### Poisson Summation Formula

Consider the summation of N complex sinusoids having frequencies uniformly spaced around the unit circle [264]:

where .

Setting (the FFT hop size) gives

 (9.26)

where (harmonics of the frame rate).

Let us now consider these equivalent signals as inputs to an LTI system, with an impulse response given by , and frequency response equal to .

Looking across the top of Fig.8.16, for the case of input signal we have

 (9.27)

Looking across the bottom of the figure, for the case of input signal

 (9.28)

we have the output signal

 (9.29)

This second form follows from the fact that complex sinusoids are eigenfunctions of linear systems--a basic result from linear systems theory [264,263].

Since the inputs were equal, the corresponding outputs must be equal too. This derives the Poisson Summation Formula (PSF):

 (9.30)

Note that the PSF is the Fourier dual of the sampling theorem [270], [264, Appendix G].

The continuous-time PSF is derived in §B.15.

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