Special 10:music algorithm for spatial spectra

Gui.

Time: 2017-09-20-07:43:12

Links: http://www.cnblogs.com/xingshansi/p/7553746.html

not to be continued ...

Objective

The MUSIC (multiple Signal classification) algorithm is typically used to estimate the arrival angle (Doa,direction of arrival).

I. Introduction to the principle of music

Based on the previous analysis, the model is still based on the narrowband signal:

X is the receiving array, F is the incident signal, A is the corresponding guide vector, and W is the noise. Can be directly recorded as matrix form

Usually solved with correlation matrix:

In fact, the correlation matrix can not be derived, generally based on stochastic signal 1) stationarity, 2) ergodic hypothesis, approximate estimation of correlation matrix:

The eigenvalue/singular value decomposition of the correlation matrix is

Suppose 1) The noise is not correlated with the signal, 2) The noise is white noise. With the resulting eigenvector, you can use the music algorithm to solve the angle:

The specific principle can refer to the subspace algorithm article.

Second, the coherence situation analysis

Take two signals as an example

Finding correlation matrices

If the correlation coefficient ρ of the two signals satisfies:

1) ρ=0, it is considered that the two signals are not correlated;

2) 0<ρ<1, the two signals are considered relevant;

3) ρ= 1, the two signals are coherent.

When two signals are coherent, ρ=1, for the correlation matrix:

Rank is 1, which causes the rank deficit, and the spatial spectral estimation algorithm for subspace is no longer applicable.

Can also be understood in a different way:

When two signals are coherent, there is, at this time

b is known as generalized array popular or generalized guided vectors. It can be seen that it does not usually correspond to the direction of two waves, but the vector superposition direction of both. The general idea is to restore the rank deficit.

Relationship between eigenvalue and peak value

One view is that the correlation matrix can be decomposed into:

And for guide vectors are:

So for guide vector A (theta):

Ahs∑sha

Should not be affected by the ∑ eigenvalue change? Why do multiple signals have different theta corresponding to a (theta), which can make the peaks approximate equal? Or, why is the maximum/minimum energy corresponding to the true angle?

Ahs∑sha can be further disassembled as:

Ahs∑sha = Aha[,0;0,]aha+m

M is the number of arrays, which is constant for any direction and negligible. With two signals as an example, the simplified expression is:

Simulation verification: The signal from [ -45°,45°], the power is approximately equal:

Special 10:music algorithm for spatial spectra