# MUSIC algorithm for frequency and DOA estimation

$\lambda$ denotes the wavelength of the incident sinusoid, $\theta$ the direction of arrival (DOA), and x the distance between the sensor array elements. The signal is $x_{1}(n)....x_{k}(n)$ How do I proof/explain that estimating the DOA($\theta$) is the same as estimating the frequency of the signal (where $x_{1}(n)=x_{1}(m)$ instead where m is the time instant.

I understand that the MUSIC algorithm can be used to estimate the frequency of a signal, and the MUSIC algorithm can be used to estimate the DOA of a signal as well. They are both related as they are basically finding the peak of the power spectrum, but I'm not entirely sure how to explain this.

• have you followed the derivation of either method? The only difference, really, is in how the covariance matrix input is estimated, and how the output is interpreted, and that construction directly follows from the physics of how the matrices are defined. – Marcus Müller Mar 11 '18 at 0:08
• I have but I don't really quite know how to explain the similarities. – Jacob Mar 11 '18 at 1:00
• OK; then write down the algorithm to calculate the autocovariance estimate matrix for the spectrum estimate case, and write down the algorithm to calculate the crosscovariance estimate matrix. These are similar, right? – Marcus Müller Mar 11 '18 at 1:03
• autocovariance from the input sequence? – Jacob Mar 11 '18 at 4:21