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Root MUSIC algorithm

* W* = rootmusic(

`X`

`P`

[

`W`

`POW`

`X`

`P`

[

`F`

`POW`

`Fs`

[

`W`

`POW`

returns
the frequencies in radians/sample for the * W* = rootmusic(

`X`

`P`

`P`

`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

`[`

returns
the estimated absolute value squared amplitudes of the sinusoids at
the frequencies * W*,

`POW`

`X`

`P`

`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 λ_{min}, the smallest estimated eigenvalue of the signal's correlation matrix. Eigenvalues below the threshold λ_{min}`*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.

`[`

returns
the vector of frequencies * F*,

`POW`

`Fs`

`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.`[`

forces the input argument * W*,

`POW`

`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'`

option
anywhere after the `P`

You can use the output of `corrmtx`

to
generate such an array `X`

.

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.

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.