lec 2 sampling
TRANSCRIPT
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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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Instantaneous (Ideal) Sampling
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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Instantaneous (Ideal) Sampling
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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Instantaneous (Ideal) Sampling
( ) ( ) ( )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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Instantaneous (Ideal) Sampling
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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Instantaneous (Ideal) Sampling
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 Considerations
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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Practical Considerations
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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Practical Considerations
A sampling rate of 8000 samples/s is used fordigital telephone systemsfor telephone quality
speech with bandwidth of 4 kHz.