Optimization of adaptive coded modulation schemes for maximum average spectral efficiency H. Holm, G. E. Øien, M.-S. Alouini, D. Gesbert, and K. J. Hole Joint BEATS-Wireless IP workshop Hotel Alexandra, Loen, Norway June 4-6, 2003

Adaptive coded modulation (ACM) Adaptation of transmitted information rate to temporally and/or spatially varying channel conditions on wireless/mobile channels Goal: –Increase average spectral efficiency (ASE) of information transmission, i.e. number of transmitted information bits/s per Hz available bandwidth. Tool: –Let transmitter switch between N different channel codes/modulation constellations of varying rates R 1 < R 2 < … R N [bits/channel symbol] according to estimated channel state information (CSI). ASE (assuming transmission at Nyquist rate) is ASE =  R n P n where P n is probability of using code n (n=1,..,N).

Generic ACM block diagram Estimate channel state Wireless channel Demodu- lation and decoding Information streamInformation stream Information about channel state and which code/modula- tion used Information about channel state Adaptive choice of error control coding and modulation schemes according to information about channel state Coded information + pilot symbols

Maximization of ASE Usually: –Codes (code rates) have been chosen more or less ad hoc, and system performance subsequently analyzed for different channel models Now: –For given channel model, we would like to find codes (rates) to maximize system throughput. Approach: –Find approachable upper bound on ASE, assuming capacity- achieving codes available for any rate –Find the optimal set of rates to use –Introduce system margin to account for deviations from ideal code performance

A little bit of information theory For an Additive White Gaussian (AWGN) channel of channel signal-to-noise ratio (CSNR) , the channel capacity C [information bits/s/Hz] is [Shannon, 1948] C = log 2 (1+  ) Interpretation: –For any AWGN channel of CSNR  , there exist codes that can be used to transmit information reliably (i.e., with arbitrarily low BER) at any rate R < C. NB: –This result assumes that infinitely long codewords and gaussian code alphabets are available.

Application of AWGN capacity to ACM With ACM, a (slowly) fading channel is in essence approximated by a set of N AWGN channels. Within each fading region n, rates up to the capacity of an AWGN channel of the lowest CSNR - s n - may be used.

ASE maximization, cont’d For a given set of switching levels s 1, s 2, … s N, (an approachable upper bound on) the maximal ASE in ACM (MASA) for arbitrarily low BER is thus MASA =  log 2 (1+  ) · s n  s n+1 p  (  )d  where p  (  ) is the pdf of the CSNR (e.g., exponential for Rayleigh fading channels). We may now maximize the MASE w.r.t. s = [s 1, s 2, … s N ] by setting  s MASA = 0.

Assumptions Wide-sense stationary (WSS) fading, single-link channel. Frequency-flat fading with known probability distribution. AWGN of known power spectral density. Constant average transmit power. Symbol period  Channel coherence time (i.e., slow fading). Perfect CSI available at transmitter.

ASE maximization: Rayleigh fading case Maximization procedure leads to closed-form recursive solution (cf. IEEE SPAWC-2003 paper by Holm, Øien, Alouini, Gesbert & Hole for details): –find s 1 –find s 2 as function of s 1 –find s n as function of s n-1 and s n-2 for n=3,…, N. Optimal component code rates can then be found as R 1 =log 2 (1+s 1 ), …, R N = log2(1+s N ).

MASA optimum w.r.t. CSNR level 1

Optimal switching levels for CSNR (N=1,2,4)

Individual optimized information rates (N=1,2,4)

Capacity comparison: AWGN + Rayleigh (N=1,2,4,8)

Probability of “outage”

Extensions and applications (1) Practical codes do not reach channel capacity: –May introduce CSNR margin 0 < < 1 in achievable code rates: Replace log 2 (1+  ) by log 2 (1+  ) [slight, straightforward modification of formulas]. –Other possible approach: Use cut-off rate instead of capacity. [yields performance limit with sequential decoding] Worst-case (over all rates  [0,4] bits/s/Hz, at BER 0 = ) theoretical margins for some given codeword lengths n [Dolinar, Divsalar & Pollinara 1998]:

Extensions and applications (2) CSI is not perfect: –Analytical methods exist for adjustment of switching levels to take this into account [done independently of level optimization]. For a Rayleigh fading channel with H receive antennas combined by maximum ratio combining (MRC), we have that barbar Pr (  n   p)  (p) n ) = Q H ( H  n /  bar (1-  ), H   p) n /  bar (1-  ) ) bar where Q H (x,y) is the generalized Marcum-Q function,  bar is the expected CSNR, and  the correlation coefficient between true CSNR   and predicted CSNR   p)  This may be exploited to adjust switching levels {  (p) n } for   p)  to obtain any desired certainty for  n, given   p)   (p) n   ASE- robustness trade-off]

Extensions and applications (3) True channels are not wide-sense stationary –Path loss and shadowing will imply variations in expected CSNR –May potentially be used for adaptation also with respect to expected CSNR E.g., in cellular systems: Use different code sets (and number of codes) within a cell, depending on distance from user to base station. Rates may also be optimized w.r.t. shadowing and interference conditions. Dividing a cell into M > 1 regions and using N codes per region is better than using MN codes over the whole cell [Bøhagen 2003].

Conclusions We have derived a method for optimization of switching thresholds and corresponding code rates in ACM - to maximize the ASE. Corresponds to “optimal discretization” of channel capacity expression (analogous to pdf-optimization of quantizers). Analytical solution for Rayleigh fading channels. Performance close to Shannon limit for small number of optimal codes (for a given average CSNR). Results can be easily augmented to take implementation losses and imperfect CSI into account. Adaptivity with respect to nonstationary channel models and cellular networks possible. NB: Results do not prescribe a certain type of codes.