I want to calculate $$f(l): = \frac{1}{{{N^2}}}\sum\limits_{{k_1},{k_2} = 0}^{N - 1} {{e^{2\pi ikl'/N}}H(k)}, $$ where $l=(l_1,l_2)\in\{0,\ldots,N-1\}^2$.
ifft2 in MATLAB can be used to calculate $f(l)$. Someone wrote that $$f(l)=\mbox{ifft2}(H)(l).$$
What is the meaning of $\mbox{ifft2}(H)(l)$? Here $H$ is a $N\times N$-martrix. iffft2 also gives a $N\times N$-martrix, let's say $\hat H$. Then does $\mbox{ifft2}(H)(l)$ mean that the element of $\hat H$ at $l$? Thanks.
This step is embedded in my big project. But I think there is no bug in other parts. I am not sure whether there is some problem with this ifft2. I did not find the original formulas of ifft2 in MATLAB.
Two points:
1) Do we agree on the right definition of a 2D DFT (Discrete Fourier Transform) either direct or inverse (which, in Matlab, cooreponf to fft2 and ifft2)?
It should be, with $k=(k_1,k_2)$ and $l=(l_1,l_2)$:
$$h(l): = \frac{1}{{{N^2}}}\sum\limits_{{k_1} = 0}^{N - 1}\sum\limits_{{k_2} = 0}^{N - 1} {{e^{2\pi ik(l.k)/N}}H(k)}$$
with $l.k$ = dot product, (what you denote $lk'$) i.e., developed:
$$h(l_1,l_2): = \frac{1}{{{N^2}}}\sum\limits_{{k_1} = 0}^{N - 1}\sum\limits_{{k_2} = 0}^{N - 1} {{e^{2\pi ik(l_1k_1+l_2k_2)/N}}H(k_1,k_2)}$$
2) My answer to the question is that $ifft2(H)$ being an $N \times N$ array, the only meaning that can be given to $ifft2(H)(l)$ is that $l$ is a certain index set that extracts something out of the array, for example a first column like in the following example: m=rand(3),I=[1,2,3],m(I)'.