**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* þ *Ng* � 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 u*ð Þ ¼ 1, 2, 3, … , *U* users, and U represents the index set of use full subcarriers with size A, means that among *N* subcarriers, the *uth* user is assigned to a subset of <sup>A</sup>*<sup>u</sup>* subcarriers with index set: <sup>U</sup>*<sup>u</sup>* <sup>¼</sup> <sup>U</sup>*<sup>u</sup>* 1U*<sup>u</sup>* 2U*<sup>u</sup>* <sup>3</sup> … U*<sup>u</sup>* A*<sup>u</sup>* n o, either contiguous or interleaved where ð Þ*: <sup>u</sup>* denotes the *uth* user. If *xm*,*<sup>u</sup>* is the *<sup>m</sup>th* frequency domain symbols sent by the *uth* user, then *kth* entry of it, say *X<sup>m</sup>*,*<sup>u</sup> <sup>k</sup>* is nonzero if *k*ЄU*u*. Thereupon discrete-time baseband signal of the *uth* user using IFFT can be represented as:

$$S\_{k,u}^{m,u} = \frac{1}{N} \sum\_{k \in \mathcal{U}\_u} X\_k^{m,u} \boldsymbol{\sigma}^{j2\pi \frac{ku}{N}}, 0 \le n \le N - 1. \tag{12}$$

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 *Lu* between the *<sup>u</sup>th* user and the uplink receiver be denoted by *<sup>g</sup><sup>u</sup>*ð Þ*<sup>l</sup>* and the channel frequency response on the *kth* subcarrier of *uth* user's channel be denoted by *hu <sup>k</sup>*, then we have:

$$h\_k^u = \sum\_{l=0}^{L\_k - 1} g^u(l) e^{-j2\pi \frac{kl}{N}} \tag{13}$$

Denoting the discrete-time transmitter PHN process, receiver PHN process, and AWGN impairing to the *uth* user by *θ<sup>u</sup> <sup>T</sup>*,*<sup>n</sup>*, *θ<sup>u</sup> <sup>R</sup>*,*<sup>n</sup>*, and *wn*, respectively, the received OFDMA symbol after downconversion and CP removal can be written as:

$$\boldsymbol{\sigma}\_{n} = \sum\_{\boldsymbol{\mu}=1}^{U} \left[ \left( \mathbf{S}\_{k,n}^{\boldsymbol{\mu}} \boldsymbol{e}^{i\boldsymbol{\theta}\_{T,n}^{\boldsymbol{\mu}}} \right) \otimes \mathbf{g}^{\boldsymbol{\mu}}(l) \right] \boldsymbol{e}^{i\boldsymbol{\theta}\_{R,n}^{\boldsymbol{\mu}}} + \boldsymbol{w}\_{n}. \tag{14}$$

After taking the FFT, the frequency domain received symbol on the *kth* subcarrier is:

$$\mathcal{Y}\_k = p\_0^u h\_k^u X\_k^u + \sum\_{i=1}^U \sum\_{\substack{q \in \mathcal{U}\_i \\ q \neq k}} p\_{k-q}^i h\_q^i X\_q^i + \mathcal{W}\_k. \tag{15}$$

As *h* is a circulant matrix, we can effectively map the transmitter PHN as receiver PHN, and by writing *θ<sup>u</sup> <sup>T</sup>*,*<sup>n</sup>* <sup>þ</sup> *<sup>θ</sup><sup>u</sup> <sup>R</sup>*,*<sup>n</sup>* <sup>¼</sup> *<sup>θ</sup><sup>u</sup> <sup>n</sup>*, we have *pu <sup>q</sup>* <sup>¼</sup> <sup>1</sup> *N* P*N*�<sup>1</sup> *<sup>n</sup>*¼<sup>0</sup> *<sup>e</sup>jθ<sup>u</sup> <sup>n</sup> e*�*<sup>j</sup>* 2*πnq <sup>N</sup>* , and *Wk* 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<sup>u</sup>* <sup>0</sup>) 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:

$$y\_k = \sum\_{u=1}^{U} \sum\_{q=0}^{N-1} p\_{k-q}^u h\_q^u \mathbf{X}\_q^u + \mathbf{W}\_k. \tag{16}$$

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 *uth* user, on his *kth* subcarrier, is given as:

$$y\_k = p\_0^u h\_k^u X\_k^u + \sum\_{\substack{q \in \mathcal{U}\_u \\ q \neq k}} p\_{k-q}^u h\_q^u X\_q^u + \sum\_{i=1}^U \sum\_{\substack{q \in \mathcal{U}\_i \\ i \neq u \quad q \neq k}} p\_{k-q}^i h\_q^i X\_q^i + W\_k. \tag{17}$$

First to characterize the phase noise strength in OFDMA transmission, we adopt a parameter widely used in literature, which is the relative PHN bandwidth, Δ*PN* ¼ 2ð Þ *PHN* 3*dB*�*BW* <sup>Δ</sup>*f subcarrier spacing* ð Þ. Having the desired advantages of OFDM transmission over singlecarrier 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 PHN 3dB ð Þ � BW 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 *<sup>U</sup>* � 1 users will produce the MUI for *<sup>u</sup>th* user but only those who will satisfy the following inequality will be the disruptive users for *uth* (user)

$$\sum\_{a=1}^{N-1} E\left[\left|p\_a^{\mu}\right|^2\right] < \sum\_{a=1}^{N-1} E\left[\left|p\_a^j\right|^2\right] \text{for} \, j = 1 \text{ to } \\ \text{Uand} \\ j \neq u. \tag{18}$$

Here we define a subset of users for the *uth* user *Iu*, ∀*j*∈ *I*<sup>u</sup> with size Iu. 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 *uth* user are likely to be most disruptive users. Keeping this valid, Eq. (17) can be rewritten while using Eq. (18) as:

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

$$y\_k = p\_0^u h\_k^u X\_k^u + \sum\_{\substack{q \in \mathcal{U}\_u \\ q \neq k}} p\_{k-q}^u h\_q^u X\_q^u + \sum\_{i \in I\_u} \sum\_{\substack{q \in \mathcal{U}\_i \\ q \neq k}} q\_{\mathbb{C}\mathcal{U}\_i} p\_{k-q}^i h\_q^i X\_q^i + W\_k \tag{19}$$

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].

### **Acknowledgements**

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.

*Multiplexing - Recent Advances and Novel Applications*
