1 Sampling and Signal Reconstruction.pdf

analog coomunication

1. Sampling
&
Signal Reconstruction
with Applications
Dr. Ali Hussein Muqaibel
ver. 4.5
1
Analog to Digital
Conversion (ADC)
Part I
Digital TransmissionPart II
Digital Communications

2. Class Objectives
•Why Digital?
•Analog to Digital Conversion (ADC)
•Sampling
•Sampling Theorem.
•What is the spectrum of sampled
signals?
•Signal Reconstruction.
•The interpolation function (sinc)
•Zero hold and first order hold
approximation
•Practical Sampling
Dr. Ali Hussein Muqaibel 2

3. Advantages of Digital Communication over
Analog Communication
➢Immunity to Noise (possibility of regenerating the original digital signal if signal power
to noise power ratio (SNR) is relatively high by using of devices called repeatersalong
the path of transmission).
➢Efficient use of communication bandwidth (through use of techniques like
compression).
➢Digital communication provides higher security(data encryption).
➢Error Control Coding : the ability to detecterrors and correctthem if necessary.
➢Designand manufacturing of electronics for digital communication systems is much
easier and much cheaper than the design and manufacturing of electronics for analog
communication systems.
Dr. Ali Hussein Muqaibel 3
What is the price for going digital ?

4. Analog to Digital Conversion
PCM: Pulse Coded Modulation
Source of
continuous
time message
signal
Low-pass
filter
Sampler Quantizer Encoder
01101110101
PCM signal applied
to channel input
Dr. Ali Hussein Muqaibel
Continuous-time vs. Discrete time signal.
Digital vs. Analog
4
Distorted PCM signal
produced at channel
output
Regenerative
repeater
……
Regenerative
repeater
Regenerated PCM
signal applied to the
receiver
Final channel
output
Regeneration
circuit
Decoder
Reconstruction
filter
Destination

5. Sampling Theorem
A signal whose spectrum is band limited to ���
[�(�)=0���|�|>�]can be reconstructed exactly from its samples taken
uniformly at a rate ??????>2���(samples/ sec). i.e??????
&#3627408454;<
1
2??????
Minimum sampling frequency is &#3627408467;
&#3627408480;=2&#3627408437;&#3627408443;&#3627408487;??????&#3627408486;&#3627408478;&#3627408482;??????&#3627408480;&#3627408481;????????????&#3627408481;&#3627408466;
??????
&#3627408480;=
1
2??????
Nyquist interval
The &#3627408475;
&#3627408481;ℎ
impulse located at &#3627408481;=&#3627408475;??????
&#3627408480;has a strength &#3627408468;&#3627408475;??????
&#3627408480;the value of &#3627408468;(&#3627408481;)at &#3627408481;=&#3627408475;??????
&#3627408480;t
s s s s s ss s s s s s
g(t)
Dr. Ali Hussein Muqaibel
??????
&#3627408480;=1/&#3627408467;
&#3627408480;sampling interval
5

6. Math Representation of Sampling
•&#3627408468;(&#3627408481;)is a continuous–time signal with bandwidth &#3627408437;&#3627408443;&#3627408487;or(2&#3627408437;&#3627408479;??????&#3627408465;/&#3627408480;).
•Sampling is equivalent to multiplying &#3627408468;(&#3627408481;)by a train consisting of unit impulses repeating periodically
every ??????
&#3627408480;second. A train of delta function 
??????
&#3627408480;
(&#3627408481;)that occur every ??????
&#3627408480;is given by

??????
&#3627408480;
&#3627408481;=෍
??????=−∞
+∞
??????(&#3627408481;−&#3627408475;??????
&#3627408480;)
•The sampled signal ҧ&#3627408468;(&#3627408481;)
ҧ&#3627408468;&#3627408481;=&#3627408468;(&#3627408481;)෍
??????=−∞
+∞
??????(&#3627408481;−&#3627408475;??????
&#3627408480;)
=෍
??????=−∞
+∞
&#3627408468;(&#3627408481;)??????(&#3627408481;−&#3627408475;??????
&#3627408480;)
=෍
??????=−∞
+∞
&#3627408468;(????????????
&#3627408532;)??????(&#3627408481;−&#3627408475;??????
&#3627408480;)
Dr. Ali Hussein Muqaibel 6
??????
&#3627408455;(&#3627408481;)t
s s s s s ss s s s s s
g(t)

7. Spectrum of Sampled Signalt
s s s s s ss s s s s s
g(t) t
s s s s s ss s s s s s
g(t) G()
+2B

2B 
s

s

s

s 
s

s
A G()
+2B

2B 
s

s

s

s 
s

s
A/T
s
......

s
+2B
s
–2B
s
+2B
s
–2B
ҧ&#3627408468;&#3627408481;=෍
??????=−∞
+∞
&#3627408468;????????????
&#3627408532;??????&#3627408481;−&#3627408475;??????
&#3627408480;=
&#3627409359;
??????
&#3627408532;
෍
−∞
+∞
&#3627408468;&#3627408481;&#3627408466;
&#3627408471;??????2??????????????????&#3627408481;
ҧ&#3627408442;&#3627408467;=
&#3627409359;
??????
&#3627408532;
෍
−∞
+∞
&#3627408442;(&#3627408467;−&#3627408475;&#3627408467;
&#3627408480;)
Dr. Ali Hussein Muqaibel 7
Using Fourier series representation:
??????
??????&#3627408532;
&#3627408533;=෍
??????=−∞
+∞
??????(&#3627408481;−&#3627408475;??????
&#3627408480;)=
&#3627409359;
??????
&#3627408532;
෍
−∞
+∞
&#3627408466;
&#3627408471;??????2??????????????????&#3627408481;

8. Signal ReconstructionG()
+2B

2B s sss ss
A/Ts
......
s+2Bs–2Bs+2Bs–2B
LPF for reconstructing the origianl
signal from the sampled signal
Reconstructed Signal
+2B2B s sss ss
A/Ts
Ts
Magnitude of LPF should be Ts to cancel
the scaling factor caused by sampling
s > 2(2B)  No interference between Images

Dr. Ali Hussein Muqaibel 8

9. Signal Reconstruction (Interpolation)
•Interpolation: the process of reconstructing continuous-time signal from its samples.
•Use of lowpassfilter of BW of &#3627408437;&#3627408443;&#3627408487;(ideal)
&#3627408443;&#3627408467;=??????
&#3627408454;Π
&#3627408467;
2&#3627408437;
•In time domain (IFT)
ℎ&#3627408481;=&#3627409360;????????????
&#3627408532;&#3627408480;??????&#3627408475;&#3627408464;2??????&#3627408437;&#3627408481;
•At Nyquist rate &#3627409360;????????????
&#3627408532;=&#3627409359;
ℎ&#3627408481;=&#3627408480;??????&#3627408475;&#3627408464;2??????&#3627408437;&#3627408481;
Observe ℎ&#3627408481;=0at all Nyquist sampling interval &#3627408481;=±
??????
2??????
&#3627408466;&#3627408485;&#3627408464;&#3627408466;&#3627408477;&#3627408481;??????&#3627408481;&#3627408481;=0
Dr. Ali Hussein Muqaibel 9

10. •The output of the reconstruction system (discrete convolution)
&#3627408468;&#3627408481;=෍
&#3627408472;
&#3627408468;????????????
&#3627408480;ℎ&#3627408481;−????????????
&#3627408480;
&#3627408468;(&#3627408481;)=෍
&#3627408472;
&#3627408468;????????????
&#3627408480;&#3627408480;??????&#3627408475;&#3627408464;2??????&#3627408437;&#3627408481;−????????????
&#3627408480;
•At Nyquist rate ??????
&#3627408480;=
1
2??????
&#3627408468;(&#3627408481;)=෍
&#3627408472;
&#3627408468;????????????
&#3627408480;&#3627408480;??????&#3627408475;&#3627408464;2??????&#3627408437;&#3627408481;−????????????
•Interpolation formula yields the value of &#3627408468;(&#3627408481;)between samples as a
weighted sum of all sample values.
Signal Reconstruction
Reconstruction
system
ℎ&#3627408481;=&#3627408480;??????&#3627408475;&#3627408464;2??????&#3627408437;&#3627408481;
Dr. Ali Hussein Muqaibel 10

11. Example
Find a signal &#3627408468;(&#3627408481;)that is band-limited to &#3627408437;&#3627408443;&#3627408487;& whose samples are
??????&#3627409358;=&#3627409359;??????&#3627408475;&#3627408465;&#3627408468;±??????
&#3627408480;=&#3627408468;±2??????
&#3627408480;=&#3627408468;±3??????
&#3627408480;=⋯=0
where the sampling interval ??????
&#3627408480;is the Nyquist interval for &#3627408468;&#3627408481;, that is ??????
&#3627408480;=1/2&#3627408437;.
&#3627408468;(&#3627408481;)=෍
&#3627408472;
&#3627408468;????????????
&#3627408480;&#3627408480;??????&#3627408475;&#3627408464;2??????&#3627408437;&#3627408481;−????????????
&#3627408480;
&#3627408468;(&#3627408481;)=&#3627408480;??????&#3627408475;&#3627408464;2??????&#3627408437;&#3627408481;
Dr. Ali Hussein Muqaibel
&#3627408468;(&#3627408481;)
11

12. Order of Signal Reconstruction (Reconstruction Filters)
Zero–Order Hold
Dr. Ali Hussein MuqaibelT
s
g(t) T
s
t
h
0
(t)
1 T
s
g(t) 1
T
s
t
h
1
(t)
–T
s T
s
g(t)
First–Order Hold–2T
s
2T
s
T
s
–T
s
1
t
h
OO
(t)
SincFilter (Infinite–Order Hold) T
s
g(t)
12
&#3627408486;&#3627408481;=ℎ&#3627408481;∗ҧ&#3627408468;&#3627408481;=ℎ&#3627408481;∗[&#3627408468;&#3627408481;??????
&#3627408455;??????
&#3627408481;]

13. Interpolation and hold Circuits in Frequency
Dr. Ali Hussein Muqaibel 13
Zero–Order Hold SincFilter (Infinite–Order Hold)
Interpolation function

14. Practical Sampling Pulses
•Finite width practical pulses
•Because it is periodic,using Fourier
series representation
??????
&#3627408455;
??????
=෍
−∞
+∞
&#3627408438;
??????&#3627408466;
&#3627408471;2??????????????????
??????
•Not
??????
??????
&#3627408532;
&#3627408533;=
&#3627409359;
??????
&#3627408532;
෍
−∞
+∞
&#3627408466;
&#3627408471;??????2????????????
??????&#3627408481;
Dr. Ali Hussein Muqaibel 14

15. Practical Difficulty in Signal Reconstruction
•To avoid the need for ideal filter &#3627408467;
&#3627408480;>2&#3627408437;,we may use a filter with gradual cutoff
characteristics.
•Also we want the filter to be zero outside...(Impossible by Paley-Wiener criterion)
but closely approximated.
Dr. Ali Hussein Muqaibel 15G()
+2B

2B 
s

s

s

s 
s

s
A/T
s
......

s
+2B
s
–2B
s
+2B
s
–2B
LPF for reconstructing the origianl
signal from the sampled signal
Reconstructed Signal
+2B2B 
s

s

s

s 
s

s
A/T
s
T
s
Magnitude of LPF should be Ts to cancel
the scaling factor caused by sampling

s
> 2(2B)  No interference between Images
