### 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):

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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Frequency-Domain COLA Constraints

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Summary of Overlap-Add FFT Processing