Sampling is a process of converting a signal (for example, a function of continuous time or space) into a sequence of values (a function … See more

When there is no overlap of the copies (also known as "images") of $${\displaystyle X(f)}$$, the $${\displaystyle k=0}$$ term of Eq.1 can be recovered by the … See more

The sampling theorem is usually formulated for functions of a single variable. Consequently, the theorem is directly applicable to time-dependent signals and is normally formulated in that context. However, the sampling theorem can be extended in … See more

The sampling theory of Shannon can be generalized for the case of nonuniform sampling, that is, samples not taken equally spaced in time. The Shannon sampling theory for … See more

When $${\displaystyle x(t)}$$ is a function with a Fourier transform $${\displaystyle X(f)}$$:
$${\displaystyle X(f)\ \triangleq \ \int _{-\infty }^{\infty }x(t)\ e^{-i2\pi ft}\ {\rm {d}}t,}$$
Then the samples, See more

Poisson shows that the Fourier series in Eq.1 produces the periodic summation of $${\displaystyle X(f)}$$, regardless of $${\displaystyle f_{s}}$$ See more

As discussed by Shannon:
A similar result is true if the band does not start at zero frequency but at some higher value, and can be proved by a linear translation (corresponding physically to single-sideband modulation) of the zero-frequency case. In … See more