Phase Noise in OFDM

Abstract

Orthogonal frequency division multiplexing (OFDM) technique provides high data rate with high spectral efficiency for operating close to the Shanon capacity bounds. With the advantages of simple channel equalization, robustness against frequency selectivity of the channel, and efficient implementation, this is a widely deployed technique. Orthogonal frequency division multiplexing access (OFDMA), the multiple access technique using OFDM, has the great potential for providing high spectral efficiency due to its integrated space-frequency and multiuser diversity. Besides all the advantages, OFDM/A is very susceptible to transceiver’s impairments such as phase noise (PHN), carrier frequency offset, and in-quadrature phase imbalance effect. Phase noise is the random fluctuation in phase of the sinusoidal waveform used for frequency up/down conversion of baseband signals to/from RF (radio frequency). This occurs due to the inherent imperfections of oscillators used for this purpose. This chapter addresses the orthogonal frequency division multiplexing/multiple access system performance under the impact of transceiver oscillator phase noise.

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, 2, 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, 7, 8, 9, 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, 12, 13]. Thus, the output of a practical oscillator is noisy and can be written as:

$$s\left(t\right)=\left[A+a\left(t\right)\right]\mathit{sin}\left[{\omega }_{c}t+\theta \left(t\right)\right]$$E1

where $A$ and ${\omega }_c=2\pi f_c$ are amplitude and angular frequency, respectively, and $\mathrm{a}\left(\mathrm{t}\right)$ is amplitude fluctuation, which can be kept in limit by using an automatic gain control (AGC). $\theta \left(t\right)$, 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 $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$f$}\right.$. 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, 12, 13, 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 ($\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$f^2$}\right.$) spectra are resulting with white noise sources and flicker PHN ($\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$f$}\right.$) and flicker FM PHN ($\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$f^3$}\right.$) spectra are resulting with color noise sources.

Figure 1.

PSD of PHN in oscillator output.

For FRO:

$${\theta }_{n+1}={\theta }_n+{\varphi }_n$$E2

which is Wiener process [15] with mean zero and variance, ${{\sigma }_{{\varphi }_n}}^2={\sigma }_{\varphi }^2=2\mathit{\pi \beta }{T}_s/N$ where $\beta =2∆f_{3\mathit{dB}}$, double of 3 dB bandwidth.

For PLL VCO [16].

$${\theta }_{n+1}={\theta }_n{e}^{-\phi \frac{T_s}{N}}+{{\varphi }_{\mathit{PLL}}}_n$$E3

which is celebrated O-U process where ${{\varphi }_{\mathit{PLL}}}_n$ is a sequence of identically and independently distributed (iid) random variables with mean zero and variance:

$${{\sigma }_{{{\varphi }_{\mathit{PLL}}}_n}}^2=4{\pi }^2{f_c}^2\left(C_{\mathit{RO}}\frac{T_s}{N}+2\sum _{i=1}^2\left({\xi }_i+{\zeta }_i\right)\left(1-e^{-{\lambda }_i\frac{T_s}{N}}\right)\right).$$

where:

$${\lambda }_{1,2}=\frac{{\omega }_{\mathit{lpf}}\pm \surd \left({{\omega }_{\mathit{lpf}}}^2-4{\omega }_{\mathit{lpf}}\sqrt{C_{\mathit{PLL}}}\right)}{2},$$

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

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

and

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

where ${\mathit{f}}_{\mathit{c}}$ is the center frequency of VCO in Hz, ${\mathit{\omega }}_{\mathit{lpf}}$ is the angular corner frequency of the low-pass filter in rad/sec, and $\sqrt{{\mathit{C}}_{\mathit{PLL}}}$ is the PLL bandwidth in Hz. ${\mathit{C}}_{\mathit{RO}}$ and ${\mathit{C}}_{\mathit{VCO}}$ 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 $\mathit{\beta }$ only, PLL VCO requires more parameter to characterize such as given in Table 1, assuming that the VCO is noisier than reference oscillator.

Figure 2.

PHN time samples for FRO and PLL VCO.

${\mathit{f}}_{\mathit{c}}$	$5\mathrm{GHz}$
$\mathit{\beta }$	$20\mathrm{kHz}$
${\mathit{f}}_{\mathit{lpf}}$	$20\mathrm{kHz}$
${\mathit{C}}_{\mathit{RO}}$	${10}^{-25}\mathrm{s}$
${\mathit{C}}_{\mathit{VCO}}$	${10}^{-19}\mathrm{s}$
${\mathit{C}}_{\mathit{PLL}}$	$4\ast {10}^8/{\mathrm{s}}^2$

Table 1.

PHN modeling parameters.

3. OFDM

OFDM is a low complex modulation/multiplexing multicarrier (MC) technique to modulate $N$ orthogonal sub carriers with $N$ complex-valued source symbols $X_k,k=0,1,\dots ,N-1$, efficiently by using digital signal processing. The source symbol is achieved after source coding, interleaving, and channel coding if applicable. The source symbol duration $T_d$ of the serial data symbol results in the OFDM symbol duration: $T_s=N{T}_d$.

From the Figure 3, the frequency domain received signal on the $k^{\mathit{th}}$ subcarrier of the $m^{\mathit{th}}$ symbol is without ISI and ICI and is given by:

Figure 3.

OFDM modulation and demodulation.

$$y_k^m=X_k^m{h}_k+W_k^{m}0\le k\le N-1$$E4

where $X_k^m$ is $k^{\mathit{th}}$ element of symbol vector ${\mathrm{X}}^m$, $h_k$ is the $k^{\mathit{th}}$ element of channel vector $\mathrm{h}={\left[h_0,h_1,h_2,\dots ,h_{N-1}\right]}^T$, $W_k^m$ is AWGN in frequency domain. It is preferable to represent the signal model in matrix form as:

$${\mathbf{Y}}^m={\mathbf{D}}^m\mathbf{Fg}+{\mathbf{W}}^m$$E5

where ${\mathbf{Y}}^m={\left[y_0^m,y_1^m,\dots ,y_{N-1}^m\right]}^T$, $\mathbf{F}$ is the $N\times L$ DFT matrix with $F\left(n,l\right)=\mathit{exp}\left(-\frac{j2\mathit{\pi nl}}{N}\right)$, ${\mathbf{D}}^m=\mathit{diag}\left[X_0^m,X_1^m,\dots ,X_{N-1}^m\right]$ and $\mathbf{g}={\left[g\left(0\right),g\left(1\right),\dots ,g\left(L-1\right)\right]}^T$ is the time domain channel vector. ${\mathbf{W}}^m={\left[W_0^m,W_1^m,\dots ,W_{N-1}^m\right]}^T$, is an uncorrelated white noise vector distributed as, $Pr\left({\mathbf{W}}^m\right)=C\mathcal{N}\left(0,2{\sigma }_{\omega }^2\mathbf{I}\right)$ with mean zero and covariance matrix $2{\sigma }_{\omega }^2\mathbf{I}$, which says:

$$Pr\left({\mathbf{W}}^m\right)=\frac{1}{{\left(2\pi \right)}^N{\sigma }_{\omega }^{2N}}exp\left(\frac{-1}{2{\sigma }_{\omega }^2}{{\mathbf{W}}^m}^H{\mathbf{W}}^m\right).$$E6

4. Phase noise in OFDM

Further to model an OFDM system with receiver PHN consisting of $N$ subcarriers with sampling instant $\raisebox{1ex}{$T_s$}\!\left/ \!\raisebox{-1ex}{$N$}\right.$, we denote the discrete-time receiver PHN impairment to the $n^{\mathit{th}}$ subcarrier of the $m^{\mathit{th}}$ symbol by ${\theta }_n^m$ than the received OFDM signal after down conversion and CP removal can be written as [17, 18, 19]:

$$r_n^m=\left[S_n^m\otimes g\left(n\right)\right]e^{{\mathit{j\theta }}_n^m}+w_n^m,0\le n\le N-1.$$E7

If ${\mathrm{\theta }}^m={\left[{\theta }_0^m,{\theta }_1^m,\dots {\theta }_{N-1}^m\right]}^T$, is the PHN vector for the $m^{\mathit{th}}$ OFDM symbol, then:

$${\mathrm{P}}^m={\left[p_{-\frac{N}{2}}^m,p_{-\frac{N}{2}+1}^m,\dots ,p_0^m,\dots ,p_{\frac{N}{2}-2}^m,p_{\frac{N}{2}-1}^m\right]}^T$$E8

defines a vector of the DFT coefficients of one realization of $e^{j{\theta }_n}$ during $m^{\mathit{th}}$ OFDM symbol where:

$$p_k^m=\frac{1}{N}\sum _{n=0}^{N-1}{e}^{j{\theta }_n^m}{e}^{\frac{-j2\mathit{\pi nk}}{N}},-\frac{N}{2}\le k\le \frac{N}{2}-1$$E9

After taking the FFT of $r_n^m$, the frequency domain received signal on the $k^{\mathit{th}}$ subcarrier of the $m^{\mathit{th}}$ symbol is:

$$y_k^m=\sum _{q=0}^{N-1}{X}_q^m{h}_q{p}_{⟨k-q⟩}^m+W_k^m,0\le k\le N-1$$E10

where $X_q^m$ is $q^{\mathit{th}}$ element of symbol vector ${\mathrm{X}}^m$, $h_q$ is the $q^{\mathit{th}}$ element of channel vector $\mathrm{h}={\left[h_0,h_1,h_2,\dots ,h_{N-1}\right]}^T$, $W_k^m$ is AWGN in frequency domain, and $p_{⟨k-q⟩}^m$ is the ${\left(k-q\right)}^{\mathit{th}}$ spectral component of PHN spectral component vector, ${\mathrm{P}}^m$, with modulo $N$ indexing. Further note that with modulo $N$ indexing, the lower-order spectral components of PHN are given by $p_0,p_1,p_{N-1},p_2,p_{N-2}$, etc. For convenience of the later analysis, it is preferable to represent the signal model in matrix form as:

$${\mathbf{Y}}^m={\mathbf{H}}^m{\mathbf{P}}^m+{\mathbf{W}}^m$$E11

where

${\mathbf{Y}}^m={\left[y_0^m,y_1^m,\dots ,y_{N-1}^m\right]}^T$, ${\mathbf{P}}^m={\left[p_0^m,p_1^m,\dots ,p_{N-1}^m\right]}^T$, $\mathbf{h}={\left[h_0,h_1,h_2,\dots ,h_{N-1}\right]}^T$,

${\mathbf{X}}^m={\left[X_0^m,X_1^m,\dots ,X_{N-1}^m\right]}^T$ and ${\mathbf{H}}^m$ is a column-wise circulant matrix whose first column is vector ${\left[h_0{X}_0^m,h_1{X}_1^m,\dots ,h_{N-1}{X}_{N-1}^m\right]}^T$. ${\mathbf{W}}^m={\left[W_0^m,W_1^m,\dots ,W_{N-1}^m\right]}^T$, is an uncorrelated white noise vector distributed as $Pr\left({\mathbf{W}}^m\right)=C\mathcal{N}\left(0,2{\sigma }_{\omega }^2\mathbf{I}\right)$ as given in Eq. (6).

4.1 Common phase error

In single carrier (SC) systems, the phase noise merely causes simple random rotation in the symbol constellation known as common phase error (CPE). Figure 4a shows the received signal constellation of an SC, 16 -QAM modulation over an AWGN channel (SNR = 30 dB), whereas the effect of PHN from an FRO (PHN variance = .06 rad2), on received signal constellation, is shown in Figure 4b.

Figure 4.

Effect of phase noise in SC communication system (random rotation in constellation).

4.2 Intercarrier interference

In OFDM systems, in addition to the rotational effect, PHN also causes ICI. The ICI is present because PHN causes energy of individual subcarriers to spread on the top of all the other subcarriers [20, 21, 22, 23]. Figure 5 shows two systems with the bandwidth of 22 MHz where first system employs the ideal oscillator without PHN with carrier frequency 2420.5 MHz, whereas second system uses a noisy FRO with carrier frequency 2433.5 MHz, which causes spectral regrowth and results in power leakage to the first band, producing the intercarrier interference (ICI).

Figure 5.

Effect of phase noise in MC communication system (spectral regrowth (in-band-ICI)(out-of-band-MUI)).

Figure 6a shows the received signal constellation of an OFDM system with 64 subcarriers, which are 16-QAM modulated over AWGN (SNR = 35 dB) with receiver PHN (both CPE and ICI) from an FRO (PHN variance = .06 rad²), whereas the effect of receiver as well as transmitter PHN from an FRO (PHN variance = .06 rad²) on received signal constellation is shown in Figure 6b. The constellation rotation is produced because of the CPE, whereas the cloudy constellation is impact of ICI.

Figure 6.

Received constellations in 16-QAM OFDM system with (a) receiver phase noise and (b) transceiver phase noise.

The effect of PHN on BER of the OFDM system is shown in Figure 7 for receiver FRO PHN (PHN variance = .06 rad2) only and for transceiver FRO PHN (PHN variance = .06 rad2) and is compared against the BER of pure AWGN channel.

Figure 7.

16-QAM OFDM BER performance in AWGN channel with phase noise.

OFDM symbols are generated using 16-quadrature amplitude modulation (QAM) and 64-point IFFT and then prepended by CP of length 16 samples before transmitting over the channel. The 64-point FFT of the received signal is taken after CP removal. Simulation model is based on IEEE 802.11 g like system with parameters given in Table 2.

$F_s$	20 MHz
$N$	64
FFT Size	64
$N_g$	16 Samples
Mapping	16-QAM

Table 2.

OFDM modeling parameters.

5. OFDMA

In OFDMA system, both the time and frequency resources are used to separate the multiple users. As OFDMA is typically used with burst transmission, a burst consists of many OFDM symbols. In an OFDM symbol, there are many subcarriers. So, a subcarrier in frequency domain and symbol duration in time domain are the finest units. The combination of a time unit and a frequency unit, i.e., a symbol period and a subcarrier, is the finest slot, which or group of which is allocated to one of the multiple users. Figure 8 shows the time frequency and power grid of OFDMA [24, 25].

Figure 8.

OFDMA time-frequency power grids [24].

Practically, in frequency domain, the allocation is not done at the level of subcarriers but on the group of subcarriers. This subcarrier’s allocation is known as subchannelization. To explain the basic principle of OFDMA transceiver, we are considering here that one user is using one subcarrier in the given time slot, i.e., number of users ($\mathit{U}$)= $\mathit{N}$. With this the simplest OFDMA uplink scheme is illustrated in Figure 9. At the transmitter side (mobile terminal), each user is having individual transmitters. At the receiver side (base station), the received signal is the sum of $\mathrm{U}$ users’signal, which acts as an OFDM signal. Because of this in OFDMA receiver, a single MC demodulator (OFDM demodulator) is required than $\mathrm{U}$ demodulators as in case of conventional frequency division multiple access (FDMA) system. At the transmitter side, a single transmitter consists of symbol generator and OFDMA modulator. The symbol is generated with applicable channel coding and mapping. These symbols are then OFDMA modulated with subchannelization and SC modulator (in case of $U=N$) or OFDM modulator in case a single user is using group of the subcarriers.

Figure 9.

OFDMA uplink.

An exact clock and carrier synchronization is must for an OFDMA system to ensure orthogonality between the $\mathrm{u}$modulated signals from different mobile terminals. This is achieved by transmitting synchronization signals from the receiver to all mobile terminals instantly. Each terminal OFDM modulator drives the carrier frequency and clock signal from these downlink signals. In case of coherent detection, simple carrier and clock recovery circuits are sufficient in the demodulator to extract this information from the received signal as the clock and carrier frequencies are available at the base station. This factor simplifies the OFDMA demodulator.

6. Phase noise in OFDMA

Like all other challenges of OFDM, i.e., time and frequency synchronization, high PAPR, CFO, and IQ imbalance, OFDMA also faces the challenge of transceiver RF impairment because of time-varying PHN. In OFDMA, CPE and in-band ICI are not the only source of interference that should be considered. The multiplexing of several users in an OFDMA scenario introduces out-of-band interference from one user on another in the OFDMA symbol. This MUI is induced by the spectral spread of the energy of each user’s subcarriers on the top of other users’ subcarriers. The spread is more severe in case of uplink, when there are unequal power levels as well as unequal transmitter 2(PHN 3-dB BW) for different users due to different path loss effects and different oscillator nonidealities, respectively, in an uplink scenario. Additionally, in the case of transmitter PHN, ICI results not only from the higher order components of PHN but also because of the loss of the cyclic nature and so the orthogonality is destroyed of the transmitted signal as the transmitter PHN affects the samples of the CP differently than the corresponding samples in the actual OFDMA signal part [26]. Further, the transmitter PHN impairing the CP also tends to produce ICI and hence not only $N-1$ but $N+N_g-1$ samples of PHN realization should be considered for PHN mitigation.

In regard of PHN impaired OFDMA modeling, we consider the uplink of an OFDMA with $U\left(u=1,2,3,\dots ,U\right)$ users, and $\mathcal{U}$ represents the index set of use full subcarriers with size $\mathfrak{A}$, means that among $N$ subcarriers, the $u^{\mathit{th}}$ user is assigned to a subset of ${\mathfrak{A}}_u$ subcarriers with index set: ${\mathcal{U}}_u=\left\{{\mathcal{U}}_1^u{\mathcal{U}}_2^u{\mathcal{U}}_3^u\dots {\mathcal{U}}_{{\mathfrak{A}}_u}^u\right\}$, either contiguous or interleaved where ${\left(.\right)}^u$ denotes the $u^{\mathit{th}}$ user. If $x^{m,u}$ is the $m^{\mathit{th}}$ frequency domain symbols sent by the $u^{\mathit{th}}$ user, then $k^{\mathit{th}}$ entry of it, say $X_k^{m,u}$ is nonzero if $k\mathrm{Є}{\mathcal{U}}_u$. Thereupon discrete-time baseband signal of the $u^{\mathit{th}}$ user using IFFT can be represented as:

$$S_{k,n}^{m,u}=\frac{1}{N}\sum _{k\mathrm{Є}{\mathcal{U}}_u}{X}_k^{m,u}{e}^{j2\pi \frac{\mathit{kn}}{N}},0\le n\le N-1.$$E12

As there is no ISI in between the windows of $N$ samples, and that the whole processing can be done in a symbol-to-symbol manner, we drop the OFDMA symbol index $m$ hereafter. After this the signal is transformed back to the serial form and is upconverted to RF with noisy transmitter oscillator and finally is sent over the channel. Let the discrete-time composite channel impulse response with order $L^u$ between the $u^{\mathit{th}}$ user and the uplink receiver be denoted by $g^u\left(l\right)$ and the channel frequency response on the $k^{\mathit{th}}$ subcarrier of $u^{\mathit{th}}$ user’s channel be denoted by $h_k^u$, then we have:

$$h_k^u=\sum _{l=0}^{L_k-1}{g}^u\left(l\right)e^{-j2\pi \frac{\mathit{kl}}{N}}$$E13

Denoting the discrete-time transmitter PHN process, receiver PHN process, and AWGN impairing to the $u^{\mathit{th}}$ user by ${\theta }_{T,n}^u$, ${\theta }_{R,n}^u$, and $w_n$, respectively, the received OFDMA symbol after downconversion and CP removal can be written as:

$$r_n=\sum _{u=1}^U\left[\left(S_{k,n}^u{e}^{j{\theta }_{T,n}^u}\right)\otimes g^u\left(l\right)\right]e^{j{\theta }_{R,n}^u}+w_n.$$E14

After taking the FFT, the frequency domain received symbol on the $k^{\mathit{th}}$ subcarrier is:

$$y_k=p_0^u{h}_k^u{X}_k^u+\sum _{i=1}^U\sum _{\begin{array}{c}q\mathrm{Є}{\mathcal{U}}_i\\ q\ne k\end{array}}{p}_{k-q}^i{h}_q^i{X}_q^i+W_k.$$E15

As $h$ is a circulant matrix, we can effectively map the transmitter PHN as receiver PHN, and by writing ${\theta }_{T,n}^u+{\theta }_{R,n}^u={\theta }_n^u$, we have $p_q^u=\frac{1}{N}{\sum }_{n=0}^{N-1}{e}^{j{\theta }_n^u}{e}^{-j\frac{2\mathit{\pi nq}}{N}}$, and $W_k$ is the AWGN noise in frequency domain.

From Eq. (15), we find the effect of phase noise in OFDMA to be different from that of single-user OFDM. First of all the CPE term ($p_0^u$) varies according to the index $u$, means that each user suffers from different CPE, and they need to be considered separately for each user to estimate and mitigate.

Secondly, the summative term, called ICI, includes the user’s “in-band” ICI (self-interference (SI)) and ICI caused by MUI. While including the frequency domain dummy symbols transmitted by each active user in Eq. (15), a unified frequency domain signal model can be given by:

$$y_k=\sum _{u=1}^U\sum _{q=0}^{N-1}{p}_{k-q}^u{h}_q^u{X}_q^u+W_k.$$E16

Splitting the summative (ICI) term is important for our analysis purpose, as MUI takes in to account the significance of the power level of users as well as the transmitter 2(PHN 3-dB BW) as these two will be significantly different for different users precisely in case of OFDMA uplink. So the signal for $u^{\mathit{th}}$ user, on his $k^{\mathit{th}}$ subcarrier, is given as:

$$y_k=p_0^u{h}_k^u{X}_k^u+\sum _{\begin{array}{c}q\mathrm{Є}{\mathcal{U}}_u\\ q\ne k\end{array}}{p}_{k-q}^u{h}_q^u{X}_q^u+\sum _{\begin{array}{c}i=1\\ i\ne u\end{array}}^U\sum _{\begin{array}{c}q\mathrm{Є}{\mathcal{U}}_i\\ q\ne k\end{array}}{p}_{k-q}^i{h}_q^i{X}_q^i+W_k.$$E17

First to characterize the phase noise strength in OFDMA transmission, we adopt a parameter widely used in literature, which is the relative PHN bandwidth, ${∆}_{\mathit{PN}}=\frac{2\left(\mathit{PHN}3\mathit{dB}-\mathit{BW}\right)}{∆f\left(\mathit{\text{subcarrier spacing}}\right)}$. Having the desired advantages of OFDM transmission over single-carrier transmission with “slow” PHN model restricts to have low of this ratio, which makes the assumption of complex Gaussian distribution of the ICI false, even with higher number of subcarriers. Secondly a higher $2\left(\mathrm{PHN}3\mathrm{dB}-\mathrm{BW}\right)$ of the PHN process and the higher value of power level can also lead to more energy in the MUI factor of ICI terms. Considering these two facts and the OFDMA uplink scenario, not all the $U-1$ users will produce the MUI for $u^{\mathit{th}}$ user but only those who will satisfy the following inequality will be the disruptive users for $u^{\mathit{th}}$ (user)

$$\sum _{a=1}^{N-1}E\left[{\left|p_a^u\right|}^2\right]<\sum _{a=1}^{N-1}E\left[{\left|p_a^j\right|}^2\right]\mathit{\text{for}}j=1\mathit{\text{toUandj}}\ne u.$$E18

Here we define a subset of users for the $u^{\mathit{th}}$ user $I_u$, $\forall j\in I_{\mathrm{u}}$ with size ${\mathrm{I}}_{\mathrm{u}}$. Since the PSD of phase noise tapers off rapidly beyond the loop bandwidth, most of the energy in a phase noise sequence is contained in the frequency components corresponding to the first few orders. Hence, the largest contribution to interference on a particular subcarrier is likely to come from users occupying adjacent subcarriers. As a result, disruptive users who are occupying subcarriers adjacent to the $u^{\mathit{th}}$ user are likely to be most disruptive users. Keeping this valid, Eq. (17) can be rewritten while using Eq. (18) as:

$$y_k=p_0^u{h}_k^u{X}_k^u+\sum _{\begin{array}{c}q\mathrm{Є}{\mathcal{U}}_u\\ q\ne k\end{array}}{p}_{k-q}^u{h}_q^u{X}_q^u+{\sum }_{i\in I_u}{\sum }_{\begin{array}{c}q\mathrm{Є}{\mathcal{U}}_i\\ q\ne k\end{array}}{p}_{k-q}^i{h}_q^i{X}_q^i+W_k$$E19

where the second term is SI, and third term is MUI.

7. Conclusion

Analyzing the impact of transceiver PHN necessitates the accurate mathematical modeling of generated PHN. As the FRO model is easy to simulate mathematically and PLL is widely used in practice for digital communication systems, the PHN modeling for both of the oscillators is presented. With the white noise sources in the oscillator circuitry, the PHN is modeled as Wiener process and celebrated O-U process, for FRO and PLL VCO, respectively.

OFDM, as a low complex modulation technique, became the potential contender for MC transmission to combat the frequency selectivity of the channel. The synchronization unit (including the time and frequency synchronization units) of OFDM demodulator is performing the robust digital synchronization and channel estimation with digital algorithms. The presence of transceiver PHN degrades the OFDM system performance because of the rotational effect CPE and spectral regrowth ICI.

Being effective in mitigating the hostile channel selectivity with adaptive subchannelization and resource allocation, the OFDMA technique has gained much more interest in recent years. With transceiver PHN in OFDMA, CPE and in-band ICI are not the only sources of interference like OFDM but the multiplexing of several users introduces out-of-band interference from one user on another known as MUI [27].

Acknowledgments

Acknowledgment is made to the Department of Electronics and Communication Engineering, BIET, and Editors, IntechOpen for the support to make this book chapter possible.

Conflict of interest

The authors declare no conflict of interest.

Notes/thanks/other declarations

Thanks to Prof. R. P. Yadav for his valuable suggestions and comments that helped to improve the presentation of the book chapter.