# dlistft

Deep learning inverse short-time Fourier transform

Since R2024a

## Syntax

``x = dlistft(y)``
``x = dlistft(y,Name=Value)``

## Description

example

````x = dlistft(y)` returns the inverse short-time Fourier transform (ISTFT) of the one-sided short-time Fourier transform `y`. The `dlistft` function requires Deep Learning Toolbox™.```

example

````x = dlistft(y,Name=Value)` specifies additional options using name-value arguments. Options include the spectral window and the FFT length. For example, `DataFormat="CBT"` specifies the data format of `y` as `CBT`.```

## Examples

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Create a random signal with three channels and 1024 samples representing a batch of 5. Save the signal as a `dlarray` in `"CTB"` format. Display the dimension sizes and data format of the array. `dlarray` permutes the array dimensions to the `"CBT"` shape expected by a deep learning network.

```x = dlarray(randn([3 1024 5]),"CTB"); size(x)```
```ans = 1×3 3 5 1024 ```
`dims(x)`
```ans = 'CBT' ```

Compute the short-time Fourier transform (STFT) of the signal using the default `dlstft` function values. The STFT is a `dlarray` in `"SCBT"` format.

```ydl = dlstft(x); dims(ydl)```
```ans = 'SCBT' ```

Compute the deep learning inverse short-time Fourier transform using the default `dlistft` function values. The output is a `dlarray` in `"CBT"` format.

```X = dlistft(ydl); dims(X)```
```ans = 'CBT' ```

Generate an N-by-2 matrix. Each column is a quadratic chirp sampled at 8192 Hz for two seconds.

```Fs = 8192; t = 0:1/Fs:2-1/Fs; Nt = numel(t); x = repmat(chirp(t,250,1,500,"quadratic")',1,2); size(x)```
```ans = 1×2 16384 2 ```

Save the matrix as an unformatted `dlarray` containing a batch of two single-channel signals. Use the `reshape` function to insert the third dimension.

```x = dlarray(reshape(x,Nt,1,[])); size(x)```
```ans = 1×3 16384 1 2 ```

Obtain the deep learning short-time Fourier transform. Specify a 512-sample Hann window, an FFT length of 600, and an overlap of 384 samples. Specify the format of the input `dlarray` as `"TCB"`. The output is an unformatted `dlarray`.

```nfft = 600; winLen = 512; win = hann(winLen); noverlap = 384; fmt = "TCB"; [ydl,fd1,tdl] = dlstft(x,Fs,Window=win,OverlapLength=noverlap, ... FFTLength=nfft,DataFormat=fmt);```

Compute the deep learning inverse short-time Fourier transform. Specify the data format as `"SCBT"`. To obtain perfect reconstruction, use the same name-value arguments for the ISTFT computation as for the STFT computation. The expected number of channels and samples in the output are `1` and `Nt`, respectively. The dimension order of the reconstruction is `"CBT"`.

```outsize = [1 Nt]; invfmt = "SCBT"; X = dlistft(ydl,Window=win,OverlapLength=noverlap, ... FFTLength=nfft,DataFormat=invfmt, ... ExpectedOutputSize=outsize); size(X)```
```ans = 1×3 1 2 16384 ```

Plot the difference between a single-channel signal and its reconstruction. Except for the end points, the reconstruction is perfect.

```idx = 1; sig = extractdata(x(:,1,idx)); Xp = permute(X,[3 1 2]); rec = extractdata(Xp(:,1,idx)); plot(t,sig-rec) title("Difference Between Signal and Reconstruction") xlabel("Time (s)")```

To remove edge effects, zero-pad the original data on both sides along the time dimension. The length of the zero-pad is the window length. Then compute the STFT.

```xPadded = paddata(x,Nt+2*winLen,dimension=1,side="both"); ydl = dlstft(xPadded,Fs,Window=win,OverlapLength=noverlap, ... FFTLength=nfft,DataFormat=fmt);```

Compute the deep learning inverse short-time Fourier transform. The expected number of samples in the output is `Nt+2*winLen`.

```outsize = [1 Nt+2*winLen]; X = dlistft(ydl,Window=win,OverlapLength=noverlap, ... FFTLength=nfft,DataFormat=invfmt, ... ExpectedOutputSize=outsize); size(X)```
```ans = 1×3 1 2 17408 ```

Trim both sides of the ISTFT output along the time dimension to length `Nt`.

```X = trimdata(X,Nt,dimension=3,side="both"); size(X)```
```ans = 1×3 1 2 16384 ```

Plot the difference between a single-channel signal and its reconstruction. Because zero-padding the data removes edge effects, the reconstruction is perfect.

```idx = 2; sig = extractdata(x(:,1,idx)); Xp = permute(X,[3 1 2]); rec = extractdata(Xp(:,1,idx)); plot(t,sig-rec) title({"Difference Between Signal and Reconstruction", ... "When Zero-Padding"}) xlabel("Time (s)")```

This example shows how to concatenate the real and imaginary parts of the STFT input to the `dlistft` function.

Generate a 3-by-160(-by-1) array containing one batch of a three-channel, 160-sample sinusoidal signal. The normalized sinusoid frequencies are π/4 rad/sample, π/2 rad/sample, and 3π/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. The STFT output is in `"SCBT"` format. Confirm the output is complex-valued. Display the array dimension sizes of the output.

```y = dlstft(x,Window=rectwin(64),FFTLength=1024); ~isreal(y)```
```ans = logical 1 ```
`size(y)`
```ans = 1×4 513 3 1 7 ```

Concatenate the real and imaginary parts of the output along the second dimension.

```yr = real(y); yi = imag(y); yc = cat(2,yr,yi); isreal(yc)```
```ans = logical 1 ```
`size(yc)`
```ans = 1×4 513 6 1 7 ```

Obtain the deep learning ISTFT of the concatenated `dlarray`. Confirm perfect reconstruction of the original data.

```xrec = dlistft(yc,Window=rectwin(64),FFTLength=1024); max(abs(x(:)-xrec(:)))```
```ans = 1x1 dlarray 3.4377e-14 ```

## Input Arguments

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One-sided short-time Fourier transform, specified as a formatted or unformatted `dlarray` (Deep Learning Toolbox) object, or a numeric array.

• If `y` is a formatted `dlarray`, it must be in the `"SCBT"` or `"CBT"` format.

• If `y` is in the `"CBT"` format, the size of the `"C"` dimension must be divisible by `floor(FFTLength/2)+1`.

• If `y` is in the `"SCBT"` format, the size of the `"S"` dimension must equal `floor(FFTLength/2)+1`.

• If `y` is in the `"CBT"` format, the `dlistft` function unflattens `y` to have shape `"SCBT"`. The `"S"` dimension corresponds to frequency.

• If `y` is real-valued, the number of channels `C` must be even because the `dlistft` function assumes the true number of channels is `C`/2. For more information, see Concatenating Real and Imaginary Parts of dlistft Input.

• If `y` is an unformatted `dlarray` or a numeric array, it must be compatible with the `"SCBT"` or `"CBT"` formats and you must set `DataFormat`.

Data Types: `double` | `single`
Complex Number Support: Yes

### 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.

Example: `Window=hamming(100),OverlapLength=50,FFTLength=128,Method="wola"` uses a 100-sample Hamming window, with 50 samples of overlap between adjoining segments and a 128-point DFT.

Input data format, specified as a character vector or string scalar. This argument is valid only if `y` is unformatted.

Each character in this argument must be one of these labels:

• `"S"` —Spatial (frequency) dimension

• `"C"` — Channel

• `"B"` — Batch observations

• `"T"` — Time

The `dlistft` function accepts any permutation of `"CBT"` or `"SCBT"`. You can specify at most one of each of the `"S"`, `"C"`, `"B"`, and `"T"` labels.

Each element of the argument labels the matching dimension of `y`. When you specify a data format, `dlistft` implicitly permutes both the argument and the data to match this order:

• `"S"`

• `"C"`

• `"B"`

• `"T"`

How the function stores the data remains the same.

Example: `"CBT"` specifies the format channel-batch-time.

Windowing function, specified as a vector or a `dlarray` object. The length of `Window` must be greater than or equal to 2. For a list of available windows, see Windows.

For perfect time-domain reconstruction, the window for the ISTFT computation must match the window for the STFT computation. Use the function `iscola` to check a window/overlap combination for constant overlap-add (COLA) compliance. COLA compliance is a requirement for perfect reconstruction for non-modified spectra. For more information, see Constant Overlap-Add (COLA) Constraint. If the window is a `dlarray` object, extract data from the `dlarray` before using the `iscola` function.

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

Data Types: `double` | `single`

Number of overlapped samples between adjoining segments, specified as a nonnegative integer smaller than the length of `Window`. If you omit `OverlapLength` or specify it as empty, the function sets it to the largest integer less than 75% of the window length, which turns out to be 96 samples for the default Hann window.

Data Types: `double` | `single`

Number of DFT points, specified as a positive integer. `FFTLength` must be greater than or equal to the window length. You must specify the same number of DFT points for the STFT and ISTFT.

Data Types: `double` | `single`

Method of overlap-add, specified as one of these:

• `"wola"` — Weighted overlap-add

• `"ola"` — Overlap-add

Expected number of channels and samples in the reconstructed signal, specified as a two-element vector of positive integers. The first element is the expected number of channels and the second element is the expected number of time samples.

By default, `dlistft` does not check the size of the output. To perform size validation, set `ExpectedOutputSize`. When you set `ExpectedOutputSize`, `dlistft` validates the expected size before computing the ISTFT. The function errors if the validation does not pass. For more information, see Algorithms.

Data Types: `single` | `double`

## Output Arguments

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Reconstructed signal, returned as a formatted or unformatted `dlarray` object.

• If `y` is a formatted `dlarray`, then `x` is a `"CBT"` formatted `dlarray`.

• If `y` is an unformatted `dlarray`, then `x` is an unformatted `dlarray`. The dimension order in `x` is `"CBT"`.

The `dlistft` function computes the inverse short-time Fourier transform along the `"S"` dimension of `y`. For perfect reconstruction for non-modified spectra, the window must be COLA compliant. For more information, see Constant Overlap-Add (COLA) Constraint.

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### Inverse Short-Time Fourier Transform

The inverse short-time Fourier transform is computed by taking the IFFT of each DFT vector of the STFT and overlap-adding the inverted signals. The ISTFT is calculated as follows.

`$\begin{array}{c}x\left(n\right)=\underset{-1/2}{\overset{1/2}{\int }}\sum _{m=-\infty }^{\infty }{X}_{m}\left(f\right){e}^{j2\pi fn}df\\ =\sum _{m=-\infty }^{\infty }\underset{-1/2}{\overset{1/2}{\int }}{X}_{m}\left(f\right){e}^{j2\pi fn}df\\ =\sum _{m=-\infty }^{\infty }{x}_{m}\left(n\right),\end{array}$`

where $R$ is the hop size between successive DFTs, ${X}_{m}$ is the DFT of the windowed data centered about time $mR,$ and ${x}_{m}\left(n\right)=x\left(n\right)\text{ }\text{\hspace{0.17em}}g\left(n-mR\right)$. The inverse STFT is a perfect reconstruction of the original signal as long as $\sum _{m=-\infty }^{\infty }{g}^{a+1}\left(n-mR\right)=c\text{\hspace{0.17em}}\forall n\in ℤ,$ where the analysis window $g\left(n\right)$ was used to window the original signal and $c$ is a constant. This figure depicts the steps in reconstructing the original signal.

To ensure successful reconstruction of nonmodified spectra, the analysis window must satisfy the COLA constraint. In general, if the analysis window satisfies the condition $\sum _{m=-\infty }^{\infty }{g}^{a+1}\left(n-mR\right)=c\text{\hspace{0.17em}}\forall n\in ℤ,$, the window is considered to be COLA-compliant. Additionally, COLA compliance can be described as either weak or strong.

• Weak COLA compliance implies that the Fourier transform of the analysis window has zeros at frame-rate harmonics such that

`$G\left({f}_{k}\right)=0,\text{ }\text{ }k=1,2,\dots ,R-1,\text{ }\text{ }{f}_{k}\triangleq \frac{k}{R}.$`

Alias cancellation is disturbed by spectral modifications. Weak COLA relies on alias cancellation in the frequency domain. Therefore, perfect reconstruction is possible using weakly COLA-compliant windows as long as the signal has not undergone any spectral modifications.

• For strong COLA compliance, the Fourier transform of the window must be bandlimited consistently with downsampling by the frame rate such that

`$G\left(f\right)=0,\text{ }\text{ }f\ge \frac{1}{2R}.$`

This equation shows that no aliasing is allowed by the strong COLA constraint. Additionally, for strong COLA compliance, the value of the constant $c$ must equal 1. In general, if the short-time spectrum is modified in any way, a stronger COLA compliant window is preferred.

You can use the `iscola` function to check for weak COLA compliance. The number of summations used to check COLA compliance is dictated by the window length and hop size. In general, it is common to use $a=1$ in $\sum _{m=-\infty }^{\infty }{g}^{a+1}\left(n-mR\right)=c\text{\hspace{0.17em}}\forall n\in ℤ,$ for weighted overlap-add (WOLA), and $a=0$ for overlap-add (OLA). By default, `istft` uses the WOLA method, by applying a synthesis window before performing the overlap-add method.

In general, the synthesis window is the same as the analysis window. You can construct useful WOLA windows by taking the square root of a strong OLA window. You can use this method for all nonnegative OLA windows. For example, the root-Hann window is a good example of a WOLA window.

### 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

must be an integer. In the equation, Nx is the length of the signal, M is the length of the window, and L is the overlap length.

• 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.

## Algorithms

If you set `ExpectedOutputSize`, the `dlistft` function computes the number of channels and samples in the reconstruction before performing the ISTFT. If the values do not match `ExpectedOutputSize`, the function errors. The size of the reconstruction depends on the dimensions, and data format of the input STFT, the length of the windowing function, the number of overlapped samples and the number of DFT points.

`dlistft` determines the number of channels and samples as follows. Define the hop size as ```hopSize = length(Window)-OverlapLength```.

• If `y` is a `"SCBT"` formatted `dlarray`, or an unformatted `dlarray` compatible with `"SCBT"` format:

• If `y` is complex-valued, the number of channels is `size(y,2)`. Otherwise, the number of channels is `size(y,2)/2`.

• The number of samples is `length(Window)+(nseg-1)*hopSize`, where `nseg = size(y,4)`.

• If `y` is a `"CBT"` formatted `dlarray`, or an unformatted `dlarray` compatible with `"CBT"` format:

• If `y` is complex-valued, the number of channels is `size(y,1)/nfreq`, where ```nfreq = floor(FFTLength/2)+1```. Otherwise, the number of channels is `size(y,1)/(2*nfreq)`.

• The number of samples is `length(Window)+(nseg-1)*hopSize`, where `nseg = size(y,3)`.

## Version History

Introduced in R2024a