Hi,
I want to know how to get SNRinstantaneous/SNRaverage for SISO (1x1) and MIMO (4x4) during simulation in HFSS or CST to plot the Cumulative Distribution Function (CDF) and get the diversity gain.
Is there any formula for this?
With Regards,
Suhas D
I don’t think you can extract the diversity gain from the CDF in a simple way. The standard approach is to plot the BER (or outage probability) versus the SNR, and then look at the slope at high SNR. It is determining the slope. The curve can either be characterized numerically (Monte Carlo simulation) or analytically. Here is a video about the basics: https://youtu.be/IuLYGIfqCA0
There are three methods from which you can extract the diversity order of a communication system for any fading model.
Method 1(Analytical, the best): Derive the CDF or the SER of the system. Define the instantaneous SNR= SNR_0 |h|^2, where SNR_0 is the SNR without fading or average SNR. Keeping SNR_0 is the most important aspect. Do not normalize it to 1. Considering Rayleigh fading as an example with the CDF F(SNR)= 1-exp(-SNR/SNR_0). Now, find an expression of the CDF at high SNR. We need to be careful which SNR should be high SNR or SNR_0. It is SNR_0 and not SNR. Actually, for the outage probability, SNR becomes the threshold SNR. Let us take a limit of F(SNR) when SNR_0 tends to infinity. For diversity analysis, it is important to find an expression with each term as a function of SNR_0. Using exp(-x)= 1-x for x tends to zero, we get F(SNR, SNR_0 to infinity)= SNR/SNR_0. Taking the negative exponent, of SNR_0 the diversity order for Rayleigh fading is 1. In general, representing the CDF in terms of SNR_0 at high SNR and taking the powers of a dominant term gives the diversity order. For example, let us consider for a general fading F(SNR, SNR_0 to infinity)= a*(SNR/SNR_0)^M+ b*(SNR/SNR_0)^N, the diversity order is min (M, N).
Method 2(Simulation experiments, Slope): Plot the CDF versus SNR_0 at a given SNR. Let us consider a general CDF representation F(SNR, SNR_0)= function(a, b, M, N, SNR, SNR_0). Suppose the diversity order is min(M, N). Plot 1: For a fixed M, N, an increase/decrease in the parameters a, b may shift the plots but the slope will remain the same. Plot2: For fixed values of a and b, and MN, the slope is determined by the N.
Method3 (Code, Slope: Answer to the Original Question): Generate CDF (at a given SNR threshold) versus SNR_0. Keep SNR_0 as high as possible. Save data of SNR_0 and corresponding CDF in a vector. Express the data in log-scale, and then find the slope. A Matlab code is here:
Vector_SNR_0=10*log10[{SNR_0}];
Vector_CDF= 10*log10[{CDF}];
Slope=( Vector_SNR_0(end)- Vector_SNR_0(end-1))/(Vector_CDF(end)-Vector_CDF(end-1)).
The estimate of diversity order is accurate as long as you keep SNR_0 as high as possible and the resolution of data points is as low as possible.
Thanks for reading the long post. The explanation as given above is in general nature and may clarify few doubts on the diversity order.
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