Short-time Fourier transform
stft(___) with no output arguments plots the
magnitude of the STFT in the current figure window. The STFT is plotted as two-sided and
Generate two seconds of a voltage controlled oscillator output, controlled by a sinusoid sampled at 10 kHz.
fs = 10e3; t = 0:1/fs:2; x = vco(sin(2*pi*t),[0.1 0.4]*fs,fs);
Compute and plot the STFT of the signal. Use a Kaiser window of length 256 and shape parameter . Specify the length of overlap as 220 samples and DFT length as 512 points. Plot the STFT with default colormap and view.
Change the view to display the STFT as a waterfall plot. Set the colormap to
view(-45,65) colormap jet
Generate a quadratic chirp sampled at 1 kHz for 2 seconds. The instantaneous frequency is 100 Hz at and crosses 200 Hz at second.
ts = 0:1/1e3:2; f0 = 100; f1 = 200; x = chirp(ts,f0,1,f1,'quadratic',,'concave');
Compute and display the STFT of the quadratic chirp with a duration of 1 ms.
d = seconds(1e-3); win = hamming(100,'periodic'); stft(x,d,'Window',win,'OverlapLength',98,'FFTLength',128);
Generate a three-channel signal consisting of three different chirps sampled at 1 kHz for one second.
The first channel consists of a concave quadratic chirp with instantaneous frequency 100 Hz at t = 0 and crosses 300 Hz at t = 1 second. It has an initial phase equal to 45 degrees.
The second channel consists of a convex quadratic chirp with instantaneous frequency 200 Hz at t = 0 and crosses 600 Hz at t = 1 second.
The third channel consists of a logarithmic chirp with instantaneous frequency 300 Hz at t = 0 and crosses 500 Hz at t = 1 second.
Compute the STFT of the multichannel signal using a periodic Hamming window of length 128 and an overlap length of 50 samples.
fs = 1e3; t = 0:1/fs:1-1/fs; x = [chirp(t,100,1,300,'quadratic',45,'concave'); chirp(t,200,1,600,'quadratic',,'convex'); chirp(t,300,1,500,'logarithmic')]'; [S,F,T] = stft(x,fs,'Window',hamming(128,'periodic'),'OverlapLength',50);
Visualize the STFT of each channel as a waterfall plot. Control the behavior of the axes using the helper function
This helper function sets the appearance and behavior of the current axes.
function helperGraphicsOpt(ChannelId) ax = gca; ax.XDir = 'reverse'; str = ['Input Channel: ',num2str(ChannelId)]; ax.Title.String = str; ax.YLabel.String = 'Frequency (Hz)'; ax.XLabel.String = 'Time (S)'; ax.View = [30 45]; end
x— Input signal
Input signal, specified as a vector, a matrix, or a MATLAB®
If the input has multiple channels, specify
x as a matrix
where each column corresponds to a channel.
For timetable input,
x must contain uniformly increasing
finite row times. If a timetable has missing or duplicate time points, you can fix
it using the tips in Clean Timetable with Missing, Duplicate, or Nonuniform Times (MATLAB).
For multichannel timetable input, specify
x as a
timetable with a single variable containing a matrix or a timetable with multiple
variables each containing a column vector. All variables must have the same
Each channel of
x must have a length greater than
the window length.
chirp(0:1/4e3:2,250,1,500,'quadratic') specifies a
timetable(rand(5,2),'SampleRate',1) specifies a
two-channel random variable sampled at 1 Hz for 4 seconds.
timetable(rand(5,1),rand(5,1),'SampleRate',1) specifies a
two-channel random variable sampled at 1 Hz for 4 seconds.
Complex Number Support: Yes
fs— Sample rate
2π(default) | positive scalar
Sample rate, specified as a positive scalar. This argument applies only when
x is a vector or a matrix.
comma-separated pairs of
the argument name and
Value is the corresponding value.
Name must appear inside quotes. You can specify several name and value
pair arguments in any order as
stft('Window',win,'OverlapLength',50,'FFTLength',128)windows the data using the window
win, with 50 samples overlap between adjoining segments and 128 point FFT.
'Window'— Spectral window
hann(128,'periodic')(default) | vector
Spectral window, specified as the comma-separated pair consisting of
'Window' and a vector. If you do not specify the window or
specify it as empty, the function uses a Hann window of length 128. The length of
Window must be greater than or equal to 2.
For a list of available windows, see Windows.
(1-cos(2*pi*(0:N)'/N))/2 both specify a Hann window of length
N + 1.
'OverlapLength'— Number of overlapped samples
75%of window length (default) | nonnegative integer
Number of overlapped samples, specified as the comma-separated pair consisting of
OverlapLength and a positive integer smaller than the length of
window. If you omit
specify it as empty, it is set to the largest integer less than 75% of the window
length, which turns out to be 96 samples for the default Hann window.
'FFTLength'— Number of DFT points
128(default) | positive integer
Number of DFT points, specified as the comma-separated pair consisting of
FFTLength and a positive integer. The value must be greater than
or equal to the window length. If the length of the input signal is less than the DFT
length, the data is padded with zeros.
'Centered'— Frequency range
Frequency range, specified as the comma-separated pair consisting of
If this option is set to
true, then the spectrum is centered and is
computed over the interval –π to π. Otherwise, the spectrum is computed over
the interval 0 to 2π.
'OutputTimeDimension'— Output time dimension
Output time dimension, specified as the comma-separated pair consisting of
downrows. Set this value to
downrows, if you
want the time dimension of
s down the rows and the frequency
dimension along the columns. Set this value to
you want the time dimension of
s across the columns and frequency
dimension along the rows. This input is ignored if the function is called without
s— Short-time Fourier transform
Short-time Fourier transform, returned as a matrix or a 3-D array. Time increases
across the columns of
s and frequency increases down the rows. The
third dimension, if present, corresponds to the input channels.
If the signal
Nx time samples, then
s has k columns, where k =
⌊(Nx–L)/(M–L)⌋, M is the length of
'Window', L is the
'OverlapLength', and the ⌊ ⌋ symbols denote the floor
The number of rows in
s is equal to the value specified in
Frequencies at which the STFT is evaluated, returned as a vector.
t— Time instants
Time instants, returned as a vector.
t contains the time values
corresponding to the centers of the data segments used to compute short-time power
The short-time Fourier transform (STFT) is used to analyze how the frequency content of a nonstationary signal changes over time.
The STFT of a signal is calculated by sliding an analysis window of length over the signal and calculating the discrete Fourier transform of the windowed data. The window hops over the original signal at intervals of samples. Most window functions taper off at the edges to avoid spectral ringing. If a nonzero overlap length is specified, overlap-adding the windowed segments compensates for the signal attenuation at the window edges. The DFT of each windowed segment is added to a matrix that contains the magnitude and phase for each point in time and frequency. The number of rows in the STFT matrix equals the number of DFT points, and the number of columns is given by
where is the length of the original signal and the ⌊⌋ symbols denote the floor function.
The STFT matrix is given by such that the th element of this matrix is
— Window function of length .
— DFT of windowed data centered about time .
— Hop size between successive DFTs. The hop size is the difference between the window length and the overlap length .
The magnitude squared of the STFT yields the
spectrogram representation of the power spectral density of the function.
In general, computing the STFT of an input signal and inverting it does not result in perfect reconstruction. If you want the output of ISTFT to match the original input signal as closely as possible, the signal and the window must satisfy the following conditions:
Input size — If you invert the output of
istft and want the result to be the same length as the
x, the value of must be an integer.
COLA compliance — Use COLA-compliant windows, assuming that you have not modified the short-time Fourier transform of the signal.
Padding — If the length of the input signal is such that the value of k is not an integer, zero-pad the signal before computing the short-time Fourier transform. Remove the extra zeros after inverting the signal.
 Mitra, Sanjit K. Digital Signal Processing: A Computer-Based Approach. 2nd Ed. New York: McGraw-Hill, 2001.
 Smith, J. O. Spectral Audio Signal Processing. https://ccrma.stanford.edu/~jos/sasp/, online book, 2011 edition, accessed Nov 2018.
 Sharpe, Bruce. Invertibility of Overlap-Add Processing. https://gauss256.github.io/blog/cola.html, accessed July 2019.
Usage notes and limitations:
Input must be a tall column vector or tall timetable with a single-channel signal.
OutputTimeDimension must be always specified and set to
For more information, see Tall Arrays (MATLAB).
Usage notes and limitations:
'ConjugateSymmetric' argument is not supported for code