Question

Using Excel, Mathcad, or any similar program, create a 256-point zero-mean random noise sample i(x).
(a) Calculate mean and variance of i(x).
(b) Calculate autocorrelation function C(Δ) for i(x). Describe how you did this and any assumptions you made. What is the relationship between C(0) and i(x)?
(c) Calculate Wiener spectrum W(u) for i(x). What is the relationship between C(0) and W(u)? How would you convert W(u) to MTF(u)?

Answer 1

a)Set (the poorly-named)  i to rand(......). Then use mean() and var() on i.

i = rand(.........................

For autocorrelation, let us consider use Conv() with flip() and 'full' option.consider that the x-axis is longer after correlation, so x represents the first element is different There was a non-linear one for removing the noise of images, which is wiener2 in Image Processing

Wiener–Khinchin theorem:-

The Wiener–Khinchin theorem, states that the autocorrelation function of a wide-sense-stationary random process has a spectral decomposition given by the power spectrum of that process.For continuous-time, the Wiener–Khinchin theorem says that if is a wide-sense stationary process such that its autocorrelation function (sometimes called autocovariance) defined in terms of statistical expected value, (the asterisk denotes complex conjugate, and of course it can be omitted if the random process is real-valued), exists and is finite at every lag , then there exists a monotone function in the frequency domain such that

The Fourier transform of does not exist in general, because stationary random functions are not generally either square-integrable or absolutely integrable. Nor is assumed to be absolutely integrable, so it need not have a Fourier transform either.

But if is absolutely continuous, for example, if the process is purely indeterministic, then is differentiable almost everywhere. In this case, one can define , the power spectral density of  , by taking the averaged derivative of . Because the left and right derivatives of existing everywhere, we can put

and the theorem simplifies to

Now one assumes that r and S satisfy the necessary conditions for Fourier inversion to be valid, the Wiener–Khinchin theorem takes the simple form of saying that r and S are a Fourier-transform pair, and