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Here's how I like to think about the expression $$ \displaystyle \frac{1}{D} \sum_{k=0}^{D-1} e^{j 2 \pi k m / D}. $$ For any value of $D$, the first term is 1, and all of the terms lie on the unit circle. Assuming $D > 1$, each term after the first is found by rotating the previous term through an angle of $2 \pi m / D$ radians. If $m$ is not a multiple ...


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$\Omega$ and $\omega$ are frequencies (in radians). Usually, $\omega$ is used, but if you need to deal with continuous-time as well as discrete-time systems, it's common to use one for discrete time and one for continuous time in order to distinguish the two. There is no real standard as to which one is which. I've seen $\Omega$ used for continuous time as ...


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Pitch is a human psychoacoustic perception phenomena, so an absolute ground truth might not be well defined enough to “solve”, except in some statistical sense, especially for a string bass, whose waveforms can evolve over time in both frequency and inharmonic as well as harmonic spectral composition. A pure perfect periodicity for many natural sounds does ...


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BOTTOM LINE UP FRONT: I think the exponential decay growth in $\left<|x(t)|^2\right>$ can be shown in the frequency domain only if the "boundary terms" are nonzero when we compute the Fourier transform of $dx(t)$ from the original SDE. I provide only a start in the work below. Since these processes seem like they could possibly be complex-...


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The power spectral density (PSD) is a natural measure of the signal's power content with respect to frequency. A central part of non-parametric signal processing is to provide a "best" estimate of the "true" PSD from knowing only one or some "realizations" with finite length. By taking into account the influence of stationary ...


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