MUSIC Algorithm for Direction of Arrival (DOA) in Acoustic Signals

Question

I have seen multiple reviews of the MUSIC algorithm for the estimation of DOA. Most of those reviews consider a complex signal input with a complex steering vector. So, how do I implement this algorithm in case I have an array of real signals?

Should I try translating the real signals to complex representation? If yes, how?

A minimal example: $$x_1(t)=\sin(\omega t) + n_1(t)$$ $$x_2(t)=\sin(\omega t + \phi) + n_2(t)$$ $$x_3(t)=\sin(\omega t + 2\phi) + n_3(t)$$ I like finding $\phi$ using the MUSIC algorithm.

Comment:

Following the suggestion of neglecting the imaginary part, the covariance matrix will become a multiplication of the unit matrix, which does not seems to help.

Just implement the algorithm and set the imaginary part of the signals to zero. — MBaz, May 06 '22 at 19:36
@MBaz, say I measure two noisy signals x1=sin(wt), x2=sin(wt + phi1). I can understand the solution if I had a complex implementation but how I solve it using your suggestion? — Gideon Genadi Kogan, May 06 '22 at 20:07
Well, if you look at the math, nothing says that the complex part of the signals cannot be zero. The math still works out. By the way, the reason all the math is complex is because it deals with lowpass equivalent signals (complex envelopes). All actual signals impinging on the array are, of course, real. And the math still works out! — MBaz, May 06 '22 at 21:04
You can perform FFT on your input signal, and loop MUSIC algorithm for each FFT bin. — ZR Han, May 06 '22 at 23:17
Maybe helpful: A real-valued MUSIC algorithm with forward/backward technique — GrapefruitIsAwesome, May 08 '22 at 00:27
@GideonGenadiKogan, We can generate an example of the MUSIC algorithm on real signals. Yet, the MUSIC algorithm is a way to differentiate signals by their DOA or Frequency. Your 3 signals has the same frequency. So please either specify different angles for each or different frequencies. — Royi, May 08 '22 at 21:01
@GrapefruitIsAwesome, I saw this now: MUSIC estimation of real-valued sine-wave frequencies By Stoica. Unfortunately, I can't access it (Paywall). — Royi, May 09 '22 at 13:10

Royi · Accepted Answer · 2022-05-10T09:43:01.310

The steering vector basically applies the delay according to the spatial location of the Uniform Linear Array (ULA).

The multiplication by the complex exponential basically applies the delay.

The answer to your question is the Hilbert Transform.
You may use it to transform the data into Analytic Signal and then apply the algorithm.

For instance, I took 3 signals as in your question (Different phase) and set them a different angle of arrival.

I calculated the signal on each element of the array and applied MUSIC to get:

As you may see the algorithm found the spatial angle quiet easily (Well, this is a basic simulation).

Applying the same process just without the Hilbert Transform yield:

As one can see while the results are still accurate they are different.
The reason for that is for a different, and interesting on its own, question.

But it shows that there is no issue running the MUSIC algorithm on real signals.

The code is available at my StackExchange Codes Signal Processing Q82918 GitHub Repository (Look at the SignalProcessing\Q82918 folder).

Remark 001: I recommend reading Marcus Miller's answer to the question DOA - 1D Music Algorithm.

Remark 002: Another motivation for using "Real Valued MUSIC" is computation. So there are method that even in the case of a complex signal, the covariance is decomposed into Real Valued matrices in order to make the computation more efficient.

MUSIC Algorithm for Direction of Arrival (DOA) in Acoustic Signals

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