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From Wikipedia, the equation for the 1D DFT is

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From a separate source, the equation for the 2D DFT is

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Notice how the 2D DFT definition features a scaling term while the first definition does not. Is this just an artifact of where the scaling takes place between the DFT and inverse DFT? Does it matter which transform contains this scaling term?

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    $\begingroup$ No it does not matter as long as you are consistent and aware... $\endgroup$
    – Fat32
    Feb 17, 2021 at 20:27
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    $\begingroup$ I agree with @Fat32, but i would say that, of course, being off by a factor of $N$ or $\sqrt{N}$ does matter. So be aware of the functional form of theorems, including the circular convolution theorem, that go with the particular scaling definition of the DFT that you are using. It's not a big deal but the DFT form with $\frac{1}{\sqrt{N}}$ as the scaler in both the forward and inverse DFT has some nice preservation of power properties, but it has ugly scalers somewhere else. I just stick with the most conventional definition. $\endgroup$ Feb 17, 2021 at 20:49
  • $\begingroup$ @robertbristow-johnson yes that's an important point... The specific scale in the Modulation, the Convolution, and the Duality properties may depend on the convention being used... Mixing different conventions may yield wrong results... $\endgroup$
    – Fat32
    Feb 17, 2021 at 21:13

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It's mostly a matter of convention.

There are three options. $1/N$ for the inverse transform, $1/N$ for the forward transform and $1/\sqrt{N}$ for both. The first is the one most commonly used, the last one has the advantage that Parseval's theorem holds directly, i.e. $$\sum |x_n|^2 = \sum |X_k|^2$$

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  • $\begingroup$ Putting the normalization in the inverse transform makes it easier to do convolution through the FFT. $\endgroup$ Feb 19, 2021 at 2:25

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