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TE312: Introduction to Digital

Telecommunications

Lecture #2

Sampling Theory


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Introduction

Points to be discussed in this lecture:

Instantaneous (Ideal) Sampling Nyquist Sampling Theorem

Natural Sampling Flat-Top Sampling

Practical Considerations


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Introduction

Reading Assignment

1. Simon Haykin, Digital Communications, JohnWiley & Sons, Inc., 1988, Chapter 4, Sec. 4.1-4.5.

2. Bernard Sklar, Digital Communications:Fundamentals and Applications, 2nd Ed., PrenticeHall PTR, 2001, Chapter 2, Sec. 2.4.

3. A. Bruce Carlson, Paul B. Crilly, and Janet C.Rutledge, Communication Systems: AnIntroduction to Signals and Noise in Electrical

Communication, 4th

Ed., McGraw Hill, 2002,Chapter 6, Sec. 6.1.


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Instantaneous (Ideal) Sampling

In most cases, message signals are generatedby analog information sources e.g., audio,

video, etc. and the signals are continuous intime and amplitude (analog).

Analog message signals are converted todigital form by an analog-to-digital (A/D)converter, which involves sampling to produce

a discrete time signal, quantizing to generate adiscrete-amplitude signal and encoding toproduce a sequence of bits.


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Consider a signal ( )sx t obtained by

instantaneous sampling a signal ( ) at a

periodic interval

x t

sT. sT is sampling period and1

ss

fT

= is sampling rate.

( )x t

t

( )sx t

ts

T sT


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The instantaneous sampled signal ( )sx t can be

expressed in time domain by

( ) ( ) ( )s s sn

x t x nT t nT

=

=

The can be obtained by taking a product of

( ) and a periodic impulse train( )sx t

x t ( )sT t

(switching function)


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( ) ( ) ( )ss T

x t t x t= ( )x t

( ) ( ) ( )

( ) ( )

ss T

s

n

x t x t t

x t t nT

=

=

=

( ) ( )sT s

n

t t nT

=

=


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Multiplication in time domain transforms toconvolution in frequency domain (frequency

convolution property). Thus

( ) ( )1 1

1 1

s

ns s

ns s

X f X f f n

T T

X f nT T

=

=

=

=

1. ( )sX f is a continuous spectrum.

2. ( )sX f is periodic with a period equal to1

sT


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Alternatively, ( )sX f can be derived using the

time-shift property of Fourier transform i.e.

( ) ( ) ( )1

exp 2s s sns

X f x nT j fnTT

=

=

This is the complex exponential Fourier seriesexpansion of the spectrum of ( ) sincesx t ( )sX f

is periodic in f with the coefficients ( )sx nT .


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Sampling Theorem

Let the spectrum of ( )x t be strictly band-limited

to HzB as shown below

( )| |X f

f

Let the sampling period 12s

TB

= , which implies

that the sampling rate 2sf B= .

B B


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Sampling Theorem

In this case,

( ) ( )2 2s nX f B X f Bn

==

f

( )sX f

B B

2B

2B 3B

3B


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Sampling Theorem

Therefore, the spectrum of ( )x t , ( )X f can be

derived from ( )sX f as

( ) ( )1

2

1 exp2 2

s

n

X f X f B f BB

n nfx j B f BB B B

=

=

=

It can be concluded that ( )X f is uniquelydetermined by sample values ( )/ 2x n B and so is

( ).x t


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Sampling Theorem

Reconstruction of ( )x t from )can be derived

from the inverse Fourier transform of

(sx t

( )X f .

( ) ( ) ( )

( )

( )

exp 2

1 exp exp 22 2

sinc 2

2

B

Bn

n

x t X f j ft df

n nfx j j ft dfB B B

nx Bt n

B

=

=

=

=

=

This is an interpolation formula.


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Sampling Theorem

Sampling theorem for band-limited signals offinite energy states that:

1. A signal of finite energy band-limited toHzB is completely described by specifying

the values of the signal at instants of timeseparated by 1/ 2Bsec.

2. A signal of finite energy band-limited toHzB may be completely recovered from its

samples taken at the rate 2B per second,

which is called theNyquist rate.


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Sampling Theorem

Reconstruction of ( )x t from )is implemented

by means of a low-pass filter (LPF) called

reconstruction filter.

(sx t

For realizable reconstruction LPF, the

sampling rate sf must be larger than .2BIf 2sf B< , frequency-shifted versions of ( )X f

overlap. This results in aliasing effect and theoriginal signal ( ) cannot be recovered from

( )x t

sx t


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Sampling Theorem

( )| |

f

( )| |sX f

B B

2B

2sf B

sX f

B B

2B

2sf B < 2sf B>f


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Natural Sampling

A natural sampled signal ( )sx t can be

expressed in time domain by (from practical

switching circuits)

( ) ( ) ( )

( )1 0

0 otherwise

s s

n

x t x t h t nT

th t

=

=

=

The switching signal )( ) ( sn

p t h t nT

=

= is a

periodic rectangular pulse train.


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Natural Sampling

( )x t ( ) ( ) ( )sx t p t x t=

( ) ( )sn

p t h t nT

==


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Natural Sampling( )x t

t

t

( )s t

t

( )p t


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Natural Sampling

Complex exponential Fourier series of theswitching signal ( )p t is expresses as:

( ) ( ) ( ) ( )exp 2 , = sinc expn s n s s sn

p t c j nf t c f n f j nf t

=

=

Using the Fourier series expression for ( )p t , it

follows that

( ) ( ) ( )exp 2s n sn

x t x t c j nf t

=

=


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Natural Sampling

From the frequency-shifting property, Fouriertransform of ( )sx t becomes

( ) ( )s n sn

X f c X f nf

=

=

Note:

(1. )consists of an infinite number of copies

of ( )shifted every

sX f

X f Hzsf .2. The copy is scaled by .thn nc


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Natural Sampling

fB B 3B3B

( )| |sX f

Reconstruction of ( )x t from )is implemented

by means of a low-pass filter (LPF) called

reconstruction filter if 2

(sx t

sf B .

Note that .0 1.0c =


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Flat-Top Sampling

Flat-top sampling of a signal ( )x t is obtained by

instantaneous sampling at every sampling

period sT and holding the sample value forduration of (e.g. sample-and-hold circuit)sec.T

The flat-top sampled signal is defined by( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

s s sn

s s

n

sn

x t x nT g t nT

g t x nT t nT

g t x t t nT

=

=

=

=

=

=


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Flat-Top Sampling

( )sx nT : n-thsample of the message signal ( )x t .

sT : sampling period

1/ 2sT B: sampling rate

( ) 1, 00, elsewhere

t Tg t =

The Fourier transform of ( )sx t is given by

( ) ( ) ( )s s sn

X f f G f X f nf

=

=

( ) ( ) ( )sinc expG f T fT j fT =


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Flat-Top Sampling

( )

( ) ( ) ( )s

s

T

x t

t x t g t

=

( )g t( )x t

( ) ( )s

T s

n

t t nT

=

=


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Flat-Top Sampling

( )X f

( )sx t

t

f( )sX f

f

1/T1/T


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Flat-Top Sampling

Recovery of ( )x t is accomplished by the

reconstruction LPF.

The reconstructed signal (LPF output) isprocessed by an equalizer to minimize the

aperture effect caused by ( )G f . The frequencyresponse of the equalizer is given by( )eqH f

( )( )sinc

seq

TH fT Tf

=


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Practical signals are time-limited which impliesthat they are not band-limited.

To avoid aliasing, an anti-aliasing (prefilter)low-pass filter processes a signal with cut-offfrequency equal to half the Nyquist rate.

Realizable filters require a nonzero transition

bandwidth which implies 2sf B> (i.e samplingrates

f must be much larger than baseband

signal bandwidth B.)


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To minimize the sampling rate, which implieslower transmission rates and less storage

memory, small transition bandwidth of filters isdesired.

A good engineering balance is to allow atransition bandwidth 20% of the basebandsignal bandwidth such that 2.2

sf B .

44,100 samples/s is used for a high qualitycompact disc (CD) digital audio system for a

music signal with bandwidth of 20 kHz.


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A sampling rate of 8000 samples/s is used fordigital telephone systemsfor telephone quality

speech with bandwidth of 4 kHz.

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