I have two signals $x(t)$ and $y(t)$ which I can sample at arbitrary $\Delta t$ and $N$. I am interested to the signals product time average $\langle x(t)y(t)\rangle_t$. In particular I want to understand when the time average is zero in term of the discrete Fourier coefficients, that is when the signals can be considered "orthogonal". For example, if $X(k)$ and $Y(k)$ are the discrete Fourier coefficients, thus the time average is well approximated by: $$ \langle x(t)y(t)\rangle_t\approx \frac{1}{N^2} \sum_{k=0}^{N-1}X(k)Y^*(k) $$ Thus vanishing cross spectrum represents a sufficient condition in order to have $\langle x(t)y(t)\rangle_t= 0$. My question is:
- when this also a necessary condition?
Thank you very much