# 2D Fourier transform of an element-wise product of two matrices

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 $$\mathcal{F}\{B\}$$.

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

• 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?
– Peter K.
Jul 7, 2023 at 15:01
• @PeterK. Yes, that helps, but still figuring out which way will be clever. Thanks a lot. Jul 7, 2023 at 17:23
• @Jazzmaniac Could you explain about $U$ you used in your answers? Jul 7, 2023 at 17:36
• $U$ is the unitary matrix of the discrete Fourier transform. Jul 7, 2023 at 17:38

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