# Is there other basis possible for DFT?

As I understand, the DFT of a signal $$x$$ is a representation of this signal in the basis

$$\{ e^{j2\pi kn/N} \}_{k = 0, 1, \dots, N-1}$$

Is it possible to form a base of such discrete complex exponentials but with different frequencies ? ($$\{ e^{j2\pi f_i n} \}$$)

Why did we choose these $$\frac{k}{N}$$ frequencies ?

In fact, my question is more about why we evaluate the DFT at these specific frequencies and not others.

• The DFT basis consists of all possible complex exponentials which have period N. Apr 10, 2022 at 16:42
• IIRC the frequencies you choose will not be orthogonal. Apr 11, 2022 at 11:08

$$k/N$$ with $$N$$ bases is the only basis which is all of:
1. Orthogonal & invertible. Means we don't lose any information. Invertibility can be seen to follow from the DFT being a square (N x N) matrix of rank N (orthogonal = all columns independent), so we can solve $$Ax = y$$ with $$x = A^{-1}y$$.
2. $$N$$-periodic. Every basis will repeat after $$N$$ uniform samples. This makes the DFT a uniform & $$N$$-periodic sampling of one period of DTFT which yields many useful properties, and enables circular convolution.
3. Minimal. Maps out the greatest continuous-time bandwidth per unit sample, meaning it's max information-dense. This follows the sampling theorem, which demands a minimum of $$N$$ samples to capture a bandwidth up to (but not including) $$N/2$$ (in physical terms, sampling at rate of $$f_s$$ to capture $$f_s/2$$). Other bases will take $$N$$ or more samples (if succeeding at all, see example).
$$2k/N$$ with $$N$$ bases is orthogonal and $$N$$-periodic, but not minimal, nor invertible. In fact it's not invertible for any number of bases, ironically due to periodicity. $$Gk/N$$, where $$G$$ is non-integer, is none of those things with $$N$$ bases. I'll also informally add,
• @AlexTP Fair, wouldn't know from top of my head though. Think it could be shown by considering every possible $N$-base cisoid basis that's orthogonal and $N$-periodic, then checking which also meets sampling theorem criteria, so a sort of system of equations. Apr 11, 2022 at 17:24