stftmag2sig

Signal reconstruction from STFT magnitude

Since R2020b

Syntax

x = stftmag2sig(s,nfft)

x = stftmag2sig(s,nfft,fs)

x = stftmag2sig(s,nfft,ts)

x = stftmag2sig(___,Name,Value)

[x,t,info] = stftmag2sig(___)

Description

x = stftmag2sig(s,nfft) returns a reconstructed time-domain real signal, x, estimated from the Short-Time Fourier Transform (STFT) magnitude, s, based on the Griffin-Lim algorithm. The function assumes s was computed using discrete Fourier transform (DFT) length nfft.

x = stftmag2sig(s,nfft,fs) returns the reconstructed signal assuming that s was sampled at rate fs.

x = stftmag2sig(s,nfft,ts) returns the reconstructed signal assuming that s was sampled with sample time ts.

x = stftmag2sig(___,Name,Value) specifies additional options using name-value pair arguments. Options include, among others, the FFT window and the method to specify initial phases. These arguments can be added to any of the previous input syntaxes. For example, 'FrequencyRange','onesided','InitializePhaseMethod','random' specifies that the signal is reconstructed from a one-sided STFT with random initial phases.

[x,t,info] = stftmag2sig(___) also returns the time instants at which the signal is reconstructed and a structure that contains information about the reconstruction process.

Examples

Reconstruct Sinusoid from STFT Magnitude

Consider 512 samples of a sinusoid with a normalized frequency of $\pi /60$ rad/sample and a DC value of 1. Compute the STFT of the signal.

n = 512;

x = cos(pi/60*(0:n-1)')+1;

S = stft(x);

Reconstruct the sinusoid from the magnitude of the STFT. Plot the original and reconstructed signals.

xr = stftmag2sig(abs(S),size(S,1));

plot(x)
hold on
plot(xr,'--','LineWidth',2)
hold off
legend('Original','Reconstructed')

Repeat the computation, but now pad the signal with zeros to decrease edge effects.

xz = circshift([x; zeros(n,1)],n/2);

Sz = stft(xz);
xr = stftmag2sig(abs(Sz),size(Sz,1));

xz = xz(n/2+(1:n));
xr = xr(n/2+(1:n));

plot(xz)
hold on
plot(xr,'--','LineWidth',2)
hold off
legend('Original','Reconstructed')

Repeat the computation, but now decrease edge effects by assuming that x is a segment of a signal twice as long.

xx = cos(pi/60*(-n/2:n/2+n-1)')+1;

Sx = stft(xx);
xr = stftmag2sig(abs(Sx),size(Sx,1));

xx = xx(n/2+(1:n));
xr = xr(n/2+(1:n));

plot(xx)
hold on
plot(xr,'--','LineWidth',2)
hold off
legend('Original','Reconstructed')

Reconstruct Audio Signal from STFT Magnitude

Load an audio signal that contains two decreasing chirps and a wideband splatter sound. The signal is sampled at 8192 Hz. Plot the STFT of the signal. Divide the waveform into 128-sample segments and window the segments using a Hamming window. Specify 64 samples of overlap between adjoining segments and 1024 FFT points.

load splat
ty = (0:length(y)-1)/Fs;

% To hear, type sound(y,Fs)

wind = hamming(128);
olen = 64;
nfft = 1024;

stft(y,Fs,'Window',wind,'OverlapLength',olen,'FFTLength',nfft)

Compute the magnitude and phase of the STFT.

s = stft(y,Fs,'Window',wind,'OverlapLength',olen,'FFTLength',nfft);

smag = abs(s);
sphs = angle(s);

Reconstruct the signal based on the magnitude of the STFT. Use the same parameters that you used to compute the STFT. By default, stftmag2sig initializes the phases to zero and uses 100 optimization iterations.

[x,tx,info] = stftmag2sig(smag,nfft,Fs,'Window',wind,'OverlapLength',olen);

% To hear, type sound(x,Fs)

Plot the original and reconstructed signals. For better comparison, offset the reconstructed signal up and to the right.

plot(ty,y,tx+500/Fs,x+1)
legend('Original','Reconstructed','Location','best')

Output the relative improvement toward convergence between the last two iterations.

impr = info.Inconsistency

impr = 0.0424

Improve the reconstruction by doubling the number of optimization iterations and setting the initial phases to the actual phases from the STFT. Plot the original and reconstructed signals. For better comparison, plot the negative of the reconstructed signal and offset it up and to the right.

[x,tx,info] = stftmag2sig(smag,nfft,Fs,'Window',wind,'OverlapLength',olen, ...
    'MaxIterations',200,'InitialPhase',sphs);

% To hear, type sound(x,Fs)

plot(ty,y,tx+500/Fs,-x+1)
legend('Original','Reconstructed','Location','best')

Output the relative improvement toward convergence between the last two iterations.

impr = info.Inconsistency

impr = 1.3874e-16

Input Arguments

s — STFT magnitude

matrix

STFT magnitude, specified as a matrix. s must correspond to a single-channel, real-valued signal.

Example: abs(stft(sin(pi/2*(0:255)),'FFTLength',128)) specifies the STFT magnitude of a sinusoid.

Example: abs(stft(chirp(0:1/1e3:1,25,1,50))) specifies the STFT magnitude of a chirp sampled at 1 kHz.

Data Types: single | double

nfft — Number of DFT points

positive integer scalar

Number of DFT points, specified as a positive integer scalar. This argument is always required.

Data Types: single | double

fs — Sample rate

2π (default) | positive numeric scalar

Sample rate, specified as a positive numeric scalar.

ts — Sample time

duration scalar

Sample time, specified as a 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.

Data Types: duration

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: 'FrequencyRange','onesided','InitializePhaseMethod','random' specifies that the signal is reconstructed from a one-sided STFT with random initial phases.

Display — Inconsistency display option

false (default) | true

Inconsistency display option, specified as the comma-separated pair consisting of 'Display' and a logical value. If this option is set to true, then stftmag2sig displays the normalized inconsistency after every 20 optimization iterations, and it also displays stopping information at the end of the run.

Data Types: logical

FrequencyRange — Frequency range of STFT magnitude

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

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

• 'centered' — Treat s as the magnitude of a two-sided, centered STFT. If nfft is even, then s is taken to have been computed over the interval (–π, π] rad/sample. If nfft is odd, then s is taken to have been 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' — Treat s as the magnitude of a two-sided STFT computed over the interval [0, 2π) rad/sample. If you specify time information, then the interval is [0, f_s) cycles/unit time.

• 'onesided' — Treat s as the magnitude of a one-sided STFT. If nfft is even, then s is taken to have been computed over the interval [0, π] rad/sample. If nfft is odd, then s is taken to have been 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.

Data Types: char | string

InconsistencyTolerance — Inconsistency tolerance of reconstruction process

1e-4 (default) | positive scalar

Inconsistency tolerance of reconstruction process, specified as the comma-separated pair consisting of 'InconsistencyTolerance' and a positive scalar. The reconstruction process stops when the Normalized Inconsistency is lower than the tolerance.

Data Types: single | double

InitializePhaseMethod — Phase initialization

'zeros' (default) | 'random'

Phase initialization, specified as the comma-separated pair consisting of 'InitializePhaseMethod' and 'zeros' or 'random'. Specify only one of 'InitializePhaseMethod' or 'InitialPhase'.

• 'zeros' — The function initializes the phases as zeros.

• 'random' — The function initializes the phases as random numbers distributed uniformly in the interval [–π, π].

Data Types: char | string

InitialPhase — Initial phases

real numeric matrix in the range [–π, π]

Initial phases, specified as the comma-separated pair consisting of 'InitialPhase' and a real numeric matrix in the range [–π, π]. The matrix must have the same size as s. Specify only one of 'InitializePhaseMethod' or 'InitialPhase'.

Example: angle(stft(randn(1000,1))) specifies the phases of the short-time Fourier transform of a random signal.

Example: 2*pi*(rand(size(stft(randn(1000,1))))-1/2) specifies a matrix of random phases distributed uniformly in the interval [–π, π]. The matrix has the same size as the short-time Fourier transform of a random signal.

.

Data Types: single | double

InputTimeDimension — Input time dimension

'acrosscolumns' (default) | 'downrows'

Input time dimension, specified as the comma-separated pair consisting of 'InputTimeDimension' and 'acrosscolumns' or 'downrows'.

• 'acrosscolumns' — The function assumes that the time dimension of s is across the columns and the frequency dimension is down the rows.

• 'downrows' — The function assumes that the time dimension of s is down the rows and the frequency dimension is across the columns.

Data Types: char | string

MaxIterations — Maximum number of optimization iterations

100 (default) | positive integer scalar

Maximum number of optimization iterations, specified as the comma-separated pair consisting of 'MaxIterations' and a positive integer scalar. The reconstruction process stops when the number of iterations is greater than 'MaxIterations'.

Data Types: single | double

Method — Signal reconstruction algorithm

'gla' (default) | 'fgla' | 'legla'

Signal reconstruction algorithm, specified as the comma-separated pair consisting of 'Method' and one of these:

• 'gla' — The original reconstruction algorithm, proposed by Griffin and Lim and described in [1].

• 'fgla' — A fast Griffin-Lim algorithm proposed by Perraudin, Balazs, and Søndergaard and described in [2].

• 'legla' — A fast algorithm proposed by Le Roux, Kameoka, Ono, and Sagayama and described in [3].

Data Types: char | string

OverlapLength — Number of overlapped samples

75% of window length (default) | nonnegative integer

Number of overlapped samples between adjoining segments, specified as the comma-separated pair consisting of 'OverlapLength' and a positive integer smaller than the length of 'Window'. Successful signal reconstruction requires 'OverlapLength' to match the number of overlapped segments used to generate the STFT magnitude. If you omit 'OverlapLength' or specify it as empty, it is set to the largest integer less than or equal to 75% of the window length, which is 96 samples for the default Hann window.

Data Types: double | single

TruncationOrder — Truncation order for 'legla' update rule

positive integer

Truncation order for 'legla' update rule, specified as the comma-separated pair consisting of 'TruncationOrder' and a positive integer. This argument applies only when 'Method' is set to 'legla' and controls the number of phase values updated in each iteration of that method. If not specified, 'TruncationOrder' is determined using an adaptive algorithm.

Data Types: single | double

UpdateParameter — Update parameter for fast Griffin-Lim algorithm

0.99 (default) | positive scalar

Update parameter for the fast Griffin-Lim algorithm, specified as the comma-separated pair consisting of 'UpdateParameter' and a positive scalar. This argument applies only when 'Method' is set to 'fgla' and specifies the parameter for that method's update rule.

Data Types: single | double
Complex Number Support: Yes

Window — Spectral window

hann(128,'periodic') (default) | vector

Spectral window, specified as the comma-separated pair consisting of 'Window' and a vector. Successful signal reconstruction requires 'Window' to match the window used to generate the STFT magnitude. 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(128,'periodic') and (1-cos(2*pi*(128:-1:1)'/128))/2 both specify the default window used by stftmag2sig.

Data Types: double | single

Output Arguments

x — Reconstructed time-domain signal

vector

Reconstructed time-domain signal, returned as a vector.

Note

If you want x and s to be the same length, the value of (length(x)-noverlap)/(length(window)-noverlap) must be an integer. Use Window to specify the length of window and OverlapLength to specify noverlap.

t — Time instants

vector

Time instants at which the signal is reconstructed, returned as a vector.

info — Reconstruction process information

structure

Reconstruction process information, returned as a structure containing these fields:

• ExitFlag — Termination flag.

  • A value of 0 indicates the algorithm stopped when it reached the maximum number of iterations.

  • A value of 1 indicates the algorithm stopped when it met the relative tolerance.

• NumIterations — Total number of iterations.

• Inconsistency — Average relative improvement toward convergence between the last two iterations.

• ReconstructedPhase — Reconstructed phase at the last iteration.

• ReconstructedSTFT — Reconstructed short-time Fourier transform at the last iteration.

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 mth 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:

$$X_m(f)={\displaystyle \sum _{n=-\infty }^{\infty }x(n)g(n-mR)e^{-j2\pi fn}}.$$

• 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.

Normalized Inconsistency

The normalized inconsistency measures the improvement toward convergence of the reconstruction process in successive optimization iterations.

The normalized inconsistency is defined as

$$\text{Inconsistency}=\frac{\Vert \mathrm{STFT}\left(\mathrm{ISTFT}\left(s_{\text{est}}\right)\right)-s_{\text{est}}\Vert }{\Vert s_{\text{est}}\Vert },$$

where s_est is the complex short-time Fourier transform estimated at each iteration, the brackets denote the matrix norm, STFT denotes the short-time Fourier transform, and ISTFT denotes its inverse. stftmag2sig uses the MATLAB^® function norm to compute matrix norms. For more information about the STFT and its inverse, see Short-Time Fourier Transform and Inverse Short-Time Fourier Transform.

References

[1] Griffin, Daniel W., and Jae S. Lim. "Signal Estimation from Modified Short-Time Fourier Transform." IEEE Transactions on Acoustics, Speech, and Signal Processing. Vol. 32, Number 2, April 1984, pp. 236–243. https://doi.org/10.1109/TASSP.1984.1164317.

[2] Perraudin, Nathanaël, Peter Balazs, and Peter L. Søndergaard. "A Fast Griffin-Lim Algorithm." In 2013 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, New Paltz, NY, October 20–23, 2013. https://doi.org/10.1109/WASPAA.2013.6701851.

[3] Le Roux, Jonathan, Hirokazu Kameoka, Nobutaka Ono, and Shigeki Sagayama. "Fast Signal Reconstruction from Magnitude STFT Spectrogram Based on Spectrogram Consistency." In Proceedings of the 13th International Conference on Digital Audio Effects (DAFx-10), Graz, Austria, September 6–10, 2010.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Arrays

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

Usage notes and limitations:

• 'legla' method is not supported.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2020b