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8/2/2019 Advance Communication System Lectures Part 3
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Chapter 6
Sampling & PCM
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Advantages of Digital Communications
Rugged: Can withstand channel noise and
distortion much better.
Use of repeaters (travels as far as needed).
Use of TDM
Can be encrypted (Security and Privacy)
Can be encoded for error correction (reliability). Easy to process, store and search.
8/2/2019 Advance Communication System Lectures Part 3
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Analog to Digital Conversion (A/D)
In converting an analog signal to an equivalentsequence of 0s and 1s, we go through threeprocesses:
Sampling:o converting continuoustime analog signals to discretetime
analog signals.
Quantization
o converting discretetime analog signals to discretetime
digital signals (finite set of amplitude levels). Coding
o Map each amplitude level to a binary sequence.
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[1] Sampling: Mathematical Representation
One sample of g(t) can be obtained from
If we want to sample g(t) periodically every Ts
sec then we
can repeat this process periodically
0 0 0( ) ( ) ( ) ( ) ( )sg t g t t t g t t t
( ) ( ) ( )
( ) ( )
( ) ( ).
s
n
s
n
s s
n
g t g t t nT
g t t nT
g nT t nT
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Sampling: Time-Domain Plot
s s s s s ss s s s s s
g(t)
s
s
s
s
s
s
s
s
s
s
s
s
g(t)
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Sampling: Frequency-Domain Analysis (1/2)
0 1 2 3
1 2 3
( ) ( )
cos( ) cos(2 ) cos(3 )
sin( ) sin(2 ) sin(3 )
sT s
n
s s s
s s s
t t nT
a a t a t a t
b t b t b t
2
s
sT
2 2
0
2 2
1 1 1( ) ( )
s s
s
s s
T T
T
T Ts s s
a t dt t dt T T T
2 2 2
0
2 2 2
2 2 2 2( ) cos( ) ( )cos( ) ( )cos(0)
s s s
s
s s s
T T T
T s s
T T Ts s s s
a t t dt t t dt t dt T T T T
2 2 2
0
2 2 2
2 2 2( ) sin( ) ( )sin( ) ( )sin(0) 0
s s s
s
s s s
T T T
T s s
T T Ts s s
b t t dt t t dt t dt T T T
)()()( ttgtgsT
an
bn
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Sampling: Frequency-Domain Analysis (2/2)
( ) ( ) ( )
1 2 2 2( ) ( )cos( ) ( )cos(2 ) ( )cos(3 ) .
s
n
s s s
s s s s
g t g t t nT
g t g t t g t t g t t T T T T
1 1 1( ) ( ) ( ) ( ) ( 2 ) ( 2 )
1
( 3 ) ( 3 ) .
1( )
s s s s
s s s
s ss
s
ns
G G G G G GT T T
G GT
G nT
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Spectrum of Sampled Function
G()
+2B
2B s
s
s
s s
s
A
G()
+2B
2B s
s
s
s s
s
A/Ts
......
s+2B
s2B
s+2B
s2B
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Recovering the Continuous-Time Signal
G()
+2B
2B s
s
s
s s
s
A/Ts
......
s+2B
s2B
s+2B
s2B
LPF for reconstructing the origianl
signal from the sampled signal
Reconstructed Signal
+2B2B 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(2B) No interference between Images
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Sampling Theorem
A baseband signal whose spectrum is band-
limited to B Hz can be reconstructed exactly
(without any error) from its samples taken
uniformly at a ratefs 2B.
fs 2B is called Nyquist Criterion of sampling.
fs = 2B is called the Nyquist rate of sampling.
Does Sampling Theorem Make Sense?
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Aliasing
Sampling a signal at a rate less that the Nyquist rate results inAliasing.
In aliasing, the higher frequency components take the identity oflower frequencies.
Real life Example: Sampling a rotating wheel.
G()
s
s
s
s
s
A/Ts
......
LPF for reconstructing the origianl
signal from the sampled signal
Reconstructed Signal
A/Ts
Ts
s < 2(2B) Interference between imageswill occur
s
s
s
Damaged part of the signal
s
s
s
s
s
s
s
s
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Time Division Multiplexing
(TDM)
Multiplexing: The process ofsending two or more signalstogether
FDM: Sending them together at thesame time over different bandsusing carrier modulation (AM &FM broadcasting)
TDM: Sending them together overthe same band by sampling thesignals and sending the samples atdifferent time instants(interleaved).
Ts
g1(t)
g2(t)
g3(t)
Ts
Ts
gTDM
(t)
Ts
Ts/3
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How to Transmit the Samples?
Pulse Modulation:
Use the samples to modulate a carrier of pulses
Pulse Amplitude Modulation (PAM)
Pulse Width Modulation (PWM)
Pulse Position Modulation (PPM)
Pulse Code Modulation (PCM)
Quantization of samples
Coding
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Pulse Amplitude Modulation (PAM)
Ts
gPAM(t)
t
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Pulse Width Modulation (PWM)
Ts
gPWM(t)
t
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Pulse Position Modulation (PPM)
Ts
gPPM(t)
t
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[2] Quantization
Analog samples with an amplitude that may take valuein a specific range are converted to a digital sampleswith an amplitude that takes one of a specific predefined set of values.
The range of possible values of the analog samples isdivide intoL levels. L is usually taken to be a power of2 (L = 2n).
The center value of each level is assigned to anysample that falls in that quantization interval.
For almost all samples, the quantized samples willdiffer from the original samples by a small amount,called the quantization error.
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Quantization: Illustration
t4TsTs 3Ts 5Ts2Ts0
mp
mp
L = 2n
L levels
n bits0
v
Qu antizer Ou tput Sam plesq
x
Qu antizer Input Samp les x
A quantization interval Corresponding quantization value
2 pmvL
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Input-Output Characteristics of Quantizer
v v v vvvvv
v/2
v/2
v/2
v/2
v/2
v/2
v/2
v/2
Quantizer
Inputx
Quantizer
Output xq
qx
x
mp
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[3] Coding
4TsTs 3Ts 5Ts2Ts0
mp
mp
L = 2n
L levels
n bits
000
001
010
111
PCM Code
n bits/sample
0
v
Quantizer Output Samples qx
Quantizer Input Samples x
A quantization interval Corresponding quantization value
011
100
101
110
001 011 100 110 110 110 100 010 010 010 100 101 101
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We want to scan and send a black-and-white image ofheight 11 inches and width 8.5 inches (Letter size paper).The resolution of the scanner is 600600 dots per inchsquare. The picture will be quantized using 256 levels.Find the size of the scanned image and the time it takesto transmit the image using a modem of speed 56 kbps.
Size of image =11(in)8.5(in)600600(samples/in2)8bits/sample
= 269280000 bits = 269 Mbits Time to transmit = 269280000 / 56,000 = 4808 sec = 80 min
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Nyquist Theorem for Transmission
Note that the larger the transmission rate
(pulses/sec) the narrower the pulse, the wider
its spectrum, the higher the channel bandwidth
required for transmission.
The minimum theoretical bandwidth
required to transmit R pulses/sec is R/2 Hz.
(To be demonstrated later)