Performance Analysis of OFDM Systems with Phase Noise

In this paper, we investigate the phase noise properties and the impacts caused by phase noise on an OFDM system, we derive the approximate expressions for BER performance in AWGN channel with various modulations techniques. The performance degradation is analyzed by graphs of BER against the number of sub-carriers. Thus the choice of the number of sub-carriers involves a tradeoff between mitigation of multi-path delay spread and reducing phase noise effects. Simulation results also show that phase noise effects performance for BPSK, QPSK & 16 PSK constellation which lead to the conclusion that a phase noise mitigation mechanism is needed to obtain acceptable performance.


Introduction
Orthogonal frequency division multiplexing (OFDM) is a multicarrier transmission technique, which divides the available spectrum into many carriers, each one being modulated by a low rate data stream.Practical implementations of OFDM technology use an inverse fast Fourier transform (IFFT) to generate a digitized version of the composite time domain signal.OFDM has been widely adopted and implemented in wire and wireless communication, such as digital subscriber line (DSL), digital terrestrial TV broadcasting (DVB-T), IEEE 802.11a wireless local area networks (WLANs), European high performance local area network (HIPERLAN/2) and possibly future 4G networks [1]- [4].The disadvantage of OFDM, however, is its sensitive to both carrier frequencies offset and phase noise [5]- [6].
Frequency offset is a deterministic phenomenon, which is usually caused by the different carrier frequencies of the transmitter and the receiver, or by Doppler shift.On the other hand, phase noise is a random process caused by frequency fluctuation at the receiver and transmitter oscillators.Phase noise causes leakage of DFT which subsequently destroys the orthogonalities among subcarrier signals, leading to significant performance degradation.The principles of analyzing the frequency offset effects have been described in many papers [5], [7], hence in our contribution, only the performance analysis of OFDM systems with phase noise problem will be discussed.
Phase noise causes significant degradation of the performance of OFDM systems.The effects of phase noise on OFDM systems have been intensively investigated in the literature [5], [8]- [9].Phase noise is normally modeled as an ideal phase modulation in the oscillator's signal.Phase noise in an OFDM system results from the instabilities in both oscillators of the transmitter and receiver.Without loss of generality, in this study the local oscillator in the receiver will be considered as the phase noise source.
Because of the random nature of phase noise, its effect is harder to be analyzed.In the literature, we will focus on the Wiener process assumption.The effect of Wiener phase noise has been expressed as a loss in the ratio between the signal power and the statistical average of the ICI [5]- [6], [10], wherein AWGN channel was considered in [10], while [5]- [6] included fading channel.However, it is both interesting and useful to find the precise relation between the bit error rate (BER) or symbol error rate (SER) and the phase noise.The original work done by Tomba [11], which is based on [12], considers only the BER for the AWGN channel and ignores the comparison with simulation results.
In this paper, we propose a new method to derive the expressions for BER performance in AWGN channel with the presence of phase noise.Simulation results validate all expressions we have derived and show clearly the necessity of phase noise mitigation mechanism.
The rest of the paper is organized as follows.We describe the system model in section 2, where phase noise and OFDM system models are presented.In section 3, different effects of phase noise are discussed.In Section 4, numerical simulation results are presented.Finally we draw some conclusions in section 5. , where β denotes the phase noise linewidth, i.e., frequency spacing between 3dB points of its Lorentzian power spectral density function [5], [12].Such noise has independent Gaussian increments [12], which, from a spectral point of view, can be presented as a Wiener process [8], [13].To better characterize phase noise, Demir et al. [14] developed a unifying theory using a nonlinear method that provides more accurate description.With such a method [14, Remark 7.1], phase noise ) (t φ is shown to converge, asymptotically with time, to a Gaussian random process having a constant mean, a variance increasing linearly with time, and a correlation function that satisfies

System Model
Furthermore, as indicated in [15], the aforementioned properties suggests a discrete Markov process which, in an OFDM system, can describe the phase noise on the nth sample of the mth symbol as where u(i)'s denote mutually independent Gaussian random variables with zero mean and variance ,and T denotes the OFDM symbol length.In particular, (3) reduces to

B. OFDM System Model
Since the effects of phase noise on OFDM systems are particular interest, perfect frequency and timing synchronizations are assumed.Thus, in the presence of phase noise, the OFDM system model is shown in figure 1.The OFDM system model over a slowly time varying multipath fading channel with phase noise can be described as are complex-valued Rayleigh fading random variables, and n(n)'s ( 0 ≤ n ≤ N-1) are independent complex-valued Gaussian random variables with zero mean and variance for both real and imaginary components, while are the time domain transmitted symbols.L is the length of the time-domain channel impulse response (CIR).After discarding the cyclic prefix (CP) and performing an FFT at the receiver, we obtain the received data frame in the frequency domain:

Effects of Phase Noise
The effect of phase noise leads to significant loss of OFDM systems.Phase noise destroys orthogonalities among OFDM subcarrier signals.
There are two major problems affected by phase noise: phase shift and ICI.For the purpose of convenience, we will drop the subscript m, i.e., our discussion is restricted in one OFDM symbol.The effects of phase noise on OFDM signals are illustrated in figure 2, where phase noise corrupts the 16-QAM constellations with both CPE and ICI.It's shown that phase noise forces the desired signal into a wrong decision area, and worsens the BER performance accordingly.
where A denotes the amplitude of CPE and θ is a random variable denoting phase shift, which is the major effect of CPE when phase noise is small.The probability density function (pdf) of θ which will be needed for analyzing its effect on BER performance, as ICI plays an important part in BER analysis.It is usually treated in AWGN channel as a Gaussian distributed random variable [11]- [12], whereas in fading channel, such approximation is not appropriate in some scenario.

Simulation Results
To demonstrate the performance analysis of OFDM systems with phase noise, computer simulations were performed.A very powerful and useful engineering software package is Matlab by MathWorks [16]- [17].It has many useful digital signal processing functions and features, which will prove to be useful in an OFDM simulation.An OFDM system was modeled using Matlab to allow various parameters of the system to be varied and tested.

A. BER Performance in AWGN Channel
In this section, We study the performance analysis in the presence of phase noise for OFDM in AWGN channel, i.e., H(k) = 1 for all k.When we transmit the signal through an AWGN channel, receive it and check the errors.The simulation is based on multiple signal-to-noise-ratio (SNR); meaning that the signal is received for various SNR values and error check is performed.The following figure shows BER VS.SNR graph under the AWGN channel conditions the results from the simulations.The simulations were performed using 200 carriers and generated using a 512-point IFFT.The results show that using QPSK the transmission can tolerate a SNR of >11-13 dB.The bit error rate BER gets rapidly worse as the SNR drops below 6 dB.However, using BPSK allows the BER to be improved in a noisy channel, at the expense of transmission data capacity.Using BPSK the OFDM transmission can tolerate a SNR of >7-8 dB.In a low noise link, using 16PSK can increase the capacity.If the SNR is >24 dB 16PSK can be used, doubling the data capacity compared with QPSK.

B. BER Performance vs Number of Sub-Carriers
The BER performance is much dependent on the choice of number of sub-carrier On the other hand, phase noise effects are worse as the number of subcarriers is increased.Thus the need for a tradeoff was realized and was examined using the simulation runs.BER vs. number of sub-carriers curves have been provided.Figure 4 shows the simulation result for the effect of the number of sub-carriers on the BER performance due to phase noise.This shows the concrete relationship between the number of subcarriers and phase noise effects on system performance.The Additive White Gaussian Noise (AWGN) and the presence of phase noise corrupted the transmitted signal and this resulted in a different received constellation than the original constellation.For small SNR values the calculated error rate was quite large and ISI was produced due the relative high power of noise.The following figure shows the simulation results for phase noise effects.The received BPSK, QPSK & 16PSK constellation is shown in Figure 5, Figure 6 and Figure 7.It is clear that as the SNR is increased the received constellation gets less affected by the noise; hence there will be fewer errors.The error, which is a symbol error, is calculated by comparing the original constellation with the received constellation.However, for low values of SNR we have ISI introduced by the noise at the receiver side.As SNR was increased the error rate was decreasing, as expected.In fact, for a SNR value greater than 8dB for BPSK, the error was zero.

Conclusions
Phase noise causes significant degradation of the performance of OFDM systems.The effects of phase noise on the performance of OFDM systems have been analytically evaluated.The expressions for CPE and ICI caused by phase noise have been derived.It is shown that both CPE and ICI depend on the overall spectrum of each weighted group of subcarriers rather than on the spectrum of each individual subcarrier.Simulation results show that the methodology presented in this work is accurate and can be used as references in practice.

Figure 1 Figure 1 Figure 1 Figure 1 .
OFDM system model in the presence of phase noise

,
and N(k) is the DFT transform of n(n).Since DFT is linear and does not change the noise energy, N(k) is still a Gaussian random variable with zero mean and variance 2 n σ .Note that common phase error (CPE) and ICI are represented by C m (0) and ICI m (k) respectively.If phase noise does not exist, Eqations (5) and (6

4 Figure 4 .
Figure 4 Figure 4 Figure 4 Figure 4.The number of sub-carriers vs. BER, when SNR=10dB C. Phase Noise Effects Performance for BPSK, QPSK & 16PSK Constellation Received BPS K Constellation for S NR 7dB