Greetings.

I seek advise regarding the following sentence as it appears pertaining to the image that appears below:

"[In Discrete Fourier Transform] it is common to express the transform as G(k) instead of G(k/N) in order to simplify the notation, but one should keep in mind that the frequency is the normalized k/N and not the integer k."



I would appreciate assistance with an explanation of what 'normalizing' refers to, and why it is required.

Look forward to a prompt response.

Best regards,
wirefree

The transform on left column is for n=1,2,3...,N. This number is mostly used when you collect N number of information.

The other transform on the right is for time within a cycle. This usually go back and forth alternately as in sine waves or cosine waves. If you know the Period of a cycle, you can divide you time, t, as you deem appropriate.
Most often, this is used in changing a discrete function into a continuous one.

If my last post was not clear, I am simplifying it here by examples.

In trigonometry, we use standard trig functions by normalizing the radius =1 unit.

In vector analysis or linear algebra, we normalize a vector by dividing it with it magnitude, converting it into unit vector.

In your case, the transform in discrete form, you sum n from 1 to N before normalizing it. After you normalizing the discrete form, you sum it between zero to one period T. That's normalized.

It cannot be more clearer than the example I gave you.