**Search Mathematics of the DFT**

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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.1}As 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.

- Introduction to Sampling
- Reconstruction from Samples--Pictorial Version
- The Sinc Function
- Reconstruction from Samples--The Math

- Aliasing of Sampled Signals

- Sampling Theorem
- Geometric Sequence Frequencies

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.

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