I have discrete signal, $x[n]$ that is imaginary and odd. I want to prove that its coefficients are real and odd. First of all, is there an intuitive understanding by this? Second, I know the following pieces of information but I'm not sure how exactly to piece it together to prove it.
Since $x[n]$ is odd, we know that $x[n] = −x[−n]$. Also, $a_k = −a_{−k}$. Thus, $a_k$ is odd in $k$. Since x[n] is imaginary, $x[n] = −x^∗[n]$. And so we have $a_k = −a^*_{−k}$. How do I piece all this together to prove that the coefficients themselves are real and odd? I'm not sure what the end product should exactly look like.
You're almost there. You've found that $a_k=-a_{-k}$ (where negative indices are taken modulo $N$), and $a_k=-a^*_{-k}$. From this it follows that $a_{-k}=a^*_{-k}$, which can only be the case if the coefficients $a_k$ are real-valued. Combining this with $a_k=-a_{-k}$ gives the result that they are real-valued and odd.