4.1 Proof

Information and signal theory Page 1 / 1

Proof of Shannon's sampling theorem

In order to recover the signal

x t

from it's samples exactly, it is necessary to sample

x t

at a rate greater than twice it's highest frequency component.

Introduction

As mentioned earlier , sampling is the necessary fundament when we want to apply digital signalprocessing on analog signals.

Here we present the proof of the sampling theorem. The proof is divided in two. First we find an expression for the spectrum of the signal resulting fromsampling the original signal $x t$ . Next we show that the signal $x t$ can be recovered from the samples. Often it is easier using the frequency domain when carrying out a proof,and this is also the case here.

Key points in the proof

We find an equation for the spectrum of the sampled signal
We find a simple method to reconstruct the original signal
The sampled signal has a periodic spectrum...
...and the period is $2 π F_{s}$

Proof part 1 - spectral considerations

By sampling $x t$ every $T_{s}$ second we obtain $x_{s} n$ . The inverse fourier transform of this time discrete signal is

x_{s} n 1 2 π ω π X_{s} ω ω n

For convenience we express the equation in terms of the real angular frequency

Ω

using

ω Ω T_{s}

.We then obtain

x_{s} n T_{s} 2 Ω π T_{s} π T_{s} X_{s} Ω T_{s} Ω T_{s} n

The inverse fourier transform of a continuous signal is

x t 12 Ω

∞ ∞ X Ω Ω t

From this equation we find an expression for

x n T_{s}

x n T_{s} 12 Ω

∞ ∞ X Ω Ω n T s

To account for the difference in region of integration we split the integration in into subintervals of length

2 π T_{s}

and then take the sum over the resulting integrals to obtain the complete area.

x n T_{s} 1 2 π k

∞ ∞ Ω 2 k 1 T s 2 k 1 T s X Ω Ω n T s

Then we change the integration variable, setting

Ω η 2 π k T_{s}

x n T_{s} 1 2 π k ∞ ∞ η T_{s} π T_{s} X η 2 π k T_{s} η 2 π k T_{s} n T_{s}

We obtain the final form by observing that

2 π k n 1

, reinserting

η Ω

and multiplying by

T_{s} T_{s}

x n T_{s} T_{s} 2 π Ω π T_{s} π T_{s} k ∞ ∞ 1 T_{s} X Ω 2 π k T_{s} Ω n T_{s}

To make

x_{s} n x n T_{s}

for all values of

n

, the integrands in and have to agreee, that is

X_{s} Ω T_{s} 1 T_{s} k ∞ ∞ X Ω 2 k T_{s}

This is a central result. We see that the digital spectrum consists of a sum of shifted versions of the original, analog spectrum. Observe the periodicity!

We can also express this relation in terms of the digital angular frequency $ω Ω T_{s}$

X_{s} ω 1 T_{s} k ∞ ∞ X ω 2 π k T_{s}

This concludes the first part of the proof. Now we want to find a reconstruction formula, so that we can recover

x t

from

x_{s} n

Proof part ii - signal reconstruction

For a bandlimited signal the inverse fourier transform is

x t 12 Ω T_{s} T_{s} X Ω Ω t

In the interval we are integrating we have:

X_{s} Ω T_{s} X Ω T_{s}

. Substituting this relation into we get

x t T_{s} 2 Ω T_{s} T_{s} X_{s} Ω T_{s} Ω t

Using the DTFT relation for

X_{s} Ω T_{s}

we have

x t T_{s} 2 Ω T_{s} T_{s} n

∞ ∞ x s n Ω n T s Ω t

Interchanging integration and summation (under the assumption of convergence) leads to

x t T_{s} 2 n

∞ ∞ x s n Ω T s T s Ω t n T s

Finally we perform the integration and arrive at the important reconstruction formula

x t n

∞ ∞ x s n T s t n T s T s t n T s

(Thanks to R.Loos for pointing out an error in the proof.)

Summary

X_{s} Ω T_{s} 1 T_{s} k ∞ ∞ X Ω 2 k T_{s}

x t n

∞ ∞ x s n T s t n T s T s t n T s

Go to

Introduction
Illustrations
Matlab Example
Hold operation
Aliasing applet
System view
Exercises

<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Information and signal theory. OpenStax CNX. Aug 03, 2006 Download for free at http://legacy.cnx.org/content/col10211/1.19

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Information and signal theory' conversation and receive update notifications?

Ask

©flickr: Dutch	Las Vegas Timeshare By Donyea Sweets Start Test
	18 AP 18 Cardiovascular System Blood MCQ By OpenStax Start Quiz
©flickr: Clive	Sports Psych. Research Methods By Dakota Bocan Start Flashcards
	5 Sociology 05 Socialization MCQ By OpenStax Start Quiz
	27 Biology 27 Animal Diversity MCQ By OpenStax Start Quiz
	2 CDL Quiz - Air Brakes Endorsement Part 2 By Jazzycazz Jackson Start Quiz
	MAPEH Test G9 (Physical Education) By Angelica Lito Start Quiz
	6 Microeconomics 06 Elasticity By OpenStax Start Flashcards
	25 AP Key Terms 25 The Urinary System By OpenStax Start Key Terms
	8 AP 08 Appendicular Skeleton MCQ By OpenStax Start Quiz