Channel Dependent Adaptive Modulation and Coding Without Channel State Information at the Transmitter Bradford D. Boyle, John MacLaren Walsh, and Steven Weber Modeling & Analysis of Networks Laboratory Drexel University Department of Electrical and Computer Engineering http://network.ece.drexel.edu odeling & nalysis of etworks May 30, 2013
Introduction P S (s) S R k Information Bits Transmitter X P Y X (y x) Y Receiver R k Received Bits k CQI (feedback) Traditional AMC adjust the rate of information flow based on CSI feedback from a receiver Feedback can consume a non-trivial amount of capacity Omniscient transmitters would select an appropriate modulation and coding scheme without feedback
AMC without Feedback P S (s) R Information Bits Transmitter X S P Y X (y x) Y Receiver R k Received Bits We consider a coding strategy for conveying a variable number of information bits across a channel with an unknown SNR Propose two metrics for comparing its performance to an omniscient transmitter and characterize an optimal strategy Investigate performance scaling in the number of unknown states and the di erence between the best and worst state
Related Work S P 1 (y x) PS (s) S R Information Bits Transmitter X PY X (y x) Y Receiver R Received Bits M Encoder Y 1 Decoder M Decoder M 1 X P B (y x) Y B B Gelfand-Pinkser [1] Approach to finite compound channels [2] Broadcast coding strategy used for Rayleigh fading AWGN channel [3] B mn B mn B mn Xn Xn Xn Encoder Yn Yn Yn Channel B Ln B Ln B Ln Decoder Variable to fixed channel capacity [4] Variable number Ln mn of bits correctly received for fixed blocklength n Provides an expression for channel capacity for AWGN channel with a continuous support set for SNR; our results match in the limit as K! 1 for uniformly distributed (in db) SNR
Problem Model + + N (0,n 1 ) N (0,N) N = n k w.p. p k + N (0,n K ) Point-to-point AWGN channel with unknown SNR drawn from a finite set of possible sets Model as a broadcast channel with virtual receivers for each possible state R k apple C k Pi<k i + 1 k! k =1,...,K [5, 6] If the channel is in state i, thereceiverwillreceiver i + + R K information bits
Gap to Omniscience Metrics 1 Expected Capacity Loss J E, ( ) = or alternatively KX p k C( k ) k=1 J E, (c) =E [C ( )] f k, KX C i=k KX f k C k=1 kx p i = P [ k] c k = i=1 2 Fractional Expected Capacity Loss J %,E ( ) = J E, ( ) E [C ( )] i Pj<i j + 1 i c k c k 1 c k 1 + 1 k lx i=1 i!!!
Gap to Omniscience Optimization The fraction of power allocated i for each channel state be chosen to minimize the expected capacity loss i should which is equivalent to minimize J E, ( ) subject to k 0 k =1,...,K KX k =1 maximize c k=1 KX f k C k=1 c k c k 1 c k 1 + 1 k! subject to c k c k 1 k =1,...,K Note: The optimization is not convex.
Gap to Omniscience General Solution Partial characterization given by c i = f i i f i+1 i+1 i i+1(f i+1 f i ) if c i 1 6= c i 6= c i+1 with c 0 =0andc K = 1. If f i s and i s are such that the above is an increasing sequence in i, then c i = apple fi i f i+1 i+1 i i+1(f i+1 f i ) 1 0 where [ ] 1 0 =max{min{, 1}, 0}. If p i = 1 /K and i = K i K with >1, then the above is an increasing sequence.
Gap to Omniscience Channels with Two States Let K = 2, 1 = 2, 1, and = 1. The optimal fraction of the power to allocate for the better SNR ( 1 )is apple 1 p1 1 = 2(1 p 1 ) 0 Properties of =0for1apple apple 1 /p 1 If p 1 < 1then is monotone increasing in If p 1 apple 2/ 2 +1, then lim =!1 p 1 2(1 p 1 ). If p 1 > 2/ 2 +1, then =1forall 1/(p 1 2 (1 p 1 )) For certain parameters (p 1, 2), fullpowerisnotallocatedforthestatewiththebetter SNR ( 1 )independentofhowmuchbetterthatstatemightbe.
Gap to Omniscience Channels with Two States Properties of Expected Capacity Loss If = 1, then J E, =0 J E, is increasing in If p 1 apple 2/ 2 +1, lim J E, ( )= p 1!1 2 log 2 and if p 1 > 2/ 2 +1 2(1 p 1 ) 1 (1 + 2 )p 1 2 log 2 (1 p 1 ) lim J E, ( )= 1 p 1!1 2 log 2 (1 + 2 ) Properties of Fractional Expected Capacity Loss J %, is increasing in for 1 apple apple 1 /p 1 J %, is non-monotonic in lim!1 J %,E ( ) =0
Results K-State Channel Results Gap to Omniscience metrics as a function of K and max SNR 1 with K =0dB, p i = 1/K, and 2,..., K 1 evenly spaced (in db) between 1 and K Exp. Cap. Loss (bits) 1.4 1.2 1 0.8 0.6 0.4 0.2 0 k =5 k =15 k =25 0 5 10 15 20 25 30 Max SNR in db Minimum Expected Capacity Loss Frac. Exp. Cap. Loss (%) 50 40 30 20 10 0 k =5 k =15 k =25 0 5 10 15 20 25 30 Max SNR in db Minimum Fractional Expected Capacity Loss J E, ( )isincreasingin 1 For small 1, J E, ( )lookstobe linear and independent of K. J %,E ( )isnot monotonic in 1. For small 1, J %,E ( )lookstobe linear and independent of K.
Two-State Channel Results Optimal Power Split 1 Power Split ( ) 0.8 0.6 0.4 0.2 0 2 = 3dB 2 =3dB 2 =0dB 0 5 10 15 20 25 30 Spacing between max and min SNR in db The optimal power split is identically 0 for < 1 /p 1 =3dB is monotonically increasing in approaches the limits 1at =6dBfor 2 = 3dB 1for 2 =0dB 0.5 for 2 =3dB
Two-State Channel Results Minimum Expected Capacity Loss Exp. Cap. Loss (bits) 0.5 0.4 0.3 0.2 0.1 2 = 3dB 2 =0dB 2 =3dB 0 0 5 10 15 20 25 30 Spacing between max and min SNR in db The minimum expected capacity loss J E, ( ) is equal to 0 for =1(0dB) is monotonically increasing in bounded by 0.1465 bits for 2 = 3dB, 0.2500 bits for 2 =0dB,and 0.3535 bits for 2 =3dB
Two-State Channel Results Minimum Fractional Capacity Loss Frac. Exp. Cap. Loss (%) 35 30 25 20 15 10 5 0 2 = 3dB 2 =0dB 2 =3dB 0 5 10 15 20 25 30 Spacing between max and min SNR in db The minimum fractional expected capacity loss J %,E ( ) is increasing in over the interval 0 db to 3 db is non-monotonic over the full range of converges rather slowly to 0.
Conclusions & Future Work K-user broadcast channel can model a point-to-point channel with K unknown states Using broadcast model to measure the loss in performance as compared to an omniscient transmitter Demonstrated several non-obvious properties of the optimal power split for K =2states J E, ( )andj %,E ( ) approach a limiting function Use calculus of variations to characterize the limiting function as K!1 The case of Rayleigh distributed channel fades in [3] The authors gratefully acknowledge AFOSR support of this research through award FA9550-12-1-0086.
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