# Number of antenna to perform MUSIC algorithm

When the MUSIC algorithm is used for direction of arrival (DOA) estimation it can identify up M-1 signals, where M is the number of array elements. MUSIC can also be used for spectral frequency estimation. Does MUSIC require more than one receiver (or more than one receive antenna) for frequency estimation of a single signal?

• No, but for spectral estimation, MUSIC needs multiple observation samples in the discrete time domain, just like the similar requirement in the spatial domain for DOA estimation. – ZR Han May 13 at 1:35
• When you say multiple observation samples, do you mean contiguous samples? Or multiple contiguous observations? For instance, if you have a continuous wave signal that is always transmitting, how would you go about constructing the correlation matrix (Rxx) with one antenna? – BigBrownBear00 May 13 at 2:53

In the field of array signal processing, the signal model is

$$\mathbf{x}(n) = [x_0(n), x_1(n), \ldots, x_{M-1}(n)]^T$$

where $$M$$ denotes the number of sensor and $$n$$ is the discrete time.

While in the field of spectral estimation, the signal model is

$$\mathbf{x}(n) = [x(n), x(n-1), \ldots, x(n-M+1)]^T$$

where $$M$$ is the number of samples of the observed signal, and $$x(n-m)$$ represents the previous signal before $$m$$ samples.

The autocorrelation matrix is

$$\mathbf{R}_{xx} = E\{ \mathbf{x}(n) \mathbf{x}^H(n) \}$$

and it is usually replaced by its estimation $$\mathbf{\hat{R}}_{xx}$$ in practice.

• How well does this work if a signal has a bandwidth (e.g., .25 normalized frequency) ? All the examples I have seen are for ideal sinusoids, which create sharp peaks in the MUSIC spectrum. – BigBrownBear00 May 13 at 11:34
• @BigBrownBear00 I think it doesn't work well for a wideband signal. Similar with DOA estimation, what if a sound source has a certain width? Theoretically, signals with a bandwidth (line sound source or surface sound source in DOA) can be decomposed into infinite single frequency components (point sources in DOA). – ZR Han May 14 at 3:54