Channel parameter can be expressed as h=x+yj. In rayleigh fading, x~N(0,1/2) and y~N(0,1/2) where N is gaussian normal distribution.
1) Is there a reason why we choose variance as 1/2?
2) If I change the variance, can we still get rayleigh fading?
3) When I would like to do a simulation in MATLAB, (considering h=x+yj where x~N(0,1/2) and y~N(0,1/2)) can I define channel parameters for rayleigh fading as h=(1/sqrt(2))*(randn(1)+j*randn(1))?
4) In MIMO case (for example 2x2), since we have 2x2 channel matrix, can we define parameters as (1/sqrt(2))*(randn(2,2)+j*randn(2,2))?
Dear Okan
I'll answer your questions one by one.
1. In principle, you can consider h as complex random variable with zero mean and unit variance of power. Since h has real and imaginary components that are independent from each other, then it is reasonable to assume that half of the power in the real component and the remaining half of power is on the imaginary component.
2. Yes, it is NOT necessary for h to be of unit variance. And still we get Rayleigh fading for the magnitude of h.
3. Yes, you can. The resulted h as you described is a complex Gaussian of zero mean and unit variance.
4. Yes, you can. Then, each element of the 2*2 MIMO matrix, will follow a complex Gaussian distribution of zero mean and unit variance.
Regards
Dear Mohannad Al-Ali,
Thanks for your answer. I would like add a question after reading your answers.
h = x + yj=a*ejϴ where a is the attenuation of the channel and ϴ is the phase. When we choose x~N(0,1/2) and y~N(0,1/2), corresponding 'a' will be rayleigh distribution
(fA(a) = 2a*e-a^2 ) and phase ϴ will be fϴ(ϴ) = 1/(2*pi) which is uniform distribution.
When we change the variance in x and y [x~N(0,var) and y~N(0,var], we will get a different rayleigh distribution for 'a'. In this case, ϴ's distribution will no longer be uniform, no?
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