Optimization of adaptive coded modulation schemes for maximum average spectral efficiency

1. 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

2. 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 R1< R2 < … RN[bits/channel symbol] according to estimated channel state information (CSI). • ASE (assuming transmission at Nyquist rate) is ASE =  RnPn where Pnis probability of using code n (n=1,..,N).

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

4. 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

5. 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 = log2(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.

6. 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 - sn - may be used.

7. ASE maximization, cont’d • For a given set of switching levels s1, s2, … sN, (an approachable upper bound on) the maximal ASE in ACM (MASA) for arbitrarily low BER is thus MASA =  log2(1+g) ·snsn+1pg(g )dg where pg(g ) is the pdf of the CSNR (e.g., exponential for Rayleigh fading channels). • We may now maximize the MASE w.r.t. s = [s1, s2, … sN]by setting sMASA = 0.

8. 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.

9. 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 s1 • find s2as function of s1 • find sn as function of sn-1and sn-2for n=3,…, N. • Optimal component code rates can then be found as R1=log2(1+s1), …, RN= log2(1+sN).

10. MASA optimum w.r.t. CSNR level 1

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

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

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

14. Probability of “outage”

15. Extensions and applications (1) • Practical codes do not reach channel capacity: • May introduce CSNR margin 0 < l < 1 in achievable code rates: Replace log2(1+g) by log2(1+lg) [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 BER0 = 10-4) theoretical margins for some given codeword lengths n [Dolinar, Divsalar & Pollinara 1998]:

16. 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 Pr(g > gn|g(p) = g(p)n) = QH(Hgn/gbar(1-r), Hg(p)n/gbar(1-r)) where QH(x,y) is the generalized Marcum-Q function, gbar is the expected CSNR, and r the correlation coefficient between true CSNR g and predicted CSNR g(p). • This may be exploited to adjust switching levels {g(p)n} for g(p)to obtain any desired certainty for g > gn, given g(p)  g(p)n. [ASE-robustness trade-off]

17. 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].

18. 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.