# 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

## 1 Answer

The estimation of frequency for one dimensional signal (dimension is time) is same as one dimensional space data for narrow band signal. To make it clear, consider you have n time samples (Nyquist criteria is met) data. The problem of frequency estimation from this n sample is same as the following case. Consider you have a n element linear array (equal spacing) and you have received a time stamp data from a narrow band signal i.e you have n samples collected at the same time from the n element array. Now once you demodulate or remove the carrier effect from the n samples then the direction finding problem is same as frequency estimation problem mentioned in case of time domain data.

The above is true when there are less then 'n' frequencies present in the signals. Why this is true: the phase difference of the received signal on the array directly related to the angle of arrival. Kindly recall that phase of a sinusoidal is constant. So time domain data with "fixed sampling frequency" is same as samples on Linear array with equal spacing element(fixed time) after demodulation .

Please understand it is true for narrow-band signal only not broad-band chirp like signal.

Hope it helps.