# dlstft

Deep learning short-time Fourier transform

## Syntax

## Description

`[`

returns the deep learning Short-Time Fourier Transform (STFT) of
`yr`

,`yi`

] = dlstft(`x`

)`x`

. `dlstft`

requires Deep Learning Toolbox™.

`[`

specifies additional options using name-value arguments. Options include the spectral
window and the FFT length. These arguments can be added to any of the previous input
syntaxes. For example, `yr`

,`yi`

] = dlstft(___,`Name,Value`

)`'DataFormat','CBT'`

specifies the data format of
`x`

as `CBT`

.

## Examples

### Deep Learning Short-Time Fourier Transform of Chirp

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 `'CTB'`

format.

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')

### Deep Learning Short-Time Fourier Transform of Sinusoid

Generate a 3-by-160(-by-1) array containing one batch of a three-channel, 160-sample sinusoidal signal. The normalized sinusoid frequencies are $\pi /4$ rad/sample, $\pi /2$ rad/sample, and $$3\pi /4$$ 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')

## Input Arguments

`x`

— Input array

`dlarray`

object | numeric array

Input array, specified as an unformatted `dlarray`

(Deep Learning Toolbox), a
formatted `dlarray`

in `'CBT'`

format, or a numeric
array. If `x`

is an unformatted `dlarray`

or a numeric
array, you must specify the `'DataFormat'`

as some permutation of
`'CBT'`

.

**Example: **`dlarray(cos(pi./[4;2]*(0:159)),'CTB')`

and
`dlarray(cos(pi./[4;2]*(0:159))','TCB')`

both specify one batch
observation of a two-channel sinusoid in the `'CBT'`

format.

`fs`

— Sample rate

2*π* (default) | positive numeric scalar

Sample rate, specified as a positive numeric scalar.

`ts`

— Sample time

`duration`

scalar

Sample time, specified as `duration`

scalar. Specifying
`ts`

is equivalent to setting a sample rate *f*_{s} =
1/`ts`

.

**Example: **`seconds(1)`

is a `duration`

scalar
representing a 1-second time difference between consecutive signal
samples.

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**Example: **`'Window',hamming(100),'OverlapLength',50,'FFTLength',128`

windows 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

character vector | string scalar

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

The `dlstft`

function accepts any permutation of
`'CBT'`

. You can specify at most one of each of the
`C`

, `B`

, and `T`

labels.

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.

**Example: **`'CBT'`

`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.

**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 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.

**Data Types: **`double`

| `single`

## Output Arguments

`yr`

, `yi`

— Short-time Fourier transform

formatted `dlarray`

objects | unformatted `dlarray`

objects

Short-time Fourier transform, returned as two formatted `dlarray`

(Deep Learning Toolbox)
objects. `yr`

contains the real part of the transform.
`yi`

contains the imaginary part of the transform.

If

`x`

is a formatted`dlarray`

,`yr`

and`yi`

are`'SCBT'`

formatted`dlarray`

objects. The`'S'`

dimension corresponds to frequency in the short-time Fourier transform.If

`x`

is an unformatted`dlarray`

or a numeric array,`yr`

and`yi`

are unformatted`dlarray`

objects. The dimension order in`yr`

and`yi`

is`'SCBT'`

.

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, *f*_{s}/2] and [0, *f*_{s}/2), respectively, where *f*_{s} is
the effective sample rate.

`f`

— Frequencies

`dlarray`

object

Frequencies at which the deep learning STFT is computed, returned as a
`dlarray`

object.

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

`f`

contains frequencies expressed in Hz.

`t`

— Times

`dlarray`

object | `duration`

array

Times at which the deep learning STFT is computed, returned as a
`dlarray`

object or a `duration`

array.

If you do not specify time information, then

`t`

contains sample numbers.If you specify a sample rate, then

`t`

contains time values in seconds.If you specify a sample time, then

`t`

is a`duration`

array with the same time format as`x`

.

## 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 magnitude squared of the STFT is
known as the *spectrogram* time-frequency representation of the signal.
For more information about the spectrogram and how to compute it using Signal Processing Toolbox™ functions, see Spectrogram Computation with Signal Processing Toolbox.

The STFT of a signal is computed by sliding an *analysis window*
*g*(*n*) of length *M* over the signal and calculating the
discrete Fourier transform (DFT) of each segment of windowed data. The window hops over the
original signal at intervals of *R* samples, equivalent to *L* = *M* –
*R* samples of overlap between adjoining segments. Most window functions taper
off at the edges to avoid spectral ringing. The DFT of each windowed segment is added to a
complex-valued matrix that contains the magnitude and phase for each point in time and
frequency. The STFT matrix has

$$k=\lfloor \frac{{N}_{x}-L}{M-L}\rfloor $$

columns, where *N _{x}* is the length
of the signal

*x*(

*n*) and the ⌊⌋ symbols denote the floor function. The number of rows in the matrix equals

*N*

_{DFT}, the number of DFT points, for centered and two-sided transforms and an odd number close to

*N*

_{DFT}/2 for one-sided transforms of real-valued signals.

The *m*th column of the STFT matrix $$X(f)=\left[\begin{array}{ccccc}{X}_{1}(f)& {X}_{2}(f)& {X}_{3}(f)& \cdots & {X}_{k}(f)\end{array}\right]$$ contains the DFT of the windowed data centered about time *mR*:

$${X}_{m}(f)={\displaystyle \sum _{n=-\infty}^{\infty}x(n)\text{\hspace{0.17em}}g(n-mR)\text{\hspace{0.17em}}{e}^{-j2\pi fn}}.$$

The short-time Fourier transform is invertible. The inversion process overlap-adds the windowed segments to compensate for the signal attenuation at the window edges. For more information, see Inverse Short-Time Fourier Transform.

The

`istft`

function inverts the STFT of a signal.Under a specific set of circumstances it is possible to achieve "perfect reconstruction" of a signal. For more information, see Perfect Reconstruction.

The

`stftmag2sig`

returns an estimate of a signal reconstructed from the magnitude of its STFT.

## Extended Capabilities

### 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).

## Version History

**Introduced in R2021a**

## See Also

`dlarray`

(Deep Learning Toolbox) | `stft`

| `istft`

| `stftmag2sig`

| `stftLayer`

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