# Is the coherence between two signals defined at frequencies above the Nyquist frequency?

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

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.