stft

Short-time Fourier transform

Syntax

s = stft(x)

s = stft(x,fs)

s = stft(x,ts)

s = stft(___,Name,Value)

[s,f] = stft(___)

[s,f,t] = stft(___)

stft(___)

Description

s = stft(x) returns the short-time Fourier transform (STFT) of x.

s = stft(x,fs) returns the STFT of x using sample rate fs.

s = stft(x,ts) returns the STFT of x using sample time ts.

s = stft(___,Name,Value) specifies additional options using name-value pair arguments. Options include the FFT window and length. These arguments can be added to any of the previous input syntaxes.

[s,f] = stft(___) returns the frequencies f at which the STFT is evaluated.

[s,f,t] = stft(___) returns the times at which the STFT is evaluated.

stft(___) with no output arguments plots the magnitude of the STFT in the current figure window. The STFT is plotted as two-sided and centered.

Examples

3D STFT Visualization

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 $\beta =5$. Specify the length of overlap as 220 samples and DFT length as 512 points. Plot the STFT with default colormap and view.

stft(x,fs,'Window',kaiser(256,5),'OverlapLength',220,'FFTLength',512);

Change the view to display the STFT as a waterfall plot. Set the colormap to jet.

view(-45,65)
colormap jet

STFT of Quadratic Chirp

Generate a quadratic chirp sampled at 1 kHz for 2 seconds. The instantaneous frequency is 100 Hz at $\mathit{t}=0$ and crosses 200 Hz at $\mathit{t}=1$ 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);

Input Arguments

x — Input signal

vector | timetable

Input signal, specified as a vector or a MATLAB^® timetable.

Note

If you invert s using istft and want the result to be the same length as x, the value of (length(x)-noverlap)/(length(window) - noverlap) must be an integer.

• For a timetable input, x must contain only a single vector and 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 a vector input, the length of x must be greater than the window length.

Example: x = chirp(0:1/4e3:2,250,1,500,'quadratic')

Data Types: double | single
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.

Data Types: double | single

ts — Sample time

duration scalar

Sample time, specified as a duration scalar. This argument applies only when x is a vector.

Example: seconds(1) is a duration scalar representing a 1-second time difference between consecutive signal samples.

Data Types: double | single

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is 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 Name1,Value1,...,NameN,ValueN.

Example: 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.

Example: hann(N+1) and (1-cos(2*pi*(0:N)'/N))/2 both specify a Hann window of length N + 1.

Data Types: double | single

'OverlapLength' — Number of overlapped samples

75% of window length (default) | nonnegative integer

Number of overlapped samples, specified as a nonnegative integer smaller than the length of window. If you omit 'OverlapLength' or specify it as empty, it is set to the largest integer less than 75% of the window length, which is 96 samples for the default Hann window.

Data Types: double | single

'FFTLength' — Number of DFT points

128 (default) | positive integer

Number of DFT points, specified as 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.

Data Types: double | single

'Centered' — Frequency range

true (default) | false

Frequency range, specified as true or false. 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.

Output Arguments

s — Short-time Fourier transform

vector

Short-time Fourier transform, returned as a vector. Time increases across the columns of s and frequency increases down the rows.

• If the signal x has length N_x, then s has k columns, where k = ⌊(N_x – L)/(length(window) – L)⌋ where window is a vector, L is equal to 'OverlapLength', and the ⌊ ⌋ symbols denote the floor function.

• The number of rows in s is equal to the value specified in 'FFTLength'.

Data Types: double | single

f — Frequencies

vector

Frequencies at which the STFT is evaluated, returned as a vector.

Data Types: double | single

t — Time instants

vector

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 spectrum estimates.

• If a sample rate fs is provided, then the vector contains time values in seconds.

• If a sample time ts is provided, then the vector is a duration array with the same time format as the input.

• If no time information is provided, then the vector contains sample numbers.

Data Types: double | single

Algorithms

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 $$M$$ over the signal and calculating the discrete Fourier transform of the windowed data. The window hops over the original signal at intervals of $$R$$ samples. Most window functions taper off at the edges to avoid spectral ringing. If a nonzero overlap length $$L$$ 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

$$k=\lfloor \frac{N_x-L}{M-L}\rfloor ,$$

where $$N_x$$ is the length of the original signal $$x(n)$$ and the ⌊⌋ symbols denote the floor function.

The STFT matrix is given by $$X(f)=\left[\begin{array}{ccccc}{X}_1(f)& X_2(f)& X_3(f)& \cdots & X_k(f)\end{array}\right]$$ such that the $$m$$th element of this matrix is

$$X_m(f)={\displaystyle \sum _{n=-\infty }^{\infty }x(n)g(n-mR)e^{-j2\pi fn}},$$

where

• $$g(n)$$ — Window function of length $$M$$.

• $$X_m(f)$$ — DFT of windowed data centered about time $$mR$$.

• $$R$$ — Hop size between successive DFTs. The hop size is the difference between the window length $$M$$and the overlap length $$L$$.

The magnitude squared of the STFT yields the spectrogram representation of the power spectral density of the function.

References

[1] Mitra, Sanjit K. Digital Signal Processing: A Computer-Based Approach. 2nd Ed. New York: McGraw-Hill, 2001.

[2] Smith, J. O. Spectral Audio Signal Processing. https://ccrma.stanford.edu/~jos/sasp/, online book, 2011 edition, accessed Nov 2018.

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