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I had a question regarding multi-taper estimated coherence. It appears when the number of tapers is set to 1, all coherency values, per frequency, are always 1. It's obvious solving for autospectra $S_{xx}$,$S_{yy}$ and cross spectra $S_{xy}$ between two complex numbers produces this with a single taper, using $C_{xy}^2=\frac{|S_{xy}|^2}{S_{xx}S_{yy}}$. And common matlab functions also reveal this applied to real data. But I struggle to undestand intuitively why having only a single taper completely botches the estimate of coherence, such that all values are one, when expanding the number of tapers gives what could be thought of as a real estimate of the underlying values. Why does the orthogonal basis of tapers have to have $K\ge2$ to obtain a sensible values? Are there any analogies or physical metaphors as to why that is?

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For any estimate of the coherence, the cross spectra and spectra need to be averaged before forming the estimate of the coherence. If there is no averaging, the squared coherence will always be 1. You can see this by plugging in a single Fourier coefficient for X and Y: $C_{xy}^2=(\alpha_x\alpha_y)^2/(\alpha_x^2\alpha_y^2)=1$. If you instead use a sum or average of two Fourier coefficients for X and Y, the answer can be less than one.

In multitaper spectral analysis, this averaging is done by averaging spectral estimates made using the same time series but different taper windows. If you only use one taper window, there is no averaging being done.

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