chapter 3 digital transmission fundamentals€¦ · why digital transmission? l information comes...
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Chapter 3 Digital Transmission Fundamentals
School of Info. Sci. & Eng.Shandong Univ.
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Information comes in a big variety.
How does it all get encoded into signals for transmission over a physical layer channel?
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Outlinel 3.1 Digital Representation of Infol 3.2 Why Digital? Why not Analog?l The basic sinewave and pulse train carrier signalsl 3.3 Characterization of Communication Channelsl 3.4 Fundamental Limits l 3.5 Line Codingl 3.6 Modemsl 3.7 Transmission Media
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DomainNames nw.com
1
10
100
1996
1996
.519
97
1997
.519
98
1998
.519
99
1999
.520
00
2000
.520
0120
02
Mill
ions
(Log
Sca
le)
Growth of the Internet
01020304050607080
1996 1997 1998 1999 2000
Dom
ain
Nam
es (M
illio
ns)
Forecasting Internet growth from nw.com data
Linear Scale. Tough to extrapolate since it is an exponential.
Log Scale. Draw a line or use Excel LOGEST
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Role of Exponential Growth in Making
l Moore’s Law models progress in the IC chip business. Since 1960 number of bits on a chip has doubled every 18 months or so. How long will it continue?
l “Andreessen’s Law” models growth in Internet domain names. Since 1995 number of names has doubled every 18 months or so. Is this a coincidence?
l Both processes were foundations for rapid growth in jobs, products and services, and Wall St. speculation
l Can you estimate the doubling interval for optical communications from Fig. 1.9. Note the change in 1995-2000 with the invention of DWDM. Can this continue? Will there be another investment frenzy? Will there be another bust?
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Receiver
Communication channel
Transmitter
Figure 3.5
The basic communications model
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Why Digital Transmission?l Information comes in many forms, but we need to
encode it into either an analog (continuously varying) or digital (two or M-level) signal for transmission
l Currently the digital option is preferred for new systems for several reasons. Computer processing is more flexible (software) vs. analog signal processing by electronic devices. We’ll see some more on the next few slides.
l Legacy systems like AM / FM radio, NSTC TV are very difficult to change to digital – Why?
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(a) Analog transmission: all details must be reproduced accurately
Sent
Sent
Received
Received
• e.g digital telephone, CD Audio
(b) Digital transmission: only discrete levels need to be reproduced
• e.g. AM, FM, TV transmission
Figure 3.6
Analog vs. digital signal designs
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Source Repeater DestinationRepeater
Transmission segment
Figure 3.7
Often repeaters are used to send a signal over a long distance.
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Attenuated & distorted signal +
noise
EqualizerRecovered signal
+residual noise
Repeater
Amp.
Figure 3.8
At each analog repeater, we amplify the received signal and noise. We amplify both: A[as(t) + n(t)] Suppose Aa =1.
After n repeaters we have: s(t) + nAn(t);
not good
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AmplifierEqualizer
TimingRecovery
Decision Circuit.& SignalRegenerator
Figure 3.9
A digital repeater can perfectly reconstruct the transmitted signal
Murphy’s brother’s-in-law Law: “Nothing is perfect.” What can go wrong here?
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signal noise signal + noise
signal noise signal + noise
HighSNR
LowSNR
SNR = Average Signal Power
Average Noise Power
SNR (dB) = 10 log10 SNR
t t t
t t t
Figure 3.12
SNR
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∆/23∆/25∆/27∆/2
−∆/2−3∆/2−5∆/2−7∆/2
∆/23∆/25∆/27∆/2
−∆/2−3∆/2−5∆/2−7∆/2
(a)
(b)
Figure 3.2
Original and samples
Original and quantized values
Conversion of analog voice to digital.
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H
W
= + +H
W
H
W
H
W
Color
Image
Red Component Image
Green Component Image
Blue Component Image
Total bits before compression = 3xHxW pixels x B bits/pixel = 3HWB
Figure 3.1
Conversion of still pictures to digital
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(a) QCIF Videoconferencing
(b) Broadcast TV
(c) HDTV
@ 30 frames/sec =
760,000 pixels/sec
@ 30 frames/sec =
10.4 x 106 pixels/sec
@ 30 frames/sec =
67 x 106 pixels/sec
720
480
1080
1920
144
176
Figure 3.3
Conversion of video for digital transmission
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communication channeld meters
0110101... 0110101...
Figure 3.10
A digital channel
How fast we can send digital information depends on:
•Energy per bit
•Distance
•Noise
•Bandwidth
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f0 W
A(f)
(a) Lowpass and idealized lowpass channel
(b) Maximum pulse transmission rate is 2W pulses/second
0 Wf
A(f)1
Channel
tt
Figure 3.11
Effect of Bandwidth on DistortionIt is possible to send 2W pulses per second through an ILPF, using the sin x / x function. For only two possible signals, this is R =2W bps.
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Channelt t
Aincos 2πft Aoutcos (2πft + ϕ(f))
AoutAin
A(f) =
Figure 3.13
When a sinusoid (perhaps a harmonic component) is passed through a linear channel, its amplitude and phase is changed by the filter function at that frequency
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-1.5-1
-0.50
0.51
1.5
0
0.12
5
0.25
0.37
5
0.5
0.62
5
0.75
0.87
5 1
-1.5-1
-0.50
0.51
1.5
0
0.12
5
0.25
0.37
5
0.5
0.62
5
0.75
0.87
5 1
-1.5-1
-0.50
0.51
1.5
0
0.12
5
0.25
0.37
5
0.5
0.62
5
0.75
0.87
5 1
(b) 2 Harmonics
(c) 4 Harmonics
(a) 1 Harmonic
Figure 3.16
The digital signal on the previous slide contains such harmonics. This slide shows the effect of passing that signal through filters that pass only 1, 2, and 4 of these terms
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Channel
t0t
h(t)
td
Figure 3.17
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f
1A(f) = 1
1+4π2f2
Figure 3.14 - Part 1
Effect of Bandwidth on Distortion. Typical amplitude response of channel or filter.
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f
0
ϕ(f) = tan-1 2πf
-45o
-90o
1/ 2π
Figure 3.14 - Part 2
Effect of Bandwidth on Distortion. Effect of delay or phase shift on distortion
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Playout delay
Jitter due to variable delay(b)
(c)
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6
Original sequence(a)
1 2 3 4 5 6 7 8 9
Figure 3.4
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1 0 0 0 0 0 0 1
. . . . . .
t
1 ms
Figure 3.15
Fourier Theorem: Any signal can be written as a series of sinusoids. There is a constant component, a fundamental frequency component, and a series of multiples of that fundamental frequency. Show onblackboard.
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-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7t
s(t) = sin(2πWt)/ 2πWt
T T T T T T T T T T T T T T
Figure 3.18
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+A
-A0 T 2T 3T 4T 5T
1 1 1 10 0
Transmitter Filter
Comm. Channel
Receiver Filter Receiver
r(t)
Received signal
t
Figure 3.19
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-2
-1
0
1
2
-2 -1 0 1 2 3 4
-1
0
1
-2 -1 0 1 2 3 4
(a) 3 separate pulses for sequence 110
(b) Combined signal for sequence 110
t
tT T T T TT
T T T T TT
Figure 3.20
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W (1+α )W(1-α )W0 f
Figure 3.21
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4 signal levels 8 signal levels
typical noise
Figure 3.22
Noise limitation on digital transmission. If we have more than 2 possible signals sent in one interval b, we can increase the number of bits / interval. We could send the whole dictionary in one pulse, but noise keeps us from distinguishing among so many levels
m =2 bits per pulse
m = 3 bits per pulse
Now data rate R = 2Wm bits/sec
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Maxium Data Ratesl With 2 levels no noise R = 2W bpsl With m bits/pulse R= 2Wm bpsl With noise Shannon limit is C = W log2 (1 + SNR)If R<C, Shannon showed that we can get as low an
error rate as desired by more elaborate coding. Current 56Kbps modems are about equal to this limit for typical 40dB noise levels and 3.4kHz bandwidths. Therefore, don't expect more from a dial- modem.
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x
222
21 σ
σπxe−
0
Figure 3.23
Noise often has a Gaussian or normal pdf
This is the famous "bell-shaped" curve. It often appears in applications because of the Central Limit Theorem. This one hasa mean of zero, so the only parameter is sigma, the variance.
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1.00E-121.00E-111.00E-101.00E-091.00E-081.00E-071.00E-061.00E-051.00E-041.00E-031.00E-021.00E-011.00E+00
0 2 4 6 8 δ/2σ
Figure 3.24
Probability of a bit error if Gaussian noise is the limiting factor. Delta is the spacing between signal levels.
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1 0 1 0 1 1 0 01UnipolarNRZ
NRZ-Inverted(DifferentialEncoding)
BipolarEncoding
ManchesterEncoding
DifferentialManchesterEncoding
Polar NRZ
Figure 3.25
Line coding permits self-synchrononization. Common designs.
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-0.2
0
0.2
0.4
0.6
0.8
1
1.20
0.2
0.4
0.6
0.8 1
1.2
1.4
1.6
1.8 2
fT
pow
er d
ensi
ty
NRZ
Bipolar
Manchester
Figure 3.26
Spectra of these line coding designs. This is the distribution of power vs. frequency in the signals.
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ff2f1 fc0
Figure 3.27
The previous signal designs were for low pass channels. Now consider a "band pass" channel.
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Information 1 1 1 10 0
+1
-10 T 2T 3T 4T 5T 6T
AmplitudeShift
Keying
+1
-1
FrequencyShift
Keying
+1
-1
PhaseShift
Keying
(a)
(b)
(c)
0 T 2T 3T 4T 5T 6T
0 T 2T 3T 4T 5T 6T
t
t
t
Figure 3.28
3 common types of digital modulation methods
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1 1 1 10 0(a) Information
(d) 2Yi(t) cos(2πfct)
+2A
-2A
+A
-A
(c) ModulatedSignal Yi(t)
0 T 2T 3T 4T 5T 6T
+A
-A
(b) BasebandSignal Xi(t)
0 2T 3T 6T
0 T 2T 3T 4T 5T 6T
T 4T 5T
t
t
t
Figure 3.29
AM via multiplication by a sinusoid
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(a) Modulate cos(2πfct) by multiplying it by Ak for (k-1)T < t <kT:
Ak x
cos(2πfct)
Yi(t) = Ak cos(2πfct)
(b) Demodulate (recover) Ak by multiplying by 2cos(2πfct) and lowpass filtering:
x
2cos(2πfct)2Ak cos2(2πfct) = Ak {1 + cos(2π2fct)}
LowpassFilter withcutoff W Hz
Xi(t)Yi(t) = Akcos(2πfct)
Figure 3.30
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Akx
cos(2πfc t)
Yi(t) = Ak cos(2πfc t)
Bkx
sin(2πfc t)
Yq(t) = Bk sin(2πfc t)
+ Y(t)
Modulate cos(2πfct) and sin (2πfct) by multiplying them by Ak and Bk respectively for (k-1)T < t <kT:
Figure 3.31
Quadrature Amplitude Modulation QAM
A and B are odd and even symbols with half the data rate of input
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Y(t) x
2cos(2πfc t)2cos2(2πfct)+2Bk cos(2πfct)sin(2πfct)
= Ak {1 + cos(4πfct)}+Bk {0 + sin(4πfct)}
LowpassFilter withcutoff W/2 Hz
Ak
x
2sin(2πfc t)2Bk sin2(2πfct)+2Ak cos(2πfct)sin(2πfct)
= Bk {1 - cos(4πfct)}+Ak {0 + sin(4πfct)}
LowpassFilter withcutoff W/2 Hz
Bk
Figure 3.32
QAM demodulator
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Ak
Bk
16 “levels”/ pulse4 bits / pulse4W bits per second
Ak
Bk
4 “levels”/ pulse2 bits / pulse2W bits per second
2-D signal2-D signal
Figure 3.33
Signal constellations. Packing more than one bit into a single pulse to get higher data rates R.
Achieves R = 2W rate of best LPF
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Ak
Bk
4 “levels”/ pulse2 bits / pulse2W bits per second
Ak
Bk
16 “levels”/ pulse4 bits / pulse4W bits per second
Figure 3.34
Constellations used in practice
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102 104 106 108 1010 1012 1014 1016 1018 1020 1022 1024
Frequency (Hz)
Wavelength (meters)
106 104 102 10 10-2 10-4 10-6 10-8 10-10 10-12 10-14
pow
er &
tele
phon
e
broa
dcas
tra
dio
mic
row
ave
radi
o
infra
red
light
visi
ble
light
ultra
viol
et li
ght
x ra
ys
gam
ma
rays
Figure 3.35
Properties of transmission media
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t = 0t = d/c
communication channel
d meters
Figure 3.36
![Page 45: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/45.jpg)
Atte
nuat
ion
(dB
/mi)
f (kHz)
19 gauge
22 gauge
24 gauge
26 gauge
6
12
3
9
15
18
21
24
27
30
1 10 100 1000
Figure 3.37
![Page 46: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/46.jpg)
l l l l l l
Figure 3.38
![Page 47: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/47.jpg)
Centerconductor
Dielectricmaterial
Braidedouter
conductor
Outercover
Figure 3.39
Coaxial Cable
![Page 48: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/48.jpg)
35
30
10
25
20
5
15Atte
nuat
ion
(dB
/km
)
0.01 0.1 1.0 10 100 f (MHz)
2.6/9.5 mm
1.2/4.4 mm
0.7/2.9 mm
Figure 3.40
Attenuation of Coaxial Cable. Note error on f axis in book.
![Page 49: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/49.jpg)
Headend
Unidirectionalamplifier
Figure 3.41
Original Cable TV Network
![Page 50: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/50.jpg)
Headend
Upstream fiber
Downstream fiber
Fibernode
Coaxialdistribution
plant
Fibernode
BidirectionalSplit-BandAmplifier
Fiber Fiber
Figure 3.42
Modification for Computer Network Service
![Page 51: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/51.jpg)
Downstream
54 MH
z
500 MH
zUpstream
Downstream
5 MH
z
42 MH
z
54 MH
z
500 MH
z
550 MH
z
750 MH
z
(a)Currentallocation
(b) Proposedhybridfiber-coaxialallocation
Proposed downstream
Figure 3.43
Cable TV Spectrum Allocations
![Page 52: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/52.jpg)
core
cladding jacketlight
θc
(a) Geometry of optical fiber
(b) Reflection in optical fiber
Figure 3.44
Optical Fiber Cable
![Page 53: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/53.jpg)
100
50
10
5
1
0.5
0.1
0.05
0.010.8 1.0 1.2 1.4 1.6 1.8
Wavelength (µm)
Loss
(dB/
km) Infrared absorption
Rayleigh scattering
Figure 3.45
Optical Fiber Attenuation
![Page 54: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/54.jpg)
(a) Multimode fiber: multiple rays follow different paths
(b) Single mode: only direct path propagates in fiber
direct path
reflected path
Figure 3.46
Much lower attenuation possible
![Page 55: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/55.jpg)
Optical fiber
Opticalsource
ModulatorElectricalsignal Receiver Electrical
signal
Figure 3.47
![Page 56: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/56.jpg)
104 106 107 108 109 1010 1011 1012
Frequency (Hz)
Wavelength (meters)
103 102 101 1 10-1 10-2 10-3
105
satellite & terrestrialmicrowave
AM radio
FM radio & TV
LF MF HF VHF UHF SHF EHF104
Cellular& PCS
Wireless cable
Figure 3.48
![Page 57: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/57.jpg)
ChannelEncoderUserinformation
PatternChecking
All inputs to channelsatisfy pattern/condition
Channeloutput Deliver user
informationor
set error alarm
Figure 3.49
![Page 58: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/58.jpg)
Calculate check bits
Channel
Recalculate check bits
Compare
Information bits Received information bits
Checkbits
Information accepted if check bits match
Received check bits
Figure 350
![Page 59: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/59.jpg)
x = codewords o = non-codewords
x
x x
x
x
x
x
o
oo
oo
oo
o
oo
o
oxx x
x
xx
x
o o o
oo
ooooo
o
o
A code with poor distance properties A code with good distance properties(a) (b)
Figure 3.51
![Page 60: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/60.jpg)
1 0 0 1 0 0
0 1 0 0 0 1
1 0 0 1 0 0
1 1 0 1 1 0
1 0 0 1 1 1
Bottom row consists of check bit for each column
Last column consists of check bits for each row
Figure 3.52
![Page 61: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/61.jpg)
1 0 0 1 0 0
0 0 0 0 0 1
1 0 0 1 0 0
1 1 0 1 1 0
1 0 0 1 1 1
1 0 0 1 0 0
0 0 0 0 0 1
1 0 0 1 0 0
1 0 0 1 1 0
1 0 0 1 1 1
1 0 0 1 0 0
0 0 0 1 0 1
1 0 0 1 0 0
1 0 0 1 1 0
1 0 0 1 1 1
1 0 0 1 0 0
0 0 0 1 0 1
1 0 0 1 0 0
1 0 0 0 1 0
1 0 0 1 1 1
Two errors
One error
Three errors
Four errors
Arrows indicate failed check bits
Figure 3.53
![Page 62: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/62.jpg)
unsigned short cksum(unsigned short *addr, int count){
/*Compute Internet Checksum for “count” bytes * beginning at location “addr”.*/
register long sum = 0;while ( count > 1 ) {
/* This is the inner loop*/ sum += *addr++; count -=2;}
/* Add left-over byte, if any */if ( count > 0 )
sum += *addr;
/* Fold 32-bit sum to 16 bits */while (sum >>16)
sum = (sum & 0xffff) + (sum >> 16) ;
return ~sum;}
Figure 3.54
![Page 63: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/63.jpg)
(x7 + x6 +1) + (x6 + x5 ) = x7 + (1 +1)x6 + x 5 +1
= x7 + x5 +1
(x +1)(x2 + x +1) = x3 + x 2 + x + x2 + x +1 = x3 +1
Addition:
Multiplication:
Division: x3 + x + 1 ) x6 + x5
x3 + x2 + x
x6 + x4 + x3
x5 + x4 + x3
x5 + x3 + x2
x4 + x2
x4 + x2 + xx
= q(x) quotient
= r(x) remainder
divisordividend
35 ) 1223
10517
Figure 3.55
![Page 64: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/64.jpg)
Steps:1) Multiply i(x) by xn-k (puts zeros in (n-k) low order positions)
2) Divide xn-k i(x) by g(x)
3) Add remainder r(x) to xn-k i(x) (puts check bits in the n-k low order positions):
quotient remainder
transmitted codewordb(x) = xn-ki(x) + r(x)
xn-ki(x) = g(x) q(x) + r(x)
Figure 3.56
![Page 65: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/65.jpg)
Generator polynomial: g(x)= x3 + x + 1Information: (1,1,0,0) i(x) = x3 + x2
Encoding: x3i(x) = x6 + x5
1011 ) 1100000
1110
1011
11101011
10101011
010
x3 + x + 1 ) x6 + x5
x3 + x2 + x
x6 + x4 + x3
x5 + x4 + x3
x5 + x3 + x2
x4 + x2
x4 + x2 + x
xTransmitted codeword:
b(x) = x6 + x5 + xb = (1,1,0,0,0,1,0)
Figure 3.57
![Page 66: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/66.jpg)
g (x ) = x 3 + x +1
reg 0 reg 1 reg 2++
g3 = 1
i (x )
g0 = 1 g1 =1 i (x ) = x 3 + x 2
Encoder for
clock input reg 0 reg 1 reg 20 - 0 0 01 1=i3 1 0 02 1=i2 1 1 03 0=i1 0 1 14 0=i0 1 1 15 0 1 0 16 0 1 0 07 0 0 1 0
check bits: r0 = 0 r1 = 1 r2 = 0r(x) = x
Figure 3.58
![Page 67: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/67.jpg)
b(x)
e(x)
R(x)+ (Receiver)(Transmitter)
Error pattern
Figure 3.59
![Page 68: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/68.jpg)
1. Single errors: e(x) = xi 0 ≤ i ≤ n-1If g(x) has more than one term, it cannot divide e(x)
2. Double errors: e(x) = xi + xj 0 ≤ i < j ≤ n-1= xi (1 + xj-i )
If g(x) is primitive, it will not divide (1 + xj-i ) for j-i ≤ 2n-k−1
3. Odd number of errors: e(1) =1 If number of errors is odd.If g(x) has (x+1) as a factor, then g(1) = 0 and all
codewords have an even number of 1s.
Figure 3.60
![Page 69: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/69.jpg)
4. Error bursts of length b: 0000110 • • • • 0001101100 • • • 0
e(x) = xi d(x) where deg(d(x)) = L-1g(x) has degree n-k; g(x) cannot divide d(x) if deg(g(x))> deg(d(x))
l L = (n-k) or less: all will be detectedl L = (n-k+1): deg(d(x)) = deg(g(x))
i.e. d(x) = g(x) is the only undetectable error pattern,fraction of bursts which are undetectable = 1/2L-2
l L > (n-k+1): fraction of bursts which are undetectable = 1/2n-k
Lith
position
error pattern d(x)
Figure 3.61
![Page 70: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/70.jpg)
b
e
r+ (receiver)(transmitter)
error pattern
b
e
r+ (receiver)(transmitter)
error pattern
(a) Single bit input
(b) Vector input
Figure 3.62
![Page 71: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/71.jpg)
0010000
s = H e = =101
single error detected
0100100
s = H e = = + =011
double error detected100
1 0 1 1 1 0 01 1 0 1 0 1 00 1 1 1 0 0 1
1110000
s = H e = = + + = 0110
triple error notdetected
011
101
1 0 1 1 1 0 01 1 0 1 0 1 00 1 1 1 0 0 1
1 0 1 1 1 0 01 1 0 1 0 1 00 1 1 1 0 0 1
111
Figure 3.63
![Page 72: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/72.jpg)
s = H r = He
s = 0 s = 0
no errors intransmission
undetectableerrors
correctableerrors
uncorrectableerrors
(1-p)7 7p3
1-3p 3p
7p
7p(1-3p) 21p2
Figure 3.64
![Page 73: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/73.jpg)
If dmin= 2t+1, non-overlapping spheres of radius tcan be drawn around each codeword; t=2 in the figure
b1 b2o o o oset of all n-tupleswithin distance t
set of all n-tupleswithin distance t
Figure 3.66
![Page 74: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/74.jpg)
b1 b2 b3 b4 bL-3 bL-2 bL-1 bL. . .
L codewordswritten verticallyin array; thentransmitted rowby row
b1 b2 b3 b4 bL-3 bL-2 bL-1 bL
. . .
A long error burst produceserrors in two adjacent rows
Figure 3.66
![Page 75: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/75.jpg)
DTE DCE
Protective Ground (PGND)Transmit Data (TXD)
Receive Data (RXD)
Request to Send (RTS)
Clear to Send (CTS)
Data Set Ready (DSR)
Ground (G)
Carrier Detect (CD)
Data Terminal Ready (DTR)
Ring Indicator (RI)
12
34
56
78
2022
12
34
56
78
2022
(b)
(a)• • • • • • • • • • • • •
• • • • • • • • • • • •
1 13
2514
Figure 3.67
![Page 76: Chapter 3 Digital Transmission Fundamentals€¦ · Why Digital Transmission? l Information comes in many forms, but we need to encode it into either an analog (continuously varying)](https://reader034.vdocuments.mx/reader034/viewer/2022050610/5fb1ce52f1ffc47e996bced4/html5/thumbnails/76.jpg)
Startbit
Stopbit1 2 3 4 5 6 7 8
Data bits
Lineidle
3T/2 T T T T T T T
Receiver samples the bits
Figure 3.68