**4. Phase noise in OFDM**

Further to model an OFDM system with receiver PHN consisting of *N* subcarriers with sampling instant *Ts=<sup>N</sup>*, we denote the discrete-time receiver PHN impairment to the *nth* subcarrier of the *mth* symbol by *θ<sup>m</sup> <sup>n</sup>* than the received OFDM signal after down conversion and CP removal can be written as [17–19]:

$$
\sigma\_n^m = \left[ \mathbf{S}\_n^m \otimes \mathbf{g}(n) \right] \mathbf{e}^{j\theta\_n^m} + w\_n^m, \mathbf{0} \le n \le N - \mathbf{1}.\tag{7}
$$

If <sup>θ</sup>*<sup>m</sup>* <sup>¼</sup> *<sup>θ</sup><sup>m</sup>* <sup>0</sup> , *θ<sup>m</sup>* <sup>1</sup> , … *θ<sup>m</sup> N*�1 � �*<sup>T</sup>* , is the PHN vector for the *mth* OFDM symbol, then:

$$\mathbf{P}^{m} = \begin{bmatrix} p\_{-\frac{\nu}{2}}^{m}, p\_{-\frac{\nu}{2}+1}^{m}, \dots, p\_{0}^{m}, \dots, p\_{\frac{\nu}{2}-2}^{m}, p\_{\frac{\nu}{2}-1}^{m} \end{bmatrix}^{T} \tag{8}$$

defines a vector of the DFT coefficients of one realization of *e<sup>j</sup>θ<sup>n</sup>* during *mth* OFDM symbol where:

$$p\_k^m = \frac{1}{N} \sum\_{n=0}^{N-1} e^{j\theta\_n^m} e^{\frac{-j2mk}{N}}, \ -\frac{N}{2} \le k \le \frac{N}{2} - 1\tag{9}$$

After taking the FFT of *r<sup>m</sup> <sup>n</sup>* , the frequency domain received signal on the *<sup>k</sup>th* subcarrier of the *mth* symbol is:

$$\mathcal{Y}\_k^m = \sum\_{q=0}^{N-1} \mathcal{X}\_q^m h\_q p\_{\langle k-q \rangle}^m + \mathcal{W}\_k^m, \mathbf{0} \le k \le N - 1 \tag{10}$$

where *X<sup>m</sup> <sup>q</sup>* is *qth* element of symbol vector X*m*, *hq* is the *qth* 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, and *pm* h i *<sup>k</sup>*�*<sup>q</sup>* is the ð Þ *k* � *q th* spectral component of PHN spectral component vector, P*m*, with modulo *N* indexing. Further note that with modulo *N* indexing, the lower-order spectral components of PHN are given by *<sup>p</sup>*0, *<sup>p</sup>*1, *pN*�1, *<sup>p</sup>*2, *pN*�2, etc. For convenience of the later analysis, it is preferable to represent the signal model in matrix form as:

$$\mathbf{Y}^{m} = \mathbf{H}^{m}\mathbf{P}^{m} + \mathbf{W}^{m} \tag{11}$$

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> , … , *y<sup>m</sup> N*�1 � �*<sup>T</sup>* , **<sup>P</sup>***<sup>m</sup>* <sup>¼</sup> *pm* <sup>0</sup> , *pm* <sup>1</sup> , … , *p<sup>m</sup> N*�1 � �*<sup>T</sup>* , **h** ¼ ½ � *h*0, *h*1, *h*2, … , *hN*�<sup>1</sup> *T*, **<sup>X</sup>***<sup>m</sup>* <sup>¼</sup> *<sup>X</sup><sup>m</sup>* <sup>0</sup> , *X<sup>m</sup>* <sup>1</sup> , … ,*X<sup>m</sup> N*�1 � �*<sup>T</sup>* and **H***<sup>m</sup>* is a column-wise circulant matrix whose first column is vector *h*0*X<sup>m</sup>* <sup>0</sup> , *h*1*X<sup>m</sup>* <sup>1</sup> , … , *hN*�<sup>1</sup>*X<sup>m</sup> N*�1 � �*<sup>T</sup>* . **<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 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>** � � 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**.

#### **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 [20–23]. **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 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.

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.

OFDM symbols are generated using 16-quadrature amplitude modulation (QAM) and 64-point IFFT and then prepended by CP of length 16 samples before
