Optimized Power Distributions for Partitioned Signaling on Random Linear Matrix Channels

Transcription

1 Optimized Power Distributions for Partitioned Signaling on Random Linear Matrix Channels Christian Schlegel, Marat Burnashev, Dmitri Truhachev, and Lukasz Krzymien Department of Electrical and Computer Engineering University of Alberta Edmonton, AB, CANADA Abstract Iterative processing for linear matrix channels, aka turbo equalization, turbo demodulation, or turbo CDMA, has traditionally been studied as the concatenation of conventional error control codes with the linear (matrix) channel. However, in several situation, such as CDMA, the channel itself contains inherent redundancy, such as a repetition code contained in the direct-spread signature sequences of CDMA. For such systems, iterative demodulation of the linear channel exploiting the inherent code structure, followed by feed-forward conventional error control coding provides an efficient and powerful alternative, and outperform the more complex turbo CDMA methods for equal power modes (users). However, such equal-power systems are spectrally limited. In this paper optimized power distributions are derived and studied which allow arbitrary spectral efficiencies to be achieved with simple two-stage receivers. I. INTRODUCTION Message passing has recently gained much attention, first as an efficient decoding method for turbo and lowdensity parity-check codes, and later for a host of applications ranging from joint detection in multiple-access channels [12], multiple-input multiple-output (MIMO) receiver processing, intersymbol-interference (ISI) channel demodulation, ad-hoc sensor network communications, and many more. In some practically relevant cases the communications system in question is well described by a random matrix equation. This is the case for example in code-division multiple access (CDMA) using (pseudo-) random signature sequences, or in random flat MIMO channels. Such random linear matrix channels are the topic of interest here. We consider efficient demodulation of signal in such random matrix channels. Our ideas are based on a body of prior work, all of which targeting the complexity of receiver processing. It has, of course, been known for a long time that linear algebraic systems (or equations) have very efficient iterative solution methods [2], [1] such as the Gauss-Seidel method. These methods provide an attractive methodology to implement virtually any linear receiver. Furthermore, implementations of such methods lead back to very basic interference cancellation operations [16]. Linear receivers, however, are not efficient detectors for discrete signals, and many modifications to account for the discrete (binary) nature of the transmitted signals have been proposed [6]. With the advent of turbo coding, researchers have realized that and how to include an error control coding device into the processing chain of an iterative receiver [7], [3]. This is commonly referred to as turbo detection, or turbo equalization. Often, and in analogy to turbo coding, an interleaver is put in between the error control coding device and the actual channel, thus creating a serially concatenated coding systems. While such systems perform well This work was supported in part by icore Alberta, and the Alberta Ingenuity Fund.

2 2 in simulations, the choice of the error control coding device was typically arbitrary and its impact is not well understood. While EXIT [15] analysis provides a means for quantifying the limits of a given combination of code and channel, it gives no definitive direction on how to find good codes. In [11] the authors argued that conventionally good codes were intrinsically poor components in such iterative demodulation systems, and proposed to separate the receiver operation into an interference suppression stage the iterative demodulator and a error control stage using a conventional error control system. Some systems, however, such as CDMA, have an inherent built-in error control mechanism which can be exploited. Tanaka [14], for example, analyses a chip-based interference cancellation receiver, while [16] essentially had proposed a similar systems without, however, using the locally optimal tanh( ) to produce soft-bit required for cancelation. [1], on the other hand, noticed that interleaving the chips could significantly increase the interference resistance of such a system. [11] finally related these systems to turbo detection by explicitly recognizing that a signature sequence can be viewed as a repetition code. Breaking such sequences into (a few) partitions with interleaving between the partitions, [11] proposed and analyzed partition spreading (PS). PS performs at least as well as direct minimum-mean square error (MMSE) filtering of the same system, and in the high signal-to-noise ratio regime performs significantly better. In fact, for load values α = K/N < 2.8, where α is the aspect ratio of the random matrix, the performance equations coincide with those for optimal APP detection computed via statistical mechanics methods by Tanaka [13], suggesting that PS is a computationally simple, optimal cancellation detection strategy for systems with α < 2.8 and equal power distribution. If higher loads, or aspect ratios, of the channel need to be handled, equal-power partitioned cancellation can no longer be applied since convergence of the iterative system undergoes a phase change, and the final signal-to-noise ratio values are much poorer. The detector still achieves a performance equivalent to MMSE filtering, but at such large aspect ratios, the performance of MMSE detection itself is rather poor. Furthermore, at higher values of the signal to noise ratio equal-power cancellation message passing fails to come close to the Shannon capacity limit of the channel. To further increase spectral efficiency, a successively wider spread distribution of user powers is required. That is, the user population must spread its transmission powers by a factor determined by the desired spectral efficiency. Note that the received power level is relevant for decoding, thus location of different transmitters can be part of this power spreading strategy. Alternately, groups of different rate users can lead to analogous conclusions, but this is not the scope of this paper. In [11], an analysis using strong error control codes with sharp performance thresholds was used to show that a system with geometrically layered power groups can achieve the channel capacity. That result was partly unsatisfactory since the implied iterative decoding/demodulation receiver is perceived to be too complex for implementation in current technology. In this paper we study the potential of non-singular power level distributions for a two-stage demodulation/exterior decoding receiver structure, where only the inherent system repetition code participates in the iterative demodulation process. A. System Description II. SYSTEM As example system we consider a CDMA channel where K users generate independent information bit streams which are encoded by K parallel (exterior) error control encoders of block length L. The binary symbols of user k {1, 2,..., K}, v k = (v k,1, v k,2,..., v k,l ), are each spread by a direct spreading sequence s k,l,i, i = 1, 2,..., N, consisting of N randomly and independently chosen chips from { 1/ N, 1/ N}, and N is the spreading gain. Each spread symbol is then partitioned into M equal-length partitions, where M N. The partitions of all the symbols are permuted within (at least) the frame. Denoting the power of user k by P k, the transmitted signal of user k is L 1 x k (t) = l= M 1 Pk v k,l m= c k,l,m ( t lt τ k π k (m) T M ) (1)

3 3 where c k,l,m (t) = N/M 1 n= s k,l,n+m N p(t nt c ) (2) M is the m-th section of the spreading waveform for bit v k,l, T the duration of a symbol, and p(t) is the unit-energy chip pulse. Furthermore, T c = T/N, and we allow the system to be asynchronous with τ k < T. π k (m) is the permuter of the position of the mth spreading partition for the symbols of user k, and is assumed to be random. The combined signal from all users is transmitted over an additive white Gaussian channel (AWGN) with power spectral density N, giving the received signal where n(t) is the noise process. B. Two-Stage Decoding y(t) = K x k (t) + n(t) (3) k=1 While iterative decoding can be organized in many feasible schedules, a natural and low-complexity schedule is the following two-stage schedule. In a first stage iterative demodulation is applied only to the different partitions, using message passing principles and utilizing the repetition property of the partitioned symbols. This first stage layers the channel into K user subchannels, each with reduced interference. In a second stage each user s date stream is decoded with a conventional error control or correction code that meets the requirements of rate and signal-to-noise ratio threshold. More precisely, the received signal is filtered by filters matched to the spreading partitions. The received signal of the mth partition of bit v k,l is therefore Pk z m,k,l = M v k,l + I m,k,l + η m,k,l (4) and log-likelihood ratios for v k,l can be computed from observation (4) as ( ) P (vk,l = 1 z 1,k,l, z 2,k,l,..., z M,k,l ) ln = 2 M P (v k,l = 1 z 1,k,l, z 2,k,l,..., z M,k,l ) σ 2 z m,k,l (5) m where σ 2 is the variance of I m,k,l + η m,k,l. A graphical representation of the dependencies of different partitions reveals the sparse factor graph illustrated in Figure 1, i.e., with sufficient inter-partition interleaving, the expected length of cycles will grow arbitrarily large. It is evident then, if optimal node processing is performed at the different nodes, message passing in this graph is expected to approach ideal a posteriori probability (APP) demodulation of the linear interference channel. Optimal node processing at the sparsely connected partition node (the repetition code) is computationally simple, and at each demodulation iteration i :,, I, the (equality) node of bit v k,l returns a the soft bit estimate of v k,l to the channel partition m, which is calculated as ṽ k,l,m (i) = tanh 1 M σi 1 2 z k,l,m (i 1) (6) m m i.e., the signal on the outgoing message edge is not included. These soft bits contain all the necessary probabilistic information required.

4 4 Optimal probability processing at the partition (CDMA) nodes is not feasible due to (generally) dense node degree. These nodes therefore compute a simplified interference canceled signals for each user k and each partition m as y (i) k (t) = x k(t) + K L 1 k k l= Pk M (v k,l ṽ k,l,m )c k,l,m m=1 ( t lt τ k π k (m) T ) + n(t) (7) M from which new received signal z k,l,m (i) are generated by matched filtering again. It is straight forward to see that the combined noise and interference variance of I m,k,l + η m,k,l at iteration i, and for sufficiently large K and N, is σ 2 i = 1 N K P k σd,i,k 2 + σ2, where σd,i,k 2 = E [ (v k ṽ k,i ) 2] (8) k=1 Demodulation proceeds for I < iteration, where it can be shown that σ 2 i is monotonically decreasing with i, and therefore σ 2 σ 2 is the minimum value possible. Fig. 1. Factor graph for a partitioned CDMA system. The square nodes represent a binary modulation symbol each, the lines and circles are partitions. C. Prior Results for Equal Power Users In the ideal case of large interleavers, the performance of an equal power partitioned system is exactly described by (8), with z m,k,l Gaussian distributed. This is due to the fact that for large K, the residual interference I m,k,l is Gaussian. Analysis of the system reduces to the analysis of the iterative equation σ 2 i = ασ 2 d,i,k + σ2, where σ 2 d,i,k = E [ (v k ṽ k,i ) 2] (9) As shown in [11], the signal-to-noise ratio evolution of (9) is given by [ ( )] 2 κσ = κσ καe 1 tanh + ξ ; κ = M M 1 κσ 2 and ξ is a unit-variance zero-mean Gaussian random variable. Equation 1 agrees for κ = 1 with the results in [13] for APP decoding, and shows that partitioned spreading is essentially performing optimal APP demodulation for loads α < Furthermore, also shown in [11], the fixed point variance of (1) is upper bounded by σmmse 2 = 1 ( ) α 1 + σ 2 + (α 1 + σ 2 2 ) 2 + 4σ 2 (11) κσ 2 (1) 1 Note that the proof of (1) in [13] works only for α < 1.49.

5 5 which is identical with the performance of an MMSE filter, and therefore partitioned iterative demodulation is always at least as good as of MMSE filtering. For large signal-to-noise ratio levels, the spectral efficiency limit of α < 2.8 means that there is a growing gap between the performance of iterative equal-power partitioned spreading and ideally achievable spectral efficiencies. This gap can only be bridged by using a non-singular power level distribution. In [8] the authors show that the system aspect ratio is limited by Lemma 1: For binary codes with rates R and AWGN signal-to-noise thresholds γ code such that Rγ code > y th /2 (typically R >.5) the maximum aspect ratio achievable with J user power groups and two-stage decoding is upper bounded by α max = Moreover, for any ε > load α max ε can be achieved as σ 2. J y th g(y th ) = Jα th 2.8J (12) The achievability of the loads in Lemma 1 used a geometric distribution of the powers in the different groups of users, i.e., P j = a j 1 P for a > 1. A. How Much Can We Gain with Unequal Power III. OPTIMAL POWER PROFILES AND ACHIEVABLE LOADS Using the simplifying assumption that once P k /σi 2 µ, the signal of user k can be removed from the interfering signals, we show that 1) In the case of small system aspect ratios α, very little can be gained with respect to equal power distributions. 2) On the other hand, if α is large, large gains are possible with unequal power distributions. To formally show the first claim, note that for any distribution P 1 P 2... P K, we have P 1 µσ 2 ; µ = 2Rγ. On the other hand, for equal powers we have from (9) P = µσ 2 1 αµg(µ) Therefore for the total powers in the unequal and equal cases, P tot (P 1,..., P K ) and P tot (P ) = αp we obtain 1 P tot(p 1,..., P K ) αp αp 1 αp 1 αµg(µ) and therefore the gain is small if αµg(µ) is small. Point 2) can be shown similarly. B. On Optimal Power Distributions We consider the case when the number of allowed power levels is not bounded. For P 1 P 2... P K, we have the equations σk 2 = 1 k ( ) Pi P i g N σ 2 + σ 2, k = 2,..., K i=1 k (13) P k σk 2 = 2Rγ = µ, k = 1,..., K That is, the optimal distribution exactly achieves the threshold value µ for every power level P k, otherwise, that power level can be reduced to further decrease the total power required in the system.

6 6 Denoting P i = P (i), we may use the approximation P (k) µ = 1 k P (x)g N ( ) µp (x) dx + σ 2, P (k) k = 2,..., K Defining the user power distribution T (u) = P (un) we obtain the integral equation z ( ) µt (u) µ = T (u)g du + σ 2, for all z α (14) for the optimal power distribution = P (zn), z α. We note two simple, but useful facts about the solution of (14), and (13), respectively. Remark 1. If T (z, µ, α, σ) is the solution of (14), then T (z, µ, α, σ) = σ 2 T (z, µ, α, 1) Therefore instead of (14) we may consider a simpler equation ( ) µt (u) µ = T (u)g du + 1, for all z α (15) Remark 2. Note that for any z < α the value does not depend on α. It means that starting with T () = µσ 2 we can build the function from (15) without regards to the aspect ratio α (as long as z α, and convergence is assured). This means that if we have two different aspect ratios α < α, then the optimal, z α, for the ratio α remains optimal for α (in the region z α ). Since g(µ) g due to the monotonicity of g(µ), we obtain from (15) Note that the equation has the unique solution z g(µ) ( ) µt (u) 1, T (u) du + 1 z µ z µ = A z β T (u) du + 1, for all z α (16) T (u) du + 1 Therefore with (16) we get the following lower and upper bounds = µe Aµz (17) µe µg(µ)z µe µz (18) and the optimal power distributions have an exponential (geometric progression) form with total power bounded as (from (18)) e µg(µ)α 1 g(µ) α dz e µα 1

7 7 IV. TWO-STAGE SYSTEM Unfortunately, in arriving at (17), we made two assumptions 1) User which have reached their decoder threshold are removed from the system. 2) The aspect ratio is small enough such that all iteration point in (13) can be reached by the iterative receiver. Point 1) would imply an iterative system where the error control decoder is included in the iteration process, as in [9], [4], [17], which is not a desirable option from a practicability point of view. Point 2) means that the results in the previous section cannot give us any conclusions about the maximal achievable spectral efficiencies. We will, however, adopt the exponential power distribution derived above as model for a two-stage decoder which operates without including the error control device. That is, we use an exponential power distribution (17) in our two-stage decoder model. Considering the dynamic nature of (8), and using the same integral approximations from Section III-B, we obtain the following integral condition for convergence of the first stage of our two-stage receiver: 1 α T (u) v g ( ) T (u) v du + σ2 v, for all T () v γ th α T (u) du + σ 2 (19) where γ th is the signal-to-noise ratio minimally required by the outer error control coding device to operate. The power distribution of the different of the modes is assumed to be the exponential distribution T (u) = e au ; u α (2) with average power P = eaα 1 (21) a Equation 19 can now be analyzed in terms of its power and spectral efficiency. In particular, there exists as threshold rate parameter Aµ = a th in (17), such that for all a > a th, the spectral efficiency grows without bound as a function of power also evident in (12). We will present spectral efficiency and further results in the presentation. The average power per bit, over the noise power spectral density for this system is given by E b = eaα 1 1 N aα 2Rσ 2 (22) Relating this to the Shannon bound of the AWGN channel, using the approximations R 1 for 1/(2σ 2 ) > γ, and γ > 6dB, we obtain the approximate relationship Eb 22η 1 2 N 2η a γ (23) which would suggest that the iterative system with optimized power distributions can approach the capacity of the AWGN channel to within about 6-8dB for arbitrary spectral efficiencies. The following figure shows computed spectral efficiencies for more practical ranges of the E b /N, illustrating that close-to-shannon bound performance is feasible by spreading out the power distribution of the different signaling modes.

8 T w o - S t a g e P S D e t e c t o r a n d S h a n n o n C a p a c i t y 8 6 ] di m / bi t s [ Shannon Bound a=1.395 a=1.37 a=1.25 a=1 i t y c ap C a 2 Equal Power E b /N Fig. 2. Achievable spectral efficiencies with two-stage iterative demodulation followed by conventional forward error control coding. REFERENCES [1] O. Axelsson, Iterative Solution Methods, Cambridge University Press, [2] A. Grant and C. Schlegel, Convergence of linear interference cancellation multiuser receivers, IEEE Trans. Commun., Vol. 49, No. 1, October 21, pp [3] P.D. Alexander, A.J. Grant, and M.C. Reed, Iterative detection on code-division multiple-access with error control coding, European Transactions on Telecommunications, 9(5): , Sept. Oct [4] P.D. Alexander, M.C. Reed, J.A. Asenstorfer, and C. Schlegel, Iterative multiuser interference reduction: Turbo CDMA. IEEE Trans. Commun., Vol. 47, No. 7, July 1999, pp [5] J. Boutros and G. Caire, Iterative multiuser joint decoding: unified framework and asymptotic analysis, IEEE Trans. Inform. Theory, Vol. 48, No. 7, pp , July 22. [6] D. Raphaeli, M.K. Simon, M.K., D. Divsalar, Improved parallel interference cancellation for CDMA, IEEE T. Commun., Vol. 46, No. 2, Feb. 1998, pp [7] M.C. Reed, C. Schlegel, P.D. Alexander, and J.A. Asenstorfer, Iterative multiuser detection for CDMA with FEC: Near single user performance. IEEE Trans. Commun., Vol. 46, No. 12, December 1998, pp [8] L. Krzymien, D. Truhachev, and C. Schlegel, Coded Random CDMA with Partitioned Spreading, 44th Allerton Conference on Communication, Computing and Control, Monticello, IL, USA, Sep. 26. [9] M. Moher, An iterative multiuser decoder for near-capacity communications, IEEE Trans. Comm., vol. 47, pp , July [1] Li Ping, Interleave-division multiple access and chip-by-chip iterative multi-user detection, IEEE Comm. Mag., Vol. 43, No. 6, pp. S19 S23, June 25. [11] C. Schlegel, Z. Shi and M. Burnashev, Asymptotically Optimal Power Allocation and Code Selection for Iterative Joint Detection of Coded Random CDMA, submitted to IEEE Transactions on Information Theory, January 24. [12] C. Schlegel and A. Grant, Coordinated Multiple User Communications, Springer Publishers, 25. [13] T. Tanaka, A statistical-mechanics approach to large-system analysis of CDMA multiuser detectors, IEEE Trans. Inform. Theory, Vol. 48, No. 11, November 22. [14] T. Tanaka, and M. Okada, Approximate belief propagation, density evolution, and statistical neurodynamics for CDMA multiuser detection, IEEE T. Inform. Theory, Vol. 51, No. 2, Feb. 25, pp [15] S. ten Brink, Convergence behavior of iteratively decoded parallel concatenated codes, IEEE Trans. Comm., Vol. 49, No. 1, Oct. 21. [16] M.K. Varanasi, and B. Aazhang, Multistage detection in asynchronous code-division multiple-access communications, IEEE T. Commun., Vol. 38, No. 4, April 199, pp [17] X. Wang and H.V. Poor, Iterative (Turbo) Soft Interference Cancellation and Decoding for Coded CDMA, IEEE Trans. Commun., vol. 47, no. 7, July 1999.