I modified array configuration, to 3 antennae orthogonally placed, as:
M = 2
N = 3
rx = [0, 0 ,1];
ry = [1, 0 ,0];
rz = [0, 0 ,0];
r = [rx; ry; rz];
With azimuth and elevation defined as in my example, rx, ry and rz must be column vectors (not row vectors). Therefore, r =[rx, ry, rz].
Signals can arrive from two DOAs: ground wave and ionosferic reflected wave.
I read about MUSIC don´t work with correlated signals.
Please, advise me about this problem.
You are correct - standard MUSIC generally does not work with fully correlated signals (i.e. where the magnitude of the complex correlation coefficient is equal to 1). The reason for this is that fully correlated signals together span only 1 dimension of the complex observation space. There are a few methods to solve this problem, but they may or may not be suitable for you:
1)
Conventional spatial smoothing. Many people think spatial smoothing can only be applied to uniform linear arrays. This is not true. Spatial smoothing can be applied to any array which has two or more
identical subarrays (i.e. they shift onto one another without rotation). These subarrays can be overlapping. For example, if you had a (4 x 4) uniform grid array like this:
Code:
x x x x
x x x x
x x x x
x x x x
Then you could view this as four identical overlapping (3 x 3) grid arrays:
Code:
x x x . . x x x . . . . . . . .
x x x . . x x x x x x . . x x x
x x x . . x x x x x x . . x x x
. . . . . . . . x x x . . x x x
Then, if you take the average of the covariance matrices of these subarrays, the resulting signal correlation can generally be seen to decrease. I can describe the maths behind this in more detail if you want. Of course, one disadvantage of this approach is that the resulting array is smaller (in my example, 9 antennas instead of 16). You always lose at least 1 antenna with conventional spatial smoothing.
2)
Backward spatial smoothing. Again, many people think backward spatial smoothing can only be applied to linear arrays and, again, this is not true. Backward spatial smoothing is somewhat similar to conventional spatial smoothing, but instead of needing identical shifted subarrays, you need an array (or subarrays) that are a reflection of themself through the origin. In other words, it must be possible to transform the array manifold vector into its complex conjugate by simply reordering its elements. For example, the (4 x 4) grid array above has this property. Or, to keep the numbers simple, imagine a (2 x 2) grid array with:
Code:
[rx, ry, rz] = -0.5 -0.5 0
0.5 -0.5 0
-0.5 0.5 0
0.5 0.5 0
If we write the locations in reverse order, we get:
Code:
reversed = 0.5 0.5 0
-0.5 0.5 0
0.5 -0.5 0
-0.5 -0.5 0
which is equal to -[rx,ry,rz]. Therefore, if we average the two resulting covariance matrices, the imaginary component of the correlation coefficient is eliminated. The array does not get smaller, but only the imaginary component is removed.
Of course, if your array has identical subarrays which reflect through their own centroids, it is possible to apply both conventional and backward spatial smoothing ("forward-backward" spatial smoothing).
3)
Other signal diversity. If possible, you should try to increase the dimension of the observation space by exploiting other signal diversity. For example, if the incoming signals have different Doppler shifts/carrier offsets, then this can be exploited. Alternatively, if the signals arrive with different time delays (and the nature of the signal waveform is known in advance) then this can also be exploited. These methods require manipulation of the received data and corresponding modification to the received signal model. It is then sometimes possible to apply similar "smoothing" methods to the modified signal model, but this would take more time to describe.
4)
Other methods (not MUSIC). If you can't get a big array with nice subarray symmetry and there is no other signal diversity, then a subspace-based algorithm like MUSIC may not be suitable. Try another approach.