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.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
[50].
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

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