This document discusses sampling and related concepts in signal processing. It begins by introducing the need to convert analog signals to discrete-time signals for digital processing. It then covers the sampling theorem, which states that a band-limited signal can be reconstructed if sampled at twice the maximum frequency. The document describes three main sampling methods: ideal (impulse), natural (pulse), and flat-top sampling. It also discusses aliasing, which occurs when a signal is under-sampled. The key aspects of sampling covered are the sampling rate, reconstruction of sampled signals, and anti-aliasing filters.

1. SAMPLING
SAGAR KUMAR
16TC-20
Signal and
System
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2. Content
2
• Introduction
• Sampling Theorem
• Sampling Methods
• Ideal sampling
• Natural sampling
• Flat top sampling
• Reconstruction of Sampled Signal
• Aliasing
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3. Introduction
3
• Most of the signals that we use in our daily life are
analog in nature ( for eg: speech, weather signals etc).
• Digital system possess many advantages in comparison
to analog system such as they are immune to noise,
can be stored, processed with more efficient
algorithms, secure, more robust and cost effective etc.
• Most of the effective signal processor are digital signal
processors which needs digital information in order to
process it.
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4. Introduction
4
• Hence there arises a need to convert our analog
signal to discrete time signal in order to process
them properly through digital signal processors and
then reconvert them back to analog signals so that
we can understand them.
• Sampling is the answer to this need.
• Sampling is a way to convert a signal from
continuous time to discrete time.
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5. Sampling Theorem
5
• A Band-limited continuous time signal can be
represented by its samples and can be recovered
back when sampling frequency fs is greater than or
equal to the twice the highest frequency component
of message signal. i. e.
fs ≥ 2fm
where fm is the maximum frequency component
of the continuous time signal.
• Key words: Band Limited
Sampling Frequency
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6. Sampling Method
fs = 1/Ts is called the sampling rate or sampling
frequency.
x(t)
Analog
signal
 xs (t)  x[nTs]
Discrete
signal
6
p(t)
• Analog signal is sampled every Ts secs.
• Ts is referred to as the sampling interval.
•
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7. Sampling Methods
• There are 3 sampling methods:
• Ideal Sampling- an impulse at each sampling
instant with amplitude equals to signal at that
point of time.
Ideal Sampling
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8. Sampling Methods
• Natural Sampling- a pulse of short width with
varying amplitude
Natural Sampling
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9. Sampling Methods
• Flat-top Sampling– make use of sample and
hold circuit almost like natural but with single
amplitude value.
Flat-top Sampling
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10. Impulse Sampling
• Impulse sampling can be performed by multiplying
input signal x(t) with impulse train p(t) of period 'Ts'.
p(t)   (t nTs )
n
Here, the amplitude of impulse changes with
respect to amplitude of input signal x(t).
p(t)
x(t)
xs(t)
101
0
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11. Impulse Sampling
• The output of sampler is given by
xs (t)  x(t) p(t)

 x(t)  (t  nTs )
n
• Recall the sifting property of impulse function:
x(t)(t t0 )  x(t0 )(t t0 )
hence we can write :
xs (t)  x(t)   x(nTs )(t nTs )
n
(1)
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12. Impulse Sampling
• To take the spectrum of the sampled signal let us
take the Fourier Transform of equation (1) as we
know multiplication in time domain becomes
convolution in frequency domain we have:
where
hence
s 
2
X () X ()
1
X()*P()
X ()and P() are Fourier transform of x(t) and p(t).
s
12
T
Xs ()  X()

(n)
1  2
 X ()*
2 
s 



n
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13. Impulse Sampling
since
Hence the spectrum of sampled signal is given as:
•we can not use ideal/impulse sampling because we can not
generate the impulse train practically.
1
s
Ts
Xs ()  X () X ( n )


n
s s
Ts
2
n
FT of p(t)  FT (t nT ) (n)


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14. Natural Sampling
• Natural sampling is similar to impulse sampling,
except the impulse train is replaced by pulse train of
period Ts.
n
• When we multiply input signal x(t) to pulse train p(t)
we get the signal as shown below:
• The pulse equation is being given as:
p(t)   p(t  nTs )
x(t) p(t) xs(t)

2

2
14
p(t)
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15. Natural Sampling
is given as:
where
n
• The exponential Fourier series representation of p(t)
• The output of the sampler is given as:
xs (t)  x(t) p(t)

 x(t)   p(t nTs ) (1)
n
C enjst

p(t)  
n
2
1 1
T
n s
Ts Ts
P(n)

C  p(t)e njst
dt 
s
2

Ts
(2)
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16. Natural Sampling
Now putting the value of Cn in equation (2) we have:
s
1
P(n)enjst

n Ts
p(t)  
s
Ts
P(n)enjst
n

1


1
s
Ts
P(n)enjst
n
Now putting the value of p(t) in equation (1) we have:
xs (t)  x(t) p(t)
 x(t)


s
16
Ts
P(n)x(t)enjst


1

n
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17. Natural Sampling
• Toget the spectrum of the sampled signal let us take
the Fourier Transform of both side:
• Now according to frequency shifting property of FT
we have:
s
s s
njt
P(n)x(t)eFT x (t) FT
 1

Ts





n
s
sFT x(t)enjt
   X() 
1
17
s
Ts n
P(n)FT x(t)enjst
  


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18. Natural Sampling
• Hence we can say that
• Hence the spectrum of sampled signal is given as:
s s
Ts n
P(n)FT x(t)enjst
FT x (t)
1
 


s
1
Ts
X () P(ns)X (s)


n
s
18
2ss 0
Xs ()
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19. Flat-Top Sampling
• The top of the samples are flat i.e. they have
constant amplitude. Hence, it is called as flat top
• During transmission, noise is introduced at top of the
transmission pulse which can be easily removed if
the pulse is in the form of flat top.
sampling or practical sampling.
x(t)
19
p(t)
xs(t)
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20. Flat-Top Sampling
.
xs (t)  p(t)* x(t)

xs (t)  p(t)*[  x(kTs )(t kTs )]
k
• Mathematically we can consider the flat top sampled
signal is equivalent to the convolved sequence of the
pulse p(t) and the ideal sampled signal xδ(t).
p(t)
x(t)
sx (t)

2

2
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*
2
0
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21. Flat-Top Sampling
.
P()  Fourier Transform of p(t)
X()  Fourier Transform of x(t)
Now applying Fourier Transform
Xs () P()X()
s s
Ts
X ()  P()
1
X (n)


n
s s
Ts
X () 
1
P()X (n)


n
s
21
2ss 0
Xs ()
where
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22. s 2ss 0
s 2ss 0
s 2ss 0
Reconstruction of Sampled Signal
Spectrum of a typical Sampled Signal
fs > 2fm
Oversampling
fs = 2fm
Perfect sampling
fs < 2fm
Undersampling
Xs ()
Xs ()
Aliasing
Low-Pass Filter
with transfer function H(ω)
ωm
Xs ()
-ωm
ωm-ωm
ωs-ωm
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23. Aliasing
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• Aliasing refers to the phenomenon of a high
frequency component in the spectrum of a signal
seemingly taking on the identity of a lower frequency
in the spectrum of its sampled version (under-
sampled version of the message signal)
• It is worth to be mention here that a time-limited
signal cannot be band-limited. Since all signals are
more or less time-limited, they cannot be band-
limited.
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24. Aliasing
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• Hence we must pass most of signals through low
pass filter before sampling in order to make them
band-limited. This is called an anti-aliasing filter and
are typically built into an analog to digital (A/D)
converter.
• Distortion will occur (If the signal is not band-limited)
when the signal is sampled. We refer to this
distortion as aliasing.
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25. References
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• Forouzan B. A, “Data Communications and Networking”,
McGraw-Hill, Fourth Edition
• Taub H.,Schilling D.L.,Saha G. “Taub’s Principle of
Communication Systems”, McGraw-Hill, Third edition
• Communication Systems, 3Rd Ed Simon Haykin
• B. P. Lathi, Modern Digital and Analog Communication
Systems, (3rd ed.) Oxford University press, 1998
• John G. Proakis and Masoud Salehi, Communication
Systems Engineering, Prentice Hall international edition,
1994
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