Sampling Theory

The dual of the Poisson Summation Formula is the continuous-time aliasing theorem, which lies at the foundation of elementary sampling theory [243, 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.

---

Order a Hardcopy of Spectral Audio Signal Processing

Previous: Poisson Summation Formula
Next: The Uncertainty Principle

---

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.