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I am trying to understand why we need MUSIC/ESPRIT algorithms to estimate the direction of arrivals. Why can we not use phased arrays to get the exact DOA (and not an estimate of the DOA) via beam steering?

Surely, I am missing something. What are the advantages that MUSIC/ESPRIT offer over phased array beamforming?

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    $\begingroup$ @Baddioes but compared to what? As far as I can tell, "beamforming" doesn't answer the question of how to get the DOA, but of what to do if you know it already $\endgroup$ Commented Apr 21 at 10:46
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    $\begingroup$ @MarcusMüller 100% agreed. I was meaning to respond to OP, not question your question, so sorry if it came off that way. I sense OP has a lot of underlying questions not explicitly stated. $\endgroup$
    – Baddioes
    Commented Apr 21 at 14:41
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    $\begingroup$ no need to apologize! Yeah, same feeling. $\endgroup$ Commented Apr 21 at 14:42
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    $\begingroup$ @RajaKrishnappa "beamforming" is just the process of, well, forming the beam. What you propose, searching the angle space for the angle with the highest power, is a separate algorithm that you just invented! (which is plenty cool!) But it's a worse alternative: when you say "scan", you probably mean "step through different values"; the actual DOA will be between two of the values; say you step in 1° steps, but your DOA might be 9.944…°; you'll never find the accurate angle in finite many steps. ESPRIT just actually gives you the direction, arbitrary precision of angle, in a single calculatio $\endgroup$ Commented Apr 22 at 7:54
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    $\begingroup$ The answer given by @MarcusMüller is a good one, so not trying to take away from that. Responding to your question in the comments, though, the resolution of any algorithm can be increased by increasing the number of elements, at the cost of computational complexity. Traditional Fourier methods are limited by the Rayleigh limit, whereas MUSIC/ESPRIT can beat the Rayleigh limit. Theoretically they have very fine resolution, but practically you see about twice the resolution, with some variance based on the correlation matrix accuracy. $\endgroup$
    – Baddioes
    Commented Apr 22 at 14:53

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We know the how the radiation pattern looks like. There will nulls, "mini peaks", nulls again and so on until we reach the main lobe. So, we use a high step size in the beginning to make out whether we are at the side lobes or main lobe and decrease the step size once we are in the main lobe. Also, the main lobe being like a concave function is useful. Now realize beamforming used this way is indeed an estimate of DOA.

What you describe in the comments is a complex algorithm that you invented (which, by the way, is pretty cool!), which uses beamforming as a building block, and it's not beamforming itself!

So, yes, you can certainly use the math of beamforming in an algorithm to find the DOA.

However, I'll caution that the algorithm you're proposing is pretty computationally intense; your "rough step size" would need to place at least two points per sidelobe, otherwise you can't know it's a side lobe, of the angular response of a signal coming from an unknown direction. The problem with that is that for angles that are "very wrong", the side lobes get very tight (due to the sin x/x transform when you evaluate your beamforming at non-main-direction angles). Since you don't know from which direction your beam comes, you need to do that tight raster in a first run everywhere. That's going to be many (and more antennas make it even more, instead of making it easier, so we're at least linear in the number of antennas). And that's in the noise-free case!

Then, you need to evaluate the beamforming response (which is quadratically complex in the number of antennas) for each of these. That makes quadratically complex for a linearly complex number of points – we're already at the cube of the number of antennas.

The most computationally step in ESPRIT/MUSIC are the quadratically complex estimation of the covariance matrix of the different antennas, and the cubic complexity of the SVD/EVD you need to do on that. So, your algorithm is, at least from a sufficiently large height, not easier to compute than ESPRIT. In fact, ESPRIT is probably much faster! But the real advantage of ESPRIT or ROOT-MUSIC (which comes close to your idea of using the knowledge about convexity!) over your iterative beamforming power maximum search is that the it's a nearly efficient estimator – efficient in the sense that the variance of the estimate under noise (and you'll always have noise in the signal) is close to "as low as it gets", i.e., close to the Cramér-Rao-Bound.

Also: Your algorithm falls flat when there's multiple signals from different directions. ESPRIT just needs to know how many – and it will find them.

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  • $\begingroup$ Thanks once again for the neat explanation. $\endgroup$ Commented Apr 22 at 9:47

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