Sampling Process

1. Electronics and Communication Engineering
Lecture 5
Sampling Theorem and PAM

2. ECE 2
Once the information source has been sampled there are 4 methods
commonly used for transmitting the amplitude of the sampled value.
All these methods utilizes modulating the carrier with a pulse whose
parameters are made to vary.
1. Pulse Amplitude Modulation (PAM), whereby the height of a pulse
is made proportional to the sampled value.
Pulse Time Modulation (PTM)
2. Pulse Duration / Width Modulation (PDM/PWM), whereby the
width of a pulse is varied as a function of the sampled value.
3. Pulse Position Modulation (PPM), whereby the position of a pulse
is changed as a function of the sampled value with reference to a
standard position/ clocking signal.
4. Pulse Code Modulation (PCM), whereby the sampled value is first
coded into a digital code and the code group then transmitted.
Another type – DM, ADM, etc.
In case of PAM, PWM and PPM, as the signal is not digitized/
digital , they are called analog pulse modulation.

3. ECE 3
Either instantaneous sampling (Flat-top) or
natural sampling can be used.
• The flat-top is more useful to conversion to PCM.
• The naturally sampled type is easier to generate and is
used in other application.

4. ECE 4
Consider an analog signal g(t)
that is continuous in both
time and amplitude and
having infinite duration and
finite energy. It is as in Fig
5.1(a).
Sample values at times t = 0,
Ts, 2Ts,…is denoted by the
series
Ts = Sampling period
fs = Sampling rate
= 1/ Ts
 
 
,.......
2
,
1
0,
, 


n
n
g Ts
Fig 5.1: Illustration of the ideal Sampling process.
(a) Analog signal (b) Discrete time signal

5. ECE 5
The discrete time signal g(t) , that results from sampling process as
Where = Dirac delta function at time t= nTs
From the definition of a delta function
From (5.1)
Where, is Dirac comb or ideal sampling function .
     
T
T s
n
s
n
t
n
g
t 
 






g
 
Ts
n
t 

       
T
T s
s
n
t
δ
t
g
n
t
δ
t
g 



     
   
t
T s
t
g
n
T s
n
t
t
g
t











g
 
t
Ts

(5.1)
See Fig. 5.1 (b)
(5.2)

6. ECE 6
According to (5.2), g(t) is the
output of an impulse
modulator, which operates
with g(t) as the modulating
wave and Ts(t) as the carrier
wave .
This circuit theoretic
interpretation of g(t)
is depicted in Fig 5.2.
Impulse
modulator
Modulating
wave g(t)
Instantaneously
sampled wave
g(t)
Carrier
wave Ts(t)
Fig 5.2: circuit theoretic interpretation of the
ideal sampling process as impulse modulation.

7. ECE 7
Let G(f) and G(f) denote the Fourier transforms of g(t) and g (t). Fourier transforms
of Ts (t) -
Where, F[.] = Fourier transforms operation
Transforming (5.2) into frequency domain,
(5.3)
   
















m
s
mf
f
f s
f
G
f
G 
 )
(
 
   







m
s
mf
f
f s
t
T s
F 

Where, * = Convolution
Interchanging the order
(5.4)
   
 















m
s
mf
f
G
f s
m
s
mf
f
f
G
f s
f
G 
 )
(
(5.5)
(5.6)

8. ECE 8
From (5.6), G(f) is a spectrum that is periodic in frequency f with period fs,
but not necessarily continuous. In other words, the process of uniformly
sampling a signal in the time domain results in a periodic spectrum in
frequency domain with a period equal to sampling rate. Thus G(f) is a
periodic extension of the original spectrum G(f).
If
 
   
T
T s
s
nf
j
n
t
F 


 2
exp

By taking the Fourier transform of both sides of (5.1),
     
T
T s
n
s
nf
j
n
g
f
G 

 


2
exp

5.7

9. ECE 9
Suppose, g(t) is strictly band limited, with no frequency components higher than W
hertz. So, G(f) = 0 for f  W as illustrated in Fig 5.3a.
(a) -fs
W
-W
H(f)
1/2W
(c)
(b)
Fig 5.3: (a )Spectrum signal of g(t). (b)Spectrum of sampled signal g(t) for a
sampling rate fs = 2W. (c) Ideal amplitude response of reconstruction filter.

10. ECE 10
Putting Ts = 1/2W in (5.7) ,
  




 







 

 W
nf
j
W
n
g
f
G
n
exp
2

(5.8)
Putting fs = 2W in (5.6) ,
    W
f
W
f
G
W
f
G 


 
2
1 (5.9)
From (5.8) ,
  W
f
W
W
nf
j
W
n
g
W
f
G
n








 







 


exp
2
2
1
(5.10)
If the sampled values g(n/2W) are specified, then G(f) is uniquely
determined by using (5.10). As g(t ) is related to G(f) by inverse FT, g(t )
is uniquely determined by {g(n/2W)} for -.n . So, {g(n/2W)}
contains all the information of g(t ) .

11. ECE 11
Reconstructing g(t):
Substituting (5.10) in formula of IFT
     
 
 
 








n
Wt
n
Wt
Sin
W
n
g
df
W
n
t
f
j
W
W
n
g
df
ft
j
W
nf
j
W
n
g
W
df
ft
j
f
G
t
g
W
W
W
W














































 















2
2
2
2
2
exp
2
1
2
2
exp
exp
2
2
1
2
exp
(5.11)
(5.12)

12. ECE 12
sinc function can be defined as
   
x
x
Sin
x
c



sin (5.13)
The sinc function exhibits interpolatory property as
 


 




0
1
.......
2
,
1
0
sin
x
for
x
for
x
c (5.14)
Rewriting (5.12),
   
n
Wt
c
W
n
g
t
g 






 



2
sin
2
(5.15)
(5.15) provides an interpolation formula for reconstructing the original
g(t) with sinc(2Wt) playing the role of an interpolation function.
Where, x is independent variable.

13. ECE 13
DEFINTION: If w(t) is an analog waveform bandlimited to B hertz,
the PAM signal that uses natural sampling (gating) is
ws(t) =w(t)s(t) Where
S(t) is a rectangular wave switching waveform and
fs = 1/Ts ≥ 2B.
(.) is single rectangular pulse.
S(t) may be represent by Fourier series
  




t
s
jn
e
n
c
t
S

 
s
T
cycle
Duty
d
d
n
d
n
d
n
c







sin
waveform.
switching
the
of
ts
coefficien
series
Fourier
the
,

14. ECE 14
Something
in Haykin      




 t
nf
j
T
nf
c
TA
f
t
c s
s
s 
2
exp
sin
Lathi    





1
cos
2
m
t
n
c
c
t
S s
n
o 
Where,  
s
s
s
s
n
s
o
f
nf
c
f
c
T
c





2
sin
,



sin( )
( ) F[ ( )] ( ) ( )
s s n s s
n n
nd
W f w t c W f nf d W f nf
nd


 
 
    
 
So,      
t
S
t
W
t
s
W 
     







m
mf
f
G
T
mf
c
TA
f
f
S s
s
s sin (Haykin)

15. ECE 15
w(t)
ws(t) =w(t)s(t)
s(t)

16. ECE 16
The PAM wave form with natural sampling can be generated
using a CMOS circuit consisting of a clock and analog switch
as shown.

17. ECE 17
• The duty cycle of the switching
waveform is d = τ/Ts = 1/3.
• The sampling rate is fs = 4B.
sin( )
( ) F[ ( )] ( ) ( )
s s n s s
n n
nd
W f w t c W f nf d W f nf
nd


 
 
    
 
sin( )
( ) ( )
sin( )
s s
n
nd
W f d W f nf
nd
nd
d
nd






 


18. ECE 18
 At the receiver, the original analog waveform, w(t), can be recovered
from the PAM signal, ws(t), by passing the PAM signal through a low-
pass filter where the cutoff frequency is: B <fcutoff < fs -B
 If the analog signal is under sampled fs < 2B, the effect of spectral
overlapping is called Aliasing. This results in a recovered analog
signal that is distorted compared to the original waveform.
LPF Filter
B <fcutoff < fs -B

19. ECE 19
 The analog waveform may be recovered from the
PAM signal by using product detection,

20. ECE 20
 DEFINITION: If w(t) is an analog waveform bandlimited to B Hertz, the
instantaneous sampled PAM signal is given by
Where h(t) denotes the sampling-pulse shape and, for flat-top sampling, the
pulse shape is,
THEOREM: The spectrum for a flat-top PAM signal is:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
s s s s s s
k k k
w t w kT h t kT h t w kT t kT h t w t t kT
 
  
  
 
       
 
 
  
 
1
( ) ( ) ( )
sin
( ) ( )
s s
k
s
W f H f W f nf
T
f
H f h t
f





 
 
    
 


21. ECE 21
Haykin
     
ft
j
ft
c
T
f
H 

 exp
sin
     
f
H
m
mf
f
G
f
f
S s
s 






where

22. ECE 22
Analog signal maybe recovered from the flat-top PAM signal by the use of a
LPF. LPF Response
Note that the recovered signal
has some distortions due to the
curvature of the H(f).
Distortions can be removed by
using a LPF having a response
1/H(f).

23. ECE 23
Self study:
Sampling and their frequency spectrum
L.W. Couch 130-131 p
134-135p

24. ECE 24
Recovery of PAM: Simple low pass filter at the receiving end will
bypass the pulse rate frequency and fill in the areas between the
pulses sufficiently to restore the fidelity of the message signal.
Bandwidth requirements: The transmission of either naturally of
instantaneously sample PAM over a channel requires a very wide
frequency response because of narrow pulse width. The bandwidth
requirement is very larger than that of original analog signal.
For each sample the frequency spectrum shows that the magnitude
exits in higher frequency due to harmonics which is required to
reconstruct the signal.
 is not very good for long distance transmission
 provides a means to other modulation.
 provides multiplexing (TDM)
P – 132/136(fig) OHP sheet
106

25. ECE 25

26. ECE 26
 A square wave is generated and
applied to an integrator to
produce a triangle wave at a in
both figure.
 The message signal (b) is mixed
with triangle wave in a linear
mixer (adder).
 The triangle wave should have
a minimum frequency that is an
odd multiple of the highest
message frequency .
 The summed wave are applied
to a comparator.
Fig: 2.21

27. ECE 27
Either instantaneous or natural
sampling can be used.

28. ECE 28

29. ECE 29
Expression for PWM wave: Assume information source have
Maximum frequency=fm.
Maximum permissible pulse width = 1/2fm = 
m information source multiplexed, then 1/2fm = 
any given information source , em(t),
T= Sampling period for 1 channel = 1/2fm
k= Arbitrary constant.
pulse width will vary as {1+em(t)}.
 
 
t
m
ke
m
T
t 

 1
2
 
k
m
T

1
2
 
k
m
T

1
2
If +1 em(t)  -1,
Minimum pulse width =
Maximum pulse width=

30. ECE 30
Pulse wave may be
   
 
     
















1
1
cos
sin
2
2
sin
,
cos
2
n
s
s
s
s
s
s
s
s
n
s
o
n
s
n
o
t
n
T
nf
c
T
T
T
T
t
s
f
nf
c
T
c
T
c
where
t
n
c
c
t
s








31. ECE 31
Substituting the value of t
       








1
cos
sin
2
2
2
1
n
s
s
m t
n
T
nf
c
T
T
t
e
m
k
m
t
s 
 The 1st term is a DC or average component.
 The 2nd term is the information having frequency spectrum equal to that
of information source em(t).
 The 3rd term of Bessel's function to yield frequency component at higher
frequencies.
Thus the information may be recovered by passing the PDM wave through a
low pass filter having a bandwidth equal to that of em(t).

32. ECE 32

33. ECE 33

34. ECE 34