# rootmusic

Root MUSIC algorithm

## Description

w = rootmusic(x,p) estimates the frequency content in the input signal x and returns w, a vector of frequencies in rad/sample. You can specify the signal subspace dimension using the input argument p.

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

[w,pow] = rootmusic(x,p) returns the vector of frequencies w and the corresponding signal power in the vector pow.

example

[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, x must be a square matrix, and all of its eigenvalues must be nonnegative. This syntax can include the input arguments from the previous syntax.

Note

You can place 'corr' anywhere after p.

[f,pow] = rootmusic(___,fs) returns the vector of frequencies f calculated in Hz. You supply the sampling frequency fs in Hz.

## Examples

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Estimate the amplitudes for two sinusoids in noise. The separation between the sinusoids is less than the resolution of the periodogram, $2\pi /N$ radians/sample. Use the autocorrelation matrix as the input to rootmusic.

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

s = 2*exp(1j*frqs(1)*n)+1.5*exp(1j*frqs(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 = 2×1

0.7946
0.8917

P = 2×1

4.1535
0.7797

## Input Arguments

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Input signal, specified as a vector or matrix. If x is a vector, then it is treated as one observation of the signal. If x is a matrix, 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.

For complex-valued input data x, pow and w have the same length. For real-valued input data x, the length of the corresponding power vector pow is 0.5*length(w).

Note

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

Complex Number Support: Yes

Subspace dimension, specified as a real positive integer or a two-element vector. If p is a real positive integer, then it is treated as the subspace dimension. If p is a two-element vector, 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.

Sample rate, specified as a positive scalar. You can supply the sample rate fsin Hz. If you specify fs as the empty vector [], the sample rate defaults to 1 Hz.

## Output Arguments

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Output frequencies in rad/sample, returned as a vector. The length of the vector w is the computed dimension of the signal subspace.

Signal power, returned as a vector.

Output frequencies in Hz, returned as a vector. You supply the sampling frequency fs in Hz. If you specify fs with the empty vector [], the sampling frequency defaults to 1 Hz.

## Tips

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

Real signals require an even number p of complex sinusoids.

## Algorithms

The multiple signal classification (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.