Analysis of coding on non-ergodic block-fading channels

We study coding for the non-ergodic block-fading channel. In particular, we analyze the error probability of full-diversity binary codes, and we elaborate on how to approach the outage probability limit. In so doing, we introduce the concept outage boundary region, which is a graphical way to illustrate failures in the decoding process. We show that outage achieving codes have a frame error probability which is independent of the block length. Conversely, we show that codes that do not approach the outage probability have an error probability that grows logarithmically with the block length. 1 Channel model and notation The block-fading channel is a simplified channel model that characterizes delay-constrained communication over slowly-varying fading channels [8]. Particular instances of the blockfading channel are orthogonal-frequency multiplexing modulation (OFDM) and frequencyhopping systems. We consider a block-fading channel with nc fading blocks, whose discrete-time channel output at time i is given by yi = hi xi + zi i = 1 . . .Nf (1) where Nf denotes the frame length, xi ∈ {−1,+1} is the i-th BPSK modulated symbol, zi ∼ N (0, σ) are the i.i.d. Gaussian noise samples, σ = N0/2, and hi is a real fading coefficient that belongs to the set א = {α1, α2, . . . , αnc}. (2) The set א contains the possible fading coefficients (assumed i.i.d. Rayleigh distributed), it is fixed for the whole duration of the codeword, and changes independently from codeword to codeword. With a slight abuse of notation, we will also denote א as the vector of fading coefficients. We further assume that the fading coefficients are known to the receiver and not known to the transmitter. A simple illustration for the block-fading model is given in Figure 1. Strictly speaking, the capacity of the block-fading channel is zero as there is an irreducible probability that the decoder makes an error. In the limit of large block-length, this probability is the information outage probability defined as [8] Pout = Pr{Iא ≤ R} (3) one frame fading 1 fading 2 fading n frame length = Nf bits Nf=n onse utive bits Figure 1: Block fading channel model for a binary (Nf , Kf ) code (one frame). where Iא denotes the instantaneous mutual information between the input and output of the channel for a particular channel realization א, and R is the transmission rate in bits per channel use. In [5] it is shown that for BPSK inputs, the asymptotic slope d of Pout with the signal-to-noise ratio in a log-log scale (commonly referred to as diversity) yields the optimal code design tradeoff, which is given by (Singleton bound) d = 1 + ⌊nc(1− R)⌋. (4) Therefore, in order to achieve reliable communication, we will design codes that achieve this optimal tradeoff. We consider linear binary block codes C = (Nf , Kf)2 of length Nf , dimension Kf , and rate R = Kf/Nf . A codeword of C will also be called a frame. Two constructions are considered in this paper. Construction 1 assumes that C is the direct sum of N small block codes C0(n, k)2, where Nf = N × n and Kf = N × k. This construction also includes the direct sum of N small Euclidean codes on a real channel, such as a space-time block codes on a multiple antenna channel. Construction 2 corresponds to a unique codeword per frame, such as using a parallel turbo code of interleaver size Kf and block length Nf . In this case, Nf = ncN . The codes studied in this paper are full diversity, i.e., R = 1/nc. We now introduce the notion of channel multiplexer, which is a particular type of interleaver that maps the output of the encoder to the nc different channel coefficients. Definition 1 A multiplexer is a bijection from the integer set {1 . . . Nf} to the set {1 . . . nc} , where the superscript denotes Cartesian product. For a given code C(Nf , Kf ) transmitted on a non-ergodic channel with nc states, the total number of multiplexers is (Nf )!/(N !) nc . A multiplexer is said to be ’regular’ if it is has a periodical pattern. The number of regular multiplexers of period n reduces to (n!)/((n/nc)!) nc . As an example, consider the (8, 4, 4) linear binary code. For nc = 2 channel states, a regular multiplexer is defined by the fading vector (α1, α1, α1, α1, α2, α2, α2, α2) or equivalently (11112222) applied on all (8, 4, 4) codewords inside a frame. Example of multiplexers for parallel turbo codes are given in Figure 2 h-diagonal Multiplexer s1 1 2 1 2 1 2 s2 2 1 2 1 2 1 s3 3 3 3 3 3 3 h-π-diagonal Multiplexer s1 1 2 1 2 1 2 s2 2 X 2 X 2 X s3 π( X 1 X 1 X 1 ) Figure 2: Multiplexers for near-outage performance from [1]. (Left) h-diagonal multiplexer for a rate 1/3 turbo code (nc = 3). (Right), h-π-diagonal multiplexer for a rate 1/2 turbo code. In this paper we study the error probability of binary codes in the block-fading channel. One of the objectives of this paper is to study the variation of the frame error probability with respect to the frame length, i.e. Pef = Pef(Nf) = Pef(N), under a fixed channel multiplexing. In particular, the outage formulation implies that outageapproaching codes not only should meet the Singleton bound, but also should perform close to the outage probability for large block-length. This implies that for a given channel realization, outage-approaching codes exhibit a threshold phenomenon, namely, the error probability goes to zero for σ < σ th, for some threshold noise variance σ 2 th (which depends on the fading realization) [5, 4]. In particular, we show that for classical short block codes or convolutional codes for which Pef scales linearly with N in the ergodic channel, Pef scales logarithmically with N in the non-ergodic case. In order to illustrate this effect, consider the simple case nc = 2 and C0 = (8, 4, 4) drawn in Figure 3. It is clear that Pef does not scale linearly with N (for a fixed Eb/N0). This behavior has been noticed in [1] for convolutional codes. Obviously, such codes cannot approach the outage probability.