Generating a random vector in MATLAB

Question

I'm trying to generate a random frequency spectrum and then use ifft to find the corresponding time domain vector.

I'm using the following code:

for i = 1:64
    randNum = (rand() + 1i * rand())/sqrt(2); % Needs to be normalized by sqrt(2)
    randFreq(i) = randNum;
end

randVec = ifft(randFreq);

Note that I'm aware of mvnrnd, however for technical reasons I need to use a for loop and generate each element individually. I also need to generate the random frequency spectrum and then use an inverse transform, I can't directly generate a random vector in the time domain.

If I plot the magnitude of the random vector (plot(abs(randVec))), I always get a graph of this form.

There's always a spike at n=0 and all the other elements are significantly smaller in magnitude. I was hoping for some insight into why this was happening.

My question is not a duplicate of Create random values in vector Matlab, they are completely different questions that just happen to be on the same topic. I'm specifically asking about the behaviour of my ifft. It's not a duplicate of spike in my inverse fourier transform either. In that question, the spike could be caused by some idiosyncrasy of the data, however in my case the data is completely random.

Answer 1

The way you generate the random values for the frequency components, they are not symmetrically distributed about zero. Specifically, the obtained real and imaginary parts have an ideal average of .5/sqrt(2). The actual (sample) average will be close to that.

When you apply the IDFT, the average over all frequencies corresponds to the first sample in the time domain. To see this, set n = 0 in the IDFT expression:

                

So you get a larger absolute value at that first sample because the average over frequency is larger than it "should".