Digital Signal Transmission

1
Digital Signal Transmission

• Modems and Modulation
• Modulated Signals (PSK and QAM)
• Properties and applications of digital media
• Wire and cable
• Fiber
• Radio
• Error Detection and Correction
• Parity
• CRC
• Error Correcting Codes

2
A Bandpass Channel
f
f2
f1
0
fc

• Channel passes a band of frequencies between f1
  and f2.
• Other frequencies excluded via filtering
• Arise in radio and in frequency multiplexed media
• Base band signals (NRZ, Bipolar, Manchester,
  etc.) wont fit
• All have low frequency components not limited to
  the band
• The Solution is modulation of a carrier frequency
• fc (f1f2)/2

Figure 3.27
3
Amplitude Modulation
1
1
1
0
0
1
1
1
0
0
1
cos(?ct)
?
a

• x(t)1a mn(t)cos(?ct)
• a is the modulation index, 0
• mn(t) is an n-level symmetric NRZ code, max 1
• ?c is the radian carrier frequency 2?fc.
• Low cost easy to generate, easy to detect
• Poor error performance (poor use of power)
• Extra power at carrier if a
• Recovered signal is unipolar 0 and 1 are not
  opposites
• Sensitive to non-linearities in the transmitter

4
Some Modulation Techniques
Information
1
(a)
Amplitude Shift Keying
t
-1
1
Frequency Shift Keying
(b)
t
-1
1
Phase Shift Keying
(c)
t
-1
Figure 3.28
5
Phase Shift Keying 2PSK
Amplitude of Carrier
0
Phase angle of Carrier
1

• Carrier has Constant Amplitude and frequency
• Carrier shifts in phase at signaling transitions
• Phase diagram representation
• Carrier amplitude plotted as distance from origin
• Carrier phase plotted as angle from X axis
• PSK has all points on a circle (constant
  amplitude)
• Signals are opposite (most efficient)
• Maximum 1 signal per Hz of bandwidth (less than
  Nyquist)

6
Phase Shift Keying 4-PSK
11
10
01
00
10
11
01
00
10
11

• 2 bits in each signal
• 4 different phases -- ?/4,-?/4, 3?/4, -3?/4
• 1 axis carries even bits, 1 carries odd bits.
• 90 degree separation insures no interference.
• 4PSK 2 signals/Hz (Nyquist limit)

7
Decoding Phase Modulation

• Decoding depends on Trig identities
• 2cos2(2?fct) 1 cos(4 ?fct)
• 2sin2(2?fct) 1 - sin(4 ?fct)
• 2cos(2?fct)sin(2?fct) 0 sin(4 ?fct)
• If the modulated signal is multiplied by the
  original carrier and low pass filtered
• Constant level represents the data carried
• Signals modulated 90? out of phase produce no
  signal

8
Constructing A modulated carrier

• A cos(2?fct?)
• A cos(?) cos(2?fct) - A sin(?) sin(2?fct)
• Any modulated carrier at frequency fc can
  constructed as the sum of sum modulations of the
  carrier and that signal shifted by 90º
• Allows simple way to construct the signal

9
Quadrature modulator
Modulate cos(2?fct) and sin (2?fct) by
multiplying them by Ak and Bk respectively for
(k-1)T

Simplest case Ak and Bk depend on one bit each
and are 1 or -1 (4PSK)

Multiple bits can be encoded in each channel by
using multiple levels for Ak and Bk

Figure 3.31
10
Quadrature demodulator
Lowpass Filter with cutoff W/2 Hz
x
Y(t)
Ak
2cos(2?fc t)
2cos2(2?fct)2Bk cos(2?fct)sin(2?fct) Ak
1 cos(4?fct)Bk 0 sin(4?fct)
Lowpass Filter with cutoff W/2 Hz
x
Bk
2sin(2?fc t)
2Bk sin2(2?fct)2Ak cos(2?fct)sin(2?fct)
Bk 1 - cos(4?fct)Ak 0 sin(4?fct)
Figure 3.32
11
QAM Modulator/Demodulator
Ak
Ak
Coherent Reference
Carrier Frequency
Modulated Carrier
?
90 Phase Shift
Bk
90 Phase Shift
Bk

• In general can use multi-level code for Ak and Bk
  to increase bits/Hz
• Subject to noise limits like baseband transmission

12
Simple QAM Constellations
Bk
2-D signal
Ak
16 levels/ pulse 4 bits / pulse 4W bits per
second

• Produced by linear multi-level codes.
• Ak and Bk chosen independently.

Figure 3.33
13
Non Rectangular codes.
4 levels/ pulse 2 bits / pulse 2W bits per
second

• Ak and Bk both depend on all bits sent in
  interval k.
• Constellations often chosen for error
  performance or power efficiency.

Figure 3.34
14
PSK and QAM Performance
?
? 2 sin(?/N)
Peak/Average power 1

• Error Performance depends on separation of points
  in the constellation, like multi-level NRZ codes.
• QAM has a substantial advantage in error
  performance for large N
• QAM Systems with N up to 1024 have been built
• QAM is not constant amplitude sensitive to
  non-linearities and noise
• Power efficiency depends on Peak/Average power
• PSK is constant power, QAM has variable power
• Large QAM systems with square constellation have
  low peak/average power

15
Trellis Coded Modulation

• Idea Ambiguity in the received signal can be
  resolved if only certain sequences are valid
• Receiver considers several signals before
  determining data
• Requires redundancy in the received data
• Uses more signal points than data (e.g. 8-PSK
  with 2 bits/signal)
• Sender picks which of the signal points to used
  based on a state model
• Receiver uses the state model to distinguish
  ambiguities

2
1
3
0
4
7
5
6
16
Multi-Carrier (Discreet MultiTone) Modulation

• Data is partitioned into channels
• Each channel is sent via a separate carrier
• Commonly done with DSP Synthesis
• Accommodates a channel with defects

Information Density per Subchannel
Interference (noise)
Channel Response
Frequency
Frequency
17
Propagation through a Channel

• Signal appears after a propagation delay
  dependent to channel length and propagation
  speed.
• Signal is degraded.

Figure 3.36
18
Wired and wireless media

• Wired (guided) Media
• twisted pair, coax, fiber, . . .
• Attenuation is linear (in dB) (kdB/km)
• Needs a right of way and construction of route
• Wireless (unguided) Media
• radio, free space optics, . . .
• Attenuation is logarithmic (in dB) (nlog10(d)dB)
• Needs spectrum license (radio)
• Subject to obstructions

19
Attenuation of twisted pair cable
26 gauge
30
24 gauge
27
24
22 gauge
21
18
Attenuation (dB/mi)
19 gauge
15
12
9
6
3
f (kHz)
100
1000
1
10
Figure 3.37
20
Applications of twisted pair

• Analog telephone
• Loss limits line length loading with
  inductance reduces low frequency loss
• DSL
• uses two frequency bands above voice and DMT
  modulation
• Range depends on length and interference.
• LANs (10 baseT, 100baseT)
• short distance baseband (unmodulated) signal to
  hub.
• Signal may be split over multiple pairs

21
Loaded loops
6000 ft
6000 ft
6000 ft
Equivalent Circuit of twisted pair cable
Loading coils
Trades away high frequency response for Reduced
and flatter low frequency attenuation
22
Digital Subscriber Loop
Low Pass Filter
Voice
DSL Access Mux
Copper loop
Internet Service Provider
DSL Modem
100baseT
PPP over ATM

• DSL multiplexes data on top of voice on a normal
  copper loop
• Data is sent with high frequency modulated
  signals
• The loop is connected to a DSLAM which removes
  the data before the switch
• Data signal may need to be filtered from phones
• Several types of DSLs
• Asymmetric DSL (different rates in both
  directions)
• HDSL T1/E1 rate using modulated carrier on 4
  wires
• SDSL T1/E1 on one pair
• VDSL very high speed (50Mb/s possible)
• ISDN DSL ISDN rate (128 or 144Kb/s), greater
  loop length

23
Ethernet hub

• Machines connect to the hub using twisted pair
• Hub provides the interconnection (switched or
  broadcast)
• Wiring standards (grade of cable, number of
  pairs) determine speed of operation and
  interoperability

Figure 3.38
24
Coaxial Cable
Outer cover
Center conductor
Dielectric material
Braided outer conductor

• Much higher bandwidth (lower attenuation at high
  frequencies)
• Higher cost
• Large physical size

Figure 3.39
25
Coaxial cable attenuation
35
0.7/2.9 mm
30
25
1.2/4.4 mm
Attenuation (dB/km)
20
15
2.6/9.5 mm
10
5
0.01
0.1
10
100
f (MHz)
1.0
Figure 3.40
26
Application, Analog Cable TV
Homes
Unidirectional amplifier

• Broadband signal containing all channels sent
  from head end
• Video is frequency multiplexed in 6Mhz bands
  (like broadcast TV)
• No communication upstream

Figure 3.41
27
Hybrid cable systems
Upstream fiber
Fiber
Fiber
Head end
Fiber node
Fiber node
Downstream fiber
Coaxial distribution plant
Bidirectional Split-Band Amplifier

• Bidirectional fiber used for main distribution
• Coaxes have multiple bands, some upstream

Figure 3.42
28
Cable Modems
Set Top Box
TV
Cable Modem Termination Service
ISP
Shared Cable
Cable Modem

• Cable Data is carried in 6Mhz Channels (One
  Analog TV Channel)
• Downstream one shared channel using 64 or 256
  QAM
• Slots for individual users identified by control
  fields
• Encrypted for Privacy
• Upstream Shared TDM Channel using 16PSK or
  16QAM from each Cable Modem
• Multi-access protocol with backoff for collisions
• Lower data rate
• Some systems use Telephone or satellite for
  upstream data
• Data to/from Individual Cable plants aggregated
  in an ISP
• Much like DSL service, ISP allocates internet
  addresses and provides services like mail and web
  hosting

29
Optical Fibers -- Why Optics?

• LOTS of bandwidth (Tens of Thz (1013))
• B ? v ??/ ?2
• Physically Small (micrometers)
• Long repeater spacing (20-200km)
• Interference Immunity (no RFI)
• Security (difficult to listen without detection)
• Challenges
• Splicing, Connectors, Distribution of signals
• Efficiently coupling light to the fibers
• Immature technology (Optical switches,
  amplifiers, etc.)
• Large bandwidth may imply large vulnerability

30
Optical Fibers
reflected
higher index
lower index
refracted
Jacket
Cladding
Core
absorbed
secondary
Primary (no reflections)
31
Fiber Attenuation

• Attenuation depends on operating wavelength
• Attenuation creates a distance limit between
  repeaters

32
Multimode Dispersion
Core

• Reflected rays travel greater distance and arrive
  later
• Pulses spread over time
• Difference depends on fiber design (core diameter
  and index difference)
• Result is a bandwidth times distance limit (BDP)
• 13Mbps-km typical

33
Single Mode Fibers
8?m

• Very narrow core eliminates reflected modes
• Eliminates multi-mode dispersion
• Reduced attenuation (long repeater distances)
• More difficult to couple power into the fiber.
• Very small diameter, need to align source with
  fiber

34
Chromatic Dispersion

• Speed of propagation is dependent on wavelength
  (color) and properties of glass
• Light source isnt all one wavelength
• Solutions
• Pick a source with narrow spectrum.
• Operate at a wavelength where effect is minimized
• Use narrow pulses (which have more room to
  spread)
• BDP of 250Ghz-km is possible

35
Line Codes for Fiber Optic links

• Some differences from coding for wires
• DC levels and polarity can be distinguished
• Non linear devices make multi-level codes
  unattractive
• Bandwidth is usually not an issue (exception
  submarine cables with long repeater spacings that
  approach bandwidth-distance limit)
• Balancing on/off states is important because of
  detector operation
• Density of transitions is still important for
  timing
• Coding techniques
• Manchester codes balance on/off states and have
  lots of transitions, but higher bandwidth creates
  dispersion problems
• NRZ codes Efficient, but signal needs to be
  coded to achieve balance and guarantee
  transitions mBnB codes used.

36
Example 4B5B
37
Wavelength Division Multiplexing (WDM)
TX ?n
RX ?n
. . .
. . .
RX ?1
TX ?1
PRISM
PRISM

• Multiple Channels at different wavelengths go
  through the same fiber.
• Prisms combine and separate wavelengths
• Practical systems with many wavelengths use
  diffraction gratings (easier to get close
  spacing)
• Both devices are passive and need no power
• Insertion loss of 1-7dB for multiplexing

38
Dense Wavelength Division Multiplexing (DWDM)
RX ?n
TX ?n
filter ?n
. . .
. . .
RX ?1
filter ?1
TX ?1

• Multiple wavelengths independently coded and
  combined passively
• Signal is boosted with an EFDA optical amplifier
  to overcome the loss in the splitter.
• Signal is split, then filtered with diffraction
  based filters to select a very narrow wavelegth
  range for the receiver.
• Very close wavelength spacing (0.8nm or less)
  and many wavelength systems can be achieved

39
Optical Fiber Applications

• Long distance transmission
• High bandwidth, wavelength multiplexed, long
  repeater spacings, single mode fiber.
• Local area networks (Gigahz Ethernet, FDDI)
• Short distances with lower bandwidths so focus is
  often on low cost (multi-mode fiber, inexpensive
  transmitter/receiver)

40
Whats different about a radio link?

• Frequency Range
• Transmission in wires can be baseband, 0-BW
• Radio must be modulated to use a specified
  frequency range
• Band Limiting
• Wire transmission is limited by the wire and by
  noise reduction filters in the receiver.
• Radio transmission must be band limited at the
  sender to contain the signal to a frequency
  channel.
• Modulation techniques and filters get designed
  together

41
Model of a radio link
(From Digital Telephony 3rd edition, Richard
Bellamy
42
Radio Spectrum Allocations
Frequency (Hz)
106
1012
107
108
105
104
1011
109
1010
FM radio TV
Wireless cable
AM radio
Cellular PCS
satellite terrestrial microwave
LF
MF
HF
VHF
UHF
SHF
EHF
10-1
1
102
10-2
10-3
103
101
104
Wavelength (meters)
Lower frequencies more suitable for broadcast
operation Higher frequencies follow line of sight
Figure 3.48
43
Radio Communication Applications
44
Error Detection and Correction

• Communication errors are inevitable
• Rates of 10-3 to 10-12, depending on medium and
  quality
• Multiple approaches to controlling them
• Retransmission (ARQ)
• Receiver detects the error and asks for resending
• Correction will be delayed and take extra
  bandwidth
• Forward Error Correction
• Transmitter encodes signal redundantly
• Receiver reconstructs correct signal even with
  errors
• May not correct all errors.

45
Error detection
Received information bits
Information bits
Recalculate check bits
Channel
Calculate check bits
Compare
Received check bits
Check bits
Information accepted if check bits match

• Extra bits sent for detection (overhead)
• Probability of detecting errors depends on error
  patterns and the algorithms.

Figure 350
46
Parity detection

• For each k bit word, compute the modulo 2 sum
  (exclusive or) of the bits and send it as bit
  k1.
• Overhead 1/(k1) extra bits sent.
• Probability of detection depends on the nature of
  the errors.
• Random error vector received information is a
  random sequence of bits
• Random bit error model each bit has an
  independent probability (pe) of being changed.
• Real channels can operate in both modes as well
  as others.

47
Random error vector model
x codewords o non-codewords

• 2k codewords (correct parity)
• 2(k1) total words
• Probability that a random bit pattern will be
  recognized as correct is 2k/2(k1) or ½.
• This will be the wrong value 2k/(2k-1) times
  (nearly always).

Figure 3.51
48
Random bit error model.

• If p is bit error probability
• Probability of a single error in a k bit word is
  k p(1-p)(k-1) kp
• Probability of k bit word having j errors is
  (kj)pj(1-p)(k-j)
• Parity check will fail with an even number of
  errors, which means
• (k2)p2(1-p)(k-2) (k4)p4(1-p)(k-4) . . .
• If p is small this is (k(k-1)/2)p2
• Whats the probability of error with a 160 bit
  word and a 10-3 error rate?

49
A code with good distance properties
(b)
A code with poor distance properties
(a)
x codewords o non-codewords

• Extra bits sent for detection (overhead)
• Probability of detecting errors depends on error
  patterns and the algorithms.

Figure 3.51
50
Two dimensional parity
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
Last column consists of check bits for each row
Bottom row consists of check bit for each column

• Data is organized into rows of words
• Parity computed on rows and columns.
• Detects up to 3 errors, but not all higher
  numbers
• Not the best linear code

Figure 3.52
51
Internet Checksum

• Used on Internet protocols (IP, UDP, TCP)
• Chosen for ease of software implementation
• 16 bit sum of all 16 bit words, mod 216-1
• Equivalently computed using 1s complement
  arithmetic
• Easy to do using software instructions
• Detects multi-bit errors.
• Changing one codeword to another requires
  multiple errors.

52
Computing Internet Checksum
Sum is 32 bits
Upper 16 hold the overflow
Loop, why?
Figure 3.54
53
Cyclic Redundancy Check

• Code is based on Polynomial Arithmetic
• Bits are used as coefficients in a polynomial
• CRC is the remainder of the polynomial divided by
  a generator polynomial using mod 2 arithmetic.
• Function easily implemented using shift
  registers.
• Data bits plus remainder coefficients are
  transmitted
• If generator is degree k, remainder is k-1 bits.
• Resulting codeword is divisible by the generator

54
Polynomial Arithmetic
Addition
Multiplication
q(x) quotient
x3 x2 x
Division
x3 x 1 ) x6 x5
x6 x4 x3
dividend
divisor
x5 x4 x3
x5 x3 x2
3
35 ) 122
x4 x2
105
x4 x2 x
17
x
r(x) remainder
Figure 3.55
55
CRC Generation

• 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
xn-ki(x) g(x) q(x) r(x)
transmitted codeword
b(x) xn-ki(x) r(x)
Figure 3.56
56
CRC Generation -- example

• Generator polynomial g(x) x3 x 1
• Information (1,1,0,0) i(x) x3 x2
• Encoding x3i(x) x6 x5
  

x3 x2 x
1110
x3 x 1 ) x6 x5
1011 ) 1100000
x6 x4 x3
1011
x5 x4 x3
1110
1011
x5 x3 x2
1010
x4 x2
1011
x4 x2 x
x
010
Transmitted codeword b(x) x6 x5 x b
(1,1,0,0,0,1,0)
Figure 3.57
57
Shift register implementation
Encoder for
reg 0
reg 1
reg 2

• clock input reg 0 reg 1 reg 2
• 0 - 0 0 0
• 1 1i3 1 0 0
• 2 1i2 1 1 0
• 3 0i1 0 1 1
• 4 0i0 1 1 1
• 5 0 1 0 1
• 6 0 1 0 0
• 7 0 0 1 0 check bits r0 0 r1 1 r2 0

r(x) x
Figure 3.58
58
Errors in transmission
b(x)
(Transmitter)
R(x)
(Receiver)

e(x)
Error pattern

• R(x) b(x) e(x) g(x)q(x)e(x)
• R(x) will be recognized as a legitimate code only
  if e(x) is divisible by g(x)
• g(x) can be designed to make large classes of
  errors detectable

Figure 3.59
59
Generators to detect errors

• 1. Single errors e(x) xi 0 ? i ? n-1
• If 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 (x1) as a factor, then g(1) 0 and
  all codewords have an even number of 1s.

Figure 3.60
60
Generators to detect errors
ith position
L

• 4. Error bursts of length b
  0000110 0001101100    0
• e(x) xi d(x) where deg(d(x)) L-1
• g(x) has degree n-k
• g(x) cannot divide d(x) if deg(g(x))
  deg(d(x))
• L (n-k) or less all will be detected
• L (n-k1) 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 (n-k1) fraction of bursts which are
  undetectable 1/2n-k

error pattern d(x)
Figure 3.61
61
Linear Codes

• Code bits are a linear combination of data bits
• bk1 a11b1 a12b2 a13b3 a14b4 a1kbk
• bk2 a21b1 a22b2 a23b3 a24b4 a2kbk
• bk1 a(n-k)1b1 a(n-k)2b2 a(n-k)3b3
  a(n-k)4b4 a(n-k)kbk
• k/n bits carry information
• Codes can be simply generated and checked.
• With appropriate choices of n, and aij these
  codes can correct errors.

62
Matrix representation
r1 r2 r3 r4 r5 r6 r7
a11 a12 a13 a14 1 0 0 a21 a22 a23 a24 0 1 0 a31
a32 a33 a34 0 0 1
0 0 0
s H R

• The check matrix H consists of the coefficients
  extended with the identity matrix
• If the matrix product is a vector of zeros, the
  received bits are a legal codeword.
• R Be, and H B 0, So H R H e
• Single bit errors are detected if all columns of
  H are non-zero
• Double bit errors are detected if all columns of
  H are distinct
• Hamming codes have m check bits and 2m-1 columns
  with all non-zero values. They check k 2m-1-m
  data bits.

63
0 0 1 0 0 0 0
1 0 1 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1
single error detected
1 0 1
s H e

0 1 0 0 1 0 0
1 0 1 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1
1 1 1
0 1 1
1 0 0
double error detected
s H e

1 1 1 0 0 0 0
1 0 1 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1
1 1 0
0 1 1
1 0 1
triple error not detected
s H e
0
Figure 3.63
64
Performance of Linear Codes

• Hamming Distance of errors required to change
  one valid code word into another.
• d(b1,b2) of 1 bits in (b1b2)
• dmin minimum distance over all codewords
• dmin minimum of 1s in a codeword.
• For H(7,4) dmin3
• For random error vector model
• Probability of failure 1/2n-k, where n-k is the
  number of check bits
• For random error model
• Probability of failure Ndminpdmin, where Ndmin
  is the of codewords with the least of 1s

65
Error Correction Codes

• If a received code word indicates an error
• 2k possible error vectors (which bits were
  changed) could produce the error
• Determine the most likely one (generally the one
  with the fewest errors) and use that to determine
  the original data.
• If there are 2 or more vectors with the fewest
  errors that produce the same pattern, the error
  is not correctable.
• Not a perfect process, as one of the less likely
  error patterns could in fact have caused the same
  result

66
Possible Outcomes with H(7,4)
s H r He
7p
s 0
1-3p
3p
undetectable errors
correctable errors
uncorrectable errors
no errors in transmission
7p(1-3p)
21p2
7p3
(1-p)7
Figure 3.64
67
Partitioning n-tuples by codewords
b1
b2
o
o
o
o
set of all n-tuples within distance t
set of all n-tuples within distance t

• If dmin 2t1, non-overlapping spheres of radius
  t can be drawn around each codeword t2 in the
  figure
• The spheres can be used to map the received data
  to the most likely original

Figure 3.66
68
Spreading burst errors.
L codewords written vertically in array
then transmitted row by row
b1
b2
b3
b4
bL-2
bL-1
bL
bL-3
. . .
b1
b2
b3
b4
bL-2
bL-1
bL
bL-3
A long error burst produces errors in two
adjacent rows
. . .
Figure 3.66
69
For the Next class

• Problem Set 4 (due at start of class)
• Read Chapter 4 (We may not cover all of it)
• We will go over problems from problem set 3.
• NOTE I will not be available for office hours
  on 9/25. I will have office hours 2-4PM on
  Friday 9/19 and Friday 9/26 instead.