## Sampling Theory

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

(B.60) |

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

*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*.

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