# STFT.jl

Documentation for STFT.jl

STFT.analysisFunction
analysis(x::Vector, w::Vector, L=0, N=length(w)) -> Matrix
analysis(x::Array{Vector}, w::Vector, L=0, N=length(w)) -> Array{Matrix}

stft(x::Vector, w::Vector, L=0, N=length(w)) -> Matrix
stft(x::Array{Vector}, w::Vector, L=0, N=length(w)) -> Array{Matrix}

Analyse discrete time-domain signal $\mathrm{x}[n]$ using Short-Time Fourier Transform given by

$$$\mathrm{X}[sH, \omega] = \sum_{n = -\infty}^{+\infty} \mathrm{w}[n - sH] \ \mathrm{x}[n] e^{-j\omega n},$$$

where $s$ and $\omega$ denotes segment index and angular frequency respectively, $\mathrm{w}[n]$ is a discrete time-domain signal of analysis window, and $H$ is nonnegative integer value that determine number of samples between two consecutive signal segments (also known as hop).

Parameters

• x - An array containing samples of a discrete time-domain signal.
• w - An array containing samples of a discrete time-domain analysis window.
• L - An overlap in samples between two consecutive segments. Default value is 0.
• N - A number of discrete frequency bins to be used in the DFT computation. Default value is length(w). If N < length(w), then N=length(w) is enforced to avoid loss of information.

Returns

• X - A complex matrix containing STFT-domain signal.

Note

1. Relation between $H$ and $L$ is given as $H = W - L$ where $W$ is a length of a window.
2. For real-valued (x isa Real) input signals function returns matrix is of size (N÷2+1, S) where S is a number of segments; i.e., one-sided spectrum.
3. For complex-valued (x isa Complex) input signals function returns matrix is of size (N, S) where S is a number of segments; i.e., two-sided spectrum.

Examples

import STFT

x = rand(10000) # Generate mock signal
W = 64          # Window length
w = ones(W)     # Rectangular analysis window
H = 10          # Hop
L = W - H       # Overlap

X = STFT.analysis(x, w, L)  # Analysis
using STFT

x = rand(100)   # Generate mock signal
W = 64          # Window length
w = ones(W)     # Rectangular analysis window
H = 4           # Hop
L = W - H       # Overlap

X  = stft(x, w, L)  # Analysis
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STFT.synthesisFunction
synthesis(X::Matrix, w::Vector, L=0, N=length(w)) -> Vector

istft(X::Matrix, w::Vector, L=0, N=length(w)) -> Vector

Syntesise discrete time-domain signal $y[n]$ from STFT-domain signal $Y_w[sH, n]$. An arbitrary STFT-domain signal $Y_w[sH, n]$, in general, is not a valid STFT in the sense that there is no discrete time-domian signal whoes STFT is given by $Y_w[sH, n]$ . As result, time-domain signal must be estimated using following formula :

$$$y[n] = \frac{ \sum\limits_{s=-\infty}^{+\infty} w[n - sH] \ y_w[sH, n] }{ \sum\limits_{s=-\infty}^{+\infty} w^2[n - sH] },$$$

where $w[n]$ is time-domain signal of analysis window, $y_w[sH, n]$ is time-domain representation of $Y_w[sH, n]$, and $H$ is nonnegative integer value that determine number of samples between two consecutive signal segments (also known as hop).

Parameters

• X - An matrix containing samples of a discrete STFT-domain signal.
• w - An array containing samples of a discrete time-domain analysis window.
• L - An overlap in samples between two consecutive segments. Default value is 0.
• N - A number of discrete frequency bins to be used in the inverse DFT computation. Default value is length(w). If N < length(w), then N=length(w) is enforced to avoid loss of information.

Returns

• x - A real-valued vector containing estimated time-domain signal.

Note

1. Relation between $H$ and $L$ is given as $H = W - L$ where $W$ is a length of a window.
2. This function supports only synthesis of real-valued signals from one-sided STFT-domain signal.
3. Synthesised time-domain signal might be shorter than analysed one, since only whole segments are analysed.

Example

import STFT

x = rand(100)   # Generate mock signal
W = 64          # Window length
w = ones(W)     # Rectangular analysis window
H = 4           # Hop
L = W - H       # Overlap

X  = STFT.analysis(x, w, L)  # Analysis
xr = STFT.synthesis(X, w, L) # Synthesis
using STFT

x = rand(100)   # Generate mock signal
W = 64          # Window length
w = ones(W)     # Rectangular analysis window
H = 4           # Hop
L = W - H       # Overlap

X  = stft(x, w, L)  # Analysis
xr = istft(X, w, L) # Synthesis

References

1. D. Griffin and J. Lim, “Signal estimation from modified short-time Fourier transform,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 32, no. 2, pp. 236–243, Apr. 1984, doi: 10.1109/TASSP.1984.1164317. [IEEE Xplore]
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