Inverse short-time Fourier transform

specifies additional options using name-value pair arguments. Options include the FFT
window length and number of overlapped samples. These arguments can be added to any of the
previous input syntaxes.`x`

= istft(___,`Name,Value`

)

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

The first channel consists of a concave quadratic chirp with instantaneous frequency 100Hz at

*t =*0 and crosses 300 Hz at*t =*1 second. It has an initial phase equal to 45 degrees.The second channel consists of a convex quadratic chirp with instantaneous frequency 200Hz at

*t =*0 and crosses 600 Hz at*t =*1 second.The third channel consists of a logarithmic chirp with instantaneous frequency 300Hz 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 256 and an overlap length of 15 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(256,'periodic'),'OverlapLength',15);

Plot the original and reconstructed versions of the first and second channels.

[ix,ti] = istft(S,fs,'Window',hamming(256,'periodic'),'OverlapLength',15); plot(t,x(:,1)','LineWidth',1.5) hold on plot(ti,ix(:,1)','r--') hold off legend('Original Channel 1','Reconstructed Channel 1')

plot(t,x(:,2)','LineWidth',1.5) hold on plot(ti,ix(:,2)','r--') legend('Original Channel 2','Reconstructed Channel 2')

The phase vocoder performs time stretching and pitch scaling by transforming the audio into the frequency domain. This diagram shows the operations involved in the phase vocoder implementation.

The phase vocoder takes the STFT of a signal with an analysis window of hop size ${\mathit{R}}_{1}$ and then performs an ISTFT with a synthesis window of hop size ${\mathit{R}}_{2}$. The vocoder thus takes advantage of the WOLA method. To time stretch a signal, the analysis window uses a larger number of overlap samples than the synthesis. As a result, there are more samples at the output than at the input (${\mathit{N}}_{\mathit{S},\mathrm{Out}}>{\mathit{N}}_{\mathit{S},\mathrm{In}}$), although the frequency content remains the same. Now, you can pitch scale this signal by playing it back at a higher sample rate, which produces a signal with the original duration but a higher pitch.

Load an audio file containing a fragment of Handel's "Hallelujah Chorus" sampled at 8192 Hz.

`load handel`

Design a root-Hann window of length 512. Set analysis overlap length as 192 and synthesis overlap length as 166.

```
wlen = 512;
win = sqrt(hann(wlen,'periodic'));
noverlapA = 192;
noverlapS = 166;
```

Implement the phase vocoder by using an analysis window of overlap 192 and a synthesis window of overlap 166.

S = stft(y,Fs,'Window',win,'OverlapLength',noverlapA); iy = istft(S,Fs,'Window',win,'OverlapLength',noverlapS); %To hear, type soundsc(w,Fs), pause(10), soundsc(iw,Fs);

If the analysis and synthesis windows are the same but the overlap length is changed, there will be an additional gain/loss that you will need to adjust. This is a common approach to implementing a phase vocoder.

Calculate the hop ratio and use it to adjust the gain of the reconstructed signal. Also calculate frequency of pitch-shifted data using the hop ratio.

```
hopRatio = (wlen-noverlapS)/(wlen-noverlapA);
iyg = iy*hopRatio;
Fp = Fs*hopRatio;
%To hear, type soundsc(iwg,Fs), pause(15), soundsc(iwg,Fp);
```

Plot the original signal and the time stretched signal with fixed gain.

plot((0:length(iyg)-1)/Fs,iyg,(0:length(y)-1)/Fs,y) xlabel('Time (s)') xlim([0 (length(iyg)-1)/Fs]) legend('Time Stretched Signal with Fixed Gain','Original Signal','Location','best')

Compare the time-stretched signal and the pitch shifted signal on the same plot.

plot((0:length(iy)-1)/Fs,iy,(0:length(iy)-1)/Fp,iy) xlabel('Time (s)') xlim([0 (length(iyg)-1)/Fs]) legend('Time Stretched Signal','Pitch Shifted Signal','Location','best')

To better understand the effect of pitch shifting data, consider the following sinusoid of frequency `Fs`

over 2 seconds.

t = 0:1/Fs:2; x = sin(2*pi*10*t);

Calculate the short-time Fourier transform and the inverse short-time Fourier transform with overlap lengths 192 and 166 respectively.

Sx = stft(x,Fs,'Window',win,'OverlapLength',noverlapA); ix = istft(Sx,Fs,'Window',win,'OverlapLength',noverlapS);

Plot the original signal on one plot and the time-stretched and pitch shifted signal on another.

subplot(2,1,1) plot((0:length(ix)-1)/Fs,ix,'LineWidth',2) xlabel('Time (s)') ylabel('Signal Amplitude') xlim([0 (length(ix)-1)/Fs]) legend('Time Stretched Signal') subplot(2,1,2) hold on plot((0:length(x)-1)/Fs,x) plot((0:length(ix)-1)/Fp,ix,'--','LineWidth',2) legend('Original Signal','Pitch Shifted Signal','Location','best') hold off xlabel('Time (s)') ylabel('Signal Amplitude') xlim([0 (length(ix)-1)/Fs])

Generate a complex sinusoid of frequency 1 kHz and duration 2 seconds.

fs = 1e3; ts = 0:1/fs:2-1/fs; x = exp(2j*pi*100*cos(2*pi*2*ts));

Design a periodic Hann window of length 100 and set the number of overlap samples to 75. Check the window and overlap length for COLA compliance.

```
nwin = 100;
win = hann(100,'periodic');
noverlap = 75;
tf = iscola(win,noverlap)
```

`tf = `*logical*
1

Zero-pad the signal to remove edge-effects. To avoid truncation, pad the input signal with zeros such that $\frac{\left(\mathrm{length}\left(\mathrm{xZero}\right)-\mathrm{noverlap}\right)}{\left(\mathrm{nwin}-\mathrm{noverlap}\right)}$ is an integer. Set the FFT length to 128. Compute the short-time Fourier transform of the complex signal.

xZero = [zeros(1,nwin) x zeros(1,nwin)]; fftlen = 128; s = stft(xZero,fs,'Window',win,'OverlapLength',noverlap,'FFTLength',fftlen);

Calculate the inverse short-time Fourier transform and remove the zeros for perfect reconstruction.

[is,ti] = istft(s,fs,'Window',win,'OverlapLength',noverlap,'FFTLength',fftlen); is(1:nwin) = []; is(end-nwin+1:end) = []; ti = ti(1:end-2*nwin);

Plot the real parts of the original and reconstructed signals. The imaginary part of the signal is also reconstructed perfectly.

plot(ts,real(x)) hold on plot(ti,real(is),'--') xlim([0 0.5]) xlabel('Time (s)') ylabel('Amplitude (V)') legend('Original Signal','Reconstructed Signal') hold off

Generate a sinusoid sampled at 2 kHz for 1 second.

fs = 2e3; t = 0:1/fs:1-1/fs; x = 5*sin(2*pi*10*t);

Design a periodic Hamming window of length 120. Check the COLA constraint for the window with an overlap of 80 samples. The window-overlap combination is COLA compliant.

```
win = hamming(120,'periodic');
noverlap = 80;
tf = iscola(win,noverlap)
```

`tf = `*logical*
1

Set the FFT length to 512. Compute the short-time Fourier transform.

fftlen = 512; s = stft(x,fs,'Window',win,'OverlapLength',noverlap,'FFTLength',fftlen);

Calculate the inverse short-time Fourier transform.

[X,T] = istft(s,fs,'Window',win,'OverlapLength',noverlap,'FFTLength',fftlen,'Method','ola','ConjugateSymmetric',true);

Plot the original and reconstructed signals.

plot(t,x,'b') hold on plot(T,X,'-.r') xlabel('Time (s)') ylabel('Amplitude (V)') title('Original and Reconstructed Signal') legend('Original Signal','Reconstructed Signal') hold off

`s`

— Short-time Fourier transformmatrix | 3-D array

Short-time Fourier transform, specified as a matrix or a 3-D array. For
single-channel signals, specify `s`

as a matrix with time increasing
across the columns and frequency increasing down the rows. For multichannel signals,
specify `s`

as a 3-D array with the third dimension corresponding to
the channels. The frequency and time vectors are obtained as outputs of `stft`

.

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.

**Data Types: **`double`

| `single`

**Complex Number Support: **Yes

`fs`

— Sample rate`2π`

(default) | positive scalarSample rate in hertz, specified as a positive scalar.

**Data Types: **`double`

| `single`

`ts`

— Sample timeduration scalar

Sample time, specified as a `duration`

scalar.

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

is a

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

**Data Types: **`duration`

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`

.

`istft(s,'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'`

— Windowing function`hann(128,'periodic')`

(default) | vectorWindowing function, 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 periodic 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 integerNumber 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 turns out to be 96 samples for the default Hann window.

**Data Types: **`double`

| `single`

`'FFTLength'`

— Number of DFT points`128`

(default) | positive integerNumber of DFT points, specified as the comma-separated pair consisting of
`FFTLength`

and a positive integer. To achieve perfect time-domain
reconstruction, you should set the `FFTLength`

to match that used
in `stft`

.

**Data Types: **`double`

| `single`

`'Method'`

— Method of overlap-add`'wola'`

(default) | `'ola'`

Method of overlap-add, specified as the comma-separated pair consisting of
`Method`

and one of the following:

`'wola'`

— Weighted overlap-add`'ola'`

— Overlap-add

`'ConjugateSymmetric'`

— Conjugate symmetry of original signal`false`

(default) | `true`

Conjugate symmetry of the original signal, specified as the comma-separated pair
consisting of `ConjugateSymmetric`

and `true`

or
`false`

. If this option is set to `true`

,
`istft`

assumes that the input `s`

is
symmetric, otherwise no symmetric assumption is made. When `s`

is
not exactly conjugate symmetric due to round-off error, setting the name-value pair to
`true`

ensures that the STFT is treated as if it were conjugate
symmetric. If `s`

is conjugate symmetric, then the inverse
transform computation is faster, and the output is real.

`'Centered'`

— Frequency range`true`

(default) | `false`

Frequency range, specified as the comma-separated pair consisting of
`Centered`

and `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.

`'InputTimeDimension'`

— Input time dimensionacrosscolumns (default) | downrows

Input time dimension, specified as the comma-separated pair consisting of
`InputTimeDimension`

and `acrosscolumns`

or
`downrows`

. If this value is set to `downrows`

,
`istft`

assumes that the time dimension of `s`

is down the rows and the frequency is across the columns. If this value is set to
`acrosscolumns`

, the function `istft`

assumes
that the time dimension of `s`

is across the columns and frequency
dimension is down the rows.

`x`

— Reconstructed signalvector | matrix

Reconstructed signal in the time domain, returned as a vector or a matrix.

**Data Types: **`single`

| `double`

`t`

— Time instantsvector

Time instants, returned as a vector.

**Data Types: **`double`

| `single`

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:

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

where $$R$$ is the hop size between successive DFTs, $${X}_{m}$$ is the DFT of the windowed data centered about time $$mR$$ and $${x}_{m}(n)=x(n)\text{\hspace{0.05em}}\text{\hspace{0.17em}}g(n-mR)$$. The inverse STFT is a perfect reconstruction of the original signal
as long as $$\sum _{m=-\infty}^{\infty}{g}^{a+1}(n-mR)=c\text{\hspace{0.17em}}}\forall n\in \mathbb{Z$$ where the *analysis window*
$$g(n)$$ was used to window the original signal and $$c$$ is a constant. The following figure depicts the steps followed 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 $$\sum _{m=-\infty}^{\infty}{g}^{a+1}(n-mR)=c\text{\hspace{0.17em}}}\forall n\in \mathbb{Z$$, 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

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

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

$$G(f)=0,\text{\hspace{1em}}\text{\hspace{1em}}f\ge \frac{1}{2R}.$$

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 $$\sum _{m=-\infty}^{\infty}{g}^{a+1}(n-mR)=c\text{\hspace{0.17em}}}\forall n\in \mathbb{Z$$ 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.

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 $$k\text{}=\text{}{\scriptscriptstyle \frac{(length(x)-noverlap)}{(length(window)-noverlap)}}$$ 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.

[1] Crochiere, R. E. “A
Weighted Overlap-Add Method of Short-Time Fourier Analysis/Synthesis.” *IEEE
Transactions on Acoustics, Speech and Signal Processing.* Vol. 28, No. 1, Feb
1980, pp. 99–102.

[2] Griffin, D. W., and J. S. Lim.
“Signal Estimation from Modified Short-Time Fourier Transform.” *IEEE
Transactions on Acoustics, Speech and Signal Processing.* Vol. 32, No. 2, April
1984, pp. 236–243.

[3] Portnoff, M. R.
“Time-Frequency Representation of Digital Signals and Systems Based on Short-Time
Fourier analysis.” *IEEE Transactions on Acoustics, Speech and Signal
Processing.* Vol. 28, No. 1, Feb 1980, pp. 55–69.

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

[5] A. D. Gotzen, N. Bernardini and D.
Arfib, “Traditional Implementations of a Phase-Vocoder: The Tricks of the
Trade”, *Proceedings of the COST G-6 Conference on Digital Audio Effects
(DAFX-00)*, Verona, Italy, Dec 7-9, 2000.

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

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