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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.