# Mathematics of the DFTSampling Theory

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# Sampling Theory

In this appendix, sampling theory is derived as an application of the DTFT and the Fourier theorems developed in Appendix C. First, we must derive a formula for aliasing due to uniformly sampling a continuous-time signal. Next, the sampling theorem is proved. The sampling theorem provides that a properly bandlimited continuous-time signal can be sampled and reconstructed from its samples without error, in principle.

An early derivation of the sampling theorem is often cited as a 1928 paper by Harold Nyquist, and Claude Shannon is credited with reviving interest in the sampling theorem after World War II when computers became public.D.1As a result, the sampling theorem is often called ``Nyquist's sampling theorem,'' ``Shannon's sampling theorem,'' or the like. Also, the sampling rate has been called the Nyquist rate in honor of Nyquist's contributions [48]. In the author's experience, however, modern usage of the term ``Nyquist rate'' refers instead to half the sampling rate. To resolve this clash between historical and current usage, the term Nyquist limit will always mean half the sampling rate in this book series, and the term ``Nyquist rate'' will not be used at all.

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About the Author: 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.