I am trying to compute a single coefficient of the DFT of a linearly ramping sequence, $x[n]=an$ where $a$ is a constant that changes from sequence to seqeunce. I have looked at loads of DFT transform property and pairs tables, but this is a simple case I have not been able to find an analytical expression for. I don't want to have to compute the entire DFT of the sequence just to extract a single coefficient.
My first approach was to notice that multiplication by $n$ in the time domain is like differentiation in the $k$-domain. However, since the DFT of $a$ is an impulse function at $k=0$, I haven't found a way to come up with an analytical expression to take the derivative.
In my application, the number of samples in the sequence, $N$, and the coefficient of interest, $k$, will frequently change. It's just such a simple case that, given $a$, there must be a way for me to predict the value of the DFT of the sequence at a particular value of $k$ without actually doing the transform. Is the only way to do this to numerically create a look-up table of all possible $N$ and $k$ of interest, and then scale by $a$?