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1I.1
Sampling Theorem and Pulse Amplitude Modulation(PAM)
Reference Stremler, Communication Systems, Chapter 3.15, 7.1
I.2
Sampling Theorem
Signals bearing information are either in analog form,discrete
form or digital form.
Sampling theorem determines the necessary conditionswhich allow
us to change an analog signal to a discrete one,or vice versa,
without loss of information.
Sampleand Hold
encoder
Analog-to-digital converter
10110..
-
2I.3
Sampling Theorem:A real-valued band-limited signal having no
spectralcomponents above a frequency of B Hz is determineduniquely
by its values at uniform interval spaced no greaterthan 1/(2B)
second apart.
Bt
f(t) F() fs(t)
Freq (Hz) tT
BT
21
(Hz)bandwidth signal:(second) period sampling:
BT
i.e. taking more than or equal to 2B samples in each second
I.4
Example:To convert a 10kHz sinusoidal signal to digital form,
theminimum sampling frequency is 20kHz.
To convert a voice signal (0-3.3kHz) to digital form, theminimum
sampling frequency is 6.6kHz.In practice, sampling frequency =
8kHz.
To convert an audio signal (0-20kHz) to digital form, theminimum
sampling frequency is 40kHz.In practice, sampling frequency for
encoding music into CDis 44.1kHz.
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3I.5
This is a sufficient condition such that an analog signalcan be
reconstructed completely from a set of uniformlyspaced discrete
samples in time.
ProofConsider a band-limited signal f(t) having no
spectralcomponents above B Hz.
2Bt
f(t) F()-2B
I.6
The signal is sampled using the periodic gate function pT(t).As
pT(t) is a periodic signal, it can be represented by aFourier
series.
)(tpT
T T2T t
nP
)/(
/2)(
TnSaT
P
TePtp
n
on
tjnnT
o
=
== =
oo2
xxxSa sin)( =
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4I.7
The sampled signal is
Taking the Fourier transform, we have
)(tfs
=
==
n
tjnn
Ts
oePtf
tptftf
)(
)()()(
{ })propertyn translatiofrequency ()(
)Linearity()(
)()(
on
n
tjn
nn
n
tjnns
nFP
etfFP
ePtfFF
o
o
=
=
=
=
=
=
)(tfs
t
)(sF
oo2
)(FPo )(1 oFP
Not a function of
I.8
Therefore, the spectral density of the sampled signal is,within
a constant factor, exactly the same as that of . Inaddition, it
repeats itself periodically. The spectral density ofthe original
signal can be retrieved by using a LPF on .
However, if the sampling period T >1/2B, the replicas ofwill
overlap and we cannot retrieve from .
)(tfs)(tf
)(sF
)(F )(sF)(F
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5I.9
)(sF
BTo 4/2 >=
)(FPo )(sF
BTo 4/2 ==
)(FPo
)(sF
To /2 =
)(sF
BTo 4/2
I.10
The maximum time interval T of sampling (=1/2B) is calledthe
Nyquist interval; its reciprocal (2B) is called the Nyquistsampling
frequency.
In practice, oversampling (T < 1/2B) is used. we cannot build
ideal lowpass filter. If the filter
characteristics has a finite slope at the band edges,frequency
components from the spectral replicas may betransmitted through the
filter.
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6I.11
A time-limited signal is never strictly band-limited. Whensuch a
signal is sampled, there will be some unavoidableoverlap of
spectral components. In reconstruction of thesignal, frequency
components originally located aboveone-half the sampling frequency
will appear below thispoint. This is known as aliasing.
)(sF
To /2 =
I.12
Pulse Amplitude Modulation (PAM)
In pulse amplitude modulation (PAM) the amplitude of atrain of
constant-width pulses is varied in proportional to thesample values
of the modulating signal.
f(t))(tfPAM
t t
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7I.13
Generating a PAM signal could be divided into twoprocesses:
sampling and holding
Sampling: Consider a lowpass signal that is band-limited to and
multiplied by a periodic train of verynarrow rectangular pulses .
The sampling interval T istaken as the Nyquist interval
seconds.
)(tfmf
)(tpTmf2/1
f(t) F()
mm)(tpT
t
t
)(TP
T T/2
I.14
The sampled signal is
Taking the Fourier transform, we have
TPTePtf
tptftf
non
tjnn
Ts
o /1 and /2 where)(
)()()(
====
=
{ }
{ }
=
=
=
=
=
=
=
no
tjn
nn
n
tjnn
Ts
nFT
etfFP
ePtfF
tptfFF
o
o
)(1
)(
)(
)()()(
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8I.15
fs(t)
t
)(sF
T/2
Holding (lengthening): achieved by applying the sampledsignal to
a time-invariant filter with unit impulse response
)(tf )(tq )(tfPAM)(tfs
t)(tq
I.16
)(Q
/2
)2/()( SaQ =
[ ]
)()()(
)()()(
)()()()()()(
tqnTtnTf
tqnTttf
tqtptftqtftf
n
n
T
sPAM
=
==
=
=
=
[ ]
=
=
=
=
n
n
nTtqnTf
tqnTtnTf
)()(
)()()(
)(tfPAM
t
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9I.17
The spectral density of the PAM signal is{ }
=
==
=
no
s
sPAM
QnFT
QFtqtfFF
)()(1)()(
)()()(
)(PAMF /2
( ) )(1 QFT
I.18
The spectrum obtained here is not the same as thatobtained in
I.7. In I.7 the spectrum consists of and its replicas at
multiples of the sampling frequency with only a gainvariation of
each spectral replica (i.e. ).
The present spectrum describes a point-to-pointmultiplication in
frequency so that the spectral densityhas lost its original shape
(i.e. ). Thisdistortion is dependent on the pulse shape; at
lowfrequencies it is not severe if the pulse width is
verynarrow.
)(F
)(F
)(0 FP
)()( FQ
