Pseudospectrum using eigenvector method
[S,w] = peig(x,p)
[S,w] = peig(x,p,w)
[S,w] = peig(...,nfft)
[S,f] = peig(x,p,nfft,fs)
[S,f] = peig(x,p,f,fs)
[S,f] = peig(...,'corr')
[S,f] = peig(x,p,nfft,fs,nwin,noverlap)
[...] = peig(...,freqrange)
[...,v,e] = peig(...)
peig(...)
[S,w] = peig(x,p)
implements
the eigenvector spectral estimation method and returns S
,
the pseudospectrum estimate of the input signal x
,
and w
, a vector of normalized frequencies (in rad/sample)
at which the pseudospectrum is evaluated. The pseudospectrum is calculated
using estimates of the eigenvectors of a correlation matrix associated
with the input data x
, where x
is
specified as either:
A row or column vector representing one observation of the 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
Note
You can use the output of 
You can specify the second input argument p
as
either:
A scalar integer. In this case, the signal subspace
dimension is p
.
A twoelement 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.
Note:
If the inputs to 
The extra threshold parameter in the second entry in p
provides
you more flexibility and control in assigning the noise and signal
subspaces.
S
and w
have the same
length. In general, the length of the FFT and the values of the input x
determine
the length of the computed S
and the range of the
corresponding normalized frequencies. The following table indicates
the length of S
(and w
) and
the range of the corresponding normalized frequencies for this syntax.
S Characteristics for an FFT Length of 256 (Default)
Real/Complex Input Data  Length of S and w  Range of the Corresponding Normalized Frequencies 

Realvalued  129  [0, π] 
Complexvalued  256  [0, 2π) 
[S,w] = peig(x,p,w)
returns
the pseudospectrum in the vector S
computed at
the normalized frequencies specified in vector w
,
which has two or more elements
[S,w] = peig(...,nfft)
specifies
the integer length of the FFT nfft
used to estimate
the pseudospectrum. The default value for nfft
(entered
as an empty vector []
) is 256.
The following table indicates the length of S
and w
,
and the frequency range for w
for
this syntax.
S and Frequency Vector Characteristics
Real/Complex Input Data  nfft Even/Odd  Length of S and w  Range of w 

Realvalued  Even  (  [0, π] 
Realvalued  Odd  (  [0, π) 
Complexvalued  Even or odd 
 [0, 2π) 
[S,f] = peig(x,p,nfft,fs)
returns
the pseudospectrum in the vector S
evaluated at
the corresponding vector of frequencies f
(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.
The frequency range for f
depends on nfft
, fs
,
and the values of the input x
.
The length of S
(and f
) is the
same as in the S and Frequency Vector Characteristics above. The following
table indicates the frequency range for f
for this
syntax.
S and Frequency Vector Characteristics with fs Specified
Real/Complex Input Data  nfft Even/Odd  Range of f 

Realvalued  Even  [0, 
Realvalued  Odd  [0, 
Complexvalued  Even or odd  [0, 
[S,f] = peig(x,p,f,fs)
returns
the pseudospectrum in the vector S
computed at
the frequencies specified in vector f
, which has
two or more elements
[S,f] = peig(...,'corr')
forces
the input argument x
to be interpreted as a correlation
matrix rather than matrix of signal data. For this syntax x
must
be a square matrix, and all of its eigenvalues must be nonnegative.
[S,f] = peig(x,p,nfft,fs,nwin,noverlap)
allows
you to specify nwin
, a scalar integer indicating
a rectangular window length, or a realvalued vector specifying window
coefficients. Use the scalar integer noverlap
in
conjunction with nwin
to specify the number of
input sample points by which successive windows overlap. noverlap
is
not used if x
is a matrix. The default value for nwin
is
2 × p(1)
and noverlap
is nwin
– 1.
With this syntax, the input data x
is segmented
and windowed before the matrix used to estimate the correlation matrix
eigenvalues is formulated. The segmentation of the data depends on nwin
, noverlap
,
and the form of x
. Comments on the resulting windowed
segments are described in the following table.
Windowed Data Depending on x and nwin
Input data x  Form of nwin  Windowed Data 

Data vector  Scalar  Length is 
Data vector  Vector of coefficients  Length is 
Data matrix  Scalar  Data is not windowed. 
Data matrix  Vector of coefficients 

See the table, Eigenvector Length Depending on Input Data and Syntax, for related information on this syntax.
Note
The arguments 
[...] = peig(...,freqrange)
specifies
the range of frequency values to include in f
or w
. freqrange
can
be either:
'half'
— returns half the
spectrum for a real input signal, x
. If nfft
is
even, then S
has length nfft
/2 + 1 and is computed over the interval
[0,π]. If nfft
is odd,
the length of S
is (nfft
+ 1)/2 and the frequency interval is
[0,π). When your specify fs
,
the intervals are [0,fs
/2) and [0,fs
/2]
for even and odd nfft
, respectively.
'whole'
— returns the whole
spectrum for either real or complex input, x
. In
this case, S
has length nfft
and
is computed over the interval [0,2π). When
you specify fs
, the frequency interval is [0,fs
).
'centered'
— returns the
centered whole spectrum for either real or complex input, x
.
In this case, S
has length nfft
and
is computed over the interval (–π,π]
for even nfft
and (–π,π)
for odd nfft
. When you specify fs
,
the frequency intervals are (–fs
/2,fs
/2]
and (–fs
/2,fs
/2) for
even and odd nfft
, respectively.
Note
You can put the string arguments 
[...,v,e] = peig(...)
returns
the matrix v
of noise eigenvectors, along with
the associated eigenvalues in the vector e
. The
columns of v
span the noise subspace of dimension size(v,2)
.
The dimension of the signal subspace is size(v,1)size(v,2)
.
For this syntax, e
is a vector of estimated eigenvalues
of the correlation matrix.
peig(...)
with no output
arguments plots the pseudospectrum in the current figure window.
[1] Marple, S. Lawrence. Digital Spectral Analysis. Englewood Cliffs, NJ: PrenticeHall, 1987, pp. 373–378.
[2] Schmidt, R. O. "Multiple Emitter Location and Signal Parameter Estimation." IEEE^{®} Transactions on Antennas and Propagation. Vol. AP34, March, 1986, pp. 276–280.
[3] Stoica, Petre, and Randolph L. Moses. Spectral Analysis of Signals. Upper Saddle River, NJ: Prentice Hall, 2005.