Deep learning short-time Fourier transform
Generate a signal sampled at 600 Hz for 2 seconds. The signal consists of a chirp with sinusoidally varying frequency content.
fs = 6e2; t = 0:1/fs:2; x = vco(sin(2*pi*t),[0.1 0.4]*fs,fs);
Store the signal in an unformatted deep learning array. Compute the short-time Fourier transform of the signal. Input the sample time as a
duration scalar. (Alternatively, input the sample rate as a numeric scalar.) Specify that the input array is in
dlx = dlarray(x); [yr,yi,f,t] = dlstft(dlx,seconds(1/fs),'DataFormat','CTB');
Convert the outputs to numeric arrays. Compute the magnitude of the short-time Fourier transform and display it as a waterfall plot.
yr = extractdata(yr); yi = extractdata(yi); f = extractdata(f); t = seconds(t); waterfall(f,t,squeeze(hypot(yr,yi))') ax = gca; ax.XDir = 'reverse'; view(30,45) ylabel('Time (s)') xlabel('Frequency (Hz)') zlabel('Magnitude')
Generate a 3-by-160(-by-1) array containing one batch of a three-channel, 160-sample sinusoidal signal. The normalized sinusoid frequencies are rad/sample, rad/sample, and rad/sample. Save the signal as a
dlarray, specifying the dimensions in order.
dlarray permutes the array dimensions to the
'CBT' shape expected by a deep learning network. Display the array dimension sizes.
x = dlarray(cos(pi.*(1:3)'/4*(0:159)),'CTB'); [nchan,nbtch,nsamp] = size(x)
nchan = 3
nbtch = 1
nsamp = 160
Compute the deep learning short-time Fourier transform of the signal. Specify a 64-sample rectangular window and an FFT length of 1024.
[re,im,f,t] = dlstft(x,'Window',rectwin(64),'FFTLength',1024);
dlstft computes the transform along the
'T' dimension. The output arrays are in
'SCBT' format. The
'S' dimension corresponds to frequency in the short-time Fourier transform.
Extract the data from the deep learning arrays.
re = squeeze(extractdata(re)); im = squeeze(extractdata(im)); f = extractdata(f); t = extractdata(t);
Compute the magnitude of the short-time Fourier transform. Plot the magnitude separately for each channel in a waterfall plot.
z = abs(re + 1j*im); for kj = 1:nchan subplot(nchan,1,kj) waterfall(f/pi,t,squeeze(z(:,kj,:))') view(30,45) end xlabel('Frequency (\times\pi rad/sample)') ylabel('Samples')
x— Input array
dlarrayobject | numeric array
Input array, specified as an unformatted
dlarray (Deep Learning Toolbox), a
'CBT' format, or a numeric
x is an unformatted
dlarray or a numeric
array, you must specify the
'DataFormat' as some permutation of
dlarray(cos(pi./[4;2]*(0:159))','TCB') both specify one batch
observation of a two-channel sinusoid in the
fs— Sample rate
Sample rate, specified as a positive numeric scalar.
ts— Sample time
Sample time, specified as
duration scalar. Specifying
ts is equivalent to setting a sample rate fs =
seconds(1) is a
representing a 1-second time difference between consecutive signal
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
'Window',hamming(100),'OverlapLength',50,'FFTLength',128windows the data using a 100-sample Hamming window, with 50 samples of overlap between adjoining segments and a 128-point FFT.
DataFormat— Data format of input
Data format of input, specified as a character vector or string scalar. This
argument is valid only if
x is unformatted.
Each character in this argument must be one of these labels:
C — Channel
B — Batch observations
T — Time
dlstft function accepts any permutation of
'CBT'. You can specify at most one of each of the
Each element of the argument labels the matching dimension of
x. If the argument is not in the listed order
'C' followed by
'B' and so on), then
dlstft implicitly permutes both the argument and the data
to match the order, but without changing how the data is stored.
Window— Spectral window
hann(128,'periodic')(default) | vector
Spectral window, specified as 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 a nonnegative integer smaller than the
'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.
FFTLength— Number of discrete Fourier transform (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.
yi— Short-time Fourier transform
dlarrayobjects | unformatted
Short-time Fourier transform, returned as two formatted
dlarray (Deep Learning Toolbox)
yr contains the real part of the transform.
yi contains the imaginary part of the transform.
x is a formatted
dlarray objects. The
corresponds to frequency in the short-time Fourier transform.
x is an unformatted
dlarray or a
yi are unformatted
dlarray objects. The dimension order in
If no time information is specified, then the STFT is computed over the Nyquist
range [0, π] if
'FFTLength' is even and over [0, π) if
'FFTLength' is odd. If you specify time
information, then the intervals are [0, fs/2] and [0, fs/2), respectively, where fs is
the effective sample rate.
Frequencies at which the deep learning STFT is computed, returned as a
If the input array does not contain time information, then the frequencies are in normalized units of rad/sample.
If the input array contains time information, then
contains frequencies expressed in Hz.
Times at which the deep learning STFT is computed, returned as a
dlarray object or a
If you do not specify time information, then
If you specify a sample rate, then
t contains time values
If you specify a sample time, then
t is a
duration array with the same time format as
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 columns in the STFT matrix is given by
where is the length of the original signal and the ⌊⌋ symbols denote the floor function. The number of rows in the matrix equals NDFT, the number of DFT points, for centered and two-sided transforms and ⌊NDFT/2⌋ + 1 for one-sided transforms.
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.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).