pxx = pwelch(x) returns
the power spectral density (PSD) estimate, pxx,
of the input signal, x, found using Welch's overlapped
segment averaging estimator. 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. If x is
real-valued, pxx is a one-sided PSD estimate.
If x is complex-valued, pxx is
a two-sided PSD estimate. By default, x is divided
into the longest possible sections to obtain as close to but not exceed
8 segments with 50% overlap. Each section is windowed with a Hamming
window. The modified periodograms are averaged to obtain the PSD estimate.
If you cannot divide the length of x exactly
into an integer number of sections with 50% overlap, x is
truncated accordingly.

pxx = pwelch(x,window) uses
the input vector or integer, window, to divide
the signal into sections. If window is a vector, pwelch divides
the signal into sections equal in length to the length of window.
The modified periodograms are computed using the signal sections multiplied
by the vector, window. If window is
an integer, the signal is divided into sections of length window.
The modified periodograms are computed using a Hamming window of length window.

pxx = pwelch(x,window,noverlap) uses noverlap samples
of overlap from section to section. noverlap must
be an positive integer smaller than window if window is
an integer. noverlap must be a positive integer
less than the length of window if window is
a vector. If you do not specify noverlap, or
specify noverlap as empty, the default number
of overlapped samples is 50% of the window length.

pxx = pwelch(x,window,noverlap,nfft) specifies
the number of discrete Fourier transform (DFT) points to use in the
PSD estimate. The default nfft is the greater
of 256 or the next power of 2 greater than the length of the segments.

[pxx,w] = pwelch(___)returns
the normalized frequency vector, w. 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).

[pxx,f] = pwelch(___,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] = pwelch(x,window,noverlap,w) returns
the two-sided Welch PSD estimates at the normalized frequencies specified
in the vector, w. The vector, w,
must contain at least 2 elements.

[pxx,f] = pwelch(x,window,noverlap,f,fs) returns
the two-sided Welch 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/sec (Hz).

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

[___] = pwelch(x,window,___,spectrumtype) returns
the PSD estimate if spectrumtype is specified
as 'psd' and returns the power spectrum if spectrumtype is
specified as 'power'.

Obtain the Welch PSD estimate of an input signal consisting of a discrete-time sinusoid with an angular frequency of rad/sample with additive white noise.

Create a sine wave with an angular frequency of rad/sample with additive white noise. Reset the random number generator for reproducible results. The signal is 320 samples in length.

rng default
n = 0:319;
x = cos(pi/4*n)+randn(size(n));

Obtain the Welch PSD estimate using the default Hamming window and DFT length. The default segment length is 71 samples and the DFT length is the 256 points yielding a frequency resolution of rad/sample. Because the signal is real-valued, the periodogram is one-sided and there are 256/2+1 points.

Obtain the Welch PSD estimate of an input signal consisting of a discrete-time sinusoid with an angular frequency of rad/sample with additive white noise.

Create a sine wave with an angular frequency of rad/sample with additive white noise. Reset the random number generator for reproducible results. The signal is 320 samples in length.

rng default
n = 0:319;
x = cos(pi/4*n)+randn(size(n));

Obtain the Welch PSD estimate dividing the signal into segments 100 samples in length. The signal segments are multiplied by a Hamming window 100 samples in length. The number of overlapped samples is 25. The DFT length is 256 points, yielding a frequency resolution of rad/sample. Because the signal is real-valued, the PSD estimate is one-sided and there are 256/2+1 points.

Obtain the Welch PSD estimate of an input signal consisting of a discrete-time sinusoid with an angular frequency of rad/sample with additive white noise.

Create a sine wave with an angular frequency of rad/sample with additive white noise. Reset the random number generator for reproducible results. The signal is 320 samples in length.

rng default
n = 0:319;
x = cos(pi/4*n)+randn(size(n));

Obtain the Welch PSD estimate dividing the signal into segments 100 samples in length. The signal segments are multiplied by a Hamming window 100 samples in length. The number of overlapped samples is 25. The DFT length is 256 points yielding a frequency resolution of rad/sample. Because the signal is real-valued, the PSD estimate is one-sided and there are 256/2+1 points.

Obtain the Welch PSD estimate of an input signal consisting of a discrete-time sinusoid with an angular frequency of rad/sample with additive white noise.

Create a sine wave with an angular frequency of rad/sample with additive white noise. Reset the random number generator for reproducible results. The signal is 320 samples in length.

rng default
n = 0:319;
x = cos(pi/4*n)+randn(size(n));

Obtain the Welch PSD estimate dividing the signal into segments 100 samples in length. Use the default overlap of 50%. Specify the DFT length to be 640 points so that the frequency of rad/sample corresponds to a DFT bin (bin 81). Because the signal is real-valued, the PSD estimate is one-sided and there are 640/2+1 points.

Create a signal consisting of a 100 Hz sinusoid in additive white noise. Reset the random number generator for reproducible results. The sample rate is 1 kHz and the signal is 5 seconds in duration.

rng default
fs = 1000;
t = 0:1/fs:5-1/fs;
x = cos(2*pi*100*t)+randn(size(t));

Obtain Welch's overlapped segment averaging PSD estimate of the preceding signal. Use a segment length of 500 samples with 300 overlapped samples. Use 500 DFT points so that 100 Hz falls directly on a DFT bin. Input the sample rate to output a vector of frequencies in Hz. Plot the result.

Create a signal consisting of a 100 Hz sinusoid in additive white noise. Reset the random number generator for reproducible results. The sample rate is 1 kHz and the signal is 5 seconds in duration.

rng default
fs = 1000;
t = 0:1/fs:5-1/fs;
noisevar = 1/4;
x = cos(2*pi*100*t)+sqrt(noisevar)*randn(size(t));

Obtain the DC-centered power spectrum using Welch's method. Use a segment length of 500 samples with 300 overlapped samples and a DFT length of 500 points. Plot the result.

You see that the power at �100 and 100 Hz is close to the expected power of 1/4 for a real-valued sine wave with an amplitude of 1. The deviation from 1/4 is due to the effect of the additive noise.

This example illustrates the use of confidence bounds with Welch's overlapped segment averaging (WOSA) PSD estimate. While not a necessary condition for statistical significance, frequencies in Welch's estimate where the lower confidence bound exceeds the upper confidence bound for surrounding PSD estimates clearly indicate significant oscillations in the time series.

Create a signal consisting of the superposition of 100 Hz and 150 Hz sine waves in additive white noise. The amplitude of the two sine waves is 1. The sample rate is 1 kHz. Reset the random number generator for reproducible results.

rng default
t = 0:0.001:1-0.001;
fs = 1000;
x = cos(2*pi*100*t)+sin(2*pi*150*t)+randn(size(t));

Obtain the WOSA estimate with 95%-confidence bounds. Set the segment length equal to 200 and overlap the segments by 50% (100 samples). Plot the WOSA PSD estimate along with the confidence interval and zoom in on the frequency region of interest near 100 and 150 Hz.

L = 200;
noverlap = 100;
[pxx,f,pxxc] = pwelch(x,hamming(L),noverlap,200,fs,...'ConfidenceLevel',0.95);
plot(f,10*log10(pxx))
hold on
plot(f,10*log10(pxxc),'r-.')
xlim([25 250])
xlabel('Frequency (Hz)')
ylabel('Magnitude (dB)');
title('Welch Estimate with 95%-Confidence Bounds');

At 100 and 150 Hz, the lower confidence bound exceeds the upper confidence bounds for surrounding PSD estimates.

Generate 1024 samples of a multichannel signal consisting of three sinusoids in additive white Gaussian noise. The sinusoids' frequencies are , , and rad/sample. Estimate the PSD of the signal using Welch's method and plot it.

N = 1024;
n = 0:N-1;
w = pi./[2;3;4];
x = cos(w*n)' + randn(length(n),3);
pwelch(x)

Window, specified as a row or column vector or an integer. If window is
a vector, pwelch divides x into
overlapping sections of length equal to the length of window,
and then multiplies each signal section with the vector specified
in window. If window is
an integer, pwelch is divided into sections of
length equal to the integer value, and a Hamming window of equal length
is used. If the length of
x cannot be divided exactly into an integer number
of sections with noverlap number of overlapping
samples, x is truncated accordingly. If you specify window as
empty, the default Hamming window is used to obtain eight sections
of x with noverlap overlapping
samples.

Number of overlapped samples, specified as a positive integer
smaller than the length of window. If you omit noverlap or
specify noverlap as empty, a value is used to
obtain 50% overlap between segments.

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.

If nfft is greater than the segment length,
the data is zero-padded. If nfft is less than
the segment length, the segment is wrapped using datawrap to
make the length equal to nfft.

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.

Power spectrum scaling, specified as one of 'psd' or 'power'.
Omitting the spectrumtype, or specifying 'psd',
returns the power spectral density. Specifying 'power' scales
each estimate of the PSD by the equivalent noise bandwidth of the
window. Use the 'power' option to obtain an estimate
of the power at each frequency.

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.

The periodogram is not a consistent estimator of the true power
spectral density of a wide-sense stationary process. Welch's
technique to reduce the variance of the periodogram breaks the time
series into segments, usually overlapping. Welch's method computes
a modified periodogram for each segment and then averages these estimates
to produce the estimate of the power spectral density. Because the
process is wide-sense stationary and Welch's method uses PSD
estimates of different segments of the time series, the modified periodograms
represent approximately uncorrelated estimates of the true PSD and
averaging reduces the variability.

The segments are typically multiplied by a window function,
such as a Hamming window, so that Welch's method amounts to
averaging modified periodograms. Because the segments usually overlap,
data values at the beginning and end of the segment tapered by the
window in one segment, occur away from the ends of adjacent segments.
This guards against the loss of information caused by windowing.