Comprehensive list of discrete Fourier transforms

Question

What is the state of known exact solutions for the discrete Fourier transform (DFT)?

Is there any good resource? Typically, the lists I find just have the geometric series, derivation of it, delta function and exponentials. Is this all there is?

Answer 1

Note that the $z$-transform for a causal signal $x[n]$ is given by: $$X(z)=\sum_{n=0}^{\infty}x[n]z^{-n}$$ where $z=e^{j\omega}$, and the DFT of a signal with finite (say, $N$) samples is derived from: $$X[k]=\sum_{n=0}^{N-1}x[n]\exp\left(-j\frac{2\pi n}N k\right)$$ So basically, the DFT is the $z$-transform evaluated at $z=e^{j\omega_k}$, where $\omega_k:=\frac{2\pi k}N$. i.e. $$X[k] = X(z) \bigg|_{z=e^{j\frac{2\pi}{N} k}}$$ You may also take a look at this question. Now if you are looking for a comprehensive table of DFTs, your search can be narrowed down by considering this fact and looking for a comprehensive table of $z$-transforms instead. (Also this one)