Lecture 7.

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The sampling theorem (теорема отсчетов-теоремы Котельникова-Шеннона)

Is it possible to express the continuous Analog signal by means of its discrete points? The answer is positive and found by Vladimir Kotelnikov. Because of its importance, the Western scientists based on the results obtained by Shennon. In nowadays, it is known as Kotelnikov-Shennon theorem.

The notations

sn =s(tn ) – the sampling of the continuous signal defined in a set of discrete points; tn =a+n t, t-the minimal discretization

interval. The given analog signal is located in the finite interval m= 2 Fm, [- 2 Fm, 2 Fm]

The definitions:

The first statement

The second statement

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The prove:

For better understanding please comment each

Mathematical expression!

Inverse transform

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We change the order and take into account

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The final stage of the prove of the Statement 1

Prove of the statement 2.

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This is the end of the prove of the Kotelnikov’s theorem. The similar result was obtained later by Shennon.

We want to stress here that the inverse theorem cannot be justified. Being presented in the form of the infinite series

can be refereed to the functions forming the special Hilbert space with the reproducible kernel. A similar decompositions belong to the class of narrow band High Frequency signals (Radio-signals), for stationary and some part of nonstationary signals and signals having non-equidistant spectrum.

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Mathematical justification of the Sampling-Kotelnikov’s theorem. The cardinal Edward

Whittaker's functions.

Whittaker considered the set of the functions that form some discrete set: …, f(a-2w), f(a-w), …, f(a+nw), f(a+(n+1)w), …

He supposed also that f(a+kw) at k this function is not a finite and cannot be presented by

the Fourier series. He proved the following statements:

1.From the formed discrete set that is determined by the sampling system it was possible to fit/find the infinite number of the continuous functions.

2.However, there is a single set of the functions (E. Whittaker defined it as the cardinal set of the functions) that can be drawn (может быть проведена) through this set

Then William Ferrer established the main property of the so-called as self-consistency (само-согласованность) If for the given set of sampling is it possible to derive the cardinal set of the functions

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There is another sampling w’ < w and b a another relationship has a place

X(k)!

If the sum

for the set

accepts the finite value. The most important result was obtained by his son John Whittaker . His theorem was the following

For the function f(x) having the form

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