Hi,
I am attempting to simulate rayleigh fading in the amplitude of a signal by multiplying a transmitted signal with rayleigh distributed coefficients. I create the Rayleigh random variable using two gaussian random variables of zero mean and variance 1. The problem is that a lot of these coefficients give a value greater than 1, so this causes amplification of the signal. So I was wondering if I should lower the mean of the rayleigh distribution and if so, what would be a practical value?
Values above 1 are fine. Look at the Wikipedia article on Rayleigh fading. Because you are using a Gaussian distribution for each channel there is a finite probability of a very big signal however you normalise it, but it doesn't need normalising, except to make sure that the rms power is right.
The power of channel coefficient of your Rayleigh fading is 1^2 + 1^2 = 2. Which will amplify the transmitted signal.
To get Rayleigh channel coefficient with power of one use this:
c = (1/2)^0.5 *[ N(0, 1) + j N(0,1)]. when N(0,1) is a Gaussian random variable with 0 mean and 1 variance.
You have to assume the coefficients according to multipath propagation model, this may be random variables from 0 to 1 for real and imaginary parts
Variance is computed over many experiments for a random variable. An average value computed over a long period should be same as defined in your code. Do an experiment by capturing complex Gaussian RV: h=(1/sqrt{2}) * (N(0, 1) + j N(0,1)) for t=1, t=10, t=100, t=1000, and t=10000 times. Compute mean(h) and var(h) for t=1, t=10, t=100, t=1000, and t=10000. You will see that mean and variance converge to its specified value. In essence, instantaneous values can be greater/equal/less than the specified value but the average value will converge to the desired value. Note that Rayleigh fading coefficient is abs(h).
Values output signal Rayleigh channel can be in the range from 0 to infinity, because the signal that represents the sum of sinusoids with random phases. Therefore, the probability density argument (signal level) of the channel extends from 0 (zero probability) to infinity (also zero probability).
Signal have unit energy even when amplitude of signal significantly more 1. Becouse energy of signal (E) is multiplication power of signal (P) on his time (T). (E=P*T=[(U^2)/2]*T, where U is amplitude of signal. Decreasing T we can increase U at fixed E.
can the real and imaginary part vales be negative? As I have once used Jakes model, and the Rayleigh real and imaginary parts are negative as well.
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