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.

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);

STFT Frequency Ranges

Generate a signal sampled at 5 kHz for 4 seconds. The signal consists of a set of pulses of decreasing duration separated by regions of oscillating amplitude and fluctuating frequency with an increasing trend. Plot the signal.

fs = 5000;
t = 0:1/fs:4-1/fs;

x = besselj(0,600*(sin(2*pi*(t+1).^3/30).^5));

plot(t,x)

Compute the one-sided, two-sided, and centered short-time Fourier transforms of the signal. In all cases, use a 202-sample Kaiser window with shape factor $\beta =10$ to window the signal segments. Display the frequency range used to compute each transform.

rngs = ["onesided" "twosided" "centered"];

for kj = 1:length(rngs)
    
    opts = {'Window',kaiser(202,10),'FrequencyRange',rngs(kj)};

    [~,f] = stft(x,fs,opts{:});
    subplot(length(rngs),1,kj)
    stft(x,fs,opts{:})
    title(sprintf('''%s'': [%5.3f, %5.3f] kHz',rngs(kj),[f(1) f(end)]/1000))

end

Repeat the computation, but now change the length of the Kaiser window to 203, an odd number. The 'twosided' frequency interval does not change. The other two frequency intervals become open at the higher end.

for kj = 1:length(rngs)
    
    opts = {'Window',kaiser(203,10),'FrequencyRange',rngs(kj)};

    [~,f] = stft(x,fs,opts{:});
    subplot(length(rngs),1,kj)
    stft(x,fs,opts{:})
    title(sprintf('''%s'': [%5.3f, %5.3f] kHz',rngs(kj),[f(1) f(end)]/1000))

end

STFT of Multichannel Signals

Generate a three-channel signal consisting of three different chirps sampled at 1 kHz for one second.

1. 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.

2. 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.

3. 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 helperGraphicsOpt.

waterfall(F,T,abs(S(:,:,1))')
helperGraphicsOpt(1)

waterfall(F,T,abs(S(:,:,2))')
helperGraphicsOpt(2)

waterfall(F,T,abs(S(:,:,3))')
helperGraphicsOpt(3)

This helper function sets the appearance and behavior of the current axes.

function helperGraphicsOpt(ChannelId)
ax = gca;
ax.XDir = 'reverse';
ax.ZLim = [0 30];
ax.Title.String = ['Input Channel: ' num2str(ChannelId)];
ax.XLabel.String = 'Frequency (Hz)';
ax.YLabel.String = 'Time (seconds)';
ax.View = [30 45];
end

Input Arguments

x — Input signal

vector | matrix | timetable

Input signal, specified as a vector, a matrix, 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.

• 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.

• 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 precision.

Each channel of x must have a length greater than the window length.

Example: chirp(0:1/4e3:2,250,1,500,'quadratic') specifies a single-channel chirp.

Example: timetable(rand(5,2),'SampleRate',1) specifies a two-channel random variable sampled at 1 Hz for 4 seconds.

Example: timetable(rand(5,1),rand(5,1),'SampleRate',1) specifies a two-channel random variable sampled at 1 Hz for 4 seconds.

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 or a matrix.

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 or a matrix

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

Data Types: duration

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 a 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 the comma-separated pair consisting of 'OverlapLength' and a positive 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 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.

Data Types: double | single

'FrequencyRange' — STFT frequency range

'centered' (default) | 'twosided' | 'onesided'

STFT frequency range, specified as the comma-separated pair consisting of 'FrequencyRange' and 'centered', 'twosided', or 'onesided'.

• 'centered' — Compute a two-sided, centered STFT. If nfft is even, then s is computed over the interval (–π, π] rad/sample. If nfft is odd, then s is computed over the interval (–π, π) rad/sample. If you specify time information, then the intervals are (–f_s, f_s/2] cycles/unit time and (–f_s, f_s/2) cycles/unit time, respectively, where f_s is the sample rate.

• 'twosided' — Compute a two-sided STFT over the interval [0, 2π) rad/sample. If you specify time information, then the interval is [0, f_s) cycles/unit time.

• 'onesided' — Compute a one-sided STFT. If nfft is even, then s is computed over the interval [0, π] rad/sample. If nfft is odd, then s is computed over the interval [0, π) rad/sample. If you specify time information, then the intervals are [0, f_s/2] cycles/unit time and [0, f_s/2) cycles/unit time, respectively, where f_s is the sample rate. This option is valid only for real signals.

  Note

  When this argument is set to 'onesided', stft outputs the values in the positive Nyquist range and does not conserve the total power.

For an example, see STFT Frequency Ranges.

Data Types: char | string

'OutputTimeDimension' — Output time dimension

'acrosscolumns' (default) | 'downrows'

Output time dimension, specified as the comma-separated pair consisting of 'OutputTimeDimension' and 'acrosscolumns' or 'downrows'. Set this value to 'downrows' if you want the time dimension of s down the rows and the frequency dimension across the columns. Set this value to 'acrosscolumns' if you want the time dimension of s across the columns and the frequency dimension down the rows. This input is ignored if the function is called without output arguments.

Output Arguments

s — Short-time Fourier transform

matrix | 3-D array

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 x has N_x time samples, then s has k columns, where k = ⌊(N_x–L)/(M–L)⌋, M is the length of 'Window', L is the '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

More About

Short-Time Fourier Transform

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.

Perfect Reconstruction

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 stft using istft and want the result to be the same length as the input signal x, the value of $$k=\frac{(length(x)-noverlap)}{(length(window)-noverlap)}$$ 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.

References

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

[2] Sharpe, Bruce. Invertibility of Overlap-Add Processing. https://gauss256.github.io/blog/cola.html, accessed July 2019.

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

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

Usage notes and limitations:

'OutputTimeDimension' must always be specified and set to 'downrows'.

For more information, see Tall Arrays.

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

Timetables are not supported for code generation.

GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Usage notes and limitations:

Timetables are not supported for code generation.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).