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I am dealing with a communication model described as

$$\overrightarrow{y(n)}=\pmb{H(n)}\overrightarrow{x(n)}+\overrightarrow{w(n)}$$

where $\pmb{H(n)}$ is the $M\times M$ fading coefficient matrix at time $n$. What is difference between time instant $n$ and length $M$?

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Below is a sample MATLAB/pseudocode to generate a 4x4 MIMO system, for 1000 Monte Carlo trials:

M=4; MC=1000; 
H= randn(M,M,MC); 
x=randn(M,1,MC); 
w=randn(M,1,MC); 
Y=zeros(M,1,MC); 
for j = 1:MC
   Y(:,:,j)= H(:,:,j)*x(:,:,j) + w(:,:,j); 
end

This would generate y according to the equation in the question. You can view the MC values as different time instances. Meaning different channel, transmit vector and noise matrix for each instance.

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$n$ is the time index for the samples of $y$, $x$, etc. So you can express the 1st sample, 2nd sample, etc., of each signal, with reference to the reference 0 time, for $n=1$, $n=2$, etc.

When $x$ is passing through a wireless channel, the result at the channel output is not just $x$, but there is some noise, $w$, and the wireless channel also will result in additional paths at various delays (known as multipath), and attenuated by different amounts. So you expect to see copies of the sequence $x$ that are time-shifted and scaled/attenuated by different amounts. $\bf{\it{H}}$ is simply a matrix that represents these channel effects. It gives you time shifted copies of $x$ that are scaled by different amounts.

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  • $\begingroup$ @ DSP guy, understood. Also indicate whether my understanding is correct or not. 1) in bracket first is row then column and then samples or time instant or Monte Carlo trials. 2) length of signal depends on row. $\endgroup$ – charu Jun 6 '20 at 6:03
  • $\begingroup$ @charu you wrote your comment under the wrong answer? btw, you asked "What is difference between time instant 'n' and length 'M'", so I explained what is n and what is H (which is the MxM matrix). Doesn't help? $\endgroup$ – auspicious99 Jun 6 '20 at 6:18
  • $\begingroup$ @Charu, yes, first row then column, then Monte Carlo instance...length of signal is indexed with row, yes $\endgroup$ – Dsp guy sam Jun 6 '20 at 6:21
  • $\begingroup$ @ DSP guy, noted and cleared my confusion. $\endgroup$ – charu Jun 6 '20 at 9:31
  • $\begingroup$ @auspicious99, your comment helped me.. $\endgroup$ – charu Jul 28 '20 at 12:15

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