**1. Introduction**

Orthogonal frequency division multiplexing (OFDM) is a multicarrier modulation technique to represent the information, which reduces the complexity of receiver digital processing unit while combating the deleterious effects of the channel with simple correction algorithms. It enables one-tap equalization by cyclic prefix (CP) insertion even in frequency selective channel and the use of discrete Fourier transform (DFT) and its extremely efficient and well-established fast Fourier transform (FFT) algorithm for implementation has made it amenable in terms of cost also [1–3]. However, some of the immediate consequences of these compelling benefits in OFDM are: limiting the spectral efficiency because of CP insertion, deleterious impact of high peak-to-average power ratio (PAPR), and serious sensitivity toward transceivers' impairments [4, 5]. The transceivers' impairments, such as phase noise (PHN), carrier frequency offset (CFO), and in-quadrature phase (IQ) imbalance effect, need to be addressed significantly to make the best possible use of limited radio spectrum to further increase throughput as well as user capacity.

While there are many transceivers' impairments that are to be taken into consideration in designing a digital communication system, there is a convincing reason to focus on the PHN precisely. While CFO and IQ imbalance is deterministic, PHN on the other hand is random perturbations in the phase of the carrier signal

generated by the transceiver oscillators [6–10]. Moreover, the multicarrier systems, such as OFDM, suffer a much loss in signal-to-noise ratio (SNR) due to PHN than single carrier systems. This is the result of longer duration of multicarrier symbol and the loss of orthogonality between the subcarriers. Further, PHN severely limits the performance of systems that employ dense constellations and degradation gets more pronounced in high-carrier-frequency systems.

## **2. Phase noise**

The autonomous system, oscillator provides a periodic cosinusoidal reference signal used for up/down conversion of the baseband/RF signal to/from RF/ baseband frequency. In practice, wireless digital communication systems use either oscillator in isolation, known as free-running oscillator (FRO) or phase-locked loop (PLL) oscillator because of its high stability and easy control. Either FRO or PLL voltage control oscillator (VCO), in an ideal oscillator for a perfect periodic signal: the transition of phase over a time interval should be constant, whereas practically this phase increment is a random variable. This random variation of phase is phase jitter and its instantaneous deviation is called PHN [11–13]. Thus, the output of a practical oscillator is noisy and can be written as:

$$s(t) = \left[A + a(t)\right] \sin\left[\alpha\_t t + \theta(t)\right] \tag{1}$$

where *A* and *ω<sup>c</sup>* ¼ 2*πf <sup>c</sup>* are amplitude and angular frequency, respectively, and a tð Þ is amplitude fluctuation, which can be kept in limit by using an automatic gain control (AGC). *θ*ð Þ*t* , the phase fluctuation (time-varying PHN), is very difficult to mitigate and can have major impact on system performance.

Phase fluctuations, resulting in the random shifting of oscillator frequency, have its origin in the noise sources present in the internal circuitry of an oscillator. These noise sources can be categorized into white (uncorrelated) and color (correlated) noise sources [14]. The white noise has the flat power spectral density (PSD) where the PSD of color noise is proportional to <sup>1</sup>*=f.* The generated PHN in an oscillator, because of these white and color noise sources, has two components. First is resulting from direct amplification/attenuation of the white and color noise, and the second is due to the phase change of white and color noise, which happens because of the time integration of white and color noise [11–14].

Resulting oscillator PHN spectrum is shown in **Figure 1** where PSD is plotted against frequency *f*. White PHN (flat) and white frequency-modulated (FM) PHN ( <sup>1</sup>*=f* 2) spectra are resulting with white noise sources and flicker PHN ( <sup>1</sup>*=<sup>f</sup>*) and flicker FM PHN ( <sup>1</sup>*=f* 3) spectra are resulting with color noise sources.

For FRO:

$$
\theta\_{n+1} = \theta\_n + \phi\_n \tag{2}
$$

which is Wiener process [15] with mean zero and variance, *σϕ<sup>n</sup>* <sup>2</sup> <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup> *<sup>ϕ</sup>* ¼ 2*πβTs=N* where *β* ¼ 2Δ*f* <sup>3</sup>*dB*, double of 3 dB bandwidth.

For PLL VCO [16].

$$
\theta\_{n+1} = \theta\_n e^{-\phi \frac{T\_t}{N}} + \phi\_{\text{PL.n}} \tag{3}
$$

which is celebrated O-U process where *ϕPLLn* is a sequence of identically and independently distributed (iid) random variables with mean zero and variance:

**Figure 1.** *PSD of PHN in oscillator output.*

$$
\sigma\_{\Phi\_{\rm PL,u}} \, ^2 = 4\pi^2 f\_c \, ^2 \left( \mathbf{C}\_{\rm RO} \frac{T\_s}{N} + 2 \sum\_{i=1}^2 (\xi\_i + \zeta\_i) \left( \mathbf{1} - e^{-\lambda \frac{T\_s}{N}} \right) \right).
$$

where:

$$
\lambda\_{1,2} = \frac{\alpha\_{lpf} \pm \sqrt{\left(\alpha\_{lpf}\right)^2 - 4\alpha\_{lpf}\sqrt{C\_{PLL}}}}{2},
$$

$$
\xi\_1 = \frac{C\_{RO}\lambda\_2}{\left(\lambda\_1 - \lambda\_2\right)\lambda\_1}, \xi\_2 = \frac{-C\_{RO}\lambda\_1}{\left(\lambda\_1 - \lambda\_2\right)\lambda\_2},
$$

$$
\zeta\_1 = \frac{C\_{RO} + C\_{VCO}}{\left(\lambda\_1 - \lambda\_2\right)^2} \left(\frac{\lambda\_2}{2\lambda\_1} - \frac{\lambda\_1 \lambda\_2}{2\left(\lambda\_1 + \lambda\_2\right)}\right),
$$

**Figure 2.** *PHN time samples for FRO and PLL VCO.*

#### *Multiplexing - Recent Advances and Novel Applications*


**Table 1.** *PHN modeling parameters.*

and

$$\zeta\_2 = \frac{\mathbf{C}\_{RO} + \mathbf{C}\_{VCO}}{\left(\lambda\_1 - \lambda\_2\right)^2} \left(\frac{\lambda\_1}{2\lambda\_2} - \frac{\lambda\_1 \lambda\_2}{2\left(\lambda\_1 + \lambda\_2\right)}\right)^2$$

where *f <sup>c</sup>* is the center frequency of VCO in Hz, *ωlpf* is the angular corner frequency of the low-pass filter in rad/sec, and ffiffiffiffiffiffiffiffiffiffi *CPLL* <sup>p</sup> is the PLL bandwidth in Hz. *CRO* and *CVCO* are diffusion rates of the reference oscillator (RO) and VCO, respectively.

The simulated samples of PHN modeled as Wiener process and celebrated O-U process, for FRO and PLL VCO, respectively, are shown in **Figure 2**. Though the time-varying PHN process of FRO can be characterized with *β* only, PLL VCO requires more parameter to characterize such as given in **Table 1**, assuming that the VCO is noisier than reference oscillator.

## **3. OFDM**

OFDM is a low complex modulation/multiplexing multicarrier (MC) technique to modulate *N* orthogonal sub carriers with *N* complex-valued source symbols *Xk*, *k* ¼ 0, 1, … , *N* � 1, efficiently by using digital signal processing. The source symbol is achieved after source coding, interleaving, and channel coding if applicable. The source symbol duration *Td* of the serial data symbol results in the OFDM symbol duration: *Ts* ¼ *N Td*.

From the **Figure 3**, the frequency domain received signal on the *kth* subcarrier of the *mth* symbol is without ISI and ICI and is given by:

$$\mathcal{Y}\_k^m = X\_k^m h\_k + W\_k^m \ 0 \le k \le N - 1 \tag{4}$$

where *X<sup>m</sup> <sup>k</sup>* is *<sup>k</sup>th* element of symbol vector X*<sup>m</sup>*, *hk* is the *<sup>k</sup>th* element of channel vector h ¼ ½ � *h*0, *h*1, *h*2, … , *hN*�<sup>1</sup> *<sup>T</sup>*, *W<sup>m</sup> <sup>k</sup>* is AWGN in frequency domain. It is preferable to represent the signal model in matrix form as:

$$\mathbf{Y}^{m} = \mathbf{D}^{m}\mathbf{F}\mathbf{g} + \mathbf{W}^{m} \tag{5}$$

where **<sup>Y</sup>***<sup>m</sup>* <sup>¼</sup> *<sup>y</sup><sup>m</sup>* <sup>0</sup> , *y<sup>m</sup>* <sup>1</sup> , … , *ym N*�1 � �*<sup>T</sup>* , **F** is the *N* � *L* DFT matrix with *F n*ð Þ¼ , *l exp* � *<sup>j</sup>*2*πnl N* � �, **<sup>D</sup>***<sup>m</sup>* <sup>¼</sup> *diag X<sup>m</sup>* <sup>0</sup> , *X<sup>m</sup>* <sup>1</sup> , … ,*X<sup>m</sup> N*�1 � � and **<sup>g</sup>** <sup>¼</sup> ½ � *<sup>g</sup>*ð Þ <sup>0</sup> , *<sup>g</sup>*ð Þ<sup>1</sup> , … , *g L*ð Þ � <sup>1</sup> *<sup>T</sup>* is the time domain channel vector. **<sup>W</sup>***<sup>m</sup>* <sup>¼</sup> *<sup>W</sup><sup>m</sup>* <sup>0</sup> , *W<sup>m</sup>* <sup>1</sup> , … , *W<sup>m</sup> N*�1 � �*<sup>T</sup>* , is an uncorrelated

*Phase Noise in OFDM DOI: http://dx.doi.org/10.5772/intechopen.105551*

**Figure 3.** *OFDM modulation and demodulation.*

white noise vector distributed as, Pr **<sup>W</sup>***<sup>m</sup>* ð Þ¼ *<sup>C</sup>*<sup>N</sup> 0, 2*σ*<sup>2</sup> *<sup>ω</sup>***<sup>I</sup>** � � with mean zero and covariance matrix 2*σ*<sup>2</sup> *<sup>ω</sup>***I**, which says:

$$\Pr(\mathbf{W}^{m}) = \frac{1}{\left(2\pi\right)^{N}\sigma\_{o}^{2N}} \exp\left(\frac{-1}{2\sigma\_{o}^{2}}\mathbf{W}^{mH}\mathbf{W}^{m}\right). \tag{6}$$
