figure 2, the value of Δλ = 0.023739 nm is the wavelength difference between both lasers and it corre‐

**Figure 2.** Optical spectrum corresponding to the mixed optical sources. The peaks located at λ1=1550.3197 nm and

A precise control of the difference, between the two central wavelengths and by consequence over the frequency difference, is obtained by tuning the DFB. The wavelength variation of the

, *c c <sup>c</sup> <sup>c</sup> <sup>f</sup>* l l

 ll

1 2 12

l l 2

l

l


difference from both lasers can be expressed by [1]

174 Advances in Optical Communication

sponds to the beat signal frequency of 2.8 GHz.

λ2=1550.3435 nm corresponds to the tunable and DFB lasers, respectively.

Some form of modulation is always needed in an RF system to translate a baseband signal (e.g., audio, video, data) from its original frequency bandwidth to a specified RF frequency spectrum. There are many modulation techniques, for example, amplitude modulation (AM), frequency modulation (FM), amplitude shift keying (ASK), frequency shift keying (FSK), phase shift keying (PSK), biphase shift keying (BPSK), quadriphase shift keying (QPSK), 8-phase shift keying (8-PSK), 16-phase shift keying (16-PSK), minimum shift keying (MSK), and quadrature amplitude modulation (QAM). AM and FM are classified as analog modulation techniques, and the others are digital modulation techniques [12]. In this section we describe the AM modulation and demodulation due to it was used in our proposed wireless communication system.

#### **4.1. Amplitude modulation**

Analog modulation uses the baseband signal (modulating signal) to vary one of three variables: amplitude *Ac* , electrical frequency (*ω*<sup>1</sup> −*ω*2)=*ω<sup>c</sup>* =2*π f <sup>c</sup>* ; or phase (*ϕ*<sup>1</sup> −*ϕ*2) =*Δϕ* . According to Eq. (8), the obtained carrier signal by using optical heterodyne technique can be written by

$$p(t) = A\_c \cos\left((a\_1 - a\_2)t + \phi\_1 - \phi\_2\right) = A\_c \cos\left(2\pi f\_c t + \Delta\phi\right). \tag{10}$$

Where *Ac* =2*sf <sup>c</sup> Popt* . In amplitude modulation, if we assume that *s*(*t*) is the information signal, and considering *Ac* =1 , *Δϕ* =0 , then a modulated signal can be written by

$$\mathbf{g}(t) = \mathbf{s}(t)\cos 2\pi f\_c t. \tag{11}$$

Applying the modulation property of the Fourier transform to Eq. (11), we can find the density spectral of *g*(*t*) is

$$\mathcal{G}(f) = \frac{1}{2}\mathcal{S}(f - f\_c) + \frac{1}{2}\mathcal{S}(f + f\_c). \tag{12}$$

Amplitude modulation therefore translates the frequency spectrum of a signal by ± *f <sup>c</sup>* hertz, but leaves the spectral shape unaltered. This type of amplitude modulation is called sup‐ pressed-carrier because the spectral density of *g*(*t*) has no identifiable carrier in it, although the spectrum is centered at the frequency *f <sup>c</sup>* .

#### **4.2. Amplitude demodulation**

Recovery the signal information *s*(*t*) from the signal *p*(*t*) requires another translation in frequency to shift the spectrum to its original position. This process is called demodulation or detection. Because the modulation property of the Fourier transform proved useful in translating spectra for modulation, we try it again for demodulation. Assuming that *g*(*t*)=*s*(*t*)cos2*π f <sup>c</sup>t* is the transmitted signal, we have

$$
\cos(t)\cos 2\pi f\_c t = s(t)\cos^2 2\pi f\_c t = \frac{1}{2}s(t) + \frac{1}{2}\cos 4\pi f\_c t. \tag{13}
$$

Taking the Fourier transform of both sides of Eq. (13) and using the modulation property, we get

$$\Im\left[\mathcal{G}(t)\cos 2\pi f\_c t\right] = \frac{1}{2}\mathcal{S}(f) + \frac{1}{4}\mathcal{S}(f + 2f\_c) + \frac{1}{4}\mathcal{S}(f - 2f\_c). \tag{14}$$

The mathematical process described in this section can be obtained by convolving the spectrum of the received signal *g*(*t*) with that of cos2*π f <sup>c</sup>t* (i.e., with impulses at ± *f <sup>c</sup>* ). A low-pass filter is required to separate out the double frequency terms from the original spectral components. Obviously we need a filter with a cut frequency *f cut* >2 *f <sup>m</sup>* for proper signal recovery. In this case *f <sup>m</sup>* represents the information frequency.

#### **4.3. Effects in frequency and phase variations**

**4.1. Amplitude modulation**

176 Advances in Optical Communication

Where *Ac* =2*sf <sup>c</sup>*

spectral of *g*(*t*) is

get

the spectrum is centered at the frequency *f <sup>c</sup>* .

*g*(*t*)=*s*(*t*)cos2*π f <sup>c</sup>t* is the transmitted signal, we have

p

**4.2. Amplitude demodulation**

Analog modulation uses the baseband signal (modulating signal) to vary one of three variables: amplitude *Ac* , electrical frequency (*ω*<sup>1</sup> −*ω*2)=*ω<sup>c</sup>* =2*π f <sup>c</sup>* ; or phase (*ϕ*<sup>1</sup> −*ϕ*2) =*Δϕ* . According to Eq. (8), the obtained carrier signal by using optical heterodyne technique can be written by

> p

1 1 ( ) ( ) ( ). 2 2 *Gf Sf f Sf f c c* = -+ + (12)

 p= = + (13)

ë û (14)

*Popt* . In amplitude modulation, if we assume that *s*(*t*) is the information signal,

 f*t A ft* + D (10)

(11)

( ) ( ) 1 2 12 ( ) cos ( ) cos 2 . *<sup>c</sup> c c pt A*= - +- =

( ) ( )cos 2 . *<sup>c</sup> gt st ft* =

p

Applying the modulation property of the Fourier transform to Eq. (11), we can find the density

Amplitude modulation therefore translates the frequency spectrum of a signal by ± *f <sup>c</sup>* hertz, but leaves the spectral shape unaltered. This type of amplitude modulation is called sup‐ pressed-carrier because the spectral density of *g*(*t*) has no identifiable carrier in it, although

Recovery the signal information *s*(*t*) from the signal *p*(*t*) requires another translation in frequency to shift the spectrum to its original position. This process is called demodulation or detection. Because the modulation property of the Fourier transform proved useful in translating spectra for modulation, we try it again for demodulation. Assuming that

<sup>2</sup> 1 1 ( )cos 2 ( )cos 2 ( ) cos 4 . 2 2 *cc c gt ft st ft st ft*

11 1 ( )cos 2 ( ) ( 2 ) ( 2 ). 24 4 *<sup>c</sup> c c* Á = + ++ - é ù *gt ft S f S f f S f f*

Taking the Fourier transform of both sides of Eq. (13) and using the modulation property, we

pp

 ff

ww

and considering *Ac* =1 , *Δϕ* =0 , then a modulated signal can be written by

When the local oscillator at the receiver, has a small frequency error *Δf* and a phase error *Δθ* , then this signal can be written as

$$p\_L(t) = \cos\left[2\pi\left(f\_c + \Delta f\right)t + \Delta\theta\right].\tag{15}$$

Assuming again that *g*(*t*)=*s*(*t*)cos2*π f <sup>c</sup>t* is the transmitted signal; then we have that at the receiver, the recovered signal can be written by

$$\begin{aligned} \mathrm{s}(t)\cos\left[2\pi(f\_c+\Delta f)t+\Delta\theta\right] &= \mathrm{s}(t)\cos(2\pi f\_c t)\cos\left[2\pi(f\_c+\Delta f)t+\Delta\theta\right] \\\\ &= \mathrm{s}(t)\left(\frac{\cos(2\pi\Delta f t+\Delta\theta)}{2}+\frac{\cos\left[2\pi(2f\_c+\Delta f)t+\Delta\theta\right]}{2}\right). \end{aligned} \tag{16}$$

The second term on the right hand side of Eq. (16) is centered at ±2 *f <sup>c</sup>* + *Δf* and can be filtered out by using a low pass filter. The output of this filter *sF* (*t*) will then be given by the remaining term in Eq. (16).

$$s\_F(t) = \left[\frac{s(t)}{2}(\cos 2\pi(\Delta f)t \cos(\Delta \theta) - \sin 2\pi(\Delta f)t \text{sen}(\Delta \theta))\right].\tag{17}$$

As can been from equation (17), the output signal is not *s*(*t*) <sup>2</sup> , unless both *Δf* and *Δθ* are zero. The effects of both frequency errors and random phase errors render this demodulation of the signal unsatisfactory. It is necessary, therefore, to have synchronization in both frequency and phase between the transmitter and the receiver when amplitude modulation is used. The synchronization of the carrier signals presents no major problem when the transmitter and the receiver are in close proximity. Recovering the original signal *s*(*t*) from the modulated signal *g*(*t*) using a synchronized oscillator is called coherent demodulation. In our case we take advantage of proposed optical heterodyne technique permits to obtain microwave carrier and local oscillator simultaneously in the transmitter and receiver respectively.

#### **5. Design of a patch antenna at 2.8 GHz**

The microstrip patch antenna is a popular printed resonant antenna for narrow-band micro‐ wave wireless links that require semihemispherical coverage. Due to its planar configuration and ease of integration with microstrip technology, the microstrip patch antenna has been studied heavily and is often used as an element for an array. Common microstrip antenna shapes are square, rectangular, circular, ring, equilateral triangular, and elliptical, but any continuous shape is possible [13]. Furthermore, a patch antenna is an excellent device due to its small size, low cost, and good performance [14-16]. In this chapter, a rectangular printed patch antenna is proposed. Simulation results have been obtained by using Advanced Design System (ADS) that is a computer-aided-engineering software tool. The radiating structure consists of a patch and a microstrip inset-feed line, allowing that the characteristic impedance (Zo) to be improved. Figure 4 shows the geometry and configuration of the top layer. The proposed antenna in this work was designed to operate in the band S of telecommunications (2.8 GHz). FR4 is used as a dielectric substrate exhibiting a thickness *h* =1.524*mm* , and relative dielectric constant *ε<sup>r</sup>* =4.2 . In a first step, the width (W) of the patch is computed by using [17]:

$$\mathcal{W} = \frac{c\_o}{2f\_c} \sqrt{\frac{2}{\varepsilon\_r + 1}},\tag{18}$$

where *co* is the light velocity in the free space, and *f <sup>o</sup>* is the operation frequency. Next, the value of the effective dielectric constant *εeff* is evaluated considering *W* / *h* >1 .

$$
\varepsilon\_{eff} = \frac{\varepsilon\_r + 1}{2} + \frac{\varepsilon\_r - 1}{2} \left( 1 + 12 \frac{h}{W} \right)^{-1/2} \tag{19}
$$

**Figure 4.** Layout of the patch antenna.

Border effects [17] must to be considered in the design of the antenna. For this reason, *ΔL* from Figure 4 can be evaluated as:

$$\frac{\Delta L}{h} = 0.412 \frac{\left(\varepsilon\_{\epsilon\%} + 0.3\right) \left(\frac{W}{h} + 0.264\right)}{\left(\varepsilon\_{\epsilon\% - 0.258}\right) \left(\frac{W}{h} + 0.8\right)}\tag{20}$$

This allows that the length ( *L* ) of the patch to be evaluated as:

**5. Design of a patch antenna at 2.8 GHz**

178 Advances in Optical Communication

The microstrip patch antenna is a popular printed resonant antenna for narrow-band micro‐ wave wireless links that require semihemispherical coverage. Due to its planar configuration and ease of integration with microstrip technology, the microstrip patch antenna has been studied heavily and is often used as an element for an array. Common microstrip antenna shapes are square, rectangular, circular, ring, equilateral triangular, and elliptical, but any continuous shape is possible [13]. Furthermore, a patch antenna is an excellent device due to its small size, low cost, and good performance [14-16]. In this chapter, a rectangular printed patch antenna is proposed. Simulation results have been obtained by using Advanced Design System (ADS) that is a computer-aided-engineering software tool. The radiating structure consists of a patch and a microstrip inset-feed line, allowing that the characteristic impedance (Zo) to be improved. Figure 4 shows the geometry and configuration of the top layer. The proposed antenna in this work was designed to operate in the band S of telecommunications (2.8 GHz). FR4 is used as a dielectric substrate exhibiting a thickness *h* =1.524*mm* , and relative dielectric constant *ε<sup>r</sup>* =4.2 . In a first step, the width (W) of the patch is computed by using [17]:

> <sup>2</sup> , 2 1 *o c r*

where *co* is the light velocity in the free space, and *f <sup>o</sup>* is the operation frequency. Next, the value

1 2 1 1 1 12


*h W*

è ø

<sup>=</sup> <sup>+</sup> (18)

(19)

*c*

*f* e

*W*

of the effective dielectric constant *εeff* is evaluated considering *W* / *h* >1 .

*eff*

e

**Figure 4.** Layout of the patch antenna.

e

2 2 *r r*

 e

$$L = \frac{\mathcal{L}\_o}{2f\_o\sqrt{\varepsilon\_r}} - 2\Delta L \tag{21}$$

Considering the values previously obtained, the effective dimensions ( *L eff* and *Weff* ) can be calculated, respectively as:

$$L\_{\rm eff} = L + 2\Delta L \tag{22}$$

$$\mathcal{W}\_{eff} = \mathcal{W} + \frac{t}{\pi} \left( 1 + \ln \left( \frac{2h}{t} \right) \right) \tag{23}$$

From Eq. (23), *t* is the conductor thickness and *W* / *h* >1 / 2*π* must to be considered. The ground plane dimensions are computed as:

$$\begin{aligned} \mathbf{L}\_1 &= \mathbf{\tilde{e}}h + \mathbf{L}\_{eff} \\ \mathbf{\mathcal{W}}\_1 &= \mathbf{\tilde{e}}h + \mathbf{\mathcal{W}}\_{eff} \end{aligned} \tag{24}$$

The best dimensions which assure a good matching between the impedances ( *Rin* =*Zo* =50*Ω* ) of the antenna and generator can be calculated by the use of LineCalc tool from ADS and by the next expression:

$$R\_{in}\left(y = y\_o\right) = \frac{1}{2\left(G\_1 \pm G\_{12}\right)} \cos^2\left(\frac{\pi}{L}y\_o\right) \tag{25}$$

where *G*1 and *G*<sup>12</sup> are the conductance values obtained by the cavity method. Finally, Table 1 shows a summary of the dimensions for the patch and the ground plane.


**Table 1.** Dimensions of the fabricated antenna.

Figure 5(a) shows a picture of the fabricated patch antenna where a SubMiniature version A (SMA) connector is added. Figure 5(b) illustrates simulation and experimental results corre‐ sponding to the *S*11 parameter. Electrical measurements are obtained by using a Vector Network Analyzer (VNA) (Agilent Technologies model: E8361A). It is clearly observable that experimental result is in good agreement with the simulation.

**Figure 5.** Fabricated antenna (a), Experimental and simulation return loss curve for the antenna (b).

#### **6. Transmission of TV signals by using heterodyne technique**

In order to show a potential application of optical heterodyne technique in the field of the wireless communications, we have proposed a coherent wired/wireless photonic communi‐ cation system as shown in Figure 6. This system is not a truly wireless communication system, since an optical fiber is required to deliver both microwave carrier and local oscillator for transmitting and receiving information of TV signals as an approximation to point to point indoor wireless communications systems. From the photodetector 1 in the transmitter, a microwave signal located at 2.8 GHz is obtained and mixed with an analog TV signal located at 62.25 MHz. Then the resulting signal is amplified before being applied to our fabricated microstrip antenna. After that, the obtained modulated signal as shown in Figure 7, is transmitted through a point to point wireless link by using the microstrip antenna. Finally in the receiver, another microstrip antenna is used to receive the transmitted information, which it is processed using optical heterodyne technique again to recover in this case the TV signal (66-72 MHz). From the photodetector 2 in the receiver, a local oscillator that is synchronized, in frequency as well as in phase with to that obtained from the photodetector 1, is mixed with the received signal. Then the resulting signal is filtered and the power spectral density obtained is displayed on an electrical spectrum analyzer, where it is analyzed to measure the power level of recovered information.

**Operation Frequency**

180 Advances in Optical Communication

**Table 1.** Dimensions of the fabricated antenna.

experimental result is in good agreement with the simulation.

**Figure 5.** Fabricated antenna (a), Experimental and simulation return loss curve for the antenna (b).

**6. Transmission of TV signals by using heterodyne technique**

In order to show a potential application of optical heterodyne technique in the field of the wireless communications, we have proposed a coherent wired/wireless photonic communi‐ cation system as shown in Figure 6. This system is not a truly wireless communication system, since an optical fiber is required to deliver both microwave carrier and local oscillator for transmitting and receiving information of TV signals as an approximation to point to point indoor wireless communications systems. From the photodetector 1 in the transmitter, a microwave signal located at 2.8 GHz is obtained and mixed with an analog TV signal located at 62.25 MHz. Then the resulting signal is amplified before being applied to our fabricated microstrip antenna. After that, the obtained modulated signal as shown in Figure 7, is transmitted through a point to point wireless link by using the microstrip antenna. Finally in the receiver, another microstrip antenna is used to receive the transmitted information, which

**Dimensions (cm)** *Wo Lo W L W1 L1 yo*

**2.8 GHz** 0.13 3.08 3.32 2.56 10 10 0.93

Figure 5(a) shows a picture of the fabricated patch antenna where a SubMiniature version A (SMA) connector is added. Figure 5(b) illustrates simulation and experimental results corre‐ sponding to the *S*11 parameter. Electrical measurements are obtained by using a Vector Network Analyzer (VNA) (Agilent Technologies model: E8361A). It is clearly observable that

**(Antenna)**

**Figure 6.** Wired/wireless photonic communication system for transmitting and receiving TV signals.

Figure 8 shows the frequency spectrum of an analog National Television System Committee (NTSC) TV signal at the input of the transmitter located at 67.25 MHz (before being applied to frequency mixer). In the same figure we can see the obtained analog NTSC TV signal at the output of the receiver. In order to measure the quality of the received signal, it is necessary to quantify the parameter of signal-to-noise ratio (SNR), in this case it is approximately 45 dB. The analog information is successfully transmitted from the transmitter to the receiver, and the received signal is satisfactorily reproduced on TV monitor. The differential gain and differential phase were not measured Nevertheless we demonstrated that the generated microwave signal by using optical heterodyning can be used as carrier information in a traditional communication system and we have used a TV signal of test to verify it.

**Figure 7.** Electrical spectrum of the modulated signal.

**Figure 8.** TV signals at 67.25 MHz, transmitted and recovered.

#### **7. Analytical model of the microwave photonic filter**

The scheme of the MPF is illustrated in Figure 9. Consider that the optical signal of a poly‐ chromatic source with spectrum *P*(*ω*) , centered at an optical frequency *ω<sup>n</sup>* , is launched into the input of the Mach-Zehnder intensity modulator (MZ-IM). A single spectral component of such an optical signal can be modeled by a stochastic process *e*(*t*)= *Eo* (*t*)exp( *jωnt*) , where *Eo* (*t*) is the complex amplitude and *ωn* is the optical angular frequency. If the intensity of such optical signal is externally modulated by an electrical signal *Vm* =1 + 2*m*cos(*ωmt*) , where *m* is the modulation index and *ωm* is the angular frequency of external modulation, then the optical field at the input of the optical fiber can be expressed by Eq. (26). The modulation index *m* is related to the electrical input signal amplitude, *Vm* , as: 2*m*=*π*(*Vm* / *Vπ*) , where *Vπ* is the half wave voltage of the MZ-IM [18].

$$e\_i(t) = e(t)s(t)\tag{26}$$

The optical fiber can be considered as a linear time invariant (LTI) system. If, for simplicity, the attenuation is ignored, then the transfer function of the optical link, for a given length *L* , is *H* ( *jω*)=exp(− *jβL* ) , where *β* is the propagation constant. Thus, the optical field at the end of the link is given by

$$e\_L(t) = e\_i(t) \exp\left(-j\beta L\right) \tag{27}$$

Substituting *e*(*t*) and *s*(*t*) in Eq. (26), and then replacing this in Eq. (27), it becomes:

$$\begin{split} e\_L(t) &= E\_o\left(t\right) \exp\left(j\left(\alpha\_m t - \beta L\right)\right) + E\_o\left(t\right) m \exp\left(j\left[\left(\alpha\_m - \alpha\_n\right)t - \beta L\right]\right) \\ &+ E\_o\left(t\right) m \exp\left(j\left[\left(\alpha\_m + \alpha\_n\right)t - \beta L\right]\right) \end{split} \tag{28}$$

In the frequency domain Eq. (28) can be expressed as:

**Figure 7.** Electrical spectrum of the modulated signal.

182 Advances in Optical Communication

**Figure 8.** TV signals at 67.25 MHz, transmitted and recovered.

$$\begin{split} \mathbb{E}\_{L}\left(\boldsymbol{\alpha}\right) &= \mathbb{E}\_{o}\left(\boldsymbol{\alpha} - \boldsymbol{\alpha}\_{n}\right) \exp\left(-j\beta L\right) + \mathbb{E}\_{o}\left(\boldsymbol{\alpha} - \left(\boldsymbol{\alpha}\_{n} - \boldsymbol{\alpha}\_{m}\right)\right) \exp\left(-j\beta L\right) \\ &+ \mathbb{E}\_{o}\left(\boldsymbol{\alpha} - \left(\boldsymbol{\alpha}\_{n} + \boldsymbol{\alpha}\_{m}\right)\right) \exp\left(-j\beta L\right) \end{split} \tag{29}$$

**Figure 9.** Scheme of the microwave photonic filter.

There are three spectral components. In the presence of chromatic dispersion, there is a propagation constant associated to each one of them, i.e. *β*(*ω* −*ωn*) , *β*(*ω* −(*ω<sup>n</sup>* + *ωm*)) and *β*(*ω* −(*ω<sup>n</sup>* −*ωm*)) . By denoting *W* =*ω* −*ωn* , Eq. (29) then becomes:

$$\begin{split} E\_L(o\rho) &= E\_o\left(\mathcal{W}\right) \exp\left(-j\mathcal{J}\left(\mathcal{W}\right)L\right) + E\_o\left(\mathcal{W} + o\nu\_m\right) \exp\left(-j\mathcal{J}\left(\mathcal{W} + o\nu\_m\right)L\right) \\ &+ E\_o\left(\mathcal{W} - o\nu\_m\right) \exp\left(-j\mathcal{J}\left(\mathcal{W} - o\nu\_m\right)L\right) \end{split} \tag{30}$$

Assuming that within the frequency range *ω<sup>n</sup>* −*ωm* to *ω<sup>n</sup>* + *ω<sup>m</sup>* , centered at *ωn* the propagation constant varies only slightly and gradually with *ω* , it can be approximated by the first three terms of a Taylor series expansion, and it can be shown that

$$\beta\left(\mathcal{W}\pm o\_m\right) = \beta\left(\mathcal{W}\right)\pm\beta\_1 o\_m + \beta\_2\left[\frac{1}{2}o\_m^2 \pm o\_m\left(o - o\_n\right)\right] \tag{31}$$

where *β<sup>i</sup>* <sup>=</sup> *<sup>d</sup> <sup>i</sup> <sup>β</sup>*(*ω*)/ *<sup>d</sup><sup>ω</sup> <sup>i</sup>* (*ω*=*ωn*) .

The optical intensity, *I* , is obtained by integrating the power spectral density over all the frequency range, i.e.

$$I = \int\_{-\infty}^{\infty} \left| E\_L \left( oo \right) \right|^2 d o \tag{32}$$

Considering that the MZ-IM is operating on its linear region, it is valid to note that *m*<sup>2</sup> ≈0 . On the other hand, if *ω<sup>n</sup>* > >*ωm* then *Eo* (*W* )≈*Eo*(*W* + *ωm*) ≈*Eo*(*W* −*ωm*) . Furthermore, in the fre‐ quency domain, the spectrum of the source is defined as *P*(*ω*)=*Eo* (*ω*)*Eo* \* (*ω*) . Thus, developing the product | *EL* (*ω*)| 2 in Eq. (32) and replacing Eq. (31), it is possible to demonstrate that the intensity at the end of the optical fiber is given by:

$$I = \int\_{-\alpha}^{\alpha} P(W)d\mathcal{W} + 4m\cos\left(\beta\_2 \frac{\alpha\_m^2}{2} L\right) \cos\left(\beta\_1 \alpha\_m L\right) \Re\left\{\int\_{-\alpha}^{\alpha} P(W) \exp\left(-j2\pi ZW\right)dW\right\} \tag{33}$$

where *Z* =*β*2*ωmL* / 2*π* , *W* =*ω* −*ωn* and its derivative, *dW* =*dω* . The total average intensity is *Io* =*∫* −*∞ ∞ P*(*W* )*dW* , and the integral ℜ{ *∫* −*∞ ∞ P*(*W* )exp(− *j*2*πZW* )*dW* } corresponds to the real part of

the Fourier transform of the spectrum of the optical source. This means that the optical intensity which reaches the surface of the photodetector is proportional to:

Wired/Wireless Photonic Communication Systems Using Optical Heterodyning http://dx.doi.org/10.5772/59081 185

$$F(\mathcal{W}) = \Re\left\{FT\left\{P(\mathcal{W})\right\}\right\} \tag{34}$$

A spectrum with Gaussian shape can be modeled by an analytical expression as:

There are three spectral components. In the presence of chromatic dispersion, there is a propagation constant associated to each one of them, i.e. *β*(*ω* −*ωn*) , *β*(*ω* −(*ω<sup>n</sup>* + *ωm*)) and

( ) ( ) ( ( ) ) ( ) ( ( ) )

 bw

( ) ( ) ( ) <sup>2</sup> 1 2

 b w w w w

The optical intensity, *I* , is obtained by integrating the power spectral density over all the

( ) 2

Considering that the MZ-IM is operating on its linear region, it is valid to note that *m*<sup>2</sup> ≈0 . On

the product | *EL* (*ω*)| 2 in Eq. (32) and replacing Eq. (31), it is possible to demonstrate that the

( ) ( ) ( ) ( )

æ ö ì ü ï ï <sup>=</sup> <sup>+</sup> ç ÷ - í ý ç ÷

*<sup>m</sup> I P W dW m L L P W j ZW dW*

 bw

2 1 4 cos cos exp 2 <sup>2</sup>

where *Z* =*β*2*ωmL* / 2*π* , *W* =*ω* −*ωn* and its derivative, *dW* =*dω* . The total average intensity is

the Fourier transform of the spectrum of the optical source. This means that the optical intensity

è ø ï ï î þ ò ò (33)

¥

*<sup>L</sup> IE d* w w

é ù ±= ± + ± - ê ú

*W W <sup>m</sup> m mm n*

 bw

Assuming that within the frequency range *ω<sup>n</sup>* −*ωm* to *ω<sup>n</sup>* + *ω<sup>m</sup>* , centered at *ωn* the propagation constant varies only slightly and gradually with *ω* , it can be approximated by the first three

 w

1 2  bw

ë û (31)


(*W* )≈*Eo*(*W* + *ωm*) ≈*Eo*(*W* −*ωm*) . Furthermore, in the fre‐

*P*(*W* )exp(− *j*2*πZW* )*dW* } corresponds to the real part of

(*ω*)*Eo* \*

p

(*ω*) . Thus, developing

(30)

( ) ( ( ) ) exp exp

*E E W j WL E W j W L EW j W L*

= - ++ -+

exp *L o o m m*

+- --

*o m m*

*β*(*ω* −(*ω<sup>n</sup>* −*ωm*)) . By denoting *W* =*ω* −*ωn* , Eq. (29) then becomes:

 b

terms of a Taylor series expansion, and it can be shown that

 b

quency domain, the spectrum of the source is defined as *P*(*ω*)=*Eo*

2

w b

¥ ¥


*m*

−*∞*

which reaches the surface of the photodetector is proportional to:

*∞*

intensity at the end of the optical fiber is given by:

*P*(*W* )*dW* , and the integral ℜ{ *∫*

w

w

184 Advances in Optical Communication

b

the other hand, if *ω<sup>n</sup>* > >*ωm* then *Eo*

where *β<sup>i</sup>* <sup>=</sup> *<sup>d</sup> <sup>i</sup>*

*Io* =*∫* −*∞ ∞*

frequency range, i.e.

 w

*<sup>β</sup>*(*ω*)/ *<sup>d</sup><sup>ω</sup> <sup>i</sup>* (*ω*=*ωn*) .

$$P\left(o\right) = \frac{2P\_o}{\Delta o \sqrt{\pi}} \exp\left(-\frac{4\left(o - o\_m\right)^2}{\Delta o^2}\right) \tag{35}$$

where *ω* is the angular frequency, *ω* is the central angular frequency, *ω* is the maximum power emission and *Δω* is the full width at half maximum (FWHM) of the optical source. If the emission spectrum of the optical source has a Gaussian shape, as defined in Eq. (35), then the Eq. (34) becomes:

$$F(\alpha) = \exp\left(-\left(\frac{\beta\_2 \alpha\_m L \Delta \alpha}{4}\right)^2\right) \tag{36}$$

In such case the FWHM of the frequency response can be determined equating *F* (*ω*)=0.5 , which implies:

$$
\left(\frac{\beta\_2 \alpha\_m \, \mathrm{L} \Delta \alpha}{4}\right)^2 = \ln\left(2\right) \tag{37}
$$

For finding the value of the frequency *f <sup>m</sup>* that yields that condition, it is necessary to express *ω<sup>m</sup>* in terms of *f <sup>m</sup>* , i.e. *ω<sup>m</sup>* =2*π f <sup>m</sup>* . But this, in turn, yields an expression that can be reduced by expressing *Δω* in terms of *Δλ* and *β*2 in terms of dispersion *D* . For *Δω* this is done as follows: given *dω* / *dλ* = −(2*πc* / *λ* 2) , where *c* is the speed of light in the free space and *λ* is the wave‐ length of the optical signal, it is possible to establish the following correspondence:

$$d\alpha = -\frac{2\pi c}{\lambda^2} d\lambda \iff \Lambda \alpha = -\frac{2\pi c}{\lambda^2} \Delta \lambda \tag{38}$$

Now, for the factor *β*<sup>2</sup> , given that the group velocity, *vg* = *L* / *τg* where *τ<sup>g</sup>* is the group delay, is related to *β*(*ωn*) as *τ<sup>g</sup>* / *L* =*dβ*(*ωn*) / *dω* , and its derivative is (*dτ<sup>g</sup>* / *<sup>d</sup>ω*) / *<sup>L</sup>* <sup>=</sup>*<sup>d</sup>* <sup>2</sup> *<sup>β</sup>*(*ωn*) / *<sup>d</sup><sup>ω</sup>* <sup>2</sup> <sup>=</sup>*β*2 , then (1 / *<sup>L</sup>* )(*dτg*) <sup>=</sup>*dωβ*2 . Thus, the derivative of this expres‐ sion by *dλ* is (1 / *L* )(*dτ<sup>g</sup>* / *dλ*) =(*dω* / *dλ*)*β*2 . Furthermore, the dispersion, as a function of the wavelength is defined as *<sup>D</sup>* =(1 / *<sup>L</sup>* )(*dτ<sup>g</sup>* / *<sup>d</sup>λ*) . This means that *β*<sup>2</sup> <sup>=</sup> <sup>−</sup>*D*(*<sup>λ</sup>* <sup>2</sup> / <sup>2</sup>*πc*) . Finally, by substituting *ω<sup>m</sup>* =2*π f <sup>m</sup>* , *Δω* , in Eq. (38), and the expression for *β*2 in Eq. (37), the frequency *f <sup>m</sup>* , which corresponds to the low-pass bandwidth *Δ f lp* , can be expressed as:

$$
\Delta f\_{lp} = \frac{2\sqrt{\ln\left(2\right)}}{\pi DL\Delta\lambda} \tag{39}
$$

where the dispersion *D* has units of ps nm-1 km-1, length *L* is given in km, and the FWHM of the optical source, *Δλ* , in nm. This means that in the presence of an optical source, like a super luminescent light-emitting diode (LED), the frequency response of the system is low-pass, and its bandwidth is given by Eq. (39). In the context of this chapter, the optical source is an multimode laser diode (MLD). The emission spectrum of this type of optical sources can be modeled by means of an analytical expression as expressed in Eq. (40):

$$P\left(o\right) = \frac{2P\_o}{\Delta o \sqrt{\pi}} \exp\left[-\frac{4\left(o - o\_n\right)^2}{\Delta o^2}\right] \left[\frac{2P\_o}{\sigma o \sqrt{\pi}} \exp\left(-\frac{4\left(o - o\_n\right)^2}{\sigma o^2}\right) \* \sum\_{n=-o}^{o} \mathcal{S}\left(o - n\delta o\right)\right] \tag{40}$$

where *ω* is the angular frequency, *ωn* is the central angular frequency, *Po* is the maximum power emission, *Δω* is the FWHM of the optical source, *σω* is the FWHM of each emission mode and *δω* is the free spectral range (FSR) between the emission modes. By using variables *Z* and *W* , as defined earlier, and substituting Eq. (40) in Eq. (34), it can be expressed as:

$$F(\rho) = \exp\left(-\left(\frac{\beta\_2 \alpha\_m L \Delta \rho}{4}\right)^2\right) \* \left[\exp\left(-\left(\frac{\beta\_2 \alpha\_m L \sigma \rho}{4}\right)^2\right) \frac{1}{\delta \alpha} \sum\_{n=-\alpha}^{\alpha} \delta\left(\frac{\beta\_2 \alpha\_m L}{2\pi} - \frac{n}{\delta \alpha}\right)\right] \tag{41}$$

The term between crochets indicates the presence of a periodic pattern. The frequency of the first maximum can be determined by equating:

$$\frac{\beta\_2 \alpha\_1 L}{2\pi} = \frac{1}{\delta \nu} \tag{42}$$

For finding the value of the frequency *f* <sup>1</sup> that yields that condition, it is necessary to express *δω* in terms of *f* 1 . In a similar way as in Eq. (38), it is possible to establish the following correspondence:

$$
\delta d\phi = -\frac{2\pi c}{\lambda^2} d\lambda \iff \delta \phi = -\frac{2\pi c}{\lambda^2} \delta \lambda \tag{43}
$$

thus, substituting *δω* in Eq. (42), expressing *ω*1 in terms of *<sup>f</sup>* 1 and using *β*<sup>2</sup> <sup>=</sup> <sup>−</sup>*D*(*<sup>λ</sup>* <sup>2</sup> / <sup>2</sup>*πc*) then the frequency *f* <sup>1</sup> can be expressed as:

$$f\_1 = \frac{1}{\text{DL\'\'\'\'\'}}\tag{44}$$

and, in general, the central frequency of the *n* -th band-pass lobe is given by

2 ln 2( )

 l*DL*

(39)

D

where the dispersion *D* has units of ps nm-1 km-1, length *L* is given in km, and the FWHM of the optical source, *Δλ* , in nm. This means that in the presence of an optical source, like a super luminescent light-emitting diode (LED), the frequency response of the system is low-pass, and its bandwidth is given by Eq. (39). In the context of this chapter, the optical source is an multimode laser diode (MLD). The emission spectrum of this type of optical sources can be

p

( ) ( ) ( ) ( )

2 2 2 22 1

2 1 1

 dw

For finding the value of the frequency *f* <sup>1</sup> that yields that condition, it is necessary to express *δω* in terms of *f* 1 . In a similar way as in Eq. (38), it is possible to establish the following

> 2 2 2 2 *c c d d*

thus, substituting *δω* in Eq. (42), expressing *ω*1 in terms of *<sup>f</sup>* 1 and using *β*<sup>2</sup> <sup>=</sup> <sup>−</sup>*D*(*<sup>λ</sup>* <sup>2</sup> / <sup>2</sup>*πc*) then

dl

 p

 l  dl

=- Û =- (43)

<sup>=</sup> (44)

2 b w *L* p

p

l

 l dw

1 <sup>1</sup> *<sup>f</sup> DL*

w

b w sw

æ öæ ö é ù æ ö æö æ ö <sup>D</sup> <sup>=</sup> ç ÷ç ÷ -- - ê ú ç ÷ ç÷ ç ÷

The term between crochets indicates the presence of a periodic pattern. The frequency of the

exp \* exp 4 42

*LLL <sup>n</sup> <sup>F</sup>*

exp exp \* *o o n n*

sw p

*P n*

2 2

where *ω* is the angular frequency, *ωn* is the central angular frequency, *Po* is the maximum power emission, *Δω* is the FWHM of the optical source, *σω* is the FWHM of each emission mode and *δω* is the free spectral range (FSR) between the emission modes. By using variables *Z* and *W* , as defined earlier, and substituting Eq. (40) in Eq. (34), it can be expressed as:

*m mm*

è ø èø è ø è øè ø ë û

æ öæ ö é ù - - ç ÷ç ÷ ê ú =- - - <sup>D</sup> <sup>D</sup> è øè ø ë û

2 2

w w

sw

*n*

¥

=-¥

dw

 d

*n*

¥

=-¥

b w

<sup>=</sup> (42)

 p  dw

å (41)

d w dw

å (40)

*lp f*

modeled by means of an analytical expression as expressed in Eq. (40):

2 2 4 4

*P P*

w

w w

bw

first maximum can be determined by equating:

the frequency *f* <sup>1</sup> can be expressed as:

 w

w

( )

correspondence:

w

w p

186 Advances in Optical Communication

D =

$$f\_n = \frac{n}{\text{DL\'\'\'\'}}\tag{45}$$

where *n* is a positive integer, dispersion *D* is given in ps nm-1 km-1, length *L* in km, and the FSR *δλ* in nm. The bandwidth of each of these band-pass lobes is equal to:

$$
\Delta f\_{bp} = \frac{4\sqrt{\ln\left(2\right)}}{\pi D L \Delta \lambda} \tag{46}
$$

which is twice Eq. (39). The periodic pattern in the frequency response of the system will appear only when an MLD is used in the system. This behavior will allow that microwave signals to be filtered and transmitted over a wide range of frequencies.

#### **8. Experimental setup of optical and wireless transmission**

In a first step, the MLD used in this experiment (OKI OL5200N-5) is optically characterized by means of an optical spectrum analyzer (Agilent, model 86143B). Figure 10 corresponds to the measured optical spectrum obtaining *λ<sup>o</sup>* =1553.53 nm , *Δλ* =5.65 nm , and *δλ* =1.00 nm for a driver current of 25 mA. The use of a laser diode temperature-controller (Thorlabs, model LTC100-C) allows us to guarantee the stability of the optical parameters to thermal fluctuations.

In a second step, considering a length L=20.70 km of single-mode-standard-fiber (SM-SF) exhibiting a chromatic fiber-dispersion parameter of D=16.67 ps/nm km. Eq. (45) allows us to determine the value of the central frequency corresponding to the first filtered microwave or first band-pass as

$$f\_1 = \frac{1}{DL\delta\lambda} = \frac{1}{\left(16.67 \times 10^{-12} \text{seg/nm} \cdot \text{km}\right) \cdot \left(20.70 \text{ km}\right) \cdot \left(1.0 \text{ nm}\right)} = 2.8 \text{ GHz}$$

Eq. (39) permits us to determine the value of the low-pass band as

$$\Delta f\_{lp} = \frac{2\sqrt{\ln(2)}}{\pi D L \Delta \lambda} = \frac{2\sqrt{\ln 2}}{(\pi) \cdot (16.67 \times 10^{-12} \text{seg/nm} \cdot \text{km}) \cdot (20.70 \text{ km}) \cdot (5.65 \text{ nm})} = 271.85 \text{ MHz}$$

Finally, according to Eq. (46), the corresponding bandwidth of the band-pass window is *Δ f bp* =543.70 MHz .

At this point, it is well worth highlighting the advantageous use of the chromatic dispersion parameter to obtain the filtered microwave signal. Once the main parameters are known, the topology illustrated in Figure 11 is assembled in order to evaluate the frequency response of the MPF.

**Figure 10.** Optical spectrum for the MLD used in the experiment.

**Figure 11.** Experimental microwave photonic filter.

At the output of the MLD, an optical isolator (OI) is placed in order to avoid reflections to the optical source. Since the MZ-IM (Photline MX-LN-10) is polarization-sensitive, a polarization controller (PC) is used to maximize the modulator output power. The optical signal is launched into the MZ-IM. The microwave electrical signal (RF) for modulating the optical intensity is supplied by using optical heterodyne as described in Figure 1. The registered frequency response is located from 0.01 to 4 GHz at 0 dBm. The intensity-modulated optical signal is then coupled into a 20.70 km of SM-SF coil. The length of the optical fiber is corroborated by using an optical time domain reflectometer, OTDR (EXFO, model FTB-7300E). At the end of the link, the optical signal is applied to a fast Photo-Detector (PD, Miteq DR-125G-A), and its output connected to an electrical spectrum analyzer (Anritsu, model MS2830A-044), in order to measure the frequency response of the MPF. Figure 12 corresponds to the measured experi‐ mental frequency response where a low-pass band centered at zero frequency and the presence of a band-pass band centered at 2.8 GHz are clearly appreciable.

**Figure 12.** Experimental frequency response of the filter.

**Figure 10.** Optical spectrum for the MLD used in the experiment.

188 Advances in Optical Communication

**Figure 11.** Experimental microwave photonic filter.

At the output of the MLD, an optical isolator (OI) is placed in order to avoid reflections to the optical source. Since the MZ-IM (Photline MX-LN-10) is polarization-sensitive, a polarization controller (PC) is used to maximize the modulator output power. The optical signal is launched into the MZ-IM. The microwave electrical signal (RF) for modulating the optical intensity is supplied by using optical heterodyne as described in Figure 1. The registered frequency response is located from 0.01 to 4 GHz at 0 dBm. The intensity-modulated optical signal is then coupled into a 20.70 km of SM-SF coil. The length of the optical fiber is corroborated by using an optical time domain reflectometer, OTDR (EXFO, model FTB-7300E). At the end of the link, the optical signal is applied to a fast Photo-Detector (PD, Miteq DR-125G-A), and its output connected to an electrical spectrum analyzer (Anritsu, model MS2830A-044), in order to

The bandwidth of 543.70 MHz associated to the band-pass window centered at 2.8 GHz allows us to guarantee enough bandwidth in case of fluctuations (in the order of nanometers) between mode spacing. On the other hand, a considerable increase on the length of the optical fiber due to thermal expansion is practically impossible. These considerations permit us to guarantee a good stability for the microwave photonic filter. Once the frequency response of the MPF is determined, the setup illustrated in Figure 13 is assembled for carrying out the fiber-radio transmission.

**Figure 13.** Experimental setup for optical and wireless transmission.

**Figure 14.** Electrical Spectrums for (a) Transmitted and (b) recovered TV signal.

Now, the electrical signal generator provides a signal of 2.8 GHz at 0 dBm that is used as the electrical carrier and demodulated signal. This signal is separated by using a power divider. Part of this signal is transmitted via radio frequency by the fabricated microstip antenna shown in Figure 5, and the rest is mixed with an analog NTSC TV signal of 67.25 MHz. The resulting mixed electrical signal is then applied to the electrodes of the MZ-IM for modulating the light emitted by the MLD. The modulated light is coupled into the 20.70 km SM-SF coil. At the end of the optical link, the signal is injected to a fast photo-detector (PD), and its electrical output is then amplified and launched to an electrical mixer. Another microstrip patch antenna placed at a distance of 10 meters is connected to a port of the mixer in order to recuperate the microwave signal that plays the role of the demodulated signal. Finally, by using another power divider, recovered analog TV signal can be launched to a digital oscilloscope or to the electrical spectrum analyzer in order to evaluate the quality of the recovered signal and at the same time display the TV signal on a TV monitor. Figure 14 (a) shows the measured electrical spectrum (Agilent, E4407B) corresponding to the transmitted TV signal where the SNR is 52.67 dB, whereas Figure (b) corresponds to the recovered TV signal with a SNR of 46.5 dB.

Finally, Figure 15 corresponds to a photograph of the screen of the oscilloscope where upper and lower traces are the waveforms of the transmitted and recuperated signal, respectively.

**Figure 15.** Transmitted and recovered TV signal.

## **9. Conclusions**

**Figure 14.** Electrical Spectrums for (a) Transmitted and (b) recovered TV signal.

190 Advances in Optical Communication

Now, the electrical signal generator provides a signal of 2.8 GHz at 0 dBm that is used as the electrical carrier and demodulated signal. This signal is separated by using a power divider. Part of this signal is transmitted via radio frequency by the fabricated microstip antenna shown in Figure 5, and the rest is mixed with an analog NTSC TV signal of 67.25 MHz. The resulting mixed electrical signal is then applied to the electrodes of the MZ-IM for modulating the light emitted by the MLD. The modulated light is coupled into the 20.70 km SM-SF coil. At the end of the optical link, the signal is injected to a fast photo-detector (PD), and its electrical output is then amplified and launched to an electrical mixer. Another microstrip patch antenna placed at a distance of 10 meters is connected to a port of the mixer in order to recuperate the microwave signal that plays the role of the demodulated signal. Finally, by using another power divider, recovered analog TV signal can be launched to a digital oscilloscope or to the

Wireless communication systems require compact sources for the generation of mm-wave signals, that must have high spectral purity (linewidth < 100 kHz, phase noise < 100 dBc @100 kHz offset), tuneability, low power consumption and low cost, and although optical hetero‐ dyne of two DFB lasers has phase noise of –75 dBc/Hz even at an offset frequency of 100 MHz and it does not very compact, we have demonstrated in this chapter that by using optical heterodyne technique, a TV signal was transmitted and received satisfactory as a result of our proposed communication system generates a microwave carrier and a local oscillator simul‐ taneously ensuring synchronization in frequency as well as in phase between microwave carrier and a local oscillator and avoiding in this case the use of an analog phase locked loop in the receiver to recover the TV information. The authors consider that the first proposed scheme in this chapter is not a truly wireless communication system, since an optical fiber is required to deliver the local oscillator in the receiver, however in order to obtain a wireless communication systems by using optical heterodyne technique, it is necessary to have collimated beams from optical fiber to photodetectors. On the other hand, due to the fact that the distribution of TV over microwave signals in the electrical domain presents loss associated with electrical distribution lines, the authors consider that the optical fiber is an ideal solution to fulfill this task because of its extremely broad bandwidth and low loss. In that case the distribution of TV over microwave can be directly by using optical fiber. In this way the second proposed experiment in this chapter represents a novel fiber-radio scheme to transmit an analog NTSC TV signal coded on a microwave band-pass located at 2.8 GHz. Filtering of microwave signal was achieved through the appropriate use of the chromatic fiber dispersion parameter, the physical length of the optical fiber, and the free spectral value of the multimode laser. Transmission of a TV signal was achieved over an optical link of 20.70 km, whereas a demodulated signal was transmitted via radiofrequency using the fabricated microstrip patch antennas. Although the distance between antennas was short, this distance can be lengthened if an array of antennas is used. Besides, a mathematical analysis corresponding to the micro‐ wave photonic filter was described demonstrating that the frequency response of the micro‐ wave photonic filter is proportional to the Fourier transform of the spectrum of the optical source used. The proposed microwave photonic filter represents an interesting technological alternative for transmitting information by using optoelectronic techniques. The results obtained in this work ensure that as an interesting alternative, several modulation schemes can be used for transmitting not only analog information but also digital information. Besides as optical heterodyne technique described here can generate microwaves continually tuned, we can use this feature to transmit several TV signals using frequency division multiplexing schemes FDM [19] and wavelength division multiplexing WDM techniques, not only point to point but also with bidirectional schemes by using simultaneous wired and wireless systems.

## **Acknowledgements**

This work was supported by CONACyT (grants No 102046 and 154691).

## **Author details**

Alejandro García Juárez1 , Ignacio Enrique Zaldívar Huerta2 , Antonio Baylón Fuentes3 , María del Rocío Gómez Colín4 , Luis Arturo García Delgado1 , Ana Lilia Leal Cruz1 and Alicia Vera Marquina1

1 University of Sonora, Department of Physics Research, México

2 National Institute of Astrophysics, Optics and Electronics. Department of Electronics, México

3 Inst. FEMTO-ST, Université de Franche-Comté, Besançon, France

4 University of Sonora, Department of Physics, México

#### **References**

collimated beams from optical fiber to photodetectors. On the other hand, due to the fact that the distribution of TV over microwave signals in the electrical domain presents loss associated with electrical distribution lines, the authors consider that the optical fiber is an ideal solution to fulfill this task because of its extremely broad bandwidth and low loss. In that case the distribution of TV over microwave can be directly by using optical fiber. In this way the second proposed experiment in this chapter represents a novel fiber-radio scheme to transmit an analog NTSC TV signal coded on a microwave band-pass located at 2.8 GHz. Filtering of microwave signal was achieved through the appropriate use of the chromatic fiber dispersion parameter, the physical length of the optical fiber, and the free spectral value of the multimode laser. Transmission of a TV signal was achieved over an optical link of 20.70 km, whereas a demodulated signal was transmitted via radiofrequency using the fabricated microstrip patch antennas. Although the distance between antennas was short, this distance can be lengthened if an array of antennas is used. Besides, a mathematical analysis corresponding to the micro‐ wave photonic filter was described demonstrating that the frequency response of the micro‐ wave photonic filter is proportional to the Fourier transform of the spectrum of the optical source used. The proposed microwave photonic filter represents an interesting technological alternative for transmitting information by using optoelectronic techniques. The results obtained in this work ensure that as an interesting alternative, several modulation schemes can be used for transmitting not only analog information but also digital information. Besides as optical heterodyne technique described here can generate microwaves continually tuned, we can use this feature to transmit several TV signals using frequency division multiplexing schemes FDM [19] and wavelength division multiplexing WDM techniques, not only point to point but also with bidirectional schemes by using simultaneous wired and wireless systems.

**Acknowledgements**

192 Advances in Optical Communication

**Author details**

Alejandro García Juárez1

Alicia Vera Marquina1

México

María del Rocío Gómez Colín4

This work was supported by CONACyT (grants No 102046 and 154691).

1 University of Sonora, Department of Physics Research, México

3 Inst. FEMTO-ST, Université de Franche-Comté, Besançon, France

4 University of Sonora, Department of Physics, México

, Ignacio Enrique Zaldívar Huerta2

, Luis Arturo García Delgado1

2 National Institute of Astrophysics, Optics and Electronics. Department of Electronics,

, Antonio Baylón Fuentes3

, Ana Lilia Leal Cruz1

,

and


#### **New Results in DF Relaying Schemes Using Time Diversity for Free-Space Optical Links New Results in DF Relaying Schemes Using Time Diversity for Free-space Optical Links**

[11] Stavros Iezekiel Microwave Photonics Devices and Applications. John Wiley; 2009.

[13] Ramesh Garg, R. B. Garg, P. Bhartia, Prakash Bhartia, Apisak Ittipiboon, and I. J. Bahl, Microstrip Antenna Design Handbook, New York: Artech House Inc; 2000. [14] J. Liang, C. C. Chiau, X. Chen, and C. G. Parini, "Printed Circular Disc Monopole An‐ tenna for Ultra-Wideband Applications", Electronic. Letter. Vol. 40, No.20, pp. 1246–

[15] K. L. Wong, T. C. Tseng, and P. L. Teng, "Low-profile Ultra Wideband Antenna for Mobile Phone Applications", Microwave Optical Technolgy Letter, Vol. 43, pp.7-9,

[16] K. L. Wong, L. C. Chou, and H. T. Chen, "Ultra-wideband Metal-Plate Monopole An‐ tenna For Laptop Application", Microwave Optical Technology Letter, Vol. 43, pp.

[17] Constantine A. Balanis, Antenna Theory-Analysis and Design, Third Edition. John

[18] G. Aguayo-Rodríguez, I. E. Zaldívar-Huerta, J. Rodríguez-Asomoza, A. García-Juár‐ ez and P. Alonso-Rubio "Modeling and performance analysis of an all-optical pho‐ tonic microwave filter in the frequency range of 0.01-15 GHz" International Society

[19] Alejandro García-Juárez, Ignacio E. Zaldívar-Huerta, Jorge Rodríguez-Asomoza, Ma‐ ría del Rocío Gómez-Colín "Method to transmit analog information by using a long distance photonic link with distributed feedback lasers biased in the low laser thresh‐

for Optics and Photonics, 2010. p. 76200B-76200B-12.

old current region" Opt. Eng. 51(6), 065006 (2012).

[12] Kai Chang RF and Microwave Wireless Systems. John Wiley; 2000.

1247, 2004.

194 Advances in Optical Communication

384-386, 2004.

Wiley & Sons, Inc; 2005.

2004.

Rubén Boluda-Ruiz, Beatriz Castillo-Vázquez, Carmen Castillo-Vázquez and Antonio García-Zambrana Rubén Boluda-Ruiz1, Beatriz Castillo-Vázquez1, Carmen Castillo-Vázquez1, and Antonio García-Zambrana2 Additional information is available at the end of the chapter

Additional information is available at the end of the chapter 10.5772/58996

http://dx.doi.org/10.5772/58996

#### **1. Introduction**

Atmospheric free-space optical (FSO) transmission using intensity modulation and direct detection (IM/DD) can be considered as an important alternative to consider for next generation broadband in order to support large bandwidth, unlicensed spectrum, excellent security, and quick and inexpensive setup [1]. Nonetheless, this technology is not without drawbacks, being the atmospheric turbulence one of the most impairments, producing fluctuations in the irradiance of the transmitted optical beam, which is known as *atmospheric scintillation*, severely degrading the link performance [2]. Additionally, since FSO systems are usually installed on high buildings, building sway causes vibrations in the transmitted beam, leading to an unsuitable alignment between transmitter and receiver and, hence, a greater deterioration in performance. Error control coding as well as diversity techniques can be used over FSO links to mitigate turbulence-induced fading [3–6]. In [7], the effects of atmospheric turbulence and misalignment considering aperture average effect were considered to study the outage capacity for single-input/single-output (SISO) links. In [8, 9], a wide range of turbulence conditions with gamma-gamma atmospheric turbulence and pointing errors is also considered on terrestrial FSO links, deriving closed-form expressions for the error-rate performance in terms of Meijer's G-functions. In [10, 11], comparing different diversity techniques, a significant improvement in terms of outage and error-rate performance is demonstrated when multiple-input/multiple-output (MIMO) FSO links based on transmit laser selection are adopted in the context of wide range of turbulence conditions with pointing errors. An alternative approach to provide spatial diversity in this turbulence FSO scenario without using multiple lasers and apertures is the employment of cooperative communications. Cooperative transmission can significantly improve the performance by creating diversity using the transceivers available at the other nodes of the network. This is a well known technique employed in radio-frequency (RF) systems, wherein more attention

Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2014 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

©2012 Boluda-Ruiz et al., licensee InTech. This is an open access chapter distributed under the terms of the

has been paid to the concept of user cooperation as a new form of diversity for future wireless communication systems [12]. Recently, several works have investigated the adoption of this technique in the context of FSO systems [13–19], being recognized as a very promising solution for future ad-hoc optical wireless systems. In [14, 15] a 3-way FSO communication setup is proposed to implement a cooperative protocol in order to improve spatial diversity without much increase in hardware, being evaluated the error-rate performance by using the photon-count method as well as the outage performance for both amplify-and-forward (AF) and decode-and-forward (DF) strategies. In [19], following the bit-detect-and-forward (BDF) cooperative protocol presented in [14], the analysis is extended to FSO communication systems using IM/DD over atmospheric turbulence and misalignment fading channels, considering cooperative communications with BDF relaying and equal gain combining (EGC) reception. In contrast to the BDF strategy considered in [14], it is assumed in [19] in addition to the analysis of the impact of pointing errors that all the bits detected at the relay are always resended regardless of these bits are detected correctly or incorrectly. Next, bits received directly from the source and from the relay are detected at destination node following an EGC technique. In [20], a novel BDF relaying scheme based on repetition coding with the relay and EGC is proposed, improving the robustness to impairments proper to these systems such as unsuitable alignment between transmitter and receiver as well as fluctuations in the irradiance of the transmitted optical beam due to the atmospheric turbulence, compared to the BDF relaying scheme analyzed in [19].

In this chapter, a novel closed-form approximation bit error-rate (BER) expression based on [21] is presented for a 3-way FSO communication setup when the irradiance of the transmitted optical beam is susceptible to either a wide range of turbulence conditions (weak to strong), following a gamma-gamma distribution of parameters *α* and *β*, or pointing errors, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. The resulting BER expression is shown to be very accurate in the range from low to high SNR, requiring the first two terms of the Taylor expansion of the channel probability density function (PDF). Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results. The superiority of the BDF relaying scheme using time diversity, compared with the cooperative protocol in [19], is corroborated by the obtained results since a greater robustness is provided not only to the pointing errors but also to the relay location, presenting a similar performance regardless of the source-destination link distance.

#### **2. System and channel model**

As shown in Fig. 1, we adopt a three-node cooperative system based on three separate full-duplex FSO links, assuming laser sources intensity-modulated and ideal noncoherent (direct-detection) receivers. For this 3-way FSO communication setup the cooperative protocol can be applied to achieve the spatial diversity without much increase in hardware. The BDF cooperative protocol here proposed is based on the use of repetition coding in the transmission corresponding to the source-relay link, fully exploiting a time diversity order of 2 in the atmospheric channel in a similar approach to [22]. As shown in Table 1, the cooperative strategy works in three phases or transmission frames. In the first phase, the nodes A and B send their own data to each other and the destination node C, i.e., the node A (B) transmits the same information to the nodes B (A) and C. In the second transmission frame, the nodes A and B send again the same information to each other delayed

2 Optical Communication

the BDF relaying scheme analyzed in [19].

the source-destination link distance.

**2. System and channel model**

has been paid to the concept of user cooperation as a new form of diversity for future wireless communication systems [12]. Recently, several works have investigated the adoption of this technique in the context of FSO systems [13–19], being recognized as a very promising solution for future ad-hoc optical wireless systems. In [14, 15] a 3-way FSO communication setup is proposed to implement a cooperative protocol in order to improve spatial diversity without much increase in hardware, being evaluated the error-rate performance by using the photon-count method as well as the outage performance for both amplify-and-forward (AF) and decode-and-forward (DF) strategies. In [19], following the bit-detect-and-forward (BDF) cooperative protocol presented in [14], the analysis is extended to FSO communication systems using IM/DD over atmospheric turbulence and misalignment fading channels, considering cooperative communications with BDF relaying and equal gain combining (EGC) reception. In contrast to the BDF strategy considered in [14], it is assumed in [19] in addition to the analysis of the impact of pointing errors that all the bits detected at the relay are always resended regardless of these bits are detected correctly or incorrectly. Next, bits received directly from the source and from the relay are detected at destination node following an EGC technique. In [20], a novel BDF relaying scheme based on repetition coding with the relay and EGC is proposed, improving the robustness to impairments proper to these systems such as unsuitable alignment between transmitter and receiver as well as fluctuations in the irradiance of the transmitted optical beam due to the atmospheric turbulence, compared to

In this chapter, a novel closed-form approximation bit error-rate (BER) expression based on [21] is presented for a 3-way FSO communication setup when the irradiance of the transmitted optical beam is susceptible to either a wide range of turbulence conditions (weak to strong), following a gamma-gamma distribution of parameters *α* and *β*, or pointing errors, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. The resulting BER expression is shown to be very accurate in the range from low to high SNR, requiring the first two terms of the Taylor expansion of the channel probability density function (PDF). Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results. The superiority of the BDF relaying scheme using time diversity, compared with the cooperative protocol in [19], is corroborated by the obtained results since a greater robustness is provided not only to the pointing errors but also to the relay location, presenting a similar performance regardless of

As shown in Fig. 1, we adopt a three-node cooperative system based on three separate full-duplex FSO links, assuming laser sources intensity-modulated and ideal noncoherent (direct-detection) receivers. For this 3-way FSO communication setup the cooperative protocol can be applied to achieve the spatial diversity without much increase in hardware. The BDF cooperative protocol here proposed is based on the use of repetition coding in the transmission corresponding to the source-relay link, fully exploiting a time diversity order of 2 in the atmospheric channel in a similar approach to [22]. As shown in Table 1, the cooperative strategy works in three phases or transmission frames. In the first phase, the nodes A and B send their own data to each other and the destination node C, i.e., the node A (B) transmits the same information to the nodes B (A) and C. In the second transmission frame, the nodes A and B send again the same information to each other delayed

**Figure 1.** Block diagram of the considered 3-way FSO communication system, where *dAC* is the A-C link distance and (*xB*, *yB*) represents the location of the node B.

by the expected fade duration and, hence, assuming that channel fades are independent and identically distributed (i.i.d.). In the third phase, the node B (or A) sends the received data from its partner A (or B) in previous frames to the node C. Following the BDF cooperative protocol, the relay node (A or B) detects each code bit to "0" or "1" based on repetion coding and sends the bit with the new power to the destination node C. When repetition coding is used in the source-relay link transmission during the first and second phases, the information is detected each transmission frame, combining with the same weight two noisy faded signals in a similar manner to a single-input multiple-output (SIMO) FSO scheme with EGC [23] and, this way, achieving a diversity gain of 2 for this link.


**Table 1.** BDF cooperative scheme based on repetition coding in the source-relay link transmission.

It must be noted that the symmetry for nodes A and B assumed in this FSO communication setup and the fact that one transmission frame is overlapped imply that no rate reduction is applied, i.e., the same information rate can be considered at the destination node C compared to the direct transmission link without using any cooperative strategy. As in [19], it is assumed in this paper that all the bits detected at the relay are always resended regardless of these bits are detected correctly or incorrectly. Next, bits received directly from A-C and from the relay A-B-C are detected at C following an EGC technique. Since atmospheric scintillation is a slow time varying process relative to typical symbol rates of an FSO system, having a coherence time on the order of milliseconds, we consider the time variations according to the theoretical block-fading model, where the channel fade remains constant during a block (corresponding to the channel coherence interval) and changes to a new independent value from one block to next. In other words, channel fades are assumed to be independent and identically distributed. This temporal correlation can be overcome by means of long interleavers, being usually assumed both in the analysis from the point of view of information theory and error rate performance analysis of coded FSO links [4, 5, 24]. However, as in [22, 25], we here assume that the interleaver depth can not be infinite and, hence, we can potentially benefit from a degree of time diversity limited equal to 2. This consideration is justified from the fact that the latency introduced by the interleaver is not an inconvenience for the required application. In the cooperative protocol here proposed, wherein repetition coding and a time diversity order available of 2 is assumed in the source-relay link transmission, perfect interleaving can be done by simply sending the same information delayed by the expected fade duration, as shown experimentally in [26].

For each link of the three possible links in this three-node cooperative FSO system, the instantaneous current *ym*(*t*) in the receiving photodetector corresponding to the information signal transmitted from the laser can be written as

$$y\_m(t) = \eta i\_m(t)\mathbf{x}(t) + z(t) \tag{1}$$

where *η* is the detector responsivity, assumed hereinafter to be the unity, *X x*(*t*) represents the optical power supplied by the source and *Im im*(*t*) the equivalent real-valued fading gain (irradiance) through the optical channel between the laser and the receive aperture. Additionally, the fading experienced between source-detector pairs *Im* is assumed to be statistically independent. *Z z*(*t*) is assumed to include any front-end receiver thermal noise as well as shot noise caused by ambient light much stronger than the desired signal at the detector. In this case, the noise can usually be modeled to high accuracy as AWGN with zero mean and variance *<sup>σ</sup>*<sup>2</sup> = *<sup>N</sup>*0/2, i.e. *<sup>Z</sup>* ∼ *<sup>N</sup>*(0, *<sup>N</sup>*0/2), independent of the on/off state of the received bit. Since the transmitted signal is an intensity, *X* must satisfy ∀*t x*(*t*) ≥ 0. Due to eye and skin safety regulations, the average optical power is limited and, hence, the average amplitude of *X* is limited. The received electrical signal *Ym ym*(*t*), however, can assume negative amplitude values. We use *Ym*, *X*, *Im* and *Z* to denote random variables and *ym*(*t*), *x*(*t*), *im*(*t*) and *z*(*t*) their corresponding realizations.

The irradiance is considered to be a product of three factors i.e., *Im* = *ζ<sup>m</sup> I* (*a*) *m I* (*p*) *<sup>m</sup>* where *ζm* is the deterministic propagation loss, *I* (*a*) *<sup>m</sup>* is the attenuation due to atmospheric turbulence and *I* (*p*) *<sup>m</sup>* the attenuation due to geometric spread and pointing errors. *<sup>ζ</sup><sup>m</sup>* is determined by the exponential Beers-Lambert law as *<sup>ζ</sup><sup>m</sup>* <sup>=</sup> *<sup>e</sup>*−Φ*d*, where *<sup>d</sup>* is the link distance and Φ is the atmospheric attenuation coefficient. It is given by Φ = (3.91/*V*(*km*)) (*λ*(*nm*)/550) <sup>−</sup>*<sup>q</sup>* where *<sup>V</sup>* is the visibility in kilometers, *<sup>λ</sup>* is the wavelength in nanometers and *q* is the size distribution of the scattering particles, being *q* = 1.3 for average visibility (6 km < *V* < 50 km), and *q* = 0.16*V* + 0.34 for haze visibility (1 km < *V* < 6 km) [27]. To consider a wide range of turbulence conditions, the gamma-gamma turbulence model proposed in [2] is here assumed. Regarding to the impact of pointing errors, we use the general model of misalignment fading given in [7] by Farid and Hranilovic, wherein the effect of beam width, detector size and jitter variance is considered. A closed-form expression of the combined probability density function (PDF) of *Im* was derived in [8] as

4 Optical Communication

signal transmitted from the laser can be written as

*ym*(*t*), *x*(*t*), *im*(*t*) and *z*(*t*) their corresponding realizations.

*ζm* is the deterministic propagation loss, *I*

turbulence and *I*

(3.91/*V*(*km*)) (*λ*(*nm*)/550)

The irradiance is considered to be a product of three factors i.e., *Im* = *ζ<sup>m</sup> I*

is determined by the exponential Beers-Lambert law as *<sup>ζ</sup><sup>m</sup>* <sup>=</sup> *<sup>e</sup>*−Φ*d*, where *<sup>d</sup>* is the link distance and Φ is the atmospheric attenuation coefficient. It is given by Φ =

nanometers and *q* is the size distribution of the scattering particles, being *q* = 1.3 for average visibility (6 km < *V* < 50 km), and *q* = 0.16*V* + 0.34 for haze visibility (1 km < *V* < 6 km)

regardless of these bits are detected correctly or incorrectly. Next, bits received directly from A-C and from the relay A-B-C are detected at C following an EGC technique. Since atmospheric scintillation is a slow time varying process relative to typical symbol rates of an FSO system, having a coherence time on the order of milliseconds, we consider the time variations according to the theoretical block-fading model, where the channel fade remains constant during a block (corresponding to the channel coherence interval) and changes to a new independent value from one block to next. In other words, channel fades are assumed to be independent and identically distributed. This temporal correlation can be overcome by means of long interleavers, being usually assumed both in the analysis from the point of view of information theory and error rate performance analysis of coded FSO links [4, 5, 24]. However, as in [22, 25], we here assume that the interleaver depth can not be infinite and, hence, we can potentially benefit from a degree of time diversity limited equal to 2. This consideration is justified from the fact that the latency introduced by the interleaver is not an inconvenience for the required application. In the cooperative protocol here proposed, wherein repetition coding and a time diversity order available of 2 is assumed in the source-relay link transmission, perfect interleaving can be done by simply sending the same information delayed by the expected fade duration, as shown experimentally in [26]. For each link of the three possible links in this three-node cooperative FSO system, the instantaneous current *ym*(*t*) in the receiving photodetector corresponding to the information

where *η* is the detector responsivity, assumed hereinafter to be the unity, *X x*(*t*) represents the optical power supplied by the source and *Im im*(*t*) the equivalent real-valued fading gain (irradiance) through the optical channel between the laser and the receive aperture. Additionally, the fading experienced between source-detector pairs *Im* is assumed to be statistically independent. *Z z*(*t*) is assumed to include any front-end receiver thermal noise as well as shot noise caused by ambient light much stronger than the desired signal at the detector. In this case, the noise can usually be modeled to high accuracy as AWGN with zero mean and variance *<sup>σ</sup>*<sup>2</sup> = *<sup>N</sup>*0/2, i.e. *<sup>Z</sup>* ∼ *<sup>N</sup>*(0, *<sup>N</sup>*0/2), independent of the on/off state of the received bit. Since the transmitted signal is an intensity, *X* must satisfy ∀*t x*(*t*) ≥ 0. Due to eye and skin safety regulations, the average optical power is limited and, hence, the average amplitude of *X* is limited. The received electrical signal *Ym ym*(*t*), however, can assume negative amplitude values. We use *Ym*, *X*, *Im* and *Z* to denote random variables and

*ym*(*t*) = *ηim*(*t*)*x*(*t*) + *z*(*t*) (1)

(*a*) *m I*

(*a*) *<sup>m</sup>* is the attenuation due to atmospheric

(*p*) *<sup>m</sup>* the attenuation due to geometric spread and pointing errors. *<sup>ζ</sup><sup>m</sup>*

<sup>−</sup>*<sup>q</sup>* where *<sup>V</sup>* is the visibility in kilometers, *<sup>λ</sup>* is the wavelength in

(*p*) *<sup>m</sup>* where

$$f\_{l\_m}(i) = \frac{\alpha\_m \beta\_m \varphi\_m^2}{A\_0 \zeta\_m \Gamma(a\_m) \Gamma(\beta\_m)} G\_{1,3}^{3,0}\left(\frac{\alpha\_m \beta\_m}{A\_0 \zeta\_m} i\left|\begin{matrix} \varrho\_m^2 - 1, \alpha\_m - 1, \beta\_m - 1 \end{matrix}\right.\right), \quad i \ge 0 \tag{2}$$

where *<sup>G</sup>m*,*<sup>n</sup> <sup>p</sup>*,*<sup>q</sup>* [·] is the Meijer's G-function [28, eqn. (9.301)] and <sup>Γ</sup>(·) is the well-known Gamma function. Assuming plane wave propagation, *α* and *β* can be directly linked to physical parameters through the following expresions [29]:

$$\mathfrak{a} = \left[ \exp \left( 0.49 \sigma\_{\mathbb{R}}^2 / (1 + 1.11 \sigma\_{\mathbb{R}}^{12/5})^{7/6} \right) - 1 \right]^{-1} \tag{3a}$$

$$\beta = \left[ \exp\left( 0.51 \sigma\_R^2 / (1 + 0.69 \sigma\_R^{12/5})^{5/6} \right) - 1 \right]^{-1} \tag{3b}$$

where *σ*<sup>2</sup> *<sup>R</sup>* <sup>=</sup> 1.23*C*<sup>2</sup> *<sup>n</sup>κ*7/6*d*11/6 is the Rytov variance, which is a measure of optical turbulence strength. Here, *κ* = 2*π*/*λ* is the optical wave number and *d* is the link distance in meters. *C*2 *<sup>n</sup>* stands for the altitude-dependent index of the refractive structure parameter and varies from 10−<sup>13</sup> *<sup>m</sup>*−2/3 for strong turbulence to 10−<sup>17</sup> *<sup>m</sup>*−2/3 for weak turbulence [2]. It must be emphasized that parameters *α* and *β* cannot be arbitrarily chosen in FSO applications, being related through the Rytov variance. It can be shown that the relationship *α* > *β* always holds, and the parameter *β* is lower bounded above 1 as the Rytov variance approaches ∞ [30]. In relation to the impact of pointing errors [7], assuming a Gaussian spatial intensity profile of beam waist radius, *ωz*, on the receiver plane at distance *z* from the transmitter and a circular receive aperture of radius *r*, *ϕ* = *ωzeq*/2*σ<sup>s</sup>* is the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter) at the receiver, *ω*<sup>2</sup> *zeq* = *<sup>ω</sup>*<sup>2</sup> *z* √*<sup>π</sup>*erf(*v*)/2*<sup>v</sup>* exp(−*v*2), *<sup>v</sup>* <sup>=</sup> <sup>√</sup>*πr*/ √<sup>2</sup>*ωz*, *<sup>A</sup>*<sup>0</sup> = [erf(*v*)]<sup>2</sup> and erf(·) is the error function [28, eqn. (8.250)]. Nonetheless, the PDF in Eq. (2) appears to be cumbersome to use in order to obtain simple closed-form expressions in the analysis of FSO communication systems. To overcome this inconvenience, the PDF is approximated by using the first two terms of the Taylor expansion at *i* = 0 as *fIm* (*i*) = *ami bm*−<sup>1</sup> <sup>+</sup> *cmi bm* + *O*(*i bm*+1). As proposed in [21], we adopt the approximation *fIm* (*i*) <sup>≈</sup> *ami bm*−<sup>1</sup> exp(*icm*/*am*). Different expressions for *fIm* (*i*), depending on the relation between the values of *ϕ*<sup>2</sup> and *β*, can be written as

$$f\_{I\_m}(\mathbf{i}) \approx \frac{\varrho\_m^2 (\mathfrak{a}\_m \mathfrak{f}\_m)^{\mathfrak{f}\_m} \Gamma(a\_m - \mathfrak{f}\_m)}{(A\_0 \mathfrak{f}\_m)^{\mathfrak{f}\_m} \Gamma(a\_m) \Gamma(\mathfrak{f}\_m) \left(\mathfrak{g}\_m^2 - \mathfrak{f}\_m\right)} \mathfrak{i}^{\mathfrak{f}\_m - 1} e^{\frac{a\_m \mathfrak{f}\_m (\mathfrak{a}\_m - \mathfrak{f}\_m) \left(\mathfrak{f}\_m - \mathfrak{f}\_m^2 + 1\right)}{\mathfrak{f}\_m}}, \qquad \varrho\_m^2 > \mathfrak{f}\_m \tag{4a}$$

$$f\_{I\_m}(i) \approx \frac{\varphi\_m^2 (\mathfrak{a}\_m \mathfrak{f}\_m)^{\varphi\_m^2} \Gamma\left(\mathfrak{a}\_m - \mathfrak{o}\_m^2\right) \Gamma\left(\mathfrak{f}\_m - \mathfrak{o}\_m^2\right)}{(A\_0 \mathfrak{f}\_m)^{\varphi\_m^2} \Gamma(\mathfrak{a}\_m) \Gamma(\mathfrak{b}\_m)} i^{\mathfrak{b}\_m^2 - 1}, \qquad \mathfrak{o}\_m^2 < \mathfrak{f}\_m \tag{4b}$$

It can be noted that the second term of the Taylor expansion is equal to 0 when the diversity order is not independent of the pointing error effects, i.e. *ϕ*<sup>2</sup> *<sup>m</sup>* < *<sup>β</sup>m*. In the following section, the fading coefficient *Im* for the paths A-B, A-C and and B-C is indicated by *IAB*, *IAC* and *IBC*, respectively.

#### **3. Error-rate performance analysis**

For the sake of clarity, without loss of generality, we can consider node A as source and node B as its relay for the BER evaluation since similar results hold when node B is considered as the source and node A as its relay. In addition to the BER performance evaluation corresponding to the cooperative protocol here proposed based on time diversity, we also consider the performance analysis for the direct path link (non-cooperative link A-C) to establish the baseline performance as well as BER performance corresponding to the traditional BDF cooperative protocol analyzed in [19]. Here, it is assumed that the average optical power transmitted from each node is *P*opt, being adopted an OOK signaling based on a constellation of two equiprobable points in a one-dimensional space with an Euclidean distance of *dE* <sup>=</sup> <sup>2</sup>*P*opt√*Tbξ*, where the parameter *Tb* is the bit period and *<sup>ξ</sup>* represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, as explained in a greater detail in [10, appendix]. According to Eq. (1), the statistical channel model corresponding to the A-B link assuming repetition coding with EGC during the first and second frames can be written as

$$Y\_{AB} = \frac{1}{2}X(I\_{AB\_1} + I\_{AB\_2}) + Z\_{AB\_{E\mathcal{C}'}} \quad X \in \{0, d\_E\}, \quad Z\_{AB\_{E\mathcal{C}}} \sim N(0, N\_0) \tag{5}$$

The information is detected each bit period, combining with the same weight 2 noisy faded signals in a similar manner to a SIMO FSO scheme with EGC [23] and, this way, increasing the diversity order for the source-relay link. As shown in Table 1, since in the first phase of the cooperative protocol the node A transmits the same information to the nodes B and C and the fact that in the second phase transmits again this information overlapping with the third phase corresponding to the symmetric scheme, the division by 2 is considered so as to maintain the average optical power in the air at a constant level of *P*opt, being transmitted by each laser an average optical power of *P*opt/2. Assuming channel side information at the receiver, the conditional BER at the node B is given by

$$P\_b^{AB}\left(E\mid I\_{AB\_T}\right) = \mathcal{Q}\left(\sqrt{(d\_E/2)^2 i^2 / 4N\_0}\right) = \mathcal{Q}\left(\sqrt{(\gamma/4)\xi i}\right) \tag{6}$$

where *IABT* represents the sum of variates *IABT* <sup>=</sup> *IAB*<sup>1</sup> <sup>+</sup> *IAB*<sup>2</sup> , *<sup>Q</sup>*(·) is the Gaussian-*<sup>Q</sup>* function defined as *Q*(*x*) = <sup>√</sup> 1 2*π* ∞ *<sup>x</sup> <sup>e</sup>*<sup>−</sup> *<sup>t</sup>* 2 <sup>2</sup> *dt* and *γ* = *P*<sup>2</sup> opt*Tb*/*N*<sup>0</sup> represents the received electrical SNR in absence of turbulence when the classical rectangular pulse shape is adopted for OOK formats. Since the variates *IAB*<sup>1</sup> and *IAB*<sup>2</sup> are independent, knowing that the resulting PDF of their sum *IABT* can be determined by using the moment generating function of their corresponding PDFs, obtained via single-sided Laplace and its inverse transforms, approximate expression for the PDF, *fIABT* (*i*), of the combined variates can be easily derived from Eq. (4) as

6 Optical Communication

*IBC*, respectively.

distance of *dE* = 2*P*opt

It can be noted that the second term of the Taylor expansion is equal to 0 when the diversity

the fading coefficient *Im* for the paths A-B, A-C and and B-C is indicated by *IAB*, *IAC* and

For the sake of clarity, without loss of generality, we can consider node A as source and node B as its relay for the BER evaluation since similar results hold when node B is considered as the source and node A as its relay. In addition to the BER performance evaluation corresponding to the cooperative protocol here proposed based on time diversity, we also consider the performance analysis for the direct path link (non-cooperative link A-C) to establish the baseline performance as well as BER performance corresponding to the traditional BDF cooperative protocol analyzed in [19]. Here, it is assumed that the average optical power transmitted from each node is *P*opt, being adopted an OOK signaling based on a constellation of two equiprobable points in a one-dimensional space with an Euclidean

square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, as explained in a greater detail in [10, appendix]. According to Eq. (1), the statistical channel model corresponding to the A-B link assuming repetition coding with EGC during the first

The information is detected each bit period, combining with the same weight 2 noisy faded signals in a similar manner to a SIMO FSO scheme with EGC [23] and, this way, increasing the diversity order for the source-relay link. As shown in Table 1, since in the first phase of the cooperative protocol the node A transmits the same information to the nodes B and C and the fact that in the second phase transmits again this information overlapping with the third phase corresponding to the symmetric scheme, the division by 2 is considered so as to maintain the average optical power in the air at a constant level of *P*opt, being transmitted by each laser an average optical power of *P*opt/2. Assuming channel side information at the

(*dE*/2)2*i*2/4*N*<sup>0</sup>

where *IABT* represents the sum of variates *IABT* <sup>=</sup> *IAB*<sup>1</sup> <sup>+</sup> *IAB*<sup>2</sup> , *<sup>Q</sup>*(·) is the Gaussian-*<sup>Q</sup>*

electrical SNR in absence of turbulence when the classical rectangular pulse shape is adopted for OOK formats. Since the variates *IAB*<sup>1</sup> and *IAB*<sup>2</sup> are independent, knowing that the resulting PDF of their sum *IABT* can be determined by using the moment generating function of their corresponding PDFs, obtained via single-sided Laplace and its inverse transforms, approximate expression for the PDF, *fIABT* (*i*), of the combined variates can be easily derived

<sup>2</sup> *dt* and *γ* = *P*<sup>2</sup>

*Tbξ*, where the parameter *Tb* is the bit period and *ξ* represents the

*<sup>X</sup>*(*IAB*<sup>1</sup> <sup>+</sup> *IAB*<sup>2</sup> ) + *ZABEGC* , *<sup>X</sup>* ∈ {0, *dE*}, *ZABEGC* <sup>∼</sup> *<sup>N</sup>*(0, *<sup>N</sup>*0) (5)

 = *Q* 

(*γ*/4)*ξi*

opt*Tb*/*N*<sup>0</sup> represents the received

(6)

*<sup>m</sup>* < *<sup>β</sup>m*. In the following section,

order is not independent of the pointing error effects, i.e. *ϕ*<sup>2</sup>

**3. Error-rate performance analysis**

√

receiver, the conditional BER at the node B is given by

*<sup>b</sup>* ( *<sup>E</sup>*<sup>|</sup> *IABT* ) <sup>=</sup> *<sup>Q</sup>*

1 2*π* ∞ *<sup>x</sup> <sup>e</sup>*<sup>−</sup> *<sup>t</sup>* 2

*PAB*

function defined as *Q*(*x*) = <sup>√</sup>

and second frames can be written as

*YAB* <sup>=</sup> <sup>1</sup> 2

$$f\_{I\_{AB\_T}}(i) \approx \frac{a\_{AB}^2 \Gamma(b\_{AB})^2}{\Gamma(2b\_{AB})} i^{2b\_{AB}-1} e^{i \frac{2c\_{AB}b\_{AB}\Gamma(2b\_{AB})}{a\_{AB}\Gamma(2b\_{AB}+1)}}\tag{7}$$

Hence, the average BER, *PAB <sup>b</sup>* (*E*), can be obtained by averaging *<sup>P</sup>AB <sup>b</sup>* ( *<sup>E</sup>*<sup>|</sup> *IABT* ) over the PDF as follows

$$P\_b^{AB}(E) = \int\_0^\infty Q\left(\sqrt{(\gamma/4)\xi}i\right) f\_{I\_{AB\_T}}(i) di. \tag{8}$$

To evaluate the integral in Eq. (8), we can use that the Q-function is related to the complementary error function erfc(·) by erfc(*x*) = 2*Q*( √2*x*) [28, eqn. (6.287)] and [31, eqn. (2.85.5.2)], obtaining the corresponding approximation of average BER as can be seen in

$$\begin{split} P\_{b}^{AB}(E) &\approx \sqrt{\pi} 2^{(b\_{AB}-1)} \gamma^{-b\_{AB}} \Gamma(b\_{AB})^2 a\_{AB}^2 \,\_2 \tilde{F}\_2 \left( b\_{AB}, b\_{AB} + \frac{1}{2}; \frac{1}{2}, b\_{AB} + 1; \frac{2c\_{AB}^2}{a\_{AB}^2 \gamma} \right) \\ &+ \frac{\sqrt{\pi} \gamma^{\frac{1}{2}(-2b\_{AB}-1)} \Gamma(b\_{AB})^2}{2^{-\frac{1}{2}(2b\_{AB}-1)} (a\_{AB} b\_{AB} c\_{AB})^{-1}} \,\_2F\_2 \left( b\_{AB} + \frac{1}{2}, b\_{AB} + 1; \frac{3}{2}, b\_{AB} + \frac{3}{2}; \frac{2c\_{AB}^2}{a\_{AB}^2 \gamma} \right) . \end{split} \tag{9}$$

where *pF*˜ *<sup>q</sup>*(*a*1, ··· , *ap*; *<sup>b</sup>*1, ··· , *bq*; *<sup>x</sup>*) is the generalized hypergeometric function [28, eqn. (9.14.1)] and the value of the parameters *aAB* and *bAB* depends on the relation between *ϕ*<sup>2</sup> and *β* as obtained in Eq. (4). Considering now that the PDF in Eq. (2) is approximated by using the first term of the Taylor expansion, i.e. assuming in Eq. (9) a value of *cAB* = 0, it is straightforward to show that the average BER behaves asymptotically as (Λ*cγξ*)−Λ*<sup>d</sup>* , where <sup>Λ</sup>*<sup>d</sup>* and <sup>Λ</sup>*<sup>c</sup>* denote diversity order and coding gain, respectively. At high SNR, if asymptotically the error probability behaves as (Λ*cγξ*)−Λ*<sup>d</sup>* , the diversity order <sup>Λ</sup>*<sup>d</sup>* determines the slope of the BER versus average SNR curve in a log-log scale and the coding gain Λ*<sup>c</sup>* (in decibels) determines the shift of the curve in SNR. Since *cAB* = 0 we can use in Eq. (9) that *pFq a*1,..., *ap*; *b*1,..., *bq*; 0 = 1 [28, eqn. (9.14.1)]. It is easy to deduce that *PAB <sup>b</sup>* (*E*) behaves asymptotically as 1/*γbAB* , corroborating not only that the diversity order corresponding to the source-relay link is independent of the pointing error when *ϕ*<sup>2</sup> > *β* but also the fact that the diversity order has been increased twice if compared to the BDF cooperative protocol analyzed in [19], as also shown in [20].

Once the error probability at the node B is known, two cases can be considered to evaluate the BER corresponding to the BDF cooperative protocol here proposed depending on the fact that the bit from the relay A-B-C is detected correctly or incorrectly. In this way, the statistical channel model corresponding to the BDF cooperative protocol, i.e. the bits received at C directly from A-C link and from the relay A-B-C can be written as

$$Y\_{\rm BDF} = \frac{1}{2}XI\_{A\mathcal{C}} + Z\_{A\mathcal{C}} + \frac{1}{2}X^\*I\_{B\mathcal{C}} + Z\_{B\mathcal{C}} \quad X \in \{0, d\_E\}, \quad Z\_{A\mathcal{C}}, Z\_{B\mathcal{C}} \sim N(0, N\_0/2) \tag{10}$$

where *<sup>X</sup>*<sup>∗</sup> represents the random variable corresponding to the information detected at the node B and, hence, *<sup>X</sup>*<sup>∗</sup> <sup>=</sup> *<sup>X</sup>* when the bit has been detected correctly at B and *<sup>X</sup>*<sup>∗</sup> <sup>=</sup> *dE* <sup>−</sup> *<sup>X</sup>* when the bit has been detected incorrectly. As shown in Table 1, since in the first phase of the cooperative protocol the node A transmits the same information to the nodes B and C and the fact that the third phase is overlapped with the second phase corresponding to the symmetric scheme, transmitting information from node B to nodes C and A, the division by 2 is considered so as to maintain the average optical power in the air at a constant level of *P*opt, being transmitted by each laser an average optical power of *P*opt/2.

Considering that the bit is correctly detected at B, the statistical channel model for the BDF cooperative protocol can be expressed as

$$Y\_{\text{BDF}\_0} = \frac{1}{2}X\left(I\_{A\mathbb{C}} + I\_{\text{BC}}\right) + Z\_{\text{BDF}\_{\text{ECC}'}} \quad X \in \{0, d\_{\text{E}}\}, \quad Z\_{\text{BDF}\_{\text{ECC}}} \sim N(0, N\_0) \tag{11}$$

As in previous analysis corresponding to the source-relay link, the conditional BER at the node C is given by *PBDF*<sup>0</sup> *<sup>b</sup>* ( *<sup>E</sup>*<sup>|</sup> *IT*) <sup>=</sup> *<sup>Q</sup>* (*γ*/4)*ξ<sup>i</sup>* , where *IT* represents the sum of variates *IT* = *IAC* + *IBC*. Since the variates *IAC* and *IBC* are independent, knowing that the resulting PDF of their sum *IT* can be determined by using the moment generating function of their corresponding PDFs, obtained via single-sided Laplace and its inverse transforms, approximate expression for the PDF, *fIT* (*i*), of the combined variates can be easily derived from Eq. (4) as

$$f\_{I\_{\Gamma}}(i) \approx \frac{a\_{A\mathcal{C}} a\_{\mathcal{BC}} \Gamma(b\_{A\mathcal{C}}) \Gamma(b\_{\mathcal{BC}})}{\Gamma(b\_{A\mathcal{C}} + b\_{B\mathcal{C}})} i^{b\_{A\mathcal{C}} + b\_{B\mathcal{C}} - 1} e^{i \frac{\Gamma(b\_{A\mathcal{C}} + b\_{B\mathcal{C}}) (a\_{A\mathcal{C}} b\_{B\mathcal{C}} c\_{B\mathcal{C}} + a\_{B\mathcal{C}} b\_{A\mathcal{C}} c\_{A\mathcal{C}})}{a\_{A\mathcal{C}} a\_{B\mathcal{C}} \Gamma(b\_{A\mathcal{C}} + b\_{B\mathcal{C}} + 1)}}. \tag{12}$$

From this expression, the average BER, *PBDF*<sup>0</sup> *<sup>b</sup>* (*E*), can be determined as follows

$$P\_b^{BDF\_0}(E) = \int\_0^\infty \mathcal{Q}\left(\sqrt{(\gamma/4)\xi}i\right) f\_{I\tau}(i) di. \tag{13}$$

Evaluating this integral as in Eq. (8), we can obtain the corresponding closed-form asymptotic solution for the BER as follows

$$\begin{split} &P\_{b}^{BDF\_{0}}(E) \approx \sqrt{\pi}2^{\frac{1}{2}(b\_{AC}+b\_{BC}-2)}a\_{A\mathcal{C}}a\_{B\mathcal{C}}\Gamma(b\_{AC})\Gamma(b\_{BC})\gamma^{-\frac{1}{2}(b\_{AC}+b\_{BC})}\\ &\times \,\_2F\_2\left(\frac{b\_{AC}+b\_{BC}}{2}, \frac{b\_{AC}+b\_{BC}+1}{2}; \frac{1}{2}, \frac{b\_{AC}+b\_{BC}+2}{2}; \frac{2(a\_{BC}b\_{AC}c\_{AC}+a\_{AC}b\_{BC}c\_{BC})^{2}}{a\_{AC}^{2}a\_{BC}^{2}(b\_{AC}+b\_{BC})^{2}\gamma}\right)\\ &+\sqrt{\pi}2^{\frac{1}{2}(b\_{AC}+b\_{BC}-2)}\Gamma(b\_{AC})\Gamma(b\_{BC})(a\_{AC}b\_{BC}c\_{BC}+a\_{BC}b\_{AC}c\_{AC})\gamma^{-\frac{1}{2}(b\_{AC}+b\_{BC}+1)}\\ &\times \,\_2F\_2\left(\frac{b\_{AC}+b\_{BC}+1}{2}, \frac{b\_{AC}+b\_{BC}+2}{2}; \frac{3}{2}, \frac{b\_{AC}+b\_{BC}+3}{2}; \frac{2(a\_{BC}b\_{AC}c\_{AC}+a\_{AC}b\_{BC}c\_{BC})^{2}}{a\_{AC}^{2}a\_{BC}^{2}(b\_{AC}+b\_{BC})^{2}\gamma}\right) \end{split} \tag{14}$$

From this expression, as previously commented in Eq. (9), it is easy to deduce that *PBDF*<sup>0</sup> *<sup>b</sup>* (*E*) behaves asymptotically as 1/*γ*<sup>1</sup> <sup>2</sup> (*bAC*+*bBC*) when *cAC* <sup>=</sup> 0 and *cBC* <sup>=</sup> 0 are considered, assuming now that the PDF in Eq. (2) is approximated by using only the first term of the Taylor expansion.

8 Optical Communication

where *<sup>X</sup>*<sup>∗</sup> represents the random variable corresponding to the information detected at the node B and, hence, *<sup>X</sup>*<sup>∗</sup> <sup>=</sup> *<sup>X</sup>* when the bit has been detected correctly at B and *<sup>X</sup>*<sup>∗</sup> <sup>=</sup> *dE* <sup>−</sup> *<sup>X</sup>* when the bit has been detected incorrectly. As shown in Table 1, since in the first phase of the cooperative protocol the node A transmits the same information to the nodes B and C and the fact that the third phase is overlapped with the second phase corresponding to the symmetric scheme, transmitting information from node B to nodes C and A, the division by 2 is considered so as to maintain the average optical power in the air at a constant level of

Considering that the bit is correctly detected at B, the statistical channel model for the BDF

As in previous analysis corresponding to the source-relay link, the conditional BER at

variates *IT* = *IAC* + *IBC*. Since the variates *IAC* and *IBC* are independent, knowing that the resulting PDF of their sum *IT* can be determined by using the moment generating function of their corresponding PDFs, obtained via single-sided Laplace and its inverse transforms, approximate expression for the PDF, *fIT* (*i*), of the combined variates can be easily derived

*bAC*+*bBC*−1*<sup>e</sup>*

Evaluating this integral as in Eq. (8), we can obtain the corresponding closed-form

*bAC* + *bBC* + 2 <sup>2</sup> ;

> *bAC* + *bBC* + 3 <sup>2</sup> ;

*aACaBC*Γ(*bAC*)Γ(*bBC*)*γ*<sup>−</sup> <sup>1</sup>

<sup>Γ</sup>(*bAC*)Γ(*bBC*)(*aACbBCcBC* <sup>+</sup> *aBCbACcAC*)*γ*<sup>−</sup> <sup>1</sup>

3 2 ,

1 2 , *i*

(*γ*/4)*ξi*

*<sup>b</sup>* (*E*), can be determined as follows

<sup>2</sup> (*bAC*+*bBC*)

*a*2 *ACa*<sup>2</sup>

> *a*2 *ACa*<sup>2</sup>

(*γ*/4)*ξ<sup>i</sup>*

*<sup>X</sup>* (*IAC* <sup>+</sup> *IBC*) <sup>+</sup> *ZBDFEGC* , *<sup>X</sup>* ∈ {0, *dE*}, *ZBDFEGC* <sup>∼</sup> *<sup>N</sup>*(0, *<sup>N</sup>*0) (11)

, where *IT* represents the sum of

*aACaBC*Γ(*bAC*+*bBC*+1) . (12)

*fIT* (*i*)*di*. (13)

2(*aBCbACcAC* + *aACbBCcBC*)<sup>2</sup>

<sup>2</sup> (*bAC*+*bBC*+1)

*BC*(*bAC* <sup>+</sup> *bBC*)2*<sup>γ</sup>*

2(*aBCbACcAC* + *aACbBCcBC*)<sup>2</sup>

*BC*(*bAC* <sup>+</sup> *bBC*)2*<sup>γ</sup>*

(14)

<sup>Γ</sup>(*bAC*+*bBC*)(*aACbBCcBC*+*aBCbACcAC*)

*P*opt, being transmitted by each laser an average optical power of *P*opt/2.

*<sup>b</sup>* ( *<sup>E</sup>*<sup>|</sup> *IT*) <sup>=</sup> *<sup>Q</sup>*

<sup>Γ</sup>(*bAC* <sup>+</sup> *bBC*) *<sup>i</sup>*

 <sup>∞</sup> 0 *Q* 

*fIT* (*i*) <sup>≈</sup> *aACaBC*Γ(*bAC*)Γ(*bBC*)

*PBDF*<sup>0</sup> *<sup>b</sup>* (*E*) =

*bAC* + *bBC* + 1 <sup>2</sup> ;

> *bAC* + *bBC* + 2 <sup>2</sup> ;

From this expression, the average BER, *PBDF*<sup>0</sup>

asymptotic solution for the BER as follows

<sup>2</sup> (*bAC*+*bBC*−2)

1

*bAC* + *bBC* <sup>2</sup> ,

<sup>2</sup> (*bAC*+*bBC*−2)

*bAC* + *bBC* + 1 <sup>2</sup> ,

cooperative protocol can be expressed as

2

*YBDF*<sup>0</sup> <sup>=</sup> <sup>1</sup>

the node C is given by *PBDF*<sup>0</sup>

from Eq. (4) as

*PBDF*<sup>0</sup>

× <sup>2</sup>*F*˜ 2 

<sup>+</sup> <sup>√</sup>*π*<sup>2</sup> 1

× <sup>2</sup>*F*˜ 2 

*<sup>b</sup>* (*E*) <sup>≈</sup> <sup>√</sup>*π*<sup>2</sup>

Alternatively, considering now that the bit is incorrectly detected at B, the statistical channel model for the BDF cooperative protocol can be expressed as

$$Y\_{\rm BDF\_1} = \frac{1}{2}X\left(I\_{\rm AC} - I\_{\rm BC}\right) + \frac{d\_E}{2}I\_{\rm BC} + Z\_{\rm BDF\_{\rm ECC}}, \quad X \in \{0, d\_E\}, \quad Z\_{\rm BDF\_{\rm ECC}} \sim N(0, N\_0) \tag{15}$$

Assuming channel side information at the receiver and given that the statistics corresponding to the term *dE* <sup>2</sup> *IBC* become irrelevant to the detection process, the conditional BER at the node C is given by *PBDF*<sup>1</sup> *<sup>b</sup>* ( *<sup>E</sup>*<sup>|</sup> *IAC*, *IBC*) <sup>=</sup> *<sup>Q</sup>* (*γ*/4)*ξ*(*i*<sup>1</sup> <sup>−</sup> *<sup>i</sup>*2) . Hence, the average BER, *PBDF*<sup>1</sup> *<sup>b</sup>* (*E*), can be obtained by averaging over the PDF as follows

$$P\_b^{BDF\_1}(E) = \int\_0^\infty \int\_0^\infty \mathcal{Q}\left(\sqrt{(\gamma/4)\xi}(i\_1 - i\_2)\right) f\_{I\_{\rm AC}}(i\_1) f\_{I\_{\rm BC}}(i\_2) di\_1 di\_2. \tag{16}$$

Unfortunately, the result in Eq. (16) is not dominated by the behavior of the PDF near the origin because of the argument of the Gaussian-*Q* function is not always positive [32]. As in [19], to overcome this inconvenience, we can use the expression *Q*(−*x*) = 1 − *Q*(*x*) to manipulate the negative values on the argument of the Gaussian-*Q* function in Eq. (16) together with the fact that Gaussian-*Q* function tends to 0 as *γ* → ∞, simplifying the integral in Eq. (16) as follows

$$P\_b^{BDF\_1}(E) \doteq \int\_0^\infty \int\_0^{i\_2} f\_{I\_{AC}}(i\_1) f\_{I\_{BC}}(i\_2) di\_1 di\_2. \tag{17}$$

It can be noted that the asymptotic behavior of *PBDF*<sup>1</sup> *<sup>b</sup>* (*E*) is independent of the SNR *<sup>γ</sup>*, resulting in a positive value that is upper bounded by 1. To evaluate the integral (17), we can use the Meijer's G-function [28, eqn. (9.301)], available in standard scientific software packages such as Mathematica and Maple, in order to transform the integral expression to the form in [33, eqn. (21)], expressing *Kµ*(·) [33, eqn. (14)] in terms of Meijer's G-function. In this way, a closed-form solution is derived as can be seen in

$$P\_{b}^{BDF\_{1}}(E) \doteq \frac{\left(\varphi\_{AC}^{2}\varphi\_{BC}^{2}\right)G\_{5,5}^{3A}\left(\frac{\mathbb{I}\_{BC}a\_{AC}\mathbb{A}\_{AC}a\_{B\_{BC}}}{\mathbb{I}\_{AC}a\_{BC}\mathbb{A}\_{BC}a\_{A\_{C}}}\,\Big|\,\begin{array}{l}1,1-a\_{BC,1}-\beta\_{BC,1}\,1-\varphi\_{BC,}^{2}\varphi\_{AC}^{2}+1\\ \varphi\_{AC}^{2},a\_{AC}\beta\_{AC,}-\varphi\_{BC}^{2},0\end{array}\right)}{\Gamma(a\_{AC})\Gamma(a\_{BC})\Gamma(\beta\_{AC})\Gamma(\beta\_{BC})}}.\tag{18}$$

Nonetheless, it must be emphasized that the Meijer's G-function has to be numerically calculated and, hence, the use of Monte Carlo integration to solve Eq. (17) may represent an alternative with less computational load. Once the error probability at the node B is known and considering these two cases, i.e. depending on the fact that the bit from the relay A-B-C is detected correctly or incorrectly, the BER corresponding to the BDF cooperative protocol here proposed is given by

$$P\_b^{BDF}(E) = P\_b^{BDF\_0}(E) \cdot (1 - P\_b^{AB}(E)) + P\_b^{BDF\_1}(E) \cdot P\_b^{AB}(E). \tag{19}$$

This expression can be simplified taking into account the asymptotic behavior previously obtained in Eq. (9) and Eq. (14) as follows

$$P\_b^{BDF}(E) \doteq P\_b^{BDF\_0}(E), \tag{20a} \\ \qquad \qquad \qquad b\_{A\mathbb{C}} + b\_{B\mathbb{C}} < 2b\_{AB} \tag{20a}$$

$$P\_b^{BDF}(E) \doteq P\_b^{BDF\_1}(E) \cdot P\_b^{AB}(E), \qquad \qquad b\_{A\mathbb{C}} + b\_{B\mathbb{C}} > 2b\_{AB} \tag{20b}$$

Taking into account these expressions, the adoption of the BDF cooperative protocol here analyzed translates into a diversity order gain, *Gd*, relative to the non-cooperative link A-C of

$$G\_d = \min(b\_{A\mathbb{C}} + b\_{B\mathbb{C}}, 2b\_{AB}) / b\_{A\mathbb{C}} \tag{21}$$

Comparing with [19, eqn. (25)], it must be noted that a factor of 2 is included in relation to the diversity order depending on the source-relay link because of repetition coding assumed in the cooperative protocol based on time diversity, as was shown in [20] using alternative expressions. As shown in [11, 19] by the authors, it can be deduced that the main aspect to consider in order to optimize the error-rate performance is the relation between *ϕ*<sup>2</sup> and *β*, corroborating that the diversity order corresponding to each link is independent of the pointing error when *ϕ*<sup>2</sup> > *β*. Once this condition is satisfied an analysis about how Eq. (21) can be optimized is required, evaluating if the diversity order corresponding cooperative protocol is determined by the source-destination and relay-destination links or by the source-relay link. For the better understanding of the impact of the configuration of the three-node cooperative FSO system under study, the diversity order gain *Gd* in Eq. (21) as a function of the horizontal displacement of the relay node, *xB*, is depicted in Fig. 2 for a source-destination link distance *dAC*= {3 km, 6 km} and different turbulence conditions when different relay locations along the source-destination distance are assumed. Here, the diversity order gain corresponding to the BDF cooperative protocol analyzed in [19] is also included in order to show the improvement in performance achieved when repetition coding is assumed in the source-relay link transmission. In contrast to the analysis in [19], it must be commented that the impact of the deterministic propagation loss *ζm* is here considered, assuming clear weather conditions with visibility of 16 km. Here, the parameters *α* and *β* are calculated from Eq. (3*a*) and Eq. (3*b*), *λ* = 1550 *nm* and values of *C*<sup>2</sup> *<sup>n</sup>* <sup>=</sup> 1.7 <sup>×</sup> <sup>10</sup>−<sup>14</sup> and *C*<sup>2</sup> *<sup>n</sup>* <sup>=</sup> <sup>8</sup> <sup>×</sup> <sup>10</sup>−<sup>14</sup> *<sup>m</sup>*−2/3 for moderate and strong turbulence, respectively, are adopted. In any case, the condition *ϕ*<sup>2</sup> > *β* is satisfied for each link and, hence, these results are independent of pointing errors. These curves are corresponding to the intersection of two profiles related to the expressions (*βAC* + *βBC*)/*βAC* and 2*βAB*/*βAC*, as deduced from Eq. (21). The improvement in performance can be easily corroborated by the fact that, even when the available diversity order is dependent on the relay location, this is now related to the expression 2*βAB*/*βAC*, being twice as in [19], fully exploiting the potential time-diversity available in the turbulent channel corresponding to the source-relay link. It can be noted that a diversity order greater than two is always guaranteed regardless of the relay location and source-destination link.

10 Optical Communication

here proposed is given by

*PBDF*

obtained in Eq. (9) and Eq. (14) as follows

*PBDF <sup>b</sup>* (*E*) .

*PBDF <sup>b</sup>* (*E*) .

of

and *C*<sup>2</sup>

*<sup>b</sup>* (*E*) = *<sup>P</sup>BDF*<sup>0</sup>

= *PBDF*<sup>0</sup>

= *PBDF*<sup>1</sup>

*<sup>b</sup>* (*E*) · *<sup>P</sup>AB*

alternative with less computational load. Once the error probability at the node B is known and considering these two cases, i.e. depending on the fact that the bit from the relay A-B-C is detected correctly or incorrectly, the BER corresponding to the BDF cooperative protocol

This expression can be simplified taking into account the asymptotic behavior previously

Taking into account these expressions, the adoption of the BDF cooperative protocol here analyzed translates into a diversity order gain, *Gd*, relative to the non-cooperative link A-C

Comparing with [19, eqn. (25)], it must be noted that a factor of 2 is included in relation to the diversity order depending on the source-relay link because of repetition coding assumed in the cooperative protocol based on time diversity, as was shown in [20] using alternative expressions. As shown in [11, 19] by the authors, it can be deduced that the main aspect to consider in order to optimize the error-rate performance is the relation between *ϕ*<sup>2</sup> and *β*, corroborating that the diversity order corresponding to each link is independent of the pointing error when *ϕ*<sup>2</sup> > *β*. Once this condition is satisfied an analysis about how Eq. (21) can be optimized is required, evaluating if the diversity order corresponding cooperative protocol is determined by the source-destination and relay-destination links or by the source-relay link. For the better understanding of the impact of the configuration of the three-node cooperative FSO system under study, the diversity order gain *Gd* in Eq. (21) as a function of the horizontal displacement of the relay node, *xB*, is depicted in Fig. 2 for a source-destination link distance *dAC*= {3 km, 6 km} and different turbulence conditions when different relay locations along the source-destination distance are assumed. Here, the diversity order gain corresponding to the BDF cooperative protocol analyzed in [19] is also included in order to show the improvement in performance achieved when repetition coding is assumed in the source-relay link transmission. In contrast to the analysis in [19], it must be commented that the impact of the deterministic propagation loss *ζm* is here considered, assuming clear weather conditions with visibility of 16 km. Here, the parameters *α* and *β*

are calculated from Eq. (3*a*) and Eq. (3*b*), *λ* = 1550 *nm* and values of *C*<sup>2</sup>

*<sup>n</sup>* <sup>=</sup> <sup>8</sup> <sup>×</sup> <sup>10</sup>−<sup>14</sup> *<sup>m</sup>*−2/3 for moderate and strong turbulence, respectively, are adopted. In any case, the condition *ϕ*<sup>2</sup> > *β* is satisfied for each link and, hence, these results are independent of pointing errors. These curves are corresponding to the intersection of two profiles related to the expressions (*βAC* + *βBC*)/*βAC* and 2*βAB*/*βAC*, as deduced from Eq.

*<sup>b</sup>* (*E*)) + *<sup>P</sup>BDF*<sup>1</sup>

*<sup>b</sup>* (*E*) · *<sup>P</sup>AB*

*<sup>b</sup>* (*E*), *bAC* <sup>+</sup> *bBC* <sup>&</sup>gt; <sup>2</sup>*bAB* (20b)

*<sup>b</sup>* (*E*), *bAC* <sup>+</sup> *bBC* <sup>&</sup>lt; <sup>2</sup>*bAB* (20a)

*Gd* = min(*bAC* + *bBC*, 2*bAB*)/*bAC* (21)

*<sup>b</sup>* (*E*). (19)

*<sup>n</sup>* <sup>=</sup> 1.7 <sup>×</sup> <sup>10</sup>−<sup>14</sup>

*<sup>b</sup>* (*E*) · (<sup>1</sup> <sup>−</sup> *<sup>P</sup>AB*

(b) Strong turbulence (*C*<sup>2</sup> *<sup>n</sup>* <sup>=</sup> <sup>8</sup> <sup>×</sup> <sup>10</sup>−<sup>14</sup> *<sup>m</sup>*<sup>−</sup>2/3)

**Figure 2.** Diversity order gain for source-destination link distances of *dAC* = {3, 6} km when different relay locations are assumed.

The approximate BER results corresponding to this analysis with rectangular pulse shapes and *ξ* = 1 are illustrated in Fig. 3, when different relay locations for source-destination link distances *dAC*= {3 km, 6 km} are assumed together with values of normalized beamwidth and normalized jitter of (*ωz*/*r*, *σs*/*r*)=(5, 1) and (*ωz*/*r*, *σs*/*r*)=(10, 2). Monte Carlo simulation results are furthermore included as a reference, confirming the accuracy and usefulness of the derived results. Due to the long simulation time involved, simulation results only up to BER=10−<sup>9</sup> are included. Simulation results corroborate that approximate expressions here obtained lead to simple bounds on the bit error probability that get tighter over a wider range of SNR compared to asymptotic expressions previously presented in [20]. Additionally, we also consider the performance analysis for the direct path link (non-cooperative link A-C) to establish the baseline performance as well as BER performance corresponding to the BDF cooperative protocol analyzed in [19].

As expected, it can be corroborated that these BER results are in excellent agreement with previous results shown in Fig. 2 in relation to the diversity order gain achieved for this 3-way FSO communication setup. For the moderate turbulence case, diversity gains of 3.18 and 2.6 are achieved when *dAC*=3 km and relay locations of (*xB*=1.6 km; *yB*=0.5 km) and (*xB*=2 km; *yB*=1 km), respectively, in contrast to the diversity gains of 1.84 and 1.3 achieved by the BDF cooperative protocol analyzed in [19]. Analogously, it can be seen that diversity gains of 2.07 and 2.18 are obtained when *dAC*=6 km and relay locations of (*xB*=2 km; *yB*=2.5 km) and (*xB*=4.5 km; *yB*=0.5 km), respectively, in contrast to the diversity gains of 1.31 and 1.09 achieved by the BDF cooperative protocol analyzed in [19]. For the strong turbulence case, the improvement in diversity order gain is even more significant. From previous results, it can be concluded that not only a significant improvement in performance has been obtained by increasing the diversity order but also that a greater robustness is now achieved regardless of the source-destination link distance. As shown in Fig. 2, it can be deduced from Eq. (21) that the diversity order gain, *Gd*, is lower as the value of *yB* is increased, presenting a maximum value wherein the two profiles related to the expressions (*βAC* + *βBC*)/*βAC* and 2*βAB*/*βAC* intersect.

Additionally, a greater robustness to the impact of pointing errors is provided by the BDF cooperative protocol here proposed in case that the condition *ϕ*<sup>2</sup> > *β* is not satisfied, as shown in Fig. 4 for a vertical displacement of the relay node of *yB*=0.5 km and a source-destination link distance of *dAC* = 3 km when moderate turbulence conditions are considered. In this configuration, a normalized beamwidth of *ωz*/*r* = 7 and different values of normalized jitter *σs*/*r* = {1, 2.5, 4} are assumed in order to contrast the impact of pointing errors when the condition *ϕ*<sup>2</sup> > *β* is or not satisfied for each link. It can be observed that diversity gains even greater than 3 are achieved by the BDF protocol here proposed when (*ωz*/*r*, *σs*/*r*)=(7, 1), not being affected by pointing errors, decreasing this value down to 2 as the normalized jitter increases. However, a remarkably greater deterioration in performance is displayed when the traditional BDF protocol is considered.

These conclusions are contrasted in Fig. 5, wherein BER performance for a source-destination link distance of *dAC* = 3 km and a relay location of (*xB*=1.7 km; *yB*=0.5 km) when values of normalized beamwidth of *ωz*/*r* = 7 and normalized jitter of *σs*/*r* = {1, 2.5, 4} are assumed. As before, we also consider the performance analysis for the direct path link (non-cooperative link A-C) to establish the baseline performance. These BER results are in excellent agreement with previous results shown in Fig. 4 in relation to the diversity order gain achieved for this 3-way FSO communication setup when pointing errors are present. In this way, diversity gains of 3.4, 2.3 and 2 are achieved when values of normalized jitter of *σs*/*r* = {1, 2.5, 4} are assumed, respectively.

12 Optical Communication

intersect.

assumed, respectively.

distances *dAC*= {3 km, 6 km} are assumed together with values of normalized beamwidth and normalized jitter of (*ωz*/*r*, *σs*/*r*)=(5, 1) and (*ωz*/*r*, *σs*/*r*)=(10, 2). Monte Carlo simulation results are furthermore included as a reference, confirming the accuracy and usefulness of the derived results. Due to the long simulation time involved, simulation results only up to BER=10−<sup>9</sup> are included. Simulation results corroborate that approximate expressions here obtained lead to simple bounds on the bit error probability that get tighter over a wider range of SNR compared to asymptotic expressions previously presented in [20]. Additionally, we also consider the performance analysis for the direct path link (non-cooperative link A-C) to establish the baseline performance as well as BER performance

As expected, it can be corroborated that these BER results are in excellent agreement with previous results shown in Fig. 2 in relation to the diversity order gain achieved for this 3-way FSO communication setup. For the moderate turbulence case, diversity gains of 3.18 and 2.6 are achieved when *dAC*=3 km and relay locations of (*xB*=1.6 km; *yB*=0.5 km) and (*xB*=2 km; *yB*=1 km), respectively, in contrast to the diversity gains of 1.84 and 1.3 achieved by the BDF cooperative protocol analyzed in [19]. Analogously, it can be seen that diversity gains of 2.07 and 2.18 are obtained when *dAC*=6 km and relay locations of (*xB*=2 km; *yB*=2.5 km) and (*xB*=4.5 km; *yB*=0.5 km), respectively, in contrast to the diversity gains of 1.31 and 1.09 achieved by the BDF cooperative protocol analyzed in [19]. For the strong turbulence case, the improvement in diversity order gain is even more significant. From previous results, it can be concluded that not only a significant improvement in performance has been obtained by increasing the diversity order but also that a greater robustness is now achieved regardless of the source-destination link distance. As shown in Fig. 2, it can be deduced from Eq. (21) that the diversity order gain, *Gd*, is lower as the value of *yB* is increased, presenting a maximum value wherein the two profiles related to the expressions (*βAC* + *βBC*)/*βAC* and 2*βAB*/*βAC*

Additionally, a greater robustness to the impact of pointing errors is provided by the BDF cooperative protocol here proposed in case that the condition *ϕ*<sup>2</sup> > *β* is not satisfied, as shown in Fig. 4 for a vertical displacement of the relay node of *yB*=0.5 km and a source-destination link distance of *dAC* = 3 km when moderate turbulence conditions are considered. In this configuration, a normalized beamwidth of *ωz*/*r* = 7 and different values of normalized jitter *σs*/*r* = {1, 2.5, 4} are assumed in order to contrast the impact of pointing errors when the condition *ϕ*<sup>2</sup> > *β* is or not satisfied for each link. It can be observed that diversity gains even greater than 3 are achieved by the BDF protocol here proposed when (*ωz*/*r*, *σs*/*r*)=(7, 1), not being affected by pointing errors, decreasing this value down to 2 as the normalized jitter increases. However, a remarkably greater deterioration in performance

These conclusions are contrasted in Fig. 5, wherein BER performance for a source-destination link distance of *dAC* = 3 km and a relay location of (*xB*=1.7 km; *yB*=0.5 km) when values of normalized beamwidth of *ωz*/*r* = 7 and normalized jitter of *σs*/*r* = {1, 2.5, 4} are assumed. As before, we also consider the performance analysis for the direct path link (non-cooperative link A-C) to establish the baseline performance. These BER results are in excellent agreement with previous results shown in Fig. 4 in relation to the diversity order gain achieved for this 3-way FSO communication setup when pointing errors are present. In this way, diversity gains of 3.4, 2.3 and 2 are achieved when values of normalized jitter of *σs*/*r* = {1, 2.5, 4} are

corresponding to the BDF cooperative protocol analyzed in [19].

is displayed when the traditional BDF protocol is considered.

**Figure 3.** BER performance when different relay locations for *dAC* = {3, 6} km are assumed together with values of normalized beamwidth and normalized jitter of (*ωz*/*r*, *σs*/*r*)=(5, 1) and (*ωz*/*r*, *σs*/*r*)=(10, 2), respectively, and moderate and strong turbulence conditions are considered.

**Figure 4.** Diversity order gain *Gd* for a source-destination link distance of *dAC* = 3 km and vertical displacement of the relay node of *yB* = 0.5 km when moderate turbulence conditions, values of normalized beamwidth of *ωz*/*r* = 7 and normalized jitter of *σs*/*r* = {1, 2.5, 4} are assumed.

**Figure 5.** BER performance is depicted for *dAC* = 3 km and a relay location of (*xB*=1.7 km; *yB*=0.5 km) when values of *ωz*/*r* = 7 and *σs*/*r* = {1, 2.5, 4} are assumed as well as when no pointing errors are considered. Results assuming the optimum beamwidth corresponding to these values of normalized jitter are also included.

As concluded in [11, 19] by the authors, the adoption of transmitters with accurate control of their beamwidth is especially important to satisfy the condition *ϕ*<sup>2</sup> > *β* in order to maximize the diversity order gain. Once this condition is satisfied, it can be convenient to compare with the BER performance obtained in a similar context when misalignment fading is not present. Knowing that the impact of pointing errors in our analysis can be suppressed by assuming *<sup>A</sup>*<sup>0</sup> → 1 and *<sup>ϕ</sup>*<sup>2</sup> → <sup>∞</sup> [7], the corresponding asymptotic expression for the configuration analyzed in Fig. 4 can be easily derived from Eq. (20*a*) as follows

14 Optical Communication

(*ωz*/*r*, *σs*/*r*)=(7, 1) (*ωz*/*r*, *σs*/*r*)=(7, 2.5) (*ωz*/*r*, *σs*/*r*)=(7, 4)

*ωz*/*r* = 7 and normalized jitter of *σs*/*r* = {1, 2.5, 4} are assumed.

*dAC* = 3 km

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4

<sup>10</sup>−<sup>10</sup>

*dAC* = 3 km

BDF no pointing errors

<sup>10</sup>−<sup>8</sup>

Average

 bit-error

probability

<sup>10</sup>−<sup>6</sup>

<sup>10</sup>−<sup>4</sup>

<sup>10</sup>−<sup>2</sup>

100

Diversity

 order gain,

*Gd*

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 0.8

**Figure 4.** Diversity order gain *Gd* for a source-destination link distance of *dAC* = 3 km and vertical displacement of the relay node of *yB* = 0.5 km when moderate turbulence conditions, values of normalized beamwidth of

<sup>20</sup> <sup>40</sup> 60 80 100 120 140 <sup>10</sup>−<sup>12</sup>

**Figure 5.** BER performance is depicted for *dAC* = 3 km and a relay location of (*xB*=1.7 km; *yB*=0.5 km) when values of *ωz*/*r* = 7 and *σs*/*r* = {1, 2.5, 4} are assumed as well as when no pointing errors are considered. Results assuming the optimum beamwidth corresponding to these values of normalized jitter are also included.

Average SNR, *γ* (dB)

Horizontal displacement of the relay node, *xB* (km)

*σs*/*r* = 1 *σs*/*r* = 2.5 *σs*/*r* = 4

traditional BDF BDF proposed

bound

BDF (*ωz*/*r* = 7) BDF (*ωz*/*roptimum*) direct link (*ωz*/*r* = 7)

*yB* = 0.5 km

$$P\_b^{BDF}(E) \doteq \frac{a\_{A\overline{C}}^{npc} a\_{B\overline{C}}^{npc} 2^{\frac{\beta\_{AC} + \beta\_{BC}}{2}} \Gamma(\beta\_{AC}) \Gamma(\beta\_{BC})}{2L\_{A\overline{C}}^{\beta\_{AC}} L\_{B\overline{C}}^{\beta\_{BC}} \Gamma\left(\frac{1}{2}(\beta\_{AC} + \beta\_{BC} + 2)\right)} (\gamma \xi)^{-\frac{1}{2}(\beta\_{AC} + \beta\_{BC})} \tag{22}$$

where the parameters *a npe AC* and *a npe BC* are obtained from the Taylor expansion in Eq. (4) when no pointing errors are present as follows

$$a\_m^{npe} = \frac{(\alpha\_m \beta\_m)^{\beta\_m} \Gamma(\alpha\_m - \beta\_m)}{\Gamma(\alpha\_m) \Gamma(\beta\_m)}. \tag{23}$$

In Fig. 5, BER performance in the same FSO context without pointing errors is also displayed. In the same way as concluded in [19], the impact of the pointing error effects translates into a coding gain disadvantage, *Dpe*[*dB*], relative to this 3-way FSO communication setup without misalignment fading given by

$$D\_{\rm pe}[dB] \stackrel{\Delta}{=} \frac{20}{\beta\_{\rm AC} + \beta\_{\rm BC}} \log\_{10} \left( \frac{\rho\_{\rm AC}^2}{A\_{0\_{\rm AC}}^{\delta\_{\rm AC}} (\varphi\_{\rm AC}^2 - \beta\_{\rm AC})} \frac{\rho\_{\rm BC}^2}{A\_{0\_{\rm BC}}^{\delta\_{\rm BC}} (\varphi\_{\rm BC}^2 - \beta\_{\rm BC})} \right) . \tag{24}$$

According to this expression, it can be observed in Fig. 5 that a coding gain disadvantage of

28.75 decibels is achieved for a value of (*ωz*/*r*, *σs*/*r*)=(7, 1) in the three-node cooperative FSO system here proposed. Since proper FSO transmission requires transmitters with accurate control of their beamwidth, the optimization procedure is finished by finding the optimum beamwidth, *ωz*/*r*, that gives the minimum BER performance. It can be observed that this is equivalent to minimize the expression in Eq. (24), deducing that the optimization process for the transmit laser corresponding to the source node and the optimization process corresponding to the relay node are independent. Hence, the optimum beamwidth for each transmit laser can be achieved using numerical optimization methods for different values of normalized jitter, *σs*/*r* and turbulence conditions [34] in a similar approach as reported in [11]. In this way, it can be shown that BER optimization provides numerical results for the normalized beamwidth *ωz*/*r* following a linear performance for each value of distance, where its corresponding slope is subject to the turbulence conditions. This leads to easily obtain a first-degree polynomial given by

$$
\omega\_z / r\_{optimal} \approx \left( -0.034 \beta^2 + 0.72 \beta + 2.15 \right) \sigma\_s / r\_\prime \tag{25}
$$

where the slope follows a quadratic form in *β* [11]. The use of this expression is also

shown in Fig. 5, where results assuming the optimum beamwidth corresponding to values of normalized jitter of *σs*/*r* = {1, 2.5, 4} are also included. Using Eq. (25) for a value of *σs*/*r* = 1, a coding gain disadvantage of *Dpe* = 21.6 decibels is achieved when values of *ωz*/*rAC* = 3.17 and *ωz*/*rBC* = 4.17 are assumed, improving BER performance in 7 decibels if compared to previous case wherein *ωz*/*r* = 7 is considered for source-destination and relay-destinations links. Additionally, an even greater improvement in BER performance is corroborated for values of normalized jitter of 2.5 and 4, wherein the diversity order is increased since the condition *ϕ*<sup>2</sup> > *β* is satisfied. In this way, using Eq. (25) for a value of *σs*/*r* = 2.5, a coding gain disadvantage of *Dpe* = 37.49 decibels is achieved when values of *ωz*/*rAC* = 7.93 and *ωz*/*rBC* = 10.43 are assumed. Using Eq. (25) for a value of *σs*/*r* = 4, values of *ωz*/*rAC* = 12.68 and *ωz*/*rBC* = 16.69 are obtained, achieving a coding gain disadvantage of *Dpe* = 45.65 decibels relative to this 3-way FSO communication setup without misalignment fading. Nonetheless, it must be emphasized in this case that nearly 50 decibels less in average SNR are required to guarantee a target BER of 10−<sup>8</sup> after optimizing the normalized beamwidth *ωz*/*r*.

#### **4. Conclusions**

In this chapter, a novel closed-form approximation bit error-rate (BER) expression based on [21] is presented for a 3-way FSO communication setup when the irradiance of the transmitted optical beam is susceptible to either a wide range of turbulence conditions (weak to strong), following a gamma-gamma distribution of parameters *α* and *β*, or pointing errors, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. The resulting BER expression is shown to be very accurate in the range from low to high SNR, requiring the first two terms of the Taylor expansion of the channel probability density function (PDF). Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results. Fully exploiting the potential time-diversity available in the turbulent channel, a greater diversity gain determined by *Gd* = min(*βAC* + *βBC*, 2*βAB*)/*βAC* is achieved, where *βAC*, *βBC* and *βAB* are parameters corresponding to the turbulence of the source-destination, relay-destination and source-relay links. The superiority of the BDF relaying scheme using time diversity, compared with the traditional cooperative protocol, is corroborated by the obtained results since a greater robustness is provided not only to the pointing errors but also to the relay location, presenting a similar performance regardless of the source-destination link distance. Additionally, asymptotic expressions are used to find the optimum beamwidth that minimizes the BER at different turbulence conditions as well as to determine a more favorable relay location.

#### **Acknowledgments**

The authors wish to acknowledge the financial support given by the Spanish MINECO Project TEC2012-32606.

#### **Author details**

16 Optical Communication

*ωz*/*r*.

**4. Conclusions**

favorable relay location.

**Acknowledgments**

TEC2012-32606.

*<sup>ω</sup>z*/*roptimum* ≈

−0.034*β*<sup>2</sup> + 0.72*β* + 2.15

where the slope follows a quadratic form in *β* [11]. The use of this expression is also

shown in Fig. 5, where results assuming the optimum beamwidth corresponding to values of normalized jitter of *σs*/*r* = {1, 2.5, 4} are also included. Using Eq. (25) for a value of *σs*/*r* = 1, a coding gain disadvantage of *Dpe* = 21.6 decibels is achieved when values of *ωz*/*rAC* = 3.17 and *ωz*/*rBC* = 4.17 are assumed, improving BER performance in 7 decibels if compared to previous case wherein *ωz*/*r* = 7 is considered for source-destination and relay-destinations links. Additionally, an even greater improvement in BER performance is corroborated for values of normalized jitter of 2.5 and 4, wherein the diversity order is increased since the condition *ϕ*<sup>2</sup> > *β* is satisfied. In this way, using Eq. (25) for a value of *σs*/*r* = 2.5, a coding gain disadvantage of *Dpe* = 37.49 decibels is achieved when values of *ωz*/*rAC* = 7.93 and *ωz*/*rBC* = 10.43 are assumed. Using Eq. (25) for a value of *σs*/*r* = 4, values of *ωz*/*rAC* = 12.68 and *ωz*/*rBC* = 16.69 are obtained, achieving a coding gain disadvantage of *Dpe* = 45.65 decibels relative to this 3-way FSO communication setup without misalignment fading. Nonetheless, it must be emphasized in this case that nearly 50 decibels less in average SNR are required to guarantee a target BER of 10−<sup>8</sup> after optimizing the normalized beamwidth

In this chapter, a novel closed-form approximation bit error-rate (BER) expression based on [21] is presented for a 3-way FSO communication setup when the irradiance of the transmitted optical beam is susceptible to either a wide range of turbulence conditions (weak to strong), following a gamma-gamma distribution of parameters *α* and *β*, or pointing errors, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. The resulting BER expression is shown to be very accurate in the range from low to high SNR, requiring the first two terms of the Taylor expansion of the channel probability density function (PDF). Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results. Fully exploiting the potential time-diversity available in the turbulent channel, a greater diversity gain determined by *Gd* = min(*βAC* + *βBC*, 2*βAB*)/*βAC* is achieved, where *βAC*, *βBC* and *βAB* are parameters corresponding to the turbulence of the source-destination, relay-destination and source-relay links. The superiority of the BDF relaying scheme using time diversity, compared with the traditional cooperative protocol, is corroborated by the obtained results since a greater robustness is provided not only to the pointing errors but also to the relay location, presenting a similar performance regardless of the source-destination link distance. Additionally, asymptotic expressions are used to find the optimum beamwidth that minimizes the BER at different turbulence conditions as well as to determine a more

The authors wish to acknowledge the financial support given by the Spanish MINECO Project

*σs*/*r*, (25)

Rubén Boluda-Ruiz1, Beatriz Castillo-Vázquez1, Carmen Castillo-Vázquez2, and Antonio García-Zambrana<sup>1</sup>

1 Department of Communications Engineering, University of Málaga, Spain

2 Department of Statistics and Operations Research, University of Málaga, Málaga, Spain

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