The dual of the Poisson Summation Formula is the continuous-time aliasing theorem, which lies at the foundation of elementary sampling theory [264, Appendix G]. If denotes a continuous-time signal, its sampled version , , is associated with the continuous-time signal
where denotes the (fixed) sampling interval in seconds. The sampled signal values are thus treated mathematically as coefficients of impulses at the sampling instants. Taking the Fourier transform gives
where denotes the sampling rate in radians per second. Note that is periodic with period . We see that if is bandlimited to less than radians per second, i.e., if for all , then only the term will be nonzero in the summation over , and this means there is no aliasing. The terms for are all aliasing terms.
The Uncertainty Principle
Poisson Summation Formula