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I would like to create complex normal Gaussian noise with dimensions $(M,N)$. The noise should be zero mean and the variance should be equal to one.

How can I do so?

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You should use the the standard formula:

s = randn(m, n) + 1i*randn(m, n);

And as pointed out by MBaz, the output should be scaled accordingly by $\frac{1}{\sqrt{2}}$

s = s/sqrt(2);

More on that topic you can find for example in a book Digital Media Processing by Hazarathaiah Malepati:

enter image description here

It should generate exactly what you want.

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  • $\begingroup$ You'll have to scale by $\frac{1}{\sqrt2}$ to get unit variance in the complex case. $\endgroup$ – David Jul 23 '15 at 11:45
  • $\begingroup$ @David can you please explain why that is?? $\endgroup$ – Dimitri C Jul 23 '15 at 12:46
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    $\begingroup$ @DimitriC The variance of a complex RV is equal to the variance of its real part plus the variance of its imaginary part. $\endgroup$ – MBaz Jul 23 '15 at 13:36

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