In ordinary Fourier series/transform of a continuous signal $f(t)$, fourier frequencies $\omega$ of series/transforms can be any of $\mathbb{C}$, not just $\mathbb{Z}$.

But why is it the case that discrete-time Fourier series and transforms have $N$ frequenicies, defined by the number of samples?


They're just different kinds of transforms, it depends on the kind of function you want to transform.

For a function defined on a compact interval $[a,b]$ (or a periodic function), you have a correspondence with Fourier series, indexed by $k \in \mathbb Z$.

For a function defined on the entire line, you have a correspondence via the Fourier transform, here the transform variable $\xi$ is defined on the line as well.

For sequences of finite length (a finite number of samples), you have correspondence to other sequences of finite length via the discrete Fourier transform.

For sequences of infinite length (an infinite number of samples), you have the discrete-time Fourier transform yielding a periodic function (i.e. this is just the Fourier series correspondence in reverse)


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