Performance Analysis of Diversity Combining over Rayleigh Fading Channels with Impulsive Noise

This paper investigates the performance analysis of diversity combining over flat fading channels subject to additive impulsive noise. Specifically, the impulsive noise follows the symmetric α-stable (SαS) distribution. Traditionally, this is a difficult problem since the analytical expression for the probability density function of a general SαS distribution remains unknown. In order to tackle this difficulty, we adopt Fourier power transform to obtain an exact bit error rate (BER) expression for the Genie-aided receiver as well as a BER upper bound for the conventional linear matched filter receiver, without taking any approximation. MonteCarlo simulations are performed to validate our analytical results. The simulation results show that our proposed BER expressions are valid in the whole geometric signal-noiseratio (GSNR) ranges, thus outperforming the previous results valid only at high GSNR values.


Introduction
The additive Gaussian noise model has been widely used in the analysis and design of communication systems since it yields a simple and tractable mathematical model [1].However, impulsive noises often arise due to automobile spark plugs [2], lighting discharges, underwater sonar, and network interference [3], etc., which demonstrates that the Gaussian noise model may not be accurate in many situations.For example, a power line communication (PLC) channel is mainly affected by the ever-present background noise and the occasional high-amplitude impulses.A standard model for characterizing impulses is to assume Gaussian distributed amplitude and Poisson distributed arrivals [4].Moreover, studies [5][6][7][8] showed that the multiple access interference might result in a Symmetric-Alpha-Stable (SαS) distribution in a multi-user network with the power-law path loss.The SαS distribution has been used to model the co-channel interference and the aggregate interference in different wireless networks such as cognitive radio networks, UWB networks and cooperative transmission in UWB relays [9].In fact, the α-stable distribution encompasses an important class of distributions which can successfully model a number of impulsive noise processes, because it satisfies the generalized center extreme limit theorem.In particular, the Gaussian distribution can be considered as a special case of the SαS distribution [10].Therefore, we consider the SαS distributed impulsive noise in this paper.
In order to overcome the performance degradation of communication systems over a fading channel, the receiver diversity is usually employed.Some results of diversity combining subject to non-Gaussian noise have been obtained in [11], which investigates the performance in the presence of flat fading plus spherically invariant noise.In [12], a statistical model called Middleton's Class-A noise was proposed to model the impulsive noise approximately.The approximation is given by a weighted sum of multiple Gaussian distributions with different variances and amplitudes.In [13], the authors further provided the closed-form bit-error-rate (BER) expressions for the linear diversity combining scheme where the noise follows the Middleton class-A distribution.Furthermore, in [14], [15], the authors investigated the symbol error rate for linear diversity combining schemes in high signal-noise-ratio (SNR) region under the assumption of flat fading plus symmetric non-Gaussian noise.It is noteworthy that all the aforementioned papers assumed the existence of the moments of the noise distribution, which cannot be guaranteed for α-stable probability density functions (PDFs).
In [16], the authors analyzed the BER performance for diversity combining over a flat fading channel with the additive noise modeled as α-stable distribution.The BER expressions therein are obtained under the high SNR region, in which the PDF of α-stable distribution can be approximated by a simple yet accurate function.However, the results in [16] will become inaccurate when the SNR decreases, because the approximation used in [16] will become inaccurate.To the best of our knowledge, it is still an open question to explore the analytical error rate performance of diversity combining schemes over flat fading channels, subject to αstable additive noise over entire SNR region of interest.The key difficulty lies in the unknown closed-form expression for the SαS distribution at arbitrary SNRs.Therefore, our contributions in this paper are to address this difficulty and to derive the analytical BER performance at arbitrary SNR values.Specifically, to tackle the difficulty, we exploit the special characteristic function (CF) of SαS distribution and solve an equivalent problem in the "frequency" domain via the technique of Fourier power transformation [17] for both the Maximal Ratio Combining (MRC) receiver with full sideinformation (which is called as Genie-Aided Receiver in [16], GAR ) and the Linear Matched Filter receiver (LMF) without making any assumption on the SNR region.
The rest of the paper is organized as follows.The αstable distribution and system model are introduced in Sec. 2. We obtain the exact BER expression for GAR in Sec.3.1 as well as a BER upper bound for LMF in Sec.3.2.Monte Carlo simulations are performed in Sec. 4 under high and moderate impulsive noise scenarios.It is shown that the presented analytical results are consistent with the simulation results in all SNR regions.Finally, this paper is concluded in Sec. 5.

Notion:
We use boldface type for vectors.E[•] denotes the statistical expectation; ||x|| p is the p-norm of a vector x; the distribution of a circularly symmetric complex Gaussian random variable with zero mean and variance σ 2 is defined as CN (0, σ 2 ), and ∼ means "distributed as"; {•}, {•} denote the real and imaginary part of a complex, respectively; x * denotes the conjugate of a complex number x; x y means the expression y is described as x for simplicity.
For any SαS random variable (α 2), only its p-order moments exist, where 0 ≤ p < α.Hence, the variance of SαS distributed noises, is undefined.As a result, the geometric SNR (GSNR) [19] is used in this paper, i.e., where A s , C g ≈ 1.78, and σ denote the amplitude of a modulated signal, the exponential of the Euler constant, and the geometric power of the SαS random variable, respectively.The normalization constant "2C g " is used to ensure that the definition of the GSNR corresponds to that of the standard SNR if the channel noise is Gaussian.
The SαS random variable has some useful properties as follows.Property 1 [20, p20, where C α = Γ(α) sin(πα/2)/π, and Γ(z) = ∞ 0 t z−1 e −t dt.Although a closed-form expression for the PDF, f α (x) ∼ S α (σ, β, 0), does not exist except for a few special cases, asymptotic expansion is well known as x → ∞, which is given by where we write Next we define the complex isometric SαS [16] random variables used in the rest of this paper.

System Model
The considered system model employs one transmitantenna and L receive-antennas.Specifically, the received signal corrupted by the SαS impulsive noise can be described by the following equation where ρ, h l , s and w l represent GSNR, channel fading coefficient, the transmitted signal and the impulsive noise, respectively.Specially, ρ is the GSNR defined by (3).The l th channel fading coefficient, h l , follows CN (0, 1) distribution.The transmitted signal s ∈ {−1, 1} is chosen from a binary phase shift keyed (BPSK) constellation.The l th channel impulsive noise w l follows the i.i.d.CS α (1, 0, 0) distribution since any scale is subsumed in ρ.
By Definition 1, w l can be written as where

Performance Evaluation
We evaluate the BER performance of different combination schemes in this section.Section 3.1 focuses on the genie-aided receiver and Section 3.2 focuses on the LMF receiver.

Genie-Aided Receiver
denote the instantaneous channel realizations and a-priori knowledge of the impulsive noises at the receiver, respectively.The socalled Genie-Aided Receiver (GAR) has complete a-priori knowledge of A and h [16].Therefore, the received signal in (8) can be rewritten as Since w l / √ A l ∼ CN (0, 2), the formulation in (8) reduces to diversity combining with Gaussian noise, for which, the optimum combining scheme is maximal ratio combining (MRC), with the l th combining coefficient equal to h * l / √ A l .Hence, the decision rule is given by As a result, the corresponding BER conditioned on A and h is given by where Q(•) is the Gaussian Q-function which can be expressed as [21, pp.182], Since {h l , l = 1, 2, • • • , L} are i.i.d.[16] and h l ∼ CN (0, 1), |h l | 2 follows the exponential distribution with mean 1.
The identity Using this identity and the Q(•) function in (13), we obtain the average BER of GAR with respect to h as where [16], the average BER of the GAR with respect to h and A is thus given by where random variable A represents any element of A.
Let f α/2 ( A) denote the PDF of A with S α/2 [cos(πα/4)] 2/α , 1, 0 distribution, the expectation on the right-hand side of ( 16) can be rewritten as Unfortunately, the closed-form expression of f α/2 ( A) does not exist, except for α = 1.Eq. ( 17) cannot be evaluated via direct integration, however, it can be obtained with numerical integral calculation.Under the high GSNR assumption (ρ → ∞), the authors of [16] obtained an approximated expression of (17).Substituting the PDF f α/2 ( A) as suggested by ( 6) into (17), we get As shown in Fig. 1, the greater GSNR ρ (or α), the higher the approximation accuracy.In particular, (18) gives a good approximation only in the case of the larger GSNR and moderate impulsive noise.Meanwhile, its approximation accuracy degrades very fast in the case of low GSNR or severe impulsive noise.Therefore, the approximation errors cannot be neglected if the GSNR ρ is medium or small.In order to address this problem, in this paper, we exploit the Fourier power theorem (FPT) [17] to obtain an exact expression of (17).Suppose f (t) and g(t) are two real functions, and the corresponding Fourier transforms are F (ω) and G(ω), respectively.According to the FPT, the following formula is workable, where (•) * denotes the conjugation of a complex function.
We obtain the following result via the FPT.

Proof:
The core of proof is to obtain the closed-form expression of (17).Let g( A) A sin 2 θ A sin 2 θ+ρ/2 = A A+ρ θ , then the expectation on the left-hand side of (17) can be rewritten as We cannot solve directly this integral value since there is no closed-form expression for f α/2 ( A).As we know, the PDF and the CF of a stochastic process compose a Fourier transform pair.Fortunately, the Fourier transform of f α/2 ( A) (i.e., its CF) and the Fourier transform of g( A) have closedform expressions.Therefore, we solve this problem by using the FPT and obtain the exact expression for (17), which is given as (21) For the details of the proof, please refer to Appendix 6.1.

Remark 1:
If GSNR ρ is large enough to make (18) satisfied, then the BER of GAR can be further simplified as which is consistent with the previous result in [16, see Eq. ( 10)].

Linear Matched Filtering
This subsection analyzes the BER performance of linear matched filtering (LMF).The decision rule of LMF is the maximum ratio combining when the noise is Gaussian, which is given by Since the SαS noise has infinite variance, the LMF is no longer optimal.Therefore, it is important to quantify its performance under the SαS noise.
By definition, BER of LMF is where λ LMF is the decision statistic, and is the instantaneous channel realizations.Substituting ( 8), ( 9) into ( 23), λ LMF can be written as where ||h|| α , 0, 0), and introducing a standardized SαS random variable v ∼ S α (1, 0, 0), the λ LMF can be represented in terms of v. Noting s = −1 in (24), the decision statistic, λ LMF , is rewritten as Hence, we will derive exact BER expressions for BER upper and lower bounds of LMF in the following theorem.

Theorem 2
The BER upper and lower bounds of LMF are given by where M (a, b, z) is the first kind of Kummer confluent hypergeometric function (KCHF) [23,Chap. 18.6].
Proof: λ LMF is represented in terms of a standardized SαS random variable v ∼ S α (1, 0, 0).Using the Hölder inequality and averaging over all channel realizations, we can derive the BER upper-bound as where f χ 2L (x) represents the PDF of a χ distribution with 2L degrees of freedom, and Since there is no closed-form expression for the PDF f α (v) of v, to the best of the authors' knowledge, this integral value is very difficult, if not impossible, to obtain via direct integration.In [24][25][26], some approximations to the PDF of the SαS distribution are proposed.Since the approximation uses simple formula to calculate the PDF, its accuracy can be further improved.Moreover, it is still very complicated to calculate (28) by using approximation method.Again, we turn to solve an equivalent problem in "frequency" domain by using the FPT.Noting that the integral lower limit of f α (v) is σ z x instead of 0, we cannot directly use the FPT.For the details of the proof, please refer to Appendix 6.2.KCHF belongs to an important class of special functions in the mathematical physics.By exploiting some properties of KCHF [23,Chap. 18.6], we draw the following remark.
We will show in Section 4 that the BER upper bound (27b) is in fact very tight for the LMF receiver via numerical simulations.

Simulation Results
In this section, we verify the results of Theorems 1 and 2 by performing Monte Carlo simulations.To compare with [16], the BERs of GAR and LMF receivers are calculated and compared against the theoretically predicted values/bounds given in Theorems 1 and 2. The impact of electromagnetic interference from household appliances on digital subscriber loop systems is shown to be well modeled by the SαS distribution [27].Similarly, we also assume that α = 1.43 which corresponds to the value estimated in [28] for modeling radio frequency interference in laptop receivers.To simulate a "highly impulsive" scenario which corresponds to a path loss exponent of 4 (2/α) in an environment where the interfering nodes are scattered according to a spatial Poisson point process on a two dimensional plane [3], α = 0.5 is also considered.The MATLAB code of the SαS distribution is obtained from [29].
Figure 2 shows the performance of GAR over channels with severe (or moderate) impulsive noise where the BER curves based on simulation and theoretical expression (20) are plotted.It is observed that the simulation results match with the theoretical formula (20) in the whole GSNR range.More specifically, the BER performance of GAR gets better when the number of receiver antenna L increases.Especially in the high GSNR, e.g., ρ > 10 dB, the BER drops faster.On the other hand, the performance of GAR with severe impulsive noise is orders of magnitude apart from one with moderate impulsive noise.
Figure 3 shows that the simplified theoretical expression (22) presented in Remark 1 is no longer accurate in the low GSNR range, e.g., ρ < 10dB for moderate impulsive noise or even ρ < 30dB for severe impulsive noise.It means that it needs higher GSNR to meet the simplified BER formula (22) in severe impulsive noise.
The performance of LMF is examined in Figs.4-5, where it is demonstrated that the theoretical LMF upperbound is tight in the whole GSNR range for the cases of moderate and severe impulsive noise, respectively.It is observed that the LMF upper-bound is tighter when L is smaller or α is larger.Simulation results show that increasing the number of antennas from one to two bring more performance gain as compared to increasing the antennas number from two to three in the case of moderate impulsive noise (e.g.α = 1.43), simplified BER with L=1 (or Eq.( 10) in [16]) simplified BER with L=2 (or Eq.( 10) in [16]) simplified BER with L=3 (or Eq.( 10) in [16]) simplified BER with L=1 (or Eq.( 10) in [16]) simplified BER with L=2 (or Eq.( 10) in [16]) simplified BER with L=3 (or Eq.( 10) in [16]) and the performance gap is limited when further increasing the number of antennas.In the case of severe impulsive noise (e.g.α = 0.5), however, increasing the number of receive antennas is detrimental to the system performance as shown in Fig. 4(a).The behavior of LMF is very different from that of GAR, where employing more antennas always brings higher diversity gain and always improves the system performance.
Figure 5 shows that the theoretical BER upper-bound of LMF in (27b) is tighter than that of [16] in the whole GSNR range.In the case of severe impulsive noise, although GSNR is greater than 20dB, the theoretically derived LMF bound in [16] differs seriously from the simulation results while our upper-bound is always identical to the simulation results.However, in the case of moderate impulsive noise, when GSNR is greater than 10dB, they are almost identical.

Conclusions
In this paper, we analyze the BER performance over fading channels with SαS impulsive noise.By exploiting the characteristic function of α-stable distribution and solving an equivalent problem in the frequency domain based on the Fourier power transform, we successfully derive the exact BER expressions for both GAR and LMF receivers.Extensive numerical results validate our analytical results, which outperform the existing results that are based on asymptotic approximation.

Proof of Theorem 1
Noting that f α/2 ( A) = 0 when A < 0 , we can extend the range of variable A to satisfy the condition of the Fourier power theorem, i.e., The characteristic function of f α/2 ( A), denoted by φ(ω; α/2, 1, σ, 0), is known as And the Fourier transform of g( A) is Hence, using the FPT, we obtain (35) Noting that the integrand of the imaginary term is of odd symmetry, resulting in zero imaginary part.Meanwhile, the integrand of the real term is of even symmetry, thus, )ω α 2 dω.

Proof of Theorem 2
In this Appendix, we prove Theorem 2. Defining for brevity and using the Hölder inequality given by we can derive bound Moreover, defining x √ 2||h|| 2 ∼ χ 2L , which represents a χ distribution (the square root of a χ 2 distribution) with 2L degrees of freedom, we obtain √ ρL 1 2 − 1 α x ≤ 2ρg(h) ≤ √ ρx and thus the following upper and lower bounds conditioned on h, where . Averaging over all channel realizations, with respect to the distribution of h (or x), we obtain the BER upper and lower bounds However, (41) is hard to solve directly.Therefore, we turn to solve an equivalent problem in "frequency" domain by using the FPT.Denoting σ 2 z ρL α−2 α for brevity, we obtain the BER upper bound as If GSNR ρ is large enough to make f α (v) ≈ αC α v −α−1 established, BER UB LMF ( ρ) can be obtained by direct integral from (41), yielding (43) which is consistent with Theorem 2 in [16].However, when GSNR ρ is small, the upper-bound given by (43) yields large deviation.The reason is explained as follows.
The PDF of the SαS distribution, f α (v), does not have a closed-form expression.The requirement of asymptotic expansion (5) in Property 3 is that v is large enough, v > θ as example (here, θ is a constant associated with α).Therefore, the requirement of (43) is σ z x > θ.Note that x is a random variable following a χ 2L distribution.The probability of meeting the requirement (43) is given by This means that σ z x > θ is satisfied with probability of almost 1 when GSNR ρ is large enough, while for a general ρ, f α (v) ≈ αC α v −α−1 is not true with a certain probability.Meanwhile, the true value of f α (v) is far less than αC α v −α−1 when v < θ, especially at the neighborhood of v = 0.

(Fig. 1 .
Fig. 1.The numerical integral calculation (17) (dotted line) against the approximated version (solid line) (18) with various α and ρ values, where the label of vertical axis is E A
a, z) = ∞ z t a−1 e −t dt presents the incomplete gamma function, and M (a, b, z) denotes the first kind of the Kummer confluent hypergeometric function.M (a, b, z) has the series expansion as ∞ k=0 (a) k (b) k k! z k , where (a) k = k−1 i=0 (a +i) = Γ(a + k)/Γ(a) are Pochhammer symbols with (a) 0 = 1 [23].