pxx = pmcov(x,order) returns
the power spectral density estimate, pxx, of
a discrete-time signal, x, found using the modified
covariance method. When x is a vector, it is
treated as a single channel. When x is a matrix,
the PSD is computed independently for each column and stored in the
corresponding column of pxx. pxx is
the distribution of power per unit frequency. The frequency is expressed
in units of rad/sample. order is the order of
the autoregressive (AR) model used to produce the PSD estimate.

pxx = pmcov(x,order,nfft) uses nfft points
in the discrete Fourier transform (DFT). For real x, pxx has
length (nfft/2+1) if nfft is
even, and (nfft+1)/2 if nfft is
odd. For complex–valued x, pxx always
has length nfft. pmcov uses
a default DFT length of 256.

[pxx,w] = pmcov(___) returns
the vector of normalized angular frequencies, w,
at which the PSD is estimated. w has units of
radians/sample. For real–valued signals, w spans
the interval [0,π] when nfft is even and
[0,π) when nfft is odd.
For complex–valued signals, w always
spans the interval [0,2π).

[pxx,f] = pmcov(___,fs) returns
a frequency vector, f, in cycles per unit time.
The sampling frequency, fs, is the number of
samples per unit time. If the unit of time is seconds, then f is
in cycles/sec (Hz). For real–valued signals, f spans
the interval [0,fs/2] when nfft is
even and [0,fs/2) when nfft is
odd. For complex-valued signals, f spans the
interval [0,fs).

[pxx,w] = pmcov(x,order,w) returns
the two-sided AR PSD estimates at the normalized frequencies specified
in the vector, w. The vector, w,
must contain at least 2 elements.

[pxx,f] = pmcov(x,order,f,fs) returns
the two-sided AR PSD estimates at the frequencies specified in the
vector, f. The vector, f,
must contain at least 2 elements. The frequencies in f are
in cycles per unit time. The sampling frequency, fs,
is the number of samples per unit time. If the unit of time is seconds,
then f is in cycles/s (Hz).

[___] = pmcov(x,order,___,freqrange) returns
the AR PSD estimate over the frequency range specified by freqrange.
Valid options for freqrange are: 'onesided', 'twosided',
or 'centered'.

[___,pxxc] = pmcov(___,'ConfidenceLevel',probability) returns
the probability × 100%
confidence intervals for the PSD estimate in pxxc.

Create a realization of an AR(4) wide-sense stationary random process. Estimate the PSD using the modified covariance method. Compare the PSD estimate based on a single realization to the true PSD of the random process.

Create an AR(4) system function. Obtain the frequency response and plot the PSD of the system.

Create a realization of the AR(4) random process. Set the random number generator to the default settings for reproducible results. The realization is 1000 samples in length. Assume a sampling frequency of 1 Hz. Use pmcov to estimate the PSD for a 4th-order process. Compare the PSD estimate with the true PSD.

rng default
x = randn(1000,1);
y = filter(1,A,x);
[Pxx,F] = pmcov(y,4,1024,1);
hold on
plot(F,10*log10(Pxx))
legend('True Power Spectral Density','pmcov PSD Estimate')

Create a multichannel signal consisting of three sinusoids in additive white Gaussian noise. The sinusoids' frequencies are 100 Hz, 200 Hz, and 300 Hz. The sampling frequency is 1 kHz, and the signal has a duration of 1 s.

Fs = 1000;
t = 0:1/Fs:1-1/Fs;
f = [100;200;300];
x = cos(2*pi*f*t)'+randn(length(t),3);

Estimate the PSD of the signal using the modified covariance method with a 12th-order autoregressive model. Use the default DFT length. Plot the estimate.

Number of DFT points, specified as a positive integer. For a
real-valued input signal, x, the PSD estimate, pxx has
length (nfft/2+1) if nfft is
even, and (nfft+1)/2 if nfft is
odd. For a complex-valued input signal,x, the
PSD estimate always has length nfft. If nfft is
specified as empty, the default nfft is used.

Sampling frequency, specified as a positive scalar. The sampling
frequency is the number of samples per unit time. If the unit of time
is seconds, the sampling frequency has units of hertz.

Normalized frequencies for Goertzel algorithm, specified as
a row or column vector with at least 2 elements. Normalized frequencies
are in radians/sample.

Cyclical frequencies for Goertzel algorithm, specified as a
row or column vector with at least 2 elements. The frequencies are
in cycles per unit time. The unit time is specified by the sampling
frequency, fs. If fs has
units of samples/second, then f has units of
Hz.

Frequency range for the PSD estimate, specified as a one of 'onesided', 'twosided',
or 'centered'. The default is 'onesided' for
real-valued signals and 'twosided' for complex-valued
signals. The frequency ranges corresponding to each option are

'onesided' — returns the
one-sided PSD estimate of a real-valued input signal, x.
If nfft is even, pxx will
have length nfft/2+1 and is computed over the
interval [0,π] radians/sample. If nfft is
odd, the length of pxx is (nfft+1)/2
and the interval is [0,π) radians/sample. When fs is
optionally specified, the corresponding intervals are [0,fs/2]
cycles/unit time and [0,fs/2) cycles/unit time
for even and odd length nfft respectively.

'twosided' — returns the
two-sided PSD estimate for either the real-valued or complex-valued
input, x. In this case, pxx has
length nfft and is computed over the interval
[0,2π) radians/sample. When fs is optionally
specified, the interval is [0,fs) cycles/unit
time.

'centered' — returns the
centered two-sided PSD estimate for either the real-valued or complex-valued
input, x. In this case, pxx has
length nfft and is computed over the interval
(-π,π] radians/sample for even length nfft and
(-π,π) radians/sample for odd length nfft.
When fs is optionally specified, the corresponding
intervals are (-fs/2, fs/2]
cycles/unit time and (-fs/2, fs/2)
cycles/unit time for even and odd length nfft respectively.

Coverage probability for the true PSD, specified as a scalar
in the range (0,1). The output, pxxc, contains
the lower and upper bounds of the probability × 100% interval estimate for the
true PSD.

PSD estimate, specified as a real-valued, nonnegative column
vector or matrix. Each column of pxx is the PSD
estimate of the corresponding column of x. The
units of the PSD estimate are in squared magnitude units of the time
series data per unit frequency. For example, if the input data is
in volts, the PSD estimate is in units of squared volts per unit frequency.
For a time series in volts, if you assume a resistance of 1 Ω
and specify the sampling frequency in hertz, the PSD estimate is in
watts per hertz.

Normalized frequencies, specified as a real-valued column vector.
If pxx is a one-sided PSD estimate, w spans
the interval [0,π] if nfft is even and
[0,π) if nfft is odd. If pxx is
a two-sided PSD estimate, w spans the interval
[0,2π). For a DC-centered PSD estimate, f spans
the interval (-π,π] radians/sample for even length nfft and
(-π,π) radians/sample for odd length nfft.

Cyclical frequencies, specified as a real-valued column vector.
For a one-sided PSD estimate, f spans the interval
[0,fs/2] when nfft is even
and [0,fs/2) when nfft is
odd. For a two-sided PSD estimate, f spans the
interval [0,fs). For a DC-centered PSD estimate, f spans
the interval (-fs/2, fs/2]
cycles/unit time for even length nfft and (-fs/2, fs/2)
cycles/unit time for odd length nfft .

Confidence bounds, specified as a matrix with real-valued elements.
The row size of the matrix is equal to the length of the PSD estimate, pxx. pxxc has
twice as many columns as pxx. Odd-numbered columns
contain the lower bounds of the confidence intervals, and even-numbered
columns contain the upper bounds. Thus, pxxc(m,2*n-1) is
the lower confidence bound and pxxc(m,2*n) is the
upper confidence bound corresponding to the estimate pxx(m,n).
The coverage probability of the confidence intervals is determined
by the value of the probability input.