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Since the coefficient of one frequency could be seen the length of the signal projects on that frequency basis.

And the length is $$|a|\cos\theta= \frac{a\cdot b}{|b|} $$

If I divide the coefficient by $|a|$, how to interpret the correlation cosine angle from frequency domain?

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Frequency domain data is complex valued. In $\mathcal{C^N}$ $\mathbf{a^H}\mathbf{b}$ is a complex number in general, so $$ \frac{\mathbf{a^H}\mathbf{b}}{\mid \mathbf{a} \mid \mid \mathbf{b} \mid} $$ Is bound by magnitude of 1, like a cosine but unlike the cosine isn't in general real. \

The spectral coherence has a similar form $$ \gamma=\frac{S_{xy}(f)}{\mid S_{xx}(f) \mid \mid S_{yy}(f) \mid} $$ where the quantities are statistical expectations of stationary ergodic processes. The numerator is Complex in general.

The answer in this particular frequency domain correlation metric a correlation cosine is not straightforward. If you have some other context where the spectral quantities are strictly real, the answer may lay closer to the idea of your original question.

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If your signal is a single pure tone and your basis is a single pure tone of the same frequency, then the "angle" directly corresponds to the phase angle between them. When they are totally in phase, the angle will be zero. When they are totally out of phase, the angle will be $\pi$. If the angle is $\pi/2$ then the signal is orthogonal to the basis.

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  • $\begingroup$ Is it the same meaning as "coherence"? But the norm |a| (input) in time domain is not the same as norm in frequency domain? $\endgroup$ – Yui Nov 1 '18 at 1:13

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