In a paper Multiple emitter location and signal parameter estimation, the well-known MUSIC algorithm starts by calculating covariance (autocorrelation, whatever) matrix with:
where overlines stand for expectation operation, i.e., $\overline{XX^H}\triangleq E\left[XX^H\right]$. As I understand, the MUSIC algorithm needs to observe signals multiple times to get $S$, i.e., $S=\frac{1}{N}\sum_{i}^{N}{X_{i}X_{i}^{H}}$
On the other hand, when I googled implementations of the MUSIC algorithm, most of them does not perform expectation operation. I mean, they just observe signals only one time and perform the remaining process of the algorithm, i.e., $S=XX^H$
Is there a reasonable explanation that the MUSIC algorithm does not need to observe signals multiple times, but only need to observe signals only once?
Answer
This question is very much related to this answer.
The issue is that the expectation operator to get the auto-correlation (auto-covariance) is generally replaced in signal processing with the sample auto-correlation.
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