Sinusoidal Synthesis of audio

Question

I am working on a sinusoidal encoder & decoder to compress .wav files. For now I am trying to regenerate the audio by combining all the frequencies of the original .wav file. Here is the code I am using:

[x fs] = wavread('clip.wav');
x  = mean(x,2);
NFFT = 1024;
hop = NFFT/2;
win = hann(NFFT);

[S F]= spectrogram(x, win, hop, NFFT, fs);
freq = linspace(0, fs/2, hop + 1);
numFrames = size(S,2);
mag = abs(S);
phase = angle(S);

%Decoding
output = zeros(size(x));
t = 0:(1/fs):(((NFFT-1)/fs));
sumOfCosine = zeros(1, NFFT );
frameStart = 1;
frameEnd = NFFT;

%f = frame
for f = 1:numFrames
    sumOfCosine = zeros(1, NFFT );

    %k = sinusoids in each frame
    for k = 1:hop+1
       sumOfCosine =  sumOfCosine + (mag(k,f)*cos(2*pi*F(k)*t + phase(k,f)));
    end

 sumOfCosine = sumOfCosine.*win';
 a = hop;
 % Normalize the window
 sumOfCosine = (sumOfCosine./a);
 sumOfCosine = sumOfCosine';
 output(frameStart:frameEnd) = ((output(frameStart:frameEnd)+ sumOfCosine));
 frameStart = frameStart + (hop);
 frameEnd = frameEnd + (hop);

end

When I play play output, there is still some noise in it. I tried comparing the original matrix x multiplied by the window, win to the output frame by frame, and I can see that output represented by green doesn't completely overlap with the original windowed signal. I am pasting the images for the first three frames below.

As it can be seen there is some error due to the overlapping of windows maybe? I'd appreciate help in fixing this and make the output sound cleaner. Thanks!

Answer 1

For the phases of sinusoids within the frame overlaps to sum constructively, either t has to be adjusted to be frame relative, or the fft phase result has to be adjusted to be relative to the same absolute time as t.