Nyquist–Shannon sampling theorem Totally Explained

Everything about Nyquist Shannon Sampling Theorem totally explained

The Nyquist–Shannon sampling theorem is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal (for example, a function of continuous time or space) into a numeric sequence (a function of discrete time or space). The theorem states, in the original words of Shannon (where he uses "cps" for "cycles per second" instead of the modern unit hertz):
ť If a function f(t) contains no frequencies higher than W cps, it's completely determined by giving its ordinates at a series of points spaced 1/(2W) seconds apart.

In essence this means that an analog signal that has been digitized can be perfectly reconstructed if the sampling rate was 1/(2W) seconds, where W is the highest frequency in the original signal.
   More recent statements of the theorem are sometimes careful to exclude the equality condition; that is, the condition is if f(t) contains no frequencies higher than or equal to W; this condition is equivalent to Shannon's except when the function includes a steady sinusoidal component at exactly frequency W.
   The assumptions necessary to prove the theorem form a mathematical model that's only an idealization of any real-world situation. The conclusion, that perfect reconstruction is possible, is mathematically correct for the model, but only an approximation for actual signals and actual sampling techniques.
   The theorem also leads to a formula for reconstruction of the original signal.

Introduction

A signal or function is bandlimited if it contains no energy at frequencies higher than some bandlimit or bandwidth

B,

. A signal that's bandlimited is constrained in how rapidly it changes in time, and therefore how much detail it can convey in an interval of time. The sampling theorem asserts that the uniformly spaced discrete samples are a complete representation of the signal if this bandwidth is less than half the sampling rate.
To formalize these concepts, let

x(t),

represent a continuous-time signal and

X(f),

be the continuous Fourier transform of that signal (which exists if

x(t),

is square-integrable):
ť

X(f)  equiv   int_

Exactly what "Nyquist's result" they're referring to remains mysterious.
When Shannon stated and proved the sampling theorem in his 1949 paper, according to Meijering "he referred to the critical sampling interval T = 1/2W as the Nyquist interval corresponding to the band W, in recognition of Nyquist’s discovery of the fundamental importance of this interval in connection with telegraphy." This explains Nyquist's name on the critical interval, but not on the theorem.
Similarly, Nyquist's name was attached to Nyquist rate in 1953 by Harold S. Black:
ť "If the essential frequency range is limited to B cycles per second, 2B was given by Nyquist as the maximum number of code elements per second that could be unambiguously resolved, assuming the peak interference is less half a quantum step. This rate is generally referred to as signaling at the Nyquist rate and 1/2B has been termed a Nyquist interval." (bold added for emphasis; italics as in the original)

According to the OED, this may be the origin of the term Nyquist rate. In Black's usage, it isn't a sampling rate, but a signaling rate.

Historical references

Further Information

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