Research Collection. Multi-layer coded direct sequence CDMA. Conference Paper. ETH Library

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1 Research Collection Conference Paper Multi-layer coded direct sequence CDMA Authors: Steiner, Avi; Shamai, Shlomo; Lupu, Valentin; Katz, Uri Publication Date: Permanent Link: Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library

2 Int. Zurich Seminar on Communications IZS, March 3-5, Multi-Layer Coded Direct Sequence CDMA Avi Steiner, Shlomo Shamai Shitz Technion IIT, Haifa 3, Israel Department of Electrical Engineering Valentin Lupu, Uri Katz Rafael, Israel Abstract We consider the problem of multi-user detection for randomly spread direct-sequence DS coded-division multiple access CDMA over flat fading channels. The analysis focuses on the case of many users, and large spreading sequences such that their ratio, defined as the system load, is kept fixed. Single layer and multi layer coding are analyzed in this setup. The spectral efficiency, for linear multiuser detectors, is derived for different decoding strategies. Iterative decoding of multilayered transmission with successive interference cancellation SIC is optimized, and the optimal layering power distribution is obtained. For small system loads, the achievable spectral efficiency with the broadcast approach and a matched filter detector exhibits significant gains over single layer coding. I. INTRODUCTION Consider the case of many users, and large spreading sequences, such that the system load is kept fixed. That is, K, N, and = K/N, where K denotes the number of users, and N is the spreading sequence length. The spectral efficiency of direct sequence DS CDMA with random spreading for this regime over fading multiaccess channels is studied in []. In that contribution, ergodic spectral efficiency is studied, assuming that all users are reliably decoded regardless of their received powers. The assumption is that users can adjust their rates according to their experienced fading level, using, for example, an instantaneous feedback from the receiver. Unfortunately, such a feedback and ideal tuning of transmission rates are not always feasible. Thus, these results can be achieved only on fast fading channels, where sufficient fading statistics is observed over a single transmission block. Motivated by practical considerations, decoding of strongest users on block fading channels is studied in []. In this work, it is assumed that all users transmit at equal rate and equal power. In this case the receiver can no longer guarantee reliable decoding of all active users. As a result, the receiver ranks all active users by their received power and decodes the transmission of the largest number of users, for which decoding is successful. The maximal expected sum rate is referred to as the outage capacity. In this work, we first derive the spectral efficiency of successive interference cancellation SIC detectors with iterative decoding. The main idea here is to keep on trying to decode users after every SIC stage, as the residual interference is reduced every iteration, which decreases the effective system load during decoding. This concept is adopted for multilayer multiuser successive decoding, where the optimal power distribution is derived for maximizing the achievable expected spectral efficiency. II. CHANNEL MODEL AND PRELIMINARIES We describe here the channel model and the basic assumptions. Consider the following system model, y = VHx + n where x =[x,..., x K ] is a vector of length K. An individual term x k is a sample of a layered coded signal of the k th user, and x k } are i.i.d. x k } CN,P, wherep sets the power constraint per user. V is an [N K] signature matrix i.i.d. with elements v i,j CN, N,andnis, without loss of generality, a normalized AWGN vector n CN,I N. The channel matrix H is a diagonal matrix H = diagh,h,..., h K of fading gains, which empirical distribution of s k } h k } converges a.s. to a distribution F s s such that E Fs [s] =. The channel matrix H remains fixed throughout a transmission block, which corresponds to a slowly fading channel model. Note that, since the additive noise is normalized, SNR = P. The energy per bit to noise spectral density ratio is used for evaluation of the spectral efficiency and comparison of different strategies. Its definition is = SNR N o R sum where R sum is the total spectral efficiency, i.e. the sum-rate in bits per second per Hertz. It is well known that the spectral efficiency of the optimal multiuser detector is achievable with a minimum mean square error MMSE detector, with successive decoding and cancellation. It is therefore interesting to study the spectral efficiency gain with successive decoding and practical linear detectors such as matched filter or decorrelator. For a system load = K N, the ergodic sum-rate is [], C,SNR = lim E s log + s ηsnr} 3 K,N where the expectationis taken w.r.t. the fading gain distribution F s s. The ergodic sum-rate is an upper bound, since its achievability requires an instantaneous feedback from receiver to all users. With SIC decoding, after every decoding stage the subtraction of users decreases the effective system load, therefore C SIC = E s lim K K,N j= N log +sη K j N SNR 8

3 Int. Zurich Seminar on Communications IZS, March 3-5, which converges a.s. to the following integral expression, IV. TWO CODED LAYERS A higher expected spectral efficiency may be obtained with C SIC,erg,SNR =E s dz log + s η z SNR. 4 coded layering at the transmitter for each user. We begin here with analysis of two coded layers for a MF detector at the receiver. Let every user use the following rate allocation Since the MMSE with SIC achieves optimal receiver performance, we focus on MF and decorellator detectors for spectral R =log + s αsnrηmf } +s αsnrηmf efficiency analysis. A matched filter detector efficiency is [] } 4 R =log +s mf η mf = +SNR. 5 where s,s are the layering fading gain thresholds, and For a decorrelator detector, we use a similar derivation. The detector efficiency is [] α + αf s s, αf s s +αf ss η dec =max,. 6 III. STRONGEST USERS DETECTION AND SIC Consider the case that all users transmit the same rate R, using a single layer code, R log + s th η SNR. 7 where s th is a rate allocation parameter which governs the fading gain threshold for reliable decoding. The probability of outage, in parallel decoding, is F s s th. The achievable rate at the first SIC stage is obtained by decoding in parallel all users that are not in outage with single user detectors. Hence, R s th,= F s s th R 8 and after cancelling all the reliably decoded users, there is a fraction F s s th of undecoded users. The mutual interference reduces after cancellation, and there may exist more users with fading gains s<s th who can now be decoded. The additional rate, obtainable at the next stage, is given by R s th,=f s s th F s s R 9 which expresses the expected sum-rate for parallel decoding of all users with fading levels s s<s th,wheres satisfies s η F s s th = s th η. A detailed derivation is available at [3, Proposition 8.]. The elementary difference between the MF and decorrelator This procedure continues similarly to the next stage. We can express the total achievable rate as follows decoding is that with a MF every decoded layer reduces the interference, and thus increases the effective system load. With a decorrelator, the effective system load can be reduced only after all layers of some user are reliably decoded. R out = F s s n F s s n R = F s s R n= V. THE CONTINUOUS BROADCAST APPROACH where s s th,andf s s =,and Consider a single-input single-output SISO channel, η y i = hx i + n i, 8 s n = s th, n =,,... η F s s n It can be shown [3] that there exists a limit s s th for the linear detectors, since F s s is a monotonically non increasing function, and η is a monotonically decreasing function. Hence s satisfies the following condition, s η F s s = s th η. 3 and αsnr, αsnr result from the power allocated to the first and second layers, respectively. Note α α, and α [, ]. The iterative decoding steps are as follows: Decode, using SIC, the first layer of all decodable users; Repeat the previous step for the next layer; 3 Repeat - until there are no more decodable users. The reduced system load after the first iteration is a direct result of and 3 for a single layer. The iterative SIC decoding converges a.s. to the following expected spectral efficiency R L = F s where s i s η mf αf s s η mf αf s s R + F s } satisfy the following conditions + αf s s s + αf s s s s R 5 = s η mf = s η mf A detailed derivation is available at [3]. A similar result is derived for the case a decorrelator detector is used by the receiver. The same expression for the average rate as in 5 can be obtained for a decorrelator, only the rates R,R are given by } R =log R =log + s αsnrη dec F s s +s αsnrη dec F s s +s αsnrη dec F s s } 7 where y i } are samples of the received symbols, x i } are the transmitted complex symbols, satisfying the power constraint E x P. n i } are the additive noise samples, which are complex Gaussian i.i.d with zero mean and unit variance denoted CN,, and h is the fading coefficient, which remains fixed during a transmission block, and varies over time according to a distribution density function f s h. Note 6 83

4 Int. Zurich Seminar on Communications IZS, March 3-5, that since the additive noise is normalized and E[ h ]=, SNR = P. In the continuous broadcast approach [4], every layer is associated with a channel state s = h. The incremental differential rate as function of the channel state is drs =log + sρsds +sis = sρsds +sis 9 where ρs is the transmit power density function. Thus ρsds is the transmit power of a layer parameterized by s, associated with fading state s. Information streams intended for receivers indexed by u > s are undetectable and play a role of additional interfering noise, denoted by Is. The interference for a fading power s is Is = s ρudu, which is a monotonically decreasing function of s. The total transmitted power is the overall collected power assigned to all layers I = P. The expected rate is achieved with sufficiently many transmission blocks, each experiencing an independent fading realization. Therefore, the expected rate R bs is R bs = du f s u u drsds = uρu du F s u +uiu where f s u is the pdf of the fading power, and F s u is the corresponding cdf. Optimization of R bs for maximal throughput w.r.t. the power distribution Is can be found by solving the associated constrained Eüler equation [4]. A. Matched Filter Detector For the multiuser channel model defined in, the achievable rates strongly depend on the transmission scheme and the decoding strategy. The decoding strategy which is adopted here is the iterative decoding, just like described for two coded layers. The achievable continuous layering rate is given by sη mf Gρs R sum,bs I = ds F s s +sη mf GIs dsjs, I, I where G corresponds to the remaining layers per user, which induce the mutual interference, G SNR F s sρsds dszs, I, I where ρs = I s. The optimization of w.r.t the residual interference constraint in can be solved by fixing the interference parameter G to an arbitrary value such that <G. ForsuchaG the optimization in is a standard variational problem with a residual interference constraint on top of the power constraint I = P. The optimization problem is therefore, max I s.t. dsjs, I, I G dszs, I, I We can write the Lagrangian form L = dsjs, I, I + λ G dszs, I, I 3 The Eüler-Lagrange condition for extremum can be derived, and the optimal layering power distribution can be expressed in a closed form, as summarized in the next proposition. Proposition 5.: The optimal power distribution, which maximizes the expected sum-rate of a continuous broadcast approach, with matched-filter multiuser detection and iterative SIC decoding, is given by Is = SNR s<s SNR + SNR 4λ FssSNR + η mf Gs F s s s s s λ sη mf G s>s with s is the smallest fading gain for which Is =, and the left boundary condition on s satisfies Is =SNR. The Lagrangian multiplier λ is obtained by an equality for the residual interference constraint, as specified by s s F s si sds = G SNR 4 Proof: sketch Details are available in [3, Proposition 8.4]. The first step is to explicitly write the extremum condition of the Lagrangian in 3. The extremum condition is given by the Eüler-Lagrange equation which is a necessary condition for a zero variation, J I J I s λ Z I Z I s = 5 which can be explicitly formulated λ SNR F sst + F sssη mf T F s sη mf = 6 where we defined T +sη mf I. Solving I from 6 yields the optimal power allocation. It remains to apply the subsidiary conditions on the optimal solution, such that the power constraint and the residual interference constraint are met with equality. B. Decorrelator Detector The decoding algorithm for a decorrelator multiuser detector is similar. In the continuous setting the detector efficiency is updated according to the number of users for which ALL layers are decoded. This is the reason the upper boundary of the power distribution is actually a subject for optimization. The solution is obtained by solving the corresponding variable end point variational optimization problem. The average achievable rate with a decorrelator detector, in its general form, is given by s + a s b sρsη F s s b R bs,decorr = ds F s s +sisη F s s b + F s s a R s a + F s s b R s b 7 84

5 Int. Zurich Seminar on Communications IZS, March 3-5, where Is a = SNR, and Is+ a is the remaining power allocation for the continuous and last layers. The rate of the first and last layers, respectively, is R s a =log + saηfss bsnr Is + a +s aηf ss b Is + a R s b =log +s b η F s s b Is b 8 where Is + b =. The optimal power allocation and its derivation are available in [3, Proposition 8.5]. It is shown in [3, Proposition 6.] that equal rates allocation for all users maximizes the spectral efficiency, for any number of layers. Spectral Efficiency [Bits/Sec/Hz] VI. NUMERICAL RESULTS Achievable Rates with Matched Filter Detector =. Equal Rates, No SIC Equal Rates iterative Variable Rates iter Equal Rates, Layers, iterative Single Layer MF Ergodic Bound Equal rates, broadcast approach, iterative /N o [db] Fig.. Expected spectral efficiency for a Rayleigh fading channel, receiver uses a MF multiuser detector =.. Spectral Efficiency [Bits/Sec/Hz] Achievable Rates with Decorrelator Detector =.8 Equal Rates, No SIC Equal Rates iterative Variable Rates iter Equal Rates, Layers, iterative Broadcast Approach Single Layer Ergodic Bound Optimal Det. Ergodic Bound /N o [db] Fig.. Expected spectral efficiency for a Rayleigh fading channel, receiver uses a decorrelator multiuser detector =.8. Figures and demonstrate the expected spectral efficiency for MF and decorrelator detectors, respectively. Different transmission and decoding strategies are compared. The single layer ergodic bound is given in 4. Equal fixed rate with single iteration refers to the case no SIC is used. The spectral efficiency for single layer iterative decoding is specified in, and for two layer coding with iterative decoding in 5. The broadcast approach achievable spectral efficiency, with iterative decoding, is given in for a MF detector. VII. CONCLUSION The spectral efficiency of practical linear multiuser detectors such as MF and decorrelator employing SIC receivers was derived. Single layer and multi-layer coding per user were studied. The multi-layer coding expected sum-rate, under iterative decoding with linear multiuser detectors, is optimized, and the optimal power distribution is obtained. The achievable spectral efficiency for a linear MF detector shows significant gains over the single layer coding approach. The interesting observation here is that the expected spectral efficiency exceeds the single layer ergodic sum-capacity. The ergodic bound assumes that every user transmits at a rate matched to its decoding stage and channel realization. For a single user setting the ergodic bound is always an upper bound for the broadcast approach. However, in our multiuser setting a MF detector is used for the ergodic bound, and the MF detection is information lossy. In the broadcast approach the MF detection is performed over and over for every layer according to the iterative decoding scheme. Therefore the broadcast approach can provide spectral efficiencies exceeding those of a single layer coding with channel side information, when a MF detector is used. It is worth noting that systems employing decorrelator detection, can significantly gain from using SIC, at system loads close to. For such system loads, single user detection is interference limited, and therefore achievable rate can be infinitesimally small. With layering and iterative SIC, the layers decoded first must have low rates. Gradually, the effective system load reduces, and higher expected spectral efficiencies can be achieved. ACKNOWLEDGMENT The work was supported by the European Commission in the framework of the FP7 Network of Excellence in Wireless in Communications NEWCOM++, and the Israel Science Foundation. REFERENCES [] S. Shamai and S. Verdu, The impact of frequency-flat fading on the spectral efficiency of cdma, Information Theory, IEEE Transactions on, vol. 47, no. 4, pp. 3 37, May. [] S. Shamai, B. Zaidel, and S. Verdu, Strongest-users-only detectors for randomly spread cdma, Information Theory,. Proceedings. IEEE International Symposium on, pp.,. [3] A. Steiner, S. Shamai, V. Lupu, and U. Katz, The spectral efficiency of successive cancellation with linear multiuser detection for randomly spread CDMA, submitted to IEEE transactions on Information Theory, September 9. [4] S. Shamai Shitz and A. Steiner, A broadcast approach for a single user slowly fading MIMO channel, IEEE Trans. on Info, Theory, vol. 49, no., pp , Oct