**1. Introduction**

All-optical signal processing is essential for optical communication systems transmitting high speed data signals. All-optical wavelength conversion (WC), pulse generation, orthogonal frequency-division multiplexing (OFDM), demultiplexing, regeneration, modulation format conversion are the important signal processing functions. The limiting factors in the existing ultra long haul optical communication systems are spectral efficiency (SE), fiber attenuation, insertion losses, chromatic dispersion, polarization mode dispersion (PMD), and the optical fiber nonlinearity [1]-[4]. Optical communication systems can be divided into two groups. In the first group, an electrical signal modulates the intensity of an optical carrier inside the optical transmitter, the modulated optical signal is transmitted through the optical fiber and converted to the original electrical signal by an optical receiver [1]. Such a scheme is called an intensity modulation with direct detection (IM/DD) [1], [5]. In the second group, information is transmitted by modulation of the optical carrier frequency or phase, and then the modulated optical signal can be linearly down-converted to a baseband electrical signal by heterodyne or homodyne detection [1], [5]. The phase coherence of the optical carrier is essential for the realization of the second group called coherent optical communication systems [1].

Coherent optical communication systems possess the following advantages with respect to IM/DD systems: (i) the shot-noise limited receiver can be achieved with a sufficient local oscillator (LO) power; (ii) the frequency resolution at the intermediate frequency (IF) or baseband stage is high enough in order to separate close wavelength-division multiplexed (WDM) channels in the electric domain; (iii) the phase detection improves the receiver sensitivity compared with IM/DD systems; (iv) the multilevel modulation formats can be introduced into optical communications by using phase modulation [5].

The rapid development of digital communication and digital signal processing (DSP) caused by the necessity of high SE and, at the same time, by the advance of the high-speed

© 2012 Lembrikov et al., licensee InTech. This is an open access chapter distributed under the terms of the 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. © 2012 Lembrikov et al., licensee InTech. This is a paper distributed under the terms of the 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.

electronics required advanced modulation, coding and digital equalization [6]. Recently, coherent optical communications attracted a large interest due to the feasibility of the high SE, the large bandwidth and multilevel modulation formats [5]. Coherent optical systems using multilevel modulation formats can increase SE up to *M* b/s/Hz where *M* is the number of bits per symbol for a given modulation format [5].

Coherent optical OFDM (CO-OFDM) has been recently proposed in order to increase receiver sensitivity, SE, and especially, provide the dispersion compensation at high data transmission rates tending to 100Gb/s [3], [4], [7], [8]. Generally, OFDM is a kind of multicarrier modulation (MCM) in which the data information is carried over many lower rate subcarriers [4]. In OFDM spectra of individual subcarriers overlap, but due to the orthogonality, the subcarriers can be demodulated without interference and without analog filtering for the received subcarrier separation [7]. The signal processing in the OFDM system can be carried out by using the Fast Fourier Transform (FFT)/Inverse Fast Fourier Transform (IFFT) [3], [4], [7]. In CO-OFDM systems the digital signal processing is used in order to mitigate the channel dispersion, nonlinearity and different types of noise.

CO-OFDM system combines the following advantages of coherent detection and OFDM modulation essential for the high-speed optical fiber communication systems: (i) in CO-OFDM systems, chromatic dispersion and PMD can be mitigated; (ii) the SE of CO-OFDM systems is high due to the partial overlapping of OFDM sub-carrier spectra; (iii) linearity in radio frequency (RF)-to-optical (RTO) up-conversion and optical-to-RF (OTR) downconversion; (iv) the electrical bandwidth requirements for the CO-OFDM transceiver can be greatly reduced by using direct up/down conversion which results in the low cost of the high-speed electronic circuits [3], [4], [8]. Recently, all-optical FFT scheme enabling Tbit/s real-time signal processing has been proposed [9]. The method based on only passive optical components realizes the highest speed signal processing without the power consumption where electronics cannot be used. This approach combines the advantages of the electronic high precision processing of the low bit rates and the optical processing of high bit rates [9].

However, OFDM is characterized by the inter-symbol-interference (ISI) and inter-carrierinterference (ICI) caused by a large number of subcarriers [4]. In the RF systems ISI is mainly due to multipath channel delay spread [10], [11] and ICI is mainly due to the carrier frequency offset [12]. In the case of CO-OFDM, ISI and ICI are caused by channel chromatic dispersion and PMD [3], [4]. A so-called cyclic prefix (CP), i.e. the cyclic extension of the OFDM waveform into the guard interval (GI) *<sup>G</sup>* , has been proposed in order to prevent ISI and ICI [4]. If the GI is long enough to contain the intersymbol transition, then the remaining part of the OFDM symbol satisfies the orthogonality condition and receiver crosstalk occurs only within GI [9]. The addition of CP requires an increase of a bandwidth and sampling rate of analog-to-digital converter (ADC) and digital-to-analog converter (DAC). CP appeared to be an easily recognizable feature of an OFDM system making the signal vulnerable to interception by surveillance receiver [10]. The elimination of CP reduces the probability of interception and improves SE [10].

The need for CP can be avoided if the wavelet packet transform (WPT) is used in CO-OFDM systems instead of discrete Fourier Transform (DFT) and inverse DFT (IDFT) [13]. The sinusoidal functions are infinitely long in the time domain while wavelets have finite length being localized in time and in frequency domains [13]. Wavelet signal analysis can be a base for an effective computational algorithm which is faster and simpler than FFT [14]. Wavelets have been used in optical communications for time-frequency multiplexing and ultrafast image transmission [14]. A signal may be expanded in an orthogonal set of wavelet packets (WPs) as the basis functions, each channel occupies a wavelet packet (WP), and IDWPT/ DWPT are used at the transmitter and receiver, respectively [13].

In this chapter, we consider the CO-OFDM based on WPT and its influence on the optical communication network performance. The chapter is constructed as follows. In Section 2, we review the coherent optical communication systems. In Section 3, we discuss high SE CO-OFDM system. In Section 4, we discuss the OFDM based on WPT and present the original results for the WPT-OFDM system performance. In Section 5, we present the original results concerning the simulations of the structure and operation mode of the novel passive components for all-optical signal processing based on Si-on-insulator (SOI) structure, and a novel hierarchical architecture of the 1Tb/s transmission system based on WPT-OFDM [15]. In Section 6, conclusions are presented.

#### **2. Coherent optical communication systems**

344 Optical Communication

rates [9].

electronics required advanced modulation, coding and digital equalization [6]. Recently, coherent optical communications attracted a large interest due to the feasibility of the high SE, the large bandwidth and multilevel modulation formats [5]. Coherent optical systems using multilevel modulation formats can increase SE up to *M* b/s/Hz where *M* is the number

Coherent optical OFDM (CO-OFDM) has been recently proposed in order to increase receiver sensitivity, SE, and especially, provide the dispersion compensation at high data transmission rates tending to 100Gb/s [3], [4], [7], [8]. Generally, OFDM is a kind of multicarrier modulation (MCM) in which the data information is carried over many lower rate subcarriers [4]. In OFDM spectra of individual subcarriers overlap, but due to the orthogonality, the subcarriers can be demodulated without interference and without analog filtering for the received subcarrier separation [7]. The signal processing in the OFDM system can be carried out by using the Fast Fourier Transform (FFT)/Inverse Fast Fourier Transform (IFFT) [3], [4], [7]. In CO-OFDM systems the digital signal processing is used in

order to mitigate the channel dispersion, nonlinearity and different types of noise.

CO-OFDM system combines the following advantages of coherent detection and OFDM modulation essential for the high-speed optical fiber communication systems: (i) in CO-OFDM systems, chromatic dispersion and PMD can be mitigated; (ii) the SE of CO-OFDM systems is high due to the partial overlapping of OFDM sub-carrier spectra; (iii) linearity in radio frequency (RF)-to-optical (RTO) up-conversion and optical-to-RF (OTR) downconversion; (iv) the electrical bandwidth requirements for the CO-OFDM transceiver can be greatly reduced by using direct up/down conversion which results in the low cost of the high-speed electronic circuits [3], [4], [8]. Recently, all-optical FFT scheme enabling Tbit/s real-time signal processing has been proposed [9]. The method based on only passive optical components realizes the highest speed signal processing without the power consumption where electronics cannot be used. This approach combines the advantages of the electronic high precision processing of the low bit rates and the optical processing of high bit

However, OFDM is characterized by the inter-symbol-interference (ISI) and inter-carrierinterference (ICI) caused by a large number of subcarriers [4]. In the RF systems ISI is mainly due to multipath channel delay spread [10], [11] and ICI is mainly due to the carrier frequency offset [12]. In the case of CO-OFDM, ISI and ICI are caused by channel chromatic dispersion and PMD [3], [4]. A so-called cyclic prefix (CP), i.e. the cyclic extension of the OFDM waveform into the guard interval (GI) *<sup>G</sup>* , has been proposed in order to prevent ISI and ICI [4]. If the GI is long enough to contain the intersymbol transition, then the remaining part of the OFDM symbol satisfies the orthogonality condition and receiver crosstalk occurs only within GI [9]. The addition of CP requires an increase of a bandwidth and sampling rate of analog-to-digital converter (ADC) and digital-to-analog converter (DAC). CP appeared to be an easily recognizable feature of an OFDM system making the signal vulnerable to interception by surveillance receiver [10]. The elimination of CP reduces the

of bits per symbol for a given modulation format [5].

probability of interception and improves SE [10].

The number of publications concerning the coherent optical communications is enormous. In this section, we can only relate to a limited number of the fundamental works concerning the concept of the coherent detection and the modulation formats used in digital communication system since these topics are related to high SE CO-OFDM systems.

The most advanced detection method is coherent detection based on the recovery of the full electric field containing both amplitude and phase information [16]. The concept of the coherent detection is to combine in a receiver the modulated optical signal with a continuous wave (CW) optical field generated by a narrow linewidth laser, or local oscillator (LO) before the photodetector (PD) [1]. Coherent detection requires the carrier synchronization with respect to LO that serves as an absolute phase reference [16]. For this purpose, optical systems can use two types of the phase-locked loops (PLLs): (i) an optical PLL (OPLL) that synchronizes the frequency and phase of the LO laser with the transmitter laser; (ii) an electrical PLL where down-conversion with a free-running LO laser is followed by a second-stage demodulation by an electrical voltage controlled oscillator (VCO) with the synchronized frequency and phase [16].

The basic component of coherent optical systems is a coherent optical receiver [1], [5]. Its block diagram is shown in Figure 1 [5].

The electric fields *<sup>s</sup> E t* and *LO E t* of the received optical signal and LO, respectively, are given by [5]

$$E\_s(t) = A\_s(t) \exp\left[j\left(\alpha\_0 t + \phi\_s\right)\right]; \quad E\_{LO}\left(t\right) = A\_{LO} \exp\left[j\left(\alpha\_{LO} t + \phi\_{LO}\right)\right] \tag{1}$$

**Figure 1.** Block diagram of the coherent receiver

where <sup>0</sup> , , , ,, *AtA s LO LO s LO* are the amplitudes, frequencies and phases of the received optical signal and LO, respectively. A 3db optical coupler (OC) is used that adds a phase shift to either the signal field or the LO field between the two output ports. When the signal and LO fields are co-polarized, the electric fields 1,2 *E* incident on the upper and lower diodes are given by, respectively.

$$E\_{1,2} = \frac{1}{\sqrt{2}} \left[ E\_s(t) \pm E\_{LO}(t) \right] \tag{2}$$

The output photocurrents 1,2 *I t* are given by, respectively [5]

$$I\_{1,2}\left(t\right) = \frac{R}{2}\left(P\_s + P\_{LO} \pm 2\sqrt{P\_s P\_{LO}}\cos\left(\rho\_{IF}t + \phi\_s - \phi\_{LO}\right)\right) \tag{3}$$

where 2 2 <sup>0</sup> / 2, / 2, *s s LO LO IF LO P At P A* is IF, <sup>0</sup> / *R e <sup>q</sup>* is the detector responsivity, *e* is the electron charge, *<sup>q</sup>* is the PD quantum efficiency, *h h* /2 , is the Planck's constant. The sum frequency component is neglected since it is averaged out to zero over the bandwidth of PD. Balanced detection is used in order to suppress the DC component and maximize the signal photocurrent *I t* at the balanced detector output given by [5]

$$I\left(t\right) = 2R\sqrt{P\_s\left(t\right)P\_{LO}}\cos\left(\alpha\_{IF}t + \phi\_s - \phi\_{LO}\right) \tag{4}$$

Eq. (4) demonstrates the main advantage of the coherent detection as compared to the direct detection. The photocurrent *I t* contains explicitly the signal phase *<sup>s</sup>* making possible to transmit information by modulating the phase or frequency of the carrier signal [1], [5]. There are two cases of the coherent detection: (i) the heterodyne detection when the signal carrier frequency ω₀ and the LO frequency *LO* are different, and /2 1 *IF IF f GHz* ; (ii) the homodyne detection when the signal carrier frequency ω₀ and the LO frequency *LO* coincide, and IF 0 *IF* [1]. In the heterodyne detection case, eq. (4) describes the output photocurrent. Typically, the LO power is much larger than the received signal power: *LO s P P* , and LO amplifies the received signal improving signal-to-noise ratio (SNR) [1]. In the case of the heterodyne detection, the optical signal is demodulated in two stages: it is first down-converted to IF and then to the baseband [1]. The output photocurrent *I t* (4) then takes the form.

346 Optical Communication

**Figure 1.** Block diagram of the coherent receiver

are the amplitudes, frequencies and phases of the received

<sup>2</sup> *s LO E Et E t* (2)

 

 is IF, <sup>0</sup> / *R e <sup>q</sup>* 

is the PD quantum efficiency, *h h* /2 ,

phase

(3)

(4)

is the detector

is the

making possible to

optical signal and LO, respectively. A 3db optical coupler (OC) is used that adds a

The output photocurrents 1,2 *I t* are given by, respectively [5]

<sup>0</sup> / 2, / 2, *s s LO LO IF LO P At P A*

shift to either the signal field or the LO field between the two output ports. When the signal and LO fields are co-polarized, the electric fields 1,2 *E* incident on the upper and lower

> 1,2 1

 1,2 2 cos 2 *s LO s LO IF s LO*

Planck's constant. The sum frequency component is neglected since it is averaged out to zero over the bandwidth of PD. Balanced detection is used in order to suppress the DC component and maximize the signal photocurrent *I t* at the balanced detector output

2 cos *s LO IF s LO It R P tP t*

Eq. (4) demonstrates the main advantage of the coherent detection as compared to the direct

transmit information by modulating the phase or frequency of the carrier signal [1], [5]. There are two cases of the coherent detection: (i) the heterodyne detection when the signal

 

*<sup>R</sup> I t P P PP t*

 

detection. The photocurrent *I t* contains explicitly the signal phase *<sup>s</sup>*

where <sup>0</sup> , , , ,, *AtA s LO LO s LO* 

diodes are given by, respectively.

where 2 2

given by [5]

responsivity, *e* is the electron charge, *<sup>q</sup>*

$$I\left(t\right) = 2R\sqrt{P\_s\left(t\right)P\_{LO}}\cos\left(\phi\_s - \phi\_{LO}\right) \tag{5}$$

Similarly, the increase of the average electrical power up to 20dB can occur in the case of the homodyne detection. If additionally, the LO phase is locked to the signal phase so that 0 *s LO* eq. (5) takes the form [1].

$$I\left(t\right) = 2R\sqrt{P\_s\left(t\right)P\_{LO}}\tag{6}$$

The phase difference *s LO* can be kept constant by using an OPLL. However, the implementation of OPLL makes the design of optical homodyne receivers a comparatively complicated problem [1], [5].

The coherent detection allows the greatest flexibility in modulation formats, since information can be encoded by modulating the amplitude, the phase, or the frequency of an optical carrier as it is seen from equations (1)-(6) or in both in-phase (I) and quadrature (Q) components of the carrier [1], [16]. In the case of the digital communication systems, these methods correspond to three modulation formats: amplitude-shift keying (ASK), phase- shift keying (PSK), and frequency-shift keying (FSK) [1]. The increased performance, speed, and reliability, and the reduced size and cost of integrated circuits permit to use DSP for the information recovery from the baseband signal [5]. Typically, the M-ary PSK modulation is used in SE high-speed CO-OFDM systems such as quaternary PSK (QPSK) (M=4), 8-PSK (M=8), as well as quadrature amplitude modulation (QAM) such as 4-QAM, 16-QAM, 64- QAM in single or dual polarization [4], [5]. The digital coherent receiver linearly detects incoming signal including phase and polarization diversities and converts this information to digital data by using ADCs while the digital information is processed by DSP circuits [5].

#### **3. High-speed and high SE CO-OFDM system**

In this section we present a brief review of the operation principle and architecture of CO-OFDM system. Detailed analysis of CO-OFDM in optical communication systems may be found in the book [4].

A generic optical OFDM system consists of five functional blocks: the RF OFDM transmitter, the RTO up-converter, the optical channel, the OTR down-converter, and the RF OFDM receiver [3], [4]. In such a system the following chain of events occurs [3]. The input data bits are mapped onto corresponding information symbols of the subcarriers within one OFDM symbol. The digital time domain signal is obtained by using IDFT. It is inserted with the GI Δ*G* in order to prevent ISI caused by channel dispersion and converted into the real time waveform through DAC [3]. The baseband signal is up-converted to an appropriate RF band with an IQ mixer/modulator. A linear RTO up-conversion can be achieved by using Mach-Zehnder modulator (MZM) [3]. MZM is mainly used for the bit rates of 40GB/s and higher due to its high modulation performance and the possibility of independent modulation of the electric field intensity and phase [6]. At the receiver, the OFDM signal is downconverted to baseband with an IQ demodulator, sampled by an ADC and demodulated by DFT and baseband DSP to recover the data [3]. A linear OTR down-conversion is provided by a coherent detection described in section 2. The high performance of the CO-OFDM transmission systems has been shown both theoretically and experimentally [8], [17]. A single-channel 1Tb/s CO-OFDM signal consisting of 4104 spectrally-overlapped subcarriers with SE of 3.3bit/s/Hz has been generated, transmitted over 600km standard single mode fiber (SSMF) without amplification and dispersion compensation, and successfully received [17]. However, CO-OFDM system is extremely sensitive to nonlinearity and channel dispersion. The dispersion mitigation with the dispersion compensation fiber (DCF) results in the additional noise and nonlinear effects decreasing the system performance [8].

Consider now the analytical expressions describing the CO-OFDM signals. The MCM transmitted signal *s t* is given by [3]

$$s(t) = \sum\_{i=-\infty}^{\infty} \sum\_{k=1}^{N\_{SC}} c\_{ki} s\_k \left( t - iT\_s \right) \tag{7}$$

$$s\_k(t) = \Pi(t) \exp\{j2\pi f\_k t\}; \Pi(t) = \begin{cases} 1, & 0 < t \le T\_s \\ 0, & t \le 0, t > T\_s \end{cases} \tag{8}$$

where *ki c* is the *i* th information symbol at the *k* th subcarrier, , *k k s f* are the waveform and the frequency of the *k* th subcarrier, respectively, *NSC* is the number of subcarriers, and *<sup>s</sup> T* is the symbol period. The optimum detector for each subcarrier could use a filter matched to the subcarrier waveform, or a correlator matched to subcarrier [3]. Eq. (7) shows that the modulation can be performed by IDFT of the input information signal *ki c* .

The detected information signal *ik c* at the output of the correlator has the form [3].

$$c'\_{ik} = \int\_0^{T\_s} r \left(t - iT\_s\right) \exp\left(-j2\pi f\_k t\right) dt\tag{9}$$

where *<sup>s</sup> r t iT* is the received time-domain signal. Eq. (9) shows that the demodulation is provided by DFT of the sampled received signal *r t* [3]. The classical MCM uses nonoverlapped band-limited signals. In order to prevent overlapping of the band-limited signals, a bank of a large number of oscillators and filters is necessary at the transmitter and the receiver [3]. The cost-efficient design of the filters and oscillators requires that the channel spacing should be multiple of the symbol rate. As a result, SE reduces and the required bandwidth increases [3].

The OFDM technique permits to use the overlapped signals under the condition that they are orthogonal [7]. The orthogonality condition for any two subcarriers *<sup>k</sup> s t* and *ls t* is given by [3].

$$\frac{1}{T\_s} \prod\_{0}^{T\_s} s\_k\left(t\right) s\_l^\*\left(t\right) dt = \delta\_{kl} = \begin{cases} 1, & k=l\\ 0, & k \neq l \end{cases} \tag{10}$$

Substituting expression (8) into condition (10) we obtain.

$$\exp\left[j\pi\left(f\_k - f\_l\right)T\_s\right] \frac{\sin\left[\pi\left(f\_k - f\_l\right)T\_s\right]}{\pi\left(f\_k - f\_l\right)T\_s} = \delta\_{kl}\tag{11}$$

The left-hand side of (11) vanishes when

348 Optical Communication

are mapped onto corresponding information symbols of the subcarriers within one OFDM symbol. The digital time domain signal is obtained by using IDFT. It is inserted with the GI Δ*G* in order to prevent ISI caused by channel dispersion and converted into the real time waveform through DAC [3]. The baseband signal is up-converted to an appropriate RF band with an IQ mixer/modulator. A linear RTO up-conversion can be achieved by using Mach-Zehnder modulator (MZM) [3]. MZM is mainly used for the bit rates of 40GB/s and higher due to its high modulation performance and the possibility of independent modulation of the electric field intensity and phase [6]. At the receiver, the OFDM signal is downconverted to baseband with an IQ demodulator, sampled by an ADC and demodulated by DFT and baseband DSP to recover the data [3]. A linear OTR down-conversion is provided by a coherent detection described in section 2. The high performance of the CO-OFDM transmission systems has been shown both theoretically and experimentally [8], [17]. A single-channel 1Tb/s CO-OFDM signal consisting of 4104 spectrally-overlapped subcarriers with SE of 3.3bit/s/Hz has been generated, transmitted over 600km standard single mode fiber (SSMF) without amplification and dispersion compensation, and successfully received [17]. However, CO-OFDM system is extremely sensitive to nonlinearity and channel dispersion. The dispersion mitigation with the dispersion compensation fiber (DCF) results

in the additional noise and nonlinear effects decreasing the system performance [8].

transmitted signal *s t* is given by [3]

Consider now the analytical expressions describing the CO-OFDM signals. The MCM

 1

*s t t j ft t t tT* 

where *ki c* is the *i* th information symbol at the *k* th subcarrier, , *k k s f* are the waveform and the frequency of the *k* th subcarrier, respectively, *NSC* is the number of subcarriers, and *<sup>s</sup> T* is the symbol period. The optimum detector for each subcarrier could use a filter matched to the subcarrier waveform, or a correlator matched to subcarrier [3]. Eq. (7) shows that the

*i k s t c s t iT* 

1, 0 exp 2 ; 0, 0,

*k k*

modulation can be performed by IDFT of the input information signal *ki c* .

0

*s T*

The detected information signal *ik c* at the output of the correlator has the form [3].

*ik <sup>s</sup> <sup>k</sup> c r t iT j f t dt*

where *<sup>s</sup> r t iT* is the received time-domain signal. Eq. (9) shows that the demodulation is provided by DFT of the sampled received signal *r t* [3]. The classical MCM uses nonoverlapped band-limited signals. In order to prevent overlapping of the band-limited signals, a bank of a large number of oscillators and filters is necessary at the transmitter and the receiver [3]. The cost-efficient design of the filters and oscillators requires that the

exp 2

*ki k s*

(7)

*t T*

(9)

*s*

*s*

(8)

*NSC*

$$f\_k - f\_l = \frac{m}{T\_s}; m = 1, 2, \dots \tag{12}$$

Then, the two subcarriers *<sup>k</sup> s t* and *ls t* are orthogonal and can be recovered with the matched filters according to (9) without ICI despite the signal spectral overlapping [3].

In the high speed CO-OFDM systems the problem of ISI and ICI caused by the channel dispersion is critical. ISI is caused by the interference between "slow" and "fast" subcarriers. ICI is due to the breaking of the orthogonality condition (12) for the subcarriers [3]. In order to prevent ISI and ICI, CP was proposed that is realized by cyclic extension of the OFDM waveform into GI [3]. The waveform in GI is essentially an identical copy of that in the DFT window [3]. The condition for ISI-free OFDM transmission requires that the dispersive channel time delay spread *d G t* [3].

SE is defined as the ratio of net per-channel information data rate B to WDM channel spacing *f* and measured in b/s/Hz [1], [18]. SE of CO-OFDM is given by [3]

$$\eta = 2 \frac{R\_s}{B\_{\text{OFDM}}} \tag{13}$$

where / *RN T s SC s* is the total symbol rate, *B TN t OFDM s SC s* 2 / 1 / is the OFDM bandwidth, *st* is the observation period, and the factor of 2 is taking into account two polarizations of the optical fiber modes. Typically, the subcarriers number is large: 1 *NSC* . Then eq. (13) takes the form: 2 / *s s t T* . The optical SE of 3.6bit/Hz can be achieved for QPSK modulation of subcarriers, and can be improved by using higher-order QAM modulation format [3]. However, the addition of CP requires an increase of a bandwidth *BOFDM* and sampling rate of ADC and DAC. The need for CP can be avoided if WPT is used in CO-OFDM systems instead of DFT and IDFT [13]. This approach will be discussed in the next section.

#### **4. WPT based CO-OFDM**

WPT can be used in CO-OFDM instead of the IDFT/DFT since it improves the system performance, and in particular, mitigates the channel chromatic dispersion without CP [13]. In this section we briefly discuss the main features of WPT and its applications to CO-OFDM. The theory and applications of continuous wavelet transform (CWT) and discrete WT (DWT) can be found in a large number of books and articles (see, for example, [13], [14], [19]-[22] and references therein).

CWT *W a <sup>T</sup>* , of a given function f(t) with respect to a mother wavelet (MW) ψ(t) is defined as follows [19], [20]

$$\mathcal{W}\_{\Gamma} \left( a, \tau \right) = \frac{1}{\sqrt{|a|}} \prod\_{-\upsilon}^{\upsilon} \varphi^\* \left( \frac{t - \tau}{a} \right) f \left( t \right) dt \tag{14}$$

where the real numbers *a* and are the scaling and shifting, or translation parameters, respectively, and asterisk means complex conjugation. Note that in many practically important cases MW ψ(t) is real. The functions 1/2 , / *<sup>a</sup> sa s a* are called wavelets [20]. The set of wavelets is orthogonal and can be used as a basis instead of sinusoidal functions [13]. It is possible to localize the events described by f(t) in time and frequency domains simultaneously by means of WT choosing the appropriate values of the parameters *a* and [19]. For this reason, wavelets are used in the multiresolution analysis (MRA) which decomposes a signal at different scales, or resolutions, using a basis whose elements are localized both in time and in frequency domains [14].

DWT is given by [19], [20]

$$\mathcal{W}\_{\Gamma}^{m,n}\left(a,\tau\right) = a\_0^{-m/2} \int\_{-\infty}^{\infty} \varphi^\* \left(a\_0^{-m}t - n\tau\_0\right) f\left(t\right) dt\tag{15}$$

where m,n∈Z, Z is the set of all integers, and the constants 0 0 *a* 1, 1 . Comparison of eqs. (14) and (15) shows that 0 *<sup>m</sup> a a* and 0 0 *m n a* [20]. The orthogonal wavelet series expansions can be successfully used in DSP and multiplexing when the scaling and translation parameters are discrete [14]. In such a case, a signal s(t) ∈V₀ can be represented by a smooth approximation at resolution 2*<sup>M</sup>* , obtained by combining translated versions of the basic scaling function *t* , and M details at the dyadic scales 2 , 1,2,..., 1 *<sup>l</sup> al M* obtained by combining shifted and dilated versions of the MW ψ(t) as follows [14].

$$s(t) = \sum\_{k} 2^{-M/2} c\_M \left\lceil k \right\rceil \phi \left( 2^{-M} t - k \Delta \tau \right) + \sum\_{l=1}^{M} \sum\_{k} 2^{-l/2} d\_l \left\lceil k \right\rceil \psi \left( 2^{-l} t - k \Delta \tau \right) \tag{16}$$

Here a subspace V₀∈L²(R), L²(R) is a the linear vector space of square integrable functions, /2 2 2 *l l t k* and /2 2 2 *l l t k* are the orthonormal bases for the subspaces

 <sup>2</sup> *V LR <sup>l</sup>* and <sup>2</sup> *W LR <sup>l</sup>* , respectively, *V W l l* , (l,k)∈Z, *<sup>l</sup> c k* and *<sup>l</sup> d k* are the scaling and detail coefficients, respectively, at resolution 2*<sup>l</sup>* , Δτ is the time interval coinciding with the inverse of the free spectral range (FSR). The scaling function *t* and wavelet function ψ(t) satisfy the dilation equations [14], [19], [21]

$$\phi(t) = \sqrt{2} \sum\_{k} h\left[k\right] \phi\left(2t - k\Delta\tau\right); \nu\left(t\right) = \sqrt{2} \sum\_{k} g\left[k\right] \phi\left(2t - k\Delta\tau\right) \tag{17}$$

where h[k] and g[k] are the coefficients of two half-band (HB) quadrature mirror filters (QMFs) described by the following functions H(ω) and G(ω)

$$H(\boldsymbol{\omega}) = \frac{1}{\sqrt{2}} \sum\_{k} h[\,^{}k\,] \exp\left(-j\, \boldsymbol{\omega} \boldsymbol{k} \, \boldsymbol{\omega}\tau\right); G(\boldsymbol{\omega}) = \frac{1}{\sqrt{2}} \sum\_{k} g[\,^{}k\,] \exp\left(-j\, \boldsymbol{\omega} \boldsymbol{k} \, \boldsymbol{\omega}\tau\right) \tag{18}$$

and Δτ is the inverse of their FSR. The functions H(ω) and G(ω) (18) satisfy the conditions [14], [22].

$$\left| H(o) \right|^2 + \left| G(o) \right|^2 = 1; G(o) = \exp \left( -j o k \Delta \tau \right) H^\* \left( o + \frac{\pi}{\Delta \tau} \right) \tag{19}$$

The evaluation of the discrete wavelet coefficients is equivalent to filtering the signal s(t) by a cascade of mutually orthogonal bandpass filters [21]. An optical HB filter can be realized by using Mach-Zehnder interferometers (MZIs) [14], [22].

The DWT decomposition procedure is described by the following recursive expressions for the scaling and detail coefficients , *l l cndn* [14], [22]

$$c\_l \left[ n \right] = \sum\_{k} c\_{l-1} \left[ k \right] \hbar \left[ 2n - k \right] \mu\_l d\_l \left[ n \right] = \sum\_{k} c\_{l-1} \left[ k \right] g \left[ 2n - k \right] \tag{20}$$

where

350 Optical Communication

CWT *W a <sup>T</sup>* ,

**4. WPT based CO-OFDM** 

[19]-[22] and references therein).

where the real numbers *a* and

(14) and (15) shows that 0

<sup>1</sup> , *<sup>T</sup>*

elements are localized both in time and in frequency domains [14].

where m,n∈Z, Z is the set of all integers, and the constants 0 0 *a* 1, 1

*st c k t k*

*t k*

*<sup>m</sup> a a* and 0 0

obtained by combining shifted and dilated versions of the MW ψ(t) as follows [14].

/2 /2

*k l k*

  

important cases MW ψ(t) is real. The functions 1/2 , / *<sup>a</sup> sa s a*

as follows [19], [20]

parameters *a* and

DWT is given by [19], [20]

the basic scaling function

 /2 2 2 *l l t k*

and /2 2 2 *l l*

WPT can be used in CO-OFDM instead of the IDFT/DFT since it improves the system performance, and in particular, mitigates the channel chromatic dispersion without CP [13]. In this section we briefly discuss the main features of WPT and its applications to CO-OFDM. The theory and applications of continuous wavelet transform (CWT) and discrete WT (DWT) can be found in a large number of books and articles (see, for example, [13], [14],

> *<sup>t</sup> W a f t dt a a*

 

 

 

respectively, and asterisk means complex conjugation. Note that in many practically

wavelets [20]. The set of wavelets is orthogonal and can be used as a basis instead of sinusoidal functions [13]. It is possible to localize the events described by f(t) in time and frequency domains simultaneously by means of WT choosing the appropriate values of the

(MRA) which decomposes a signal at different scales, or resolutions, using a basis whose

 , /2 0 00 , *m n <sup>m</sup> <sup>m</sup> W a a a t n f t dt <sup>T</sup>*

 

expansions can be successfully used in DSP and multiplexing when the scaling and translation parameters are discrete [14]. In such a case, a signal s(t) ∈V₀ can be represented by a smooth approximation at resolution 2*<sup>M</sup>* , obtained by combining translated versions of

2 2 22

Here a subspace V₀∈L²(R), L²(R) is a the linear vector space of square integrable functions,

*m*

1

*M M M ll M l*

 *dk t k* (16)

of a given function f(t) with respect to a mother wavelet (MW) ψ(t) is defined

 

[19]. For this reason, wavelets are used in the multiresolution analysis

(14)

 

(15)

*t* , and M details at the dyadic scales 2 , 1,2,..., 1 *<sup>l</sup> al M*

are the orthonormal bases for the subspaces

*n a* [20]. The orthogonal wavelet series

are called

. Comparison of eqs.

are the scaling and shifting, or translation parameters,

$$c\_0 = \int s(t)\phi(t - n\Lambda\tau)dt\tag{21}$$

In the DWT case only the scaling coefficients *<sup>l</sup> c n* are recursively filtered, while the detail coefficients *<sup>l</sup> d n* are not reanalyzed [14]. In the case of the WP decomposition both the scaling coefficients *<sup>l</sup> c n* and the detail coefficients *<sup>l</sup> d n* are recursively decomposed following the same filtering and subsampling scheme, and consequently, all outputs have the same number of samples span over the same frequency bandwidth [14]. The WP decomposition based on the wavelet atom functions *w t l m*, is performed as follows [14]

$$w\_{l+1,2,m}\left(t\right) = \sum\_{k} h\left[k\right] w\_{l,m}\left(t - \mathcal{Z}^l k \Delta \tau\right) \tag{22}$$

$$w\_{l+1,2m+1}\left(t\right) = \sum\_{k} \mathcal{g}\left[k\right] w\_{l,m}\left(t - \mathcal{2}^l k \Delta \tau\right) \tag{23}$$

where *l* is the decomposition level, 0 2 1 *<sup>l</sup> m* is the wavelet atom position in the tree, *w t* 0,0 and

$$w\_{l,m}\left(t\right) = \sum\_{k} f\_{l,m} \left\lceil k \right\rceil \phi\left(t - k\Lambda\tau\right) \tag{24}$$

and *l m*, *f k* is the equivalent filter from the root to the *l m*, th terminal recursively evaluated using eqs. (22), (23). The orthogonality condition for WP atoms has the form [14]

$$\int w\_{l,m} \left( t - 2^l n \tau \right) w\_{\lambda, \mu} \left( t - 2^{\lambda} k \tau \right) dt = \delta \left( l - \lambda \right) \delta \left( m - \mu \right) \delta \left( n - k \right) \tag{25}$$

where *l Z* , , 0 2 1,0 2 1 *<sup>l</sup> m* , *nk Z* , . The waveform orthogonality is used in WPT based OFDM in order to transmit multiple message signals overlapping in time and frequency domains [14]. The time and frequency localization of wavelets can mitigate the optical channel chromatic dispersion which affects only the detail coefficients, or the highpass-filtered versions of the original signal. Then, a selective reconstruction of the wavelet coefficients is necessary [14].

WPTs can provide orthogonality between OFDM subcarriers similarly to DFT, and consequently, DWPT can replace DFT in the CO-OFDM system [13]. The all-optical WPT based CO-OFDM (WPDM) system has been proposed where the digital sequences are encoded by a set of orthogonal waveforms [13], [14]. The system performance is improved due to the orthogonal properties (25) of the wavelet atoms (22)–(24) and their overlapping in time and frequency [13], [14]. Each optical pulse is transformed into the corresponding wavelet atom function at the device output under the conditions that the input bit duration *bit t* and the processing gain 2*<sup>l</sup> F* is equal to the number of simultaneous users [14]. In the WPT-OFDM system each channel occupies a WP [13]. At the transmitter, IDWPT reconstructs the time domain signal from WPs; at the receiver DWPT is used in order to decompose the time domain signal into different WPs by using successive low-pass and high-pass filtering [13]. Unlike IDFT/DFT system, in the IDWPT/DWPT OFDM system the basis function wavelets are finite in time, the inter-symbol orthogonality in WT is maintained due to the shift orthogonal property of waveforms, and symbols are overlapped in time domain [13]. As a result, the symbol duration increases, providing the tolerance with respect to the chromatic dispersion and eliminating the need of CP [13].

Consider the computational complexity *CWPT* of WPT-OFDM defined as the total required number of complex multiplications [23]. It depends on the specific type of wavelets and system configuration. The complexity of one basic block *CBB* determined by the convolution between complex input data and real QMFs, and the total complexity *CWPT* are given by, respectively [23]

$$\mathbf{C}\_{BB} = \mathbf{L}\_{\mathrm{QMF}} \colon \mathbf{C}\_{\mathrm{WP}} = \left(\mathbf{N} - \mathbf{1}\right) \mathbf{L}\_{\mathrm{QMF}} \tag{26}$$

where *QMF L* is QMF length, N is the number of subcarriers. WTP-OFDM reduces the complexity by a factor of 6 to 10 for different wavelets in the range of moderate accumulated dispersion as compared to FFT based CO-OFDM without CP [23].

352 Optical Communication

where *l Z* ,

*bit t* 

given by, respectively [23]

and

 *w t* 0,0 

where *l* is the decomposition level, 0 2 1 *<sup>l</sup> m* is the wavelet atom position in the tree,

and *l m*, *f k* is the equivalent filter from the root to the *l m*, th terminal recursively evaluated using eqs. (22), (23). The orthogonality condition for WP atoms has the form [14]

> , , 2 2 *<sup>l</sup> w t n w t k dt l m n k l m*

in WPT based OFDM in order to transmit multiple message signals overlapping in time and frequency domains [14]. The time and frequency localization of wavelets can mitigate the optical channel chromatic dispersion which affects only the detail coefficients, or the highpass-filtered versions of the original signal. Then, a selective reconstruction of the

WPTs can provide orthogonality between OFDM subcarriers similarly to DFT, and consequently, DWPT can replace DFT in the CO-OFDM system [13]. The all-optical WPT based CO-OFDM (WPDM) system has been proposed where the digital sequences are encoded by a set of orthogonal waveforms [13], [14]. The system performance is improved due to the orthogonal properties (25) of the wavelet atoms (22)–(24) and their overlapping in time and frequency [13], [14]. Each optical pulse is transformed into the corresponding wavelet atom function at the device output under the conditions that the input bit duration

and the processing gain 2*<sup>l</sup> F* is equal to the number of simultaneous users [14].

In the WPT-OFDM system each channel occupies a WP [13]. At the transmitter, IDWPT reconstructs the time domain signal from WPs; at the receiver DWPT is used in order to decompose the time domain signal into different WPs by using successive low-pass and high-pass filtering [13]. Unlike IDFT/DFT system, in the IDWPT/DWPT OFDM system the basis function wavelets are finite in time, the inter-symbol orthogonality in WT is maintained due to the shift orthogonal property of waveforms, and symbols are overlapped in time domain [13]. As a result, the symbol duration increases, providing the tolerance with

Consider the computational complexity *CWPT* of WPT-OFDM defined as the total required number of complex multiplications [23]. It depends on the specific type of wavelets and system configuration. The complexity of one basic block *CBB* determined by the convolution between complex input data and real QMFs, and the total complexity *CWPT* are

*CLC NL BB QMF WPT* ; 1 *QMF* (26)

respect to the chromatic dispersion and eliminating the need of CP [13].

 

 

 

(25)

(24)

 

, *nk Z* , . The waveform orthogonality is used

*k w t f k tk*

*l m*, ,*l m*

 

 

, 0 2 1,0 2 1 *<sup>l</sup> m*

wavelet coefficients is necessary [14].

  The performance of a digital optical communication systems is characterized by the bit error rate (BER) defined as the average probability of incorrect bit identification [1]. The simulations of the BER for WPT-OFDM and FFT based OFDM have been carried out using different wavelets, optical SNR of 20dB , chromatic dispersion parameter of 17ps/(nm⋅km), and forward error correction code (FEC) threshold of 10⁻³ [13]. The results show the chromatic dispersion tolerance of 5600 ps/nm and the longest distance of 330km for SSMF for the Johnston64 (E) wavelet [13].

We have carried out the numerical simulations of BER dependence on the transmission distance in the single polarization regime for the WPT-OFDM system without CP, with GI length of 5% of the symbol interval, and for generic IDFT/DFT systems with values of CP length from 5% up to 30% of the symbol interval. We used the single-polarization signal with the optical carrier frequency 193.1 *opt f THz* , with 128 subcarriers. An optical fiber is

characterized by the attenuation of 0.2dB/km and chromatic dispersion parameter of 17ps/(nm⋅km). We assumed that the efficient transmission can be realized with the BER less than the FEC threshold of 2⋅10⁻². PMD has not been taken into account. At the receiver, we used window synchronization Schmidl - Cox algorithm and 1 tap equalizer in frequency domain.

The BER dependence on the distance for the Haar WPT-OFDM and FFT CO-OFDM with different CPs is shown in Figure 2.

**Figure 2.** BER dependence on the transmission distance for FFT CO-OFDM with different CPs and WPT-OFDM without CP, with GI 5%

The results clearly show that WPT-OFDM provides the efficient transmission up to 500km without CP with 5% GI, while the FFT based CO-OFDM may achieve the same distance with the CP length of 25% of the symbol interval which substantially reduces SE of the communication system.
