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 R1< R2 < … RN [bits/channel symbol] according to estimated channel state information (CSI).

• ASE (assuming transmission at Nyquist rate) is

ASE = RnPn

where Pn is probability of using code n (n=1,..,N).

Generic ACM block diagram

Estimatechannelstate

Wireless channel

Demodu-lation anddecoding

Information stream

Informationabout channelstate and whichcode/modula-tion used

Information about channel state

Adaptive choice of error control codingand modulation schemes accordingto information aboutchannel 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 = 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.

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.

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+) ·sn

sn+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 = [s1, s2, … sN] 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 s1

– find s2 as function of s1

– find sn as function of sn-1 and sn-2 for n=3,…, N.

• Optimal component code rates can then be found as

R1=log2(1+s1), …, RN= log2(1+sN).

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 log2(1+) by log2(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 BER0 = 10-4) theoretical margins for some given codeword lengths n [Dolinar, Divsalar & Pollinara 1998]:

n [bits] Worst casemargin [dB]

Corre-sponding [-]

256 -1.8 dB 0.661

1024 -1.0 dB 0.794

4096 -0.6 dB 0.871

0 dB 1

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(np)(p)n) = QH(Hn/barbar(1-), Hp)

n/barbar(1-))

where QH(x,y) is the generalized Marcum-Q function, barbar 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)nASE-

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.