Inverse the accidental add of time-shifted version of my audio signal

Hello,

I have an audio signal (e.g. z(n)) which is the add of my original signal (e.g. x(n)) and a time shifted version of it (x(n-n0)) so z(n)=x(n)+x(n-n0). Can I use the properties of timeshifting in frequency domain in order to get my original signal back (get x(n) from z(n))?

I tried to change the code I found here and I tried the following code:

[data,fs] = audioread('z_signal.mp3');
no_frame = 1;
datalength = length(data);
N = floor(datalength/no_frame);  %Framesize
temp = 0;
for i = 1 : no_frame
    frames(i,:) = data(temp + 1 : temp + N,1);
    temp = temp + N;
end
i=1;               %for 1st frame
g=+1.6; %the number of seconds i suppose is the time-shift so n0=g*fs
t=g*fs; %number of sample shift
yi=frames(i,:);
yp(i,:)=fft(yi);
y(i,:) = exp(-1i*2*pi/N*(0:N-1)*t).*yp(i,:);
rslt(i,:)=ifft(y(i,:),'symmetric');

Do you have any ideas what I'm missing?

Thank you in advance

Answer 1

[Suraj Kumar]

Hi Maria,

I understand you're trying to recover your original audio signal ( x(n) ) from the combined signal ( z(n) = x(n) + x(n-n_0) ). The solution involves using the Fourier Transform, which translates time shifts into phase shifts within the frequency domain.You can refer to the below mentioned steps:

1. Initially we can convert the time-domain signal ( z(n) ) into the frequency domain using "fft" function in MATLAB.

[data, fs] = audioread('z_signal.mp3');
datalength = length(data);
g = 1.6;
n0 = round(g * fs); 
Z = fft(data);

2. Then we can observe that the time shift corresponds to a phase shift. We can use this to express the relationship between ( Z(f) ) and ( X(f) and solve for ( X(f) ) using the known relationship

N = length(Z);
f = (0:N-1)*(fs/N);
H = exp(-1i * 2 * pi * f * n0 / fs);
X = Z ./ (1 + H);
epsilon = 1e-10;
X(abs(1 + H) < epsilon) = 0;

3. Finally we can convert the isolated ( X(f) ) back to the time domain using "ifft" function in MATLAB.

x_recovered = ifft(X, 'symmetric');
x_recovered = x_recovered / max(abs(x_recovered));
audiowrite('recovered_signal.wav', x_recovered, fs);
sound(x_recovered, fs);

For more insights on the "fft" and "ifft" functions in MATLAB, you can refer to the following documentation links:

https://www.mathworks.com/help/matlab/ref/fft.html

https://www.mathworks.com/help/matlab/ref/ifft.html

Hope this helps solving your query!