# DOA - 1D Music Algorithm

I am currently working on a MUSIC algorithm for DOA of RF emitters in 3D.

At the first stage, interested to apply the 1D MUSIC using a linear array of 4 antennas, and estimate only the azimuth of the target projected on X-Y surface(and not the elevation). The code was written and the relevant simulations performed.

However, there are two issues which require further clarifications.

1. In the described setup, two solutions will be received, as plotted below:

​Wondered what may be the methods to eliminate the second solution, given a static array and a moving target.

1. Taking into account that the elevation is not being estimated (as opposed to the case of 2D MUSIC), wondered whether the estimation accuracy in azimuth is being effected, or only the 2D projection of the signal has an impact on the estimated azimuth.

Further Elaboration of the second question:

Given the following 3D scenario, and using 1D MUSIC algorithm(with 4 equally spaced antenna array for instance), can it be assumed that the estimated azimuth of arrival will be the projection on XY as can be seen below:

Thank you Envidia – as 360 degrees are to be covered, and mechanical rotating is not being considered, limiting to 90 degrees will not answer the requirements.

Thank you David, following is the presentation of the error in the azimuth, resulting from elevation.

Seems that the error in the estimated angle can be expressed as: alpha – alpha’ = atan(y0/x0)- atan(sqrt(y0^2+h^2)/x0)

Thank you Stanely, as described in “Left-right ambiguity resolution of a towed array sonar” – page 9, Assuming a non-straight array breaks the undesirable symmetry of the beam pattern. In order to answer both the ambiguity and the error in Azimuth due to elevation, evaluating the UCA structure.

In addition, considering to use simple interferometry instead of Music, as multiple target separation is not a current requirement of the system.

The desired system must however be relatively compact(4 element UCA) and low cost(single channel proffered).

Update - 18 July:

Taking into account the following origin of the error, when using ULA for 3D DOA:

Wondered what is the expression describing the estimated angle error size as a function of elevation(for a given x and y).

• For your first problem: If you can allow it, you can easily resolve the ambiguity if you limit your observation angles to be 90 degrees from boresight (assuming broadside array). Otherwise as Marcus stated, if you allow the broadside angle the entire range, you will have ambiguities using a broadside linear array. – Envidia Jul 9 '18 at 21:37
• In the 3D case the ambiguity surface for a linear array is a cone. In your figure imagine the cone formed by rotating a target at angle $\alpha$ around the x- axis. – David Jul 11 '18 at 17:16

If you have three or more antennas, simply don't put them in a linear array. Any other array setup doesn't suffer from the ambiguity. Linear array is the worst choice here.

Of course, that breaks the periodicity constraint on the linear array that makes the autocorrelation space method MUSIC So, you'd need to change that algorithm.

Luckily, you can simply put them in a circular array and change from cartesian to polar coordinates, and keep a very similar structure for your MUSIC implementation. Just remember that the resulting estimate you get might not be in angles, but in trigonometric function of angles, and needs to be converted back.

Another option is to use two different linear arrays and to combine the observation.

As long as you stick to the linear array, you simply never get any information that can tell one side from the other, and that's the end of the story :(

• Thank you Marcus. As I am currently limited to 4 channels, would check the option of using a square array(2X2), the required updates for the MUSIC algorithm and update. Wondered whether you may have insights regarding the second issue mentioned. – SenSen Jul 9 '18 at 20:50
• what kind of insights are you looking for? – Marcus Müller Jul 10 '18 at 8:53
• Hi Marcus. Have elaborated further the second question above. Would be glad to know your opinion. Thanks. – SenSen Jul 10 '18 at 14:17
• Any array with symmetry regarding its 2 sides will have ambiguity regarding the side of the emitter. Linear or not. – Royi Jul 10 '18 at 14:25
• um, no, @Royi, that's wrong as far as I can tell. Can you make an example? – Marcus Müller Jul 10 '18 at 14:28

If your array has some wiggle, there are some approaches to ambiguity resolution . The reference section to this thesis might be helpful.

Kaouri K. Left-right ambiguity resolution of a towed array sonar (Doctoral dissertation, University of Oxford).2000

In SONAR, an isotropic response of a sensor is a modestly incorrect assumption. For RF, a less reasonable assumption.

Friedlander, Benjamin. "A sensitivity analysis of the MUSIC algorithm." IEEE Transactions on acoustics, speech, and signal processing 38.10 (1990): 1740-1751.