# Probability of a random signal being approximately sparse

Given a random signal of length $N$, is there any way of estimating (or bounding) the probability of it having an approximately sparse DFT representation, with the degree of sparseness given by parameter $\delta$, and $k$ representing the number of approximately non-zero coefficients $(k = N^{\frac{1}{\delta}})$?

• It depends the statistical model of random signal. For example if signal samples are iid Gaussian, its DFT coefficients are iid Gaussian. The spareness probability can be calculated from the joint cumulative distribution function (CDF) that is multiplication of elementary CDFs thanks to the independence assumption. – AlexTP May 31 '17 at 12:14
• Possible duplicate of Relation between frequency spectrum and PDF of a random variable – Marcus Müller May 31 '17 at 14:02
• I just realized that the question I marked as duplicate is also yours – but seriously, the autocorrelation properties of the stochastic process leading to your random signal is key here, as explained in the answer over there. – Marcus Müller May 31 '17 at 14:23
• To follow up on Marcus's statement: Almost regardless of the signal sample PDF, if the noise samples are IID, then the DFT coefficients are IID Gaussian (they tend that way because of the central limit theorem). There are some cases where this is not true, but those tend to be pathological. – Peter K. May 31 '17 at 15:37
• @Peter, I don't understand what you mean by noise – Television Jun 1 '17 at 5:39