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I wonder if there is any known formula to describe a 2D Fourier transform of an element-wise product, i.e., Hadamard product, of two matrices. Let $\odot$ is the Hadamard product operator, and there are matrices $A,B\in\mathbb{C}^{N\times N}$ in the spatial domain where $N$ is a known integer (so $A\odot B\in\mathbb{C}^{N\times N}$). Let $\mathcal{F}\{\cdot\}$ is a 2D discrete Fourier transform operator (transformation from the spatial domain to the frequency domain). I want to know if I can express the below in any other way, hopefully using the frequency domain of $A$ and $B$, i.e., $\mathcal{F}\{A\}$ and $\mathcal{F}\{B\}$.

$$\mathcal{F}\{A\odot B\}$$

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  • $\begingroup$ Interesting question. Does it help if the 2D discrete Fourier transform can be written as a matrix multiplying the Hadamard product? I wondered whether something like this might be useful? $\endgroup$
    – Peter K.
    Jul 7, 2023 at 15:01
  • $\begingroup$ @PeterK. Yes, that helps, but still figuring out which way will be clever. Thanks a lot. $\endgroup$
    – Junho
    Jul 7, 2023 at 17:23
  • $\begingroup$ @Jazzmaniac Could you explain about $U$ you used in your answers? $\endgroup$
    – Junho
    Jul 7, 2023 at 17:36
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    $\begingroup$ $U$ is the unitary matrix of the discrete Fourier transform. $\endgroup$
    – Jazzmaniac
    Jul 7, 2023 at 17:38

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Yes. It's the 2D circular convolution of the Fourier transform of each matrix. I.e. $\mathcal F \{A \odot B\} = \mathcal F\{A\} * \mathcal F\{B\}$.

This is basically the same property as with the 1D Fourier transform; it works because a 2D rectilinear Fourier transform is just the row-wise Fourier transform of the Fourier transforms of the columns (or the column-wise Fourier transform of the Fourier transforms of the rows).

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  • $\begingroup$ Thank you so much. I hoped there is a easier way to calculate or express to handle a problem that I am dealing with. It seems like I should stick to spatial domain due to the complexity of the circular convolution .. haha $\endgroup$
    – Junho
    Aug 8, 2023 at 19:54

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