function y = istft(spec,parameter)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Name: istft
% Date: 03-2014
% Programmer: Jonathan Driedger
% http://www.audiolabs-erlangen.de/resources/MIR/TSMtoolbox/
%
% Computing the 'inverse' of the stft according to the paper "Signal
% Estimation from Modified Short-Time Fourier Transform" by Griffin and
% Lim.
%
% Input: spec a complex spectrogram generated by stft.
% parameter.
% synHop hop size of the synthesis window.
% win the synthesis window.
% zeroPad number of zeros that were padded to the
% window to increase the fft size and therefore
% the frequency resolution.
% numOfIter number of iterations the algorithm should
% perform to adapt the phase.
% origSigLen original length of the audio signal such that
% the output can be trimmed accordingly.
% restoreEnergy when windowing the synthesis frames, there is a
% potential for some energy loss. This option
% will rescale every windowed synthesis frame to
% compensate for this energy leakage.
% fftShift in case the stft was computed with an fftShift,
% setting this parameter to 1 will compensate for
% that by applying the same shifting operation
% again to each frame after the ifft.
%
% Output: y the time-domain signal.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Reference:
% If you use the 'TSM toolbox' please refer to:
% [DM14] Jonathan Driedger, Meinard Mueller
% TSM Toolbox: MATLAB Implementations of Time-Scale Modification
% Algorithms
% Proceedings of the 17th International Conference on Digital Audio
% Effects, Erlangen, Germany, 2014.
%
% License:
% This file is part of 'TSM toolbox'.
%
% MIT License
%
% Copyright (c) 2021 Jonathan Driedger, Meinard Mueller, International Audio
% Laboratories Erlangen, Germany.
%
% We thank the German Research Foundation (DFG) for various research grants
% that allow us for conducting fundamental research in music processing.
% The International Audio Laboratories Erlangen are a joint institution of
% the Friedrich-Alexander-Universitaet Erlangen-Nuernberg (FAU) and
% Fraunhofer Institute for Integrated Circuits IIS.
%
% Permission is hereby granted, free of charge, to any person obtaining a
% copy of this software and associated documentation files (the
% "Software"), to deal in the Software without restriction, including
% without limitation the rights to use, copy, modify, merge, publish,
% distribute, sublicense, and/or sell copies of the Software, and to permit
% persons to whom the Software is furnished to do so, subject to the
% following conditions:
%
% The above copyright notice and this permission notice shall be included
% in all copies or substantial portions of the Software.
%
% THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
% OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
% MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
% IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
% CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
% TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
% SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% check parameters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if nargin<2
parameter=[];
end
if ~isfield(parameter,'synHop')
parameter.synHop = 2048;
end
if ~isfield(parameter,'win')
parameter.win = win(4096,2); % hann window
end
if ~isfield(parameter,'zeroPad')
parameter.zeroPad = 0;
end
if ~isfield(parameter,'numOfIter')
parameter.numOfIter = 1;
end
if ~isfield(parameter,'origSigLen')
parameter.origSigLen = -1; % no trimming
end
if ~isfield(parameter,'restoreEnergy')
parameter.restoreEnergy = 0;
end
if ~isfield(parameter,'fftShift')
parameter.fftShift = 0;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% some pre calculations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
numOfFrames = size(spec,2);
numOfIter = parameter.numOfIter;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% audio calculation
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% first iteration
Yi = spec;
yi = LSEE_MSTFT(Yi,parameter);
% remaining iterations
parStft.win = parameter.win;
parStft.numOfFrames = numOfFrames;
parStft.zeroPad = parameter.zeroPad;
parStft.anaHop = parameter.synHop;
for j = 2 : numOfIter
Yi = abs(spec) .* exp(1j*angle(stft(yi,parStft)));
yi = LSEE_MSTFT(Yi,parameter);
end
y = yi;
% if the original Length of the signal is known, also remove the zero
% padding at the end
if parameter.origSigLen > 0
y = y(1:parameter.origSigLen);
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% the Griffin Lim procedure
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function x = LSEE_MSTFT(X,parameter)
% some pre calculations
w = parameter.win;
w = w(:);
zp = parameter.zeroPad;
w = [zeros(floor(zp/2),1);w;zeros(floor(zp/2),1)];
winLen = length(w);
winLenHalf = round(winLen/2);
synHop = parameter.synHop;
numOfFrames = size(X,2);
winPos = (0:numOfFrames-1) * synHop + 1;
signalLength = winPos(end) + winLen - 1;
x = zeros(signalLength,1); % resynthesized signal
ow = zeros(signalLength,1); % sum of the overlapping windows
for i = 1 : numOfFrames
currSpec = X(:,i);
% add the conjugate complex symmetric upper half of the spectrum
Xi = [currSpec;conj(currSpec(end-1:-1:2))];
xi = real(ifft(Xi));
if parameter.fftShift == 1
xi = fftshift(xi);
end
xiw = xi .* w;
if parameter.restoreEnergy == 1
xiEnergy = sum(abs(xi));
xiwEnergy = sum(abs(xiw));
xiw = xiw * (xiEnergy/(xiwEnergy+eps));
end
x(winPos(i):winPos(i)+winLen-1) = ...
x(winPos(i):winPos(i)+winLen-1) + xiw;
ow(winPos(i):winPos(i)+winLen-1) = ...
ow(winPos(i):winPos(i)+winLen-1) + w.^2;
end
ow(ow<10^-3) = 1; % avoid potential division by zero
x = x ./ ow;
% knowing the zeropads that were added in the stft computation, we can
% remove them again now. But since we do not know exactly how many
% zeros were padded at the end of the signal, it is only safe to remove
% winLenHalf zeros.
x = x(winLenHalf+1:end-winLenHalf);
end