# rootmusic

Root MUSIC algorithm

## Syntax

W = rootmusic(X,P)
[W,POW] = rootmusic(X,P)
[F, POW] = rootmusic(...,Fs)
[W,POW] = rootmusic(...,'corr')

## Description

W = rootmusic(X,P) returns the frequencies in radians/sample for the P complex exponentials (sinusoids) that make up the signal X.

The input X is specified either as:

• A row or column vector representing one realization of signal

• A rectangular array for which each row of X represents a separate observation of the signal (for example, each row is one output of an array of sensors, as in array processing), such that X'*X is an estimate of the correlation matrix

[W,POW] = rootmusic(X,P) returns the estimated absolute value squared amplitudes of the sinusoids at the frequencies W.

The second input argument, P is the number of complex sinusoids in X. You can specify P as either:

• A positive integer. In this case, the signal subspace dimension is P.

• A two-element vector. In this case, P(2), the second element of P, represents a threshold that is multiplied by λmin, the smallest estimated eigenvalue of the signal's correlation matrix. Eigenvalues below the threshold λmin*P(2) are assigned to the noise subspace. In this case, P(1) specifies the maximum dimension of the signal subspace.

The extra threshold parameter in the second entry in P provides you more flexibility and control in assigning the noise and signal subspaces.

The length of the vector W is the computed dimension of the signal subspace. For real-valued input data X, the length of the corresponding power vector POW is given by

length(POW) = 0.5*length(W)

For complex-valued input data X, POW and W have the same length.

[F, POW] = rootmusic(...,Fs) returns the vector of frequencies F calculated in Hz. You supply the sampling frequency Fs in Hz. If you specify Fs with the empty vector [], the sampling frequency defaults to 1 Hz.

[W,POW] = rootmusic(...,'corr') forces the input argument X to be interpreted as a correlation matrix rather than a matrix of signal data. For this syntax, you must supply a square matrix for X, and all of its eigenvalues must be nonnegative. You can place the 'corr' string anywhere after the P input argument.

 Note   You can use the output of corrmtx to generate such an array X.

## Examples

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### Sinusoid Amplitudes

Estimate the amplitudes for two sinusoids in noise. The separation between the sinusoids is less than the resolution of the periodogram, radians/sample. Use the autocorrelation matrix as the input to rootmusic.

rng default
n = (0:99)';
freqs = [pi/4 pi/4+0.06];

s = 2*exp(1j*freqs(1)*n)+1.5*exp(1j*freqs(2)*n)+ ...
0.5*randn(100,1)+1j*0.5*randn(100,1);

[~,R] = corrmtx(s,12,'mod');
[W,P] = rootmusic(R,2,'corr')
W =

0.7946
0.8917

P =

4.1535
0.7797

## Diagnostics

If the input signal, x is real and an odd number of sinusoids, p is specified, the following error message is displayed:

Real signals require an even number p of complex sinusoids.

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### Algorithms

The MUSIC algorithm used by rootmusic is the same as that used by pmusic. The algorithm performs eigenspace analysis of the signal's correlation matrix in order to estimate the signal's frequency content.

The difference between pmusic and rootmusic is:

• pmusic returns the pseudospectrum at all frequency samples.

• rootmusic returns the estimated discrete frequency spectrum, along with the corresponding signal power estimates.

rootmusic is most useful for frequency estimation of signals made up of a sum of sinusoids embedded in additive white Gaussian noise.

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