Issues with an Inverse Fourier Transform

I have an algorithm that I am trying to reverse. The initial algorithm involves a Fast Fourier Transform of a matrix I call d:

for m = 1:f

    D(:,m) = fft(d(:,m)/sqrt(J));

end

After stepping through the rest of the initial algorithm and running through the reverse of it, I arrive to the point where I need to Inverse Fast Fourier Transform the D matrix above that is renamed below to D_ROI:

for m = 1:f

    % Inverse fft to get to weighted measured data
    d_ROI(:,m) = (ifft(D_ROI(:,m)));

end

Essentially, I need d_ROI to equal d to continue on through the reverse of my algorithm to prove that it works. It seems as simple as the ifft and a normalization factor, but for some reason, it's not. I've attached the code with the input .csv file.

Answer 1

[Matt J]

In your forward processing you have applied a scale factor of 1/sqrt(J). In your inversion, you therefore need to apply 1/(1/sqrt(J)).

[Sean Raffetto]

Yeah, I knew that I needed the scaling factor there however it produces the incorrect data still. This is where I began to not understand what the issue was.

[Matt J]

Well, your actual code from your attachment is as below. I see no resemblance at all between the scaling 1/ TH{m,n}(j,1) in the forward direction and the reverse direction 1/v(j). Also the scaling is j-dependent unlike in your posted question where it is a global constant.

% Instead of ifft to get to h, we want to go from h to get to the
% point-by-point fft.
for m = 1:f
    for n = 1:N

          TTH_ROI{m,n} = (fft(ttH_2{m,n}));

          for j = 1:J

              % Gets back to weighted measured data
              D_ROI(j,m) = TTH_ROI{m,n}(j,1) / TH{m,n}(j,1);

          end
      end  
  end

for m = 1:f

      % Inverse fft to get to weighted measured data
      d_ROI(:,m) = (ifft(D_ROI(:,m)));

end

% Remove the wighting factor to become a pattern again
for m = 1:f
    for j = 1:J

          MeasuredH_ROI(j,m) = d_ROI(j,m) / v(j);

      end
  end

[Sean Raffetto]

So are you saying I need to scale those values as well in the reverse part of the algorithm? The forward part of the algorithm (i.e. the forward algorithm ends at Line 227 where I solve for gH and the reverse beings at ttH_2 Line 230) I have left untouched because I proved that gH is correct in the commented-out area.

[Matt J]

Anything you do in the forward part, you must undo in the "reverse".