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

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*.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University