Scaling Problem with IFFT after operating in the Fourier domain.

Hi all,

I've encountered a scaling problem when doing the IFFT.

In this code, the DFT a slope matirx of a surface function is used to reconstruct the function. A slew of operations are required in the Fourier domain before IDFT to get reconstructed surface matirx.

However, the resulting reconstructed surface is scaled by a factor. I am unable to determine any scaling standard that is suitable.

Here, the code in 1D is as follows:

%----Determination of Sampling period/space, Length of matrix etc.-------%
dX = 0.64;                    % Sampling distance/period -mm
L = 25;                       %Physical Length of Specimen -mm
M =floor(L/dX);              % Length of matrix
x = (0:M)*dX;                % space vector

N = size(x,2);               % vector size of x/phi
df=1/(dX*N);                 % frequency
q=0:M;                       % x/phi Matrix Coordinahtes (starting frm 0)

%-------Surface function, slope matrix and FFTs------------------%
phi=sin(x);                 % given surface function, phi - 1D
Fphi=fft(phi);              % DFT of phi

[Sx]=gradient(phi,dX);       % Slope matrix
FSx=(fft(Sx));              % DFT of slope

%----Operations to form IDFT of surface (in Fourier domain)------%
phiup= ((exp(1i*-2*pi.*q./N)-1).*FSx );
phido=1./(2*((sin(pi.*q./N)).^2 ));
phifi=phiup.*phido;
phifi(1)=0;                       %eliminates infinite/NaN value

%Make matrix conjugate symmetric%
for k=2:floor(N/2)

         phifi(N-k+2)= real(phifi(k))-1i*imag(phifi(k));

end

%------IDFT,final reconstructed matrix--------%
phiRe=(ifft(phifi));

%----------------PLOTS-------------------  %
figure;
subplot(2,2,1);plot(x,real(phiRe),'r',x,phi,'b'); %8
title('Reconstructed phase, comparison'); 
subplot(2,2,2);plot(x,abs(phifi),'r',x,abs(Fphi),'b');
title('alpha/DFT phase (abs), comparison');
subplot(2,2,3);plot(x,real(phifi),'r',x,real(Fphi),'b'); %7
title('alpha/DFT phase (real), comparison');
subplot(2,2,4);plot(x,imag(phifi),'r',x ,imag(Fphi),'b');
title('alpha/DFT phase (imag), comparison');legend('Reconstructed','Actual');

The resultant plot gives phiRe with a scaling of around 3. However, I need to establish a standard for Scaling. How should i scale it correctly?

I need to understand this well as it will be applied for a 2D case too. THANKS!