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Performance Analysis of OFDM Systems with Gaussian Distributed Phase and Frequency Offsets

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Performance Analysis of OFDM Systems with Gaussian Distributed Phase and Frequency Offsets International Journal of Computer Applications (0975 – 8887) Volume 47– No.1, June 2012 Performance Analysis of OFDM Systems with Gaussian Distributed Phase and Frequency Offsets Sabitha Gauni, J. Elakkiya, R. Kumar, Vidhyacharan Bhaskar Department of Electronics and Communication Engineering SRM University, Kattankulathur-603203, Tamilnadu, India. ABSTRACT Frequency offset is a deterministic phenomenon, which is In this paper, a theoretical analysis to evaluate the usually caused by different carrier frequencies of the performance of Orthogonal Frequency Division transmitter and receiver, or by Doppler shift. On the other Multiplexing (OFDM) systems under the combined hand, phase noise is a random process caused by influence of Phase Offset and Frequency Offset over frequency fluctuations at the receiver and transmitter Rayleigh, Weibull and Nakagami fading channels using oscillators. The principles of analyzing the frequency Binary Phase Shift Keying (BPSK) Modulation is offset effects have been described in [5], [6]. presented. The performance of an OFDM system is Several works discussing the error probability of OFDM degraded by both the Frequency Offset and the Phase systems with Carrier Frequency Offset (CFO) can be Offset. The degradation of Bit Error Rate (BER) caused found in literature [3]-[5]. In this context, Moose [6] gives by the presence of Phase Offset and Frequency Offset is the maximum likelihood estimator of the CFO, based on analytically evaluated. Hence, closed form expressions the observation of two consecutive and identical symbols. are derived to compute the BER with Rayleigh, Weibull The maximum offset that can be handled is 1/ 2 the and Nakagami fading channels assuming Gaussian subcarrier spacing. However, analytical expressions were probability density function of the Frequency Offset and given for the first time in [7] for Binary Phase Shift Phase Offset. BER performance results are evaluated Keying (BPSK) OFDM systems with CFO in AWGN, flat numerically following the analytical approach. and frequency selective Rayleigh fading channels. Keywords Although the BER expression derived therein is acceptable for all values of CFO for AWGN channels, the Orthogonal Frequency Division Multiplexing (OFDM); same is not true for flat and frequency selective channels. Phase Offset; Frequency Offset; Bit Error Rate (BER); This is because the expressions have been derived Probability Density Function (PDF); Signal to Noise Ratio assuming the argument of Q function in the expressions (SNR). (7) and (32) of [7] to be positive. This is not true for 1. INTRODUCTION higher values of CFO, leading to a mismatch between the theoretical and actual Bit Error Rate (BER) in such cases. OFDM has become a popular technique for transmission Performance of an OFDM system in presence of CFO and in wireless communication systems due to its high data phase noise has been analyzed in several research works rate transmission capability with high bandwidth during the last few years. The BER performance efficiency and robustness to multipath fading and delay. It evaluation of an OFDM system in the presence of CFO has also been proposed as the core technique for fourth- and phase noise is shown in [8]. generation (4G) mobile communications. As OFDM makes use of the spectrum efficiently, it has been widely The main objective of this paper is to derive accurate adopted and implemented in wired and wireless BER expressions valid for all values of Phase and communications, such as Digital Subscriber Line (DSL), Frequency Offsets for BPSK OFDM systems. We first European Digital Audio Broadcasting (DAB), Digital calculate the PDF of various fading distributions in an Video Broadcasting-Terrestrial (DVB-T), its handheld integral form and then use this PDF to evaluate the error version DVB-H, and IEEE 802.11a/g standards for rate performance for BPSK modulated OFDM system in Wireless Local Area Networks (WLANs) etc [1],[2]. Rayleigh, Weibull and Nakagami fading channels. OFDM systems have become significant because they are Theoretical analysis is presented in this work to show the robust to frequency selective fading. As OFDM systems effects of fading techniques on the BER performance of have low equalization complexity the cost of FFT an OFDM system. techniques implementation is reduced. But, the OFDM This paper consists of six sections. In Section II, the systems are more sensitive to frequency synchronization system model is described. Section III shows various errors and Phase noise than single carrier systems [3]. The fading channels with their corresponding PDFs. Section presence of Frequency Offset and Phase noise disturbs the IV discusses the performance analysis of OFDM systems orthogonality between the carriers, thereby causing Inter with Phase and Frequency Offsets. In Section V, Carrier Interference (ICI). Hence, there is a need to numerical results are presented for Rayleigh, Weibull and analyze the performance of such systems [3]. Nakagami fading channels. Finally, Section VI presents the conclusion and future work. 42 International Journal of Computer Applications (0975 – 8887) Volume 47– No.1, June 2012 2 variance . To obtain the preceding expression of probability of error, we consider the PDF of ε as, 1 2 2. SYSTEM MODEL f exp 0 2 An OFDM system with N sub-carriers is considered. Let 2 2 X(k) be the kth OFDM data block to be transmitted with N and . subcarriers. These data blocks are used to modulate N f 0 for 0 orthogonal subcarriers. Then, using IDFT the input signal 3.2 Phase Offset is modulated which can be represented as Phase noise is a random process which results from the fluctuation of the transmitter and receiver Local 1 N1 xn X kexpi2kn/ N V Oscillators (LO) with time. It also has adverse effect on N k0 system performance. Figures 1, 3 and 5 show the resulting n=0,1,….N-1. probability of error from the phase offset in an OFDM signal. The Signal to Noise Ratio (SNR) of an OFDM Due to the phenomenon of multipath propagation, when signal in the presence of phase noise is obtained. the wave hits the obstacle, the signal can be reflected, diffracted or diffused. The channel impulse response of a multipath fading channel is modeled as | hn| e j n . 3.3 Fading Channels The receiver side frequency domain signal is represented The effect of the wireless channel on the transmitted as signals is multiplicative, where the multiplicative term is a complex Gaussian random variable. The statistics for R(k) X(k)H(k) N, fading is very important to design any wireless systems [9]-[14]. where N is an independent identically distributed (i.i.d) 3.3.1 Rayleigh Fading Channel complex Gaussian noise component with zero mean and If the channel coefficient has zero mean, then such a unit variance. The amplitude of H(k) is modeled as a channel is considered as Rayleigh fading since the Rayleigh, Weibull and Nakagami fading distribution with absolute value of the received amplitude is a Rayleigh the corresponding PDFs. random variable. 3. BER analysis in the presence of Offsets The PDF of the received signal amplitude in a Rayleigh In this section, we discuss BER analysis in the presence fading channel is given as [12, page no. 15] of phase and frequency offsets in detail. x x 2 f (x) exp x 0, 2 2 2 3.1 Carrier Frequency Offset 2 f The absolute value of the actual CFO, , is either an (1) integer multiple or a fraction of subcarrier spacing, f , f f 2 or the sum of them. If is normalized to , then the where x has a Rayleigh Distribution, is the variance resulting normalized CFO of the channel can be generally of received signal amplitude and is the fading expressed as parameter. 3.3.2 Weibull Fading Channel f , The Weibull distribution is related to a number of other f probability distributions; in particular, it interpolates between the exponential distribution (k =1) and the where is an integer and 0.5 , where Rayleigh distribution (k = 2). frequency offset . The PDF of the recovered signal in Weibull fading subcarrier spacing channel is given as [12, page no. 15] k1 k The influence of an integer CFO on OFDM system is k x x different from the influence of a fractional CFO. In the f x exp x 0 , event that 0 and 0 , symbols transmitted on a certain subcarrier, e.g., subcarrier k, will shift to another (2) subcarrier k , k k mod N 1. where k 0 is the shape parameter, and 0 is As we focus on the ICI effect, we will consider the scale parameter of the distribution. normalized CFO, , since no ICI is caused by an integer 3.3.3 Nakagami Fading Channel CFO. We assume ε to be a Gaussian process, statistically independent of the input signal with zero mean and The Nakagami-m or Nakagami distribution is a probability distribution related to the Gamma 43 International Journal of Computer Applications (0975 – 8887) Volume 47– No.1, June 2012 2m1 2 In the Rayleigh case, the conditional probability of BPSK m x x 2m m modulation can be written using (6) as f x m exp x 0. m 1 0 2 x 12 P exp e 2 2 2 (3) 1 2 x 1 2 distribution. It has two parameters: a shape parameter, μ, exp . 2 and a second parameter controlling spread, . The PDF 2 0 2 of the received signal amplitude in a Nakagami fading (8) channel is given as [12, page no. 16] In the Nakagami fading channel, the fading parameter, m In (8), using integration by substitution method, we have = 1, refers to a Rayleigh fading channel. We further note the results of conditional probability of error obtained as that the system performance analysis with Nakagami fading model is usually more tractable than Rician fading.
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