Hi Benjamin,
Here's an approach for the 'valid' convolution.
Problem parameters:
x = 1:7; nx = numel(x);
h = 1:3; nh = numel(h);
nyf = nx + nh - 1;
nyv = nx - nh + 1;
Define the window function such that conv(x,h,'valid) = w*conv(x,h,'full') with appropriate zero padding as will be shown
d = nh - 1;
w = [zeros(1,d) ones(1,nyv) zeros(1,nyf - nyv - d)];
nw = numel(w);
Verify
yf = conv(x,h)
w.*yf
conv(x,h,'valid')
Now, just like we could compute yf as
yf = ifft(fft(x,nw).*fft(h,nw)) % same results as above
We can compute yv (the zero-padded form of the valid convolution) as
yv = ifft(cconv(fft(w),fft(x,nyf).*fft(h,nyf),nw)/nw) % same as above (*)
Verify the imaginary part is rounding error and get rid of it
max(abs(imag(yv)))
yv = real(yv)
From which conv(x,h,'valid') can be extracted with appropriate indexing.
If we want to recover x starting from fft(yv), it may be possible to invert (*) to get to x. But I couldn't see a way to do that. It seems clear that
fft(yv) = fft(w.*conv(x,h)) = fft(w.*ifft(fft(x,nw).*fft(h,nw)))
but I don't see a way to isolate an expression for fft(x), which could then be ifft'd.
Will be interested to see if anyone comes up with a solution.
