2
$\begingroup$

Suppose I sample from two continuous signals, $x$ and $y$, with sampling rate $f_s=1000$ Hz yielding $N$ samples.

A plot using coherence from scipy.signal shows frequencies up to $f_s/2$. Given the DFT conjugate relationship, $X[N-k]=\overline{X[k]}$, the relationship between the power spectral densities and the DFTs, $X$ and $Y$, and the definition of coherence, is it then true that the coherence is actually defined at bin $N-k$?

If so, does $\texttt{coh}_{xy}[k] = \texttt{coh}_{xy}[N-k]$ ?

$\endgroup$

1 Answer 1

3
$\begingroup$

Yes.

Per the definition of coherence between two signals $x[n]$ and $y[n]$: $$\texttt{coh}_{xy}[k] = \frac{|P_{xy}[k]|^2}{P_x[k]P_y[k]}$$ where $P_{xy}[k]$ is the Cross Power Spectral Density between $x[n]$ and $y[n]$, and $P_x[k], P_y[k]$ the Auto Power Spectral Densities of $x[n]$ and $y[n]$ respectively.
Note that both $P_x[k]$ and $P_y[k]$ are real functions.

$P_{xy}[k]$, $P_x[k]$ and $P_y[k]$ are all two-sided spectra, so $\texttt{coh}_{xy}[k]$ is two-sided as well.

In practice however, you can always discard the redundant part (assuming both signals are real).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.