DFT k values and their dependence on interval length

[SchroedingersLion]

TL;DR

Question to DFTs

Greetings!

I am getting started with Python's Fast Fourier Transform, and I noticed a missunderstanding on my part.
I always thought that the k's in the expression of the DFT resemble actual wavenumbers $\frac{2\pi}{\lambda}$ that form the waves the signal is composed of.
But the actual meaning seems to depend on the length of the x-interval that holds the signal.

Wiki:
$$
X_{k} = \sum_{n=0}^{N-1}x_n e^{-2\pi i k \frac n N} ~~~~(1) \\

x_n = \frac 1 N \sum_{k=0}^{N-1}X_k e^{2\pi i k \frac n N}~~~~ (2),
$$

where the $x_n$ are the $N$ signal values that are equidistantly positioned at the points $\frac n N$, i.e. in the interval $[0, 1]$.
The expressions are $N-$periodic in $k$, so it makes no difference whether one uses $k\in [0,N-1]$ or $k\in [-\frac N 2 +1, \frac N 2]$ (suppose $N$ is even).

In this representation, however, the $k$ are no wavenumbers. Rather, they are wavenumbers divided by $2\pi$, i.e. $k=\frac 1 \lambda$.In another derivation of a book, the points $x_n$ are equidistantly spaced across $[0, 2\pi]$.
(1) then becomes
$$ X_{k} = \sum_{n=0}^{N-1}x_n e^{-i k x_n}. $$

Here, however, it is $k=\frac {2\pi} {\lambda}$.
Now I was wondering, is it generally true that the $k$ are given by $\frac L \lambda$ with $L$ as the length of the $x-$interval?

It somehow makes sense, as the sampling interval does not enter anywhere else in these formulas.SL

 

[mathman]

k is simply an index. How it is defined depends entirely on the context, as you noticed.

 

[SchroedingersLion]

Alright, I think I understand now, thanks!

On another note, I wonder whether my intuitive understanding is correct:
Given my grid of data points, I know which wavenumbers k can be distinguished on the grid.
I want to decompose my signal into a sum over these "eigen-wavenumbers".
If I imagine the complex plane and discrete lines that go from 0 to $e^{ik}$ for the different $k$, then $X_k$ is simply a point on the line to the corresponding $k$, correct?

I then try to imagine that eq. (1) projects each data point onto that line and sums over these projections in order to find the total $X_k$. Is this idea correct? If so, I don't really see how the term $x_n e^{-2\pi ik\frac n N}$ performs that task. If I was correct, I would simply write $x_n e^{ik}$. So my view seems to be a bit oversimplified.

 

[mathman]

I can't comment any further, since I have never worked with FFT. Try https://math.stackexchange.com/

 

[marcusl]

k is not wavenumber, it’s simply an integer index, as is n. All context is lost once your data are sampled and digitized. Context about units (time, position, frequency or other variables) must be tracked with external bookkeeping.