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 $\texttt{coh}_{xy}[k] = \texttt{coh}_{xy}[N-k]$ ?


1 Answer 1



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).


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