Agenda. Background. System model. Optimal Design of Adaptive Coded Modulation Schemes for Maximum Average Spectral E$ciency

Transcription

1 Optimal Design of Adaptive Coded Modulation Schemes for Maximum Average Spectral E$ciency Henrik Holm Mohamed#Slim Alouini David Gesbert Geir E. %ien Frode B&hagen Kjell J. Hole Norwegian University of Science and Technology University of Minnesota University of Oslo Agenda Design switching levels in adaptive systems optimally independent of code choice Performance close to Ergodic Capacity limit 1 System model Background Single#link wireless system Flat, wide#sense stationary fading Slow fading Transmitter has perfect channel state information!via noiseless, zero#delay return channel" Goldsmith & Varaiya!1997": Shannon capacity of fading channels under idealized assumptions. Alouini & Goldsmith!": Capacity of Nakagami#m channel. 3

2 Background Adaptive coded modulation!acm": Set of channel codes/constellations Each code guarantees BER lower than target for sub#range of SNR Partitioning of SNR range: Outage R 1 R s 1 s s 3 s N R N Analysis of ACM systems!traditionally" Start with set of codes as a design criterion Optimal partitioning given by code set Sub#optimal BER M = M = 5 M = = M = 3M M = 1 M = 51 M = CSNR 5 Analysis of ACM systems!optimal partitioning" Analysis of ACM systems!optimal partitioning" Turn issue upside#down: No codes as design criterion Optimal partitioning Yields spectral e$ciency!se" and SNR requirements Capacity of AWGN channel: C = log (1 + γ) SNR region de'ned by lower and upper switching levels {s n, s n+1 } Can support codes with SE up to C n = log (1 + s n ) 7

3 Analysis of ACM systems!optimal partitioning" Maximum ASE for ACM!MASA" is sum of contributions from all regions/codes Probability that SNR is in region n is MASA = N N sn+1 C n P n = log (1 + s n ) f γ (γ) dγ n=1 n=1 s n P n Optimization of MASA Finding optimal set of switching levels {s n } MASA = MASA s 1. MASA s N =!Discrete#sum approximation of variable#rate constant#power Shannon capacity!goldsmith/varaiya, 1997"" 9 Optimization of MASA Yields set of N nonlinear equations ( ) 1 + sn F γ (s n+1 ) F γ (s n ) (1 + s n ) ln f γ (s n ) = 1 + s n 1 Switching levels can be expressed recursively [ ( ) ] 1 + s n+1 = Fγ 1 sn F γ (s n ) + (1 + s n ) ln f γ (s n ) 1 + s n 1 Optimization of MASA For a given s 1 : recursively MASA nats (s 1 ) = Simpli'cation s,..., s N can be expressed N ( ) 1 + sn f γ (s n )(1 + s n ) ln(1 + s n ) ln 1 + s n=1 n

4 Optimization of MASA Rayleigh Rayleigh 5.5 PDF f γ (γ) 1 γ e MASA as a function of s 1 for γ = db and N = MASA CDF F γ (γ) 1 e γ.5 Can be optimized numerically wrt. s s 1 icdf Fγ 1 (p) 1 ln( 1 p ) 13 1 Nakagami#! Rice!analytical expressions" f γ (γ) ( ) m m γ m 1 mγ Γ(m) e gampdf(γ, m, m ) f γ (γ) ) K + 1 (K+1)γ K(K + 1)γ e K I ( F γ (γ) Γ(m, m γ) 1 Γ(m) gamcdf(γ, m, m ) F γ (γ) 1 Q 1 ( K, ) K + γ F 1 γ (p) Γ 1 (m, (1 p)γ(m)) m gaminv(p, m, m ) Fγ 1 (p) ( Q 1 1 ( K, 1 p)) /(K + ) 15 1

5 Rice!Matlab expressions" Design example f γ (γ) F γ (γ) K + 1 ncxpdf( K + 1 γ,, K) ncxcdf( K + 1 γ,, K) s i, i=1,..,n (threshold values) N=1 N= N= log (1+s i ) [bits/s/hz], i=1,..,n (spectral efficiencies) 1 N=1 N= N= 5 Fγ 1 (p) ncxinv(p,, K) (K + 1) Switching Levels Spectral E$ciency 17 1 Comparison with Shannon capacity Comparison with Shannon capacity 1 Capacity of gaussian channel Capacity of a Rayleigh channel MASA Important outcome: relative signi'cance of the number of SNR regions/codes. Alouini & Goldsmith!1999": Expression for capacity C ora of constant power/optimal rate adaptive transmission on Rayleigh channel ASE [bits/s/hz] C ora N = N =

6 Probability of no transmission Probability of no transmission : Probability that smallest code cannot ful'll BER requirement = probability that SNR < Rayleigh: = s1 s 1 f γ (γ) dγ = 1 e s 1!1! N = N = 1! ! bar 1 requirement MASA with restrictions 1 <!3 Unconstrained N= Low essential in certain applications. N= Calculate s 1 as a function of Express MASA as a function of s, with s 3,..., s N calculated recursively and s 1 as a parameter. MASA N=

7 SNR margin Practical modi'cations and applications Strict upper bound requires codes that achieve capacity Accounting for real#life limitations: SNR margin λ Achievable SE = log (1 + λγ) 5 SNR margin SNR margin 1 Capacity of a Rayleigh channel MASA MASA with block length 5 New expresssion for s,..., s N : [ ( ) ] 1 + s n+1 = Fγ 1 sn /λ F γ (s n ) + λ(1 + s n /λ) ln f γ (s n ) 1 + s n 1 /λ ASE [bits/s/hz]

8 Conclusions Method for optimizing switching levels/ spectral e$ciency of ACM systems Semi#analytical solution, solved in recursive manner Performance close to Shannon limit Approach real#life limitations by introducing SNR margin 9