Assume $\mathbf{x}=[x_0, x_1,..., x_{N-1}]$ where $x_i \in \mathbb{R}$
Then (as I understand it) $\mathbf{X} = $ DFT$(x) = $ FFT$(x)$ with $\mathbf{X}=[X_0, X_1, ..., X_{N-1}]$ where $X_i \in \mathbb{C}$
Assuming $\mathbf{x}$ are signals over time we will get into the frequency domain with $\mathbf{X}$
From the definition of DFT we get that $X_k = \sum\limits_{n=0}^{N-1}{x_n\cdot e^{-i\frac{2 \pi k n}{N}}}$
According to what I've read is that subscript $k$ for $X_k$ is related to a specific frequency term. But I just don't see exactly what frequency it is and how... I mean the DFT value for $k$ is really a dot product between $\mathbf{x}$ and a complex exponential vector with different frequencies. How does this end up in a single frequency value?
The main question I have is given only $\mathbf{X}$, N and the rate of the samples $f_s$ what is the frequency term for $\mathbf{X}_k$?
Secondary how can I come to this conclusion (algebraically) given the DFT definition?
Any links/help appreciated, it is not homework...
Edit:
The "rule" for FFT is that you say that the frequency term for index $k$ is $$f_k = \frac{f_s}{2} \cdot \frac{k}{N}, k = {0, 1, ..., N-1}$$ Yet I cannot see how to deduce it algebraically...
$X_k$ is a complex number so it can be rewritten as: $$X_k = r \cdot e^{i\theta}$$ $$ln(X_k) = ln(r \cdot e^{i\theta}) = ln(r) + i \theta$$ where we can set $\theta = \phi + j\cdot 2\pi, j \in \mathbb{Z}$ due to definition of $e^{i\theta}$, so we get $$ln(X_k) = ln(r) + i \cdot (\phi+2\pi j)$$ We know that $$r=|X_k|=\sqrt{re(X_k)^2+im(X_k)^2}$$ $$\theta=arg(X_k)=atan2(im(X_k), re(X_k)) \iff \phi + 2\pi j = arg(X_k)$$ $$ln(X_k) = ln(|X_k|) + i \cdot(arg(X_k)+2\pi j)$$
But I still don't see how to the frequency term for $X_k$ gets specified.