Signal Processing Toolbox 
Power spectral density (PSD) using Welch's method
Syntax
[Pxx,w]=
pwelch(x) [Pxx,w]=
pwelch(x,window) [Pxx,w]=
pwelch(x,window,noverlap) [Pxx,w]=
pwelch(x,window,noverlap,nfft) [Pxx,f]=
pwelch(x,window,noverlap,nfft,fs) [...]=
pwelch(x,window,noverlap,...,'range
') pwelch(...)
Description
Note
pwelch computes the power spectral density, not the power spectrum. The difference between them is discussed in Spectral Analysis.

[Pxx,w]
estimates the power spectral density =
pwelch(x)
Pxx
of the input signal vector x
using Welch's averaged modified periodogram method of spectral estimation. With this syntax:
x
is segmented into eight sections of equal length, each with 50% overlap.
x
that cannot be included in the eight segments of equal length are discarded.
hamming
) that is the same length as the segment.
The power spectral density is calculated in units of power per radians per sample. The corresponding vector of frequencies w is computed in radians per sample, and has the same length as Pxx
.
A realvalued input vector x
produces a full power onesided (in frequency) PSD (by default), while a complexvalued x
produces a twosided PSD.
In general, the length N of the FFT and the values of the input x
determine the length of Pxx
and the range of the corresponding normalized frequencies. For this syntax, the (default) length N of the FFT is the larger of 256 and the next power of 2 greater than the length of the segment. The following table indicates the length of Pxx
and the range of the corresponding normalized frequencies for this syntax.
Real/Complex Input Data 
Length of Pxx 
Range of the Corresponding Normalized Frequencies 
Realvalued 
(N/2) +1 
[0, ] 
Complexvalued 
N 
[0, 2) 
[Pxx,w]
calculates the modified periodogram using either: =
pwelch(x,window)
window
for the Hamming window when window
is a positive integer
window
when window
is a vector
With this syntax, the input vector x
is divided into an integer number of segments with 50% overlap, and each segment is the same length as the window. Entries in x
that are left over after it is divided into segments are discarded. If you specify window
as the empty vector []
, then the signal data is divided into eight segments, and a Hamming window is used on each one.
[Pxx,w]
divides =
pwelch(x,window,noverlap)
x
into segments according to window
, and uses the integer noverlap
to specify the number of signal samples (elements of x
) that are common to two adjacent segments. noverlap
must be less than the length of the window you specify. If you specify noverlap
as the empty vector []
, then pwelch
determines the segments of x
so that there is 50% overlap (default).
[Pxx,w]
uses Welch's method to estimate the PSD while specifying the length of the FFT with the integer =
pwelch(x,window,noverlap,nfft)
nfft
. If you set nfft
to the empty vector []
, it adopts the default value for N listed in the previous syntax.
The length of Pxx
and the frequency range for w
depend on nfft
and the values of the input x
. The following table indicates the length of Pxx
and the frequency range for w
for this syntax.
[Pxx,f]
uses the sampling frequency =
pwelch(x,window,noverlap,nfft,fs)
fs
specified in hertz (Hz) to compute the PSD vector (Pxx
) and the corresponding vector of frequencies (f). In this case, the units for the frequency vector are in Hz. The spectral density produced is calculated in units of power per Hz. If you specify fs
as 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 Pxx
is the same as in the Table , PSD and Frequency Vector Characteristics above. The following table indicates the frequency range for f
for this syntax.
Real/Complex Input Data 
nfft Even/Odd 
Range of f 
Realvalued 
Even 
[0,fs/2] 
Realvalued 
Odd 
[0,fs/2) 
Complexvalued 
Even or odd 
[0,fs) 
[...]
specifies the range of frequency values. This syntax is useful when =
pwelch(x,window,noverlap,...,'range
')
x
is real. The string 'range
' can be either:
'twosided'
: Compute the twosided PSD over the frequency range [0,fs)
. This is the default for determining the frequency range for complexvalued x
.
fs
as the empty vector, []
, the frequency range is [0,1)
.
fs
, the frequency range is [0, 2).
'onesided'
: Compute the onesided PSD over the frequency ranges specified for real x
. This is the default for determining the frequency range for realvalued x
.
The string 'range
' can appear anywhere in the syntax after noverlap
.
pwelch(x,...)
with no output arguments plots the PSD estimate in dB per unit frequency in the current figure window.
Examples
Estimate the PSD of a signal composed of a sinusoid plus noise, sampled at 1000 Hz. Use 33sample windows with 32sample overlap, and the default FFT length, and display the twosided PSD estimate:
randn('state',0); Fs = 1000; t = 0:1/Fs:.3; x = cos(2*pi*t*200) + randn(size(t)); % 200Hz cosine plus noise pwelch(x,33,32,[],Fs,'twosided')
Algorithm
pwelch
calculates the power spectral density using Welch's method (see references):
x
is divided into k overlapping segments according to window
and noverlap
(or their default values).
x
.
nfft
point FFT is applied to the windowed data.
The number of segments k that x
is divided into is calculated as:
window
, or if you specify it as the empty vector []
In this equation, m is the length of the signal vector x
, o is the number of overlapping samples (noverlap
), and l is the length of each segment (the window length).
See Also
dspdata.msspectrum
, pburg
, pcov
, peig
, periodogram
, pmcov
, pmtm
, pmusic
, pyulear
References
[1] Hayes, M., Statistical Digital Signal Processing and Modeling, John Wiley & Sons, 1996.
[2] Stoica, P., and R.L. Moses, Introduction to Spectral Analysis, PrenticeHall, Englewood Cliffs, NJ, 1997, pp. 5254.
[3] Welch, P.D, "The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified Periodograms," IEEE Trans. Audio Electroacoustics, Vol. AU15 (June 1967), pp. 7073.
pulstran  pyulear 
© 19942005 The MathWorks, Inc.