Setting (the FFT hop size) gives
where (harmonics of the frame rate).
Looking across the top of Fig.8.16, for the case of input signal we have
Looking across the bottom of the figure, for the case of input signal
we have the output signal
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 , [264, Appendix G].
The continuous-time PSF is derived in §B.15.
Frequency-Domain COLA Constraints
Summary of Overlap-Add FFT Processing