Understand ergodic capacity of a SISO flat-fading system over fading channels. Model and simulate capacity curves in Matlab.
Channel model
In the previous post, derivation of SISO fading channel capacity was discussed. For a flat fading channel (model shown below), with the perfect knowledge of the channel at the receiver, the capacity of a SISO link was derived as
where,
Since the channel impulse response
Jensen’s inequality [1] states that for any concave function f(x), where x is a random variable,
Applying Jensen’s inequality to Ergodic capacity in equation (2),
This implies that the Ergodic capacity of a fading channel cannot exceed that of an AWGN channel with constant gain. The above equation is simulated in Matlab for a Rayleigh Fading channel with
Matlab code
%This work is licensed under a Creative Commons %Attribution-NonCommercial-ShareAlike 4.0 International License %Attribute author : Mathuranathan Viswanathan at gaussianwaves.com snrdB=-10:0.5:20; %Range of SNRs to simulate h= (randn(1,100) + 1i*randn(1,100) )/sqrt(2); %Rayleigh flat channel sigma_z=1; %Noise power - assumed to be unity snr = 10.^(snrdB/10); %SNRs in linear scale P=(sigma_z^2)*snr./(mean(abs(h).^2)); %Calculate corresponding values for P C_erg_awgn= (log2(1+ mean(abs(h).^2).*P/(sigma_z^2))); %AWGN channel capacity (Bound) C_erg = mean((log2(1+ ((abs(h).^2).')*P/(sigma_z^2)))); %ergodic capacity for Fading channel plot(snrdB,C_erg_awgn,'b'); hold on; plot(snrdB,C_erg,'r'); grid on; legend('AWGN channel capacity','Fading channel Ergodic capacity'); title('SISO fading channel - Ergodic capacity'); xlabel('SNR (dB)');ylabel('Capacity (bps/Hz)');
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References
[1] Konstantinos G. Derpanis, Jensen’s Inequality, Version 1.0,March 12, 2005.↗
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