Performance Analysis of Signal Detection using Quantized Received Signals of Linear Vector Channel

Although there have been many studies on performance analysis of digital communication systems, information-theoretic performance analyses usually assume that the received analog signals are available [1][2][3]. Previous studies [4,5], considering the quantized effect of received signals, used non optimal signal detection schemes, such as linear filters.

In this paper, we evaluate the performance of optimal signal detection using the quantized received signals of a linear vector channel, which is an extension of CDMA or multiple-input multiple-output (MIMO) channels, in the large system limit, where the dimensions of channel input and output are both sent to infinity while their ratio remains fixed.

We consider a K-input, N -output linear vector channel, defined as follows.

Let x 0 = (x 01 , . . . , x 0K ) T ∈ {+1, -1} K denote the channel input vector, and y = (y 1 , . . . , y N ) T denote the output vector, given a linear transform Hx 0 of the inputs, where H is an N × K channel matrix. Assuming the additive white Gaussian (AWGN) channel (variance σ 2 0 ), the outputs are described as

where the components of ν = (ν 1 , . . . , ν N ) T follow the normal distribution N (0, 1). To simplify the analysis, we assume a random channel matrix; the elements {H µk } are independent and identically distributed (i.i.d.) with mean zero and variance 1/N . In this study, we consider signal detection using received signals, which are quantized by an A/D converter at the receiver. We assume an A/D converter that quantizes received signal y into integer n, which satisfies the condition (n -1/2)d < y < (n + 1/2)d, where d > 0 is the quantization step size. The probability distribution of the quantized received signals n = (n 1 , . . . , n N ) T , given the input x 0 and channel matrix H, is represented as

where H T µ , which denotes the µth row of H and

The signal detection of channel input x 0 , given the quantized received signals n and the channel matrix H, can be solved by an inference scheme based on Baysian inference. We define a prior as P 0 (x 0 ), and assume P 0 (x 0 ) = 1/2 K . The detector assumes the channel model of quantization channel to be P (n|Hx) = N µ=1 ρ(n µ |H µ x) and the prior distribution to be P (x). The posterior distribution is represented as P (x|n, H) = P (n|Hx)P (x)

x P (n|Hx)P (x)

Maximizer of the Posterior Marginals (MPM) estimation xk = argmax x k x l =k P (x|n, H) is the optimal inference scheme for minimizing the component-wise estimation error probability, when the assumed channel model and prior distribution are matched to the true ones. Hereafter, we call the detector using true channel model and prior distribution as an optimal detector.

We evaluate the performance of the optimal detector using the quantized received signals in the large system limit, where K, N → ∞ while the ratio β = K/N is kept finite. This paper [6] studies the multiuser detection performance of a CDMA channel with an arbitrary memoryless channel and arbitrary distribution of channel inputs in the large system limit, using the replica method. Since the defined quantized channel (3) is an example of memoryless channels, one can apply the analysis reported in [6] to the performance evaluation of the optimal detector using quantized received signals in the large system limit. Although there is no rigorous justification for the replica method, we assume the validity of the replica method, and related techniques throughout this paper.

In the large system limit, the estimation error probability of the optimal detector

where E X [• • • ], which denotes the average with respect to X, can be evaluated as

The parameter E is determined by solving the following equations for {m, E},

where ρ0 (n| √ βm) is defined as

and ρ′ 0 (n|∆) = ∂ ∂∆ ρ0 (n|∆). For some values of the parameters (σ 0 , d, β), equations ( 7) and ( 8) have three solutions, and these solutions can be distinguished by the value of estimation error probabilities P b ; we define good, intermediate, and bad solutions from the smallest to the largest value of P b . Furthermore, we define the correct solution as one minimizes the function F , which is defined as

It is known that the correct solution corresponds to the performance of the optimal detector. Since the exact evaluation of marginal posterior probability is usually computationally infeasible, we have to resort to approximation techniques. It is also known that a bad solution corresponds to the performance of the belief propagation based on iterative algorithm [7], which produces approximate marginal posterior probabilities in low time complexity.

For

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