Understanding fourier transform normalization for continuous vs discrete

Question

I intuitively understand what the DFT is doing and what the equation means. We are essentially adding up all the points of the input after projection onto the unit circle and then taking an average (dividing by 1/N). I note that this division of 1/N is often done in the IDFT (inverse).

What I don't understand is why when we consider the continuous fourier transform (integrated from -infinity to infinity), I notice there is no average. So in essence, this is just a sum.

Why do we need this 1/N factor in the DFT equation, but it omitted from both the forward and inverse continuous equations? Without it, isn't that just a sum? How mathematically is this equivalent to an average if we aren't dividing?

So it seems there is something weird about why scaling is necessary in DFT but not in the continuous case.

For example, equations 1 and 2 on the Wikipedia page about fourier transform (under definitions), shown the continuous transform, but there is no division by number of points. But isn't that just a sum?

Further, while I understand the 1/N in the DFT case, it isn't clear why we do it only on the inverse transform and why things are still correct in the forward transform without it.

Answer 1

There is no convention to scaling a discrete or continuous Fourier transform. Different scaling factors are used for different purposes.

For example, when interested in the amplitude of a particular frequency component, typically you would scale by the length of the DFT ($N$ in your example). As a matter of fact, the correct scaling for this purpose is to scale by the sum of the samples of the window used to get the segment. In the case of a rectangular window, that’s $N$.

In the case of an impulse response and its DFT, the frequency response, you’d likely be interested in the gain at a particular frequency, in which case you wouldn’t scale the DFT.

The reason scaling is applied in the definitions of both forward and inverse transforms, whether continuous or discrete, is to conserve symmetry between them, so that applying both simultaneously to a function gives you the same function back. In that sense, the scaling factor can be $1/2\pi$ ($1/N$ in the discrete case) for the forward and $1$ for the inverse, or $1/\sqrt{2\pi} (1/\sqrt{N}$ in the discrete case) for both. Or $100$ and $1/(100N)$ if you really felt like it.

Answer 2

That factor of $\frac1N$ can be split up and distributed between the forward DFT and inverse DFT any which way you want as long as the two scaling factors multiply to be $\frac1N$.

It has to do with the fundamental derivation of the DFT.

Maybe this answer will help.