Chapter 3: Data Transmission

1
COE 341 Data Computer Communications Dr.
Radwan E. Abdel-Aal

• Chapter 3 Data Transmission

2
Remaining Six Chapters
Chapter 7 Data Link Flow and Error
control, Link management
Data Link
Chapter 8 Improved utilization Multiplexing
Chapter 6 Data Communication Synchronization,
Error detection and correction
Physical Layer
Chapter 4 Transmission Media
Transmission Medium
Chapter 5 Encoding From data to signals
Chapter 3 Signals, their representations, their
transmission over media, Resulting impairments
3
Agenda

• Concepts Terminology
• Signal representation
  Time and Frequency domains
• Bandwidth and data rate
• The decibels notation for signal strength
  (Appendix 3A)
• Fourier Analysis (Appendix B)
• Analog Digital Data Transmission
• Transmission Impairments
• Channel Capacity

4
Terminology (1)

• Transmission system Components
• Transmitter
• Receiver
• Medium
• Guided media
• e.g. twisted pair, coaxial cable, optical fiber
• Unguided media
• e.g. air, water, vacuum

5
Terminology (2)

• Link Configurations
• Direct link
• No intermediate communication devices
• (these exclude repeaters/amplifiers)
• Two types
• Point-to-point (A-B)
• Only 2 devices share link
• Multi-point (C-B)
• More than two devices share the same link,
  e.g. Ethernet bus segment

C
Amplifier
A
B
6
Terminology (3)

• Transmission Types (ANSI Definitions)
• Simplex
• Information flows only in one direction all the
  time
• e.g. Television, Radio broadcasting
• Duplex
• Information flows in both directions
• Two types
• Half duplex
• Only one direction at a time
• e.g. Walki-Talki
• Full duplex
• In both directions at the same time
• e.g. telephone

7
Frequency, Spectrum and Bandwidth

• Time domain concepts
• Analog signal
• Varies in a smooth, continuous way in both time
  and amplitude
• Digital signal
• Maintains a constant level for sometime and then
  changes to another constant level
  (i.e.
  amplitude takes only a finite number of discrete
  levels)
• Periodic signal
• Same pattern repeated over time, e.g. sine wave
  or a square wave
• Aperiodic signal
• Pattern not repeated over time

8
Analogue Digital Signals
All values on the time and amplitude axes are
allowed
Analogue
Only a few amplitude levels allowed - Binary
signal 2 levels
Digital
9
PeriodicSignals
T
Temporal Period

t
t1T
t2T
For any periodic wave S (tnT) S (t) 0 ? t
? T Where t is time over first period T is the
waveform period n is an integer
Signal behavior over one period describes
behavior at all times
10
Aperiodic (non periodic) Signals in time
s(t)
11
Continuous Versus Discrete
Availability of the signal over the horizontal
axis
(Time or Frequency)
Continuous Signal is defined at all points on
the horizontal axis
Sampling with a train of very narrow pulses
(delta function)
Discrete Signal is defined Only at certain
points on the horizontal axis
12
Sine Wave s(t) A sin(2?f t ?)
T (Period)
A sin (F)
A (Amplitude)
w

• Peak Amplitude (A)
• Peak strength of signal, volts
• Repetition Frequency (f), Cycles/s Hz
• Measures how fast the signal varies with time
• Number of cycles per second (Hz)
• f 1/ T(xx sec/cycle) yy cycles/sec yy Hz
• Angular Frequency (w), Radians/s
• w radians per second 2? f 2? /T
• Temporal (time) Period, T 1/f
• Phase Angle (?), Radians
• Determines relative position in time, radians
  (how to calculate?)

13
Varying one of the three parameters of a sine
wave carriers(t) A sin(2?ft ?) A
sin(wtF)
Can be used to convey information!
M o d u l a t I o n
AM
Varying A
Varying ?
FM
PM
Varying f
14
Sine Wave Traveling in the ive x directions(t)
A sin (k x - ? t)
? Angular Frequency 2? f 2? / T
k Wave Number 2? / ?

• Spatial Period
• Wavelength

?

• For point p on the wave
• Total phase at t 0 kx - ? (0) kx
• Total phase at t ?t k(x ?x) - ? (?t)
• Same total phase,
• kx k(x ?x) - ? (?t)
• k ?x ? ?t
• Wave propagation velocity v ?x / ?t
• v ?/k ?/T ?f

?x
p
x
Distance, x
t 0
t ?t
ive x direction
Direction of wave travel, at velocity v
? Show that the wave s(t) A sin (k x ?
t travels in the negative x direction
V is constant for a given wave type (e.g. sound)
and medium (e.g. air)
v ?f
15
Wave Propagation Velocity, v m/s

• Constant for
• A given wave type (e.g. electromagnetic, seismic,
  ultrasound, ..)
• and a given propagation medium (air, water,
  optical fiber)
• For all types of waves
• v l f
• For a given wave type and medium (given v)
  higher frequencies correspond to shorter
  wavelengths and vise versa
• Electromagnetic waves
• long wave radio (km), short wave radio (m),
  microwave (cm) light (nm)
• For electromagnetic waves
• In free space, v ? speed of light in vacuum
• v ? c 3x108 m/sec
• Over other guided media (coaxial cable, optical
  fiber, twisted pairs) v is always lower than c

Shorter wavelength ? Higher frequency
16
Wavelength, l (meters)

• Is the Spatial period of the wave
  i.e. distance between two points
  in space on the wave propagation path where the
  wave has the same total phase
• Also Distance traveled by the wave during one
  temporal (time) cycle
• dT v T (l f) T l

17
Frequency Domain Concepts

• Response of systems to a sine waves is easy to
  analyze
• But signals we deal with in practice are not all
  sine waves, e.g. Square waves
• Can we relate waves we deal with in practice to
  sine waves? YES!
• Fourier analysis shows that any signal can be
  treated as the sum of many sine wave components
  having different frequencies, amplitudes, and
  phases (Fourier Analysis Appendix B)
• This forms the basis for frequency domain
  analysis
• For a linear system, its response to a complex
  signal will be the sum of its response to the
  individual sine wave components representing that
  signal.
• Dealing with functions in the frequency domain is
  simpler than in the time domain

18
Addition of Twofrequency Components
Fundamental
A 1(4/?) frequency f

1/3 rd the Amplitude 3 times the frequency
3rd harmonic
A (1/3)(4/?) frequency 3f
Frequency Spectrum

Approaching a square wave
Fourier Series
t
3
f
Frequency Domain S(f) vs f
Time Domain s(t) vs t
Fourier Series
Discrete Function in f
Periodic function in t
19
Asymptotically approaching a square wave by
combining the fundamental an infinite number of
odd harmonics at prescribed amplitudes
Topic for a programming assignment
Adding more of the higher harmonics
What is the highest Harmonic added?
20
More Frequency Domain Representations A single
square pulse (Aperiodic signal)
Sinc(f) sin(f)/f
Fourier Transform
X
To ?
To ?
To ?
frequency
1/X
Fourier Transform
Frequency Domain S(f) vs f
Time Domain s(t) vs t
Continuous Function in f
Aperiodic function in t

• What happens to the spectrum as the pulse gets
  broader ? DC ?

• What happens to the spectrum as the pulse gets
  narrower ? spike ?

21
Spectrum Bandwidth of a signal

• Spectrum of a signal
• Range of frequencies contained in a signal
• Absolute (theoretical) Bandwidth (BW)
• Is the full width of spectrum fmax- fmin
• But in many situations, fmax ?!
• (e.g. a square wave), so
• Effective Bandwidth
• Often called just bandwidth
• Narrow band of frequencies containing most of the
  signal energy
• Somewhat arbitrary what is most?
• ? e.g. that contains say 95 of the energy of
  the signal

S(f)
.
f
5f
7f
3f
f
22
Signals with a DC Component
NO DC Component, Signal average over a period 0

_
t

1V DC Level

t
1V DC Component
DC Component ? is the component at zero
frequency ? Determines if fmin 0 or not
23
Bandwidth for these signals
(fmax- fmin)
fmin fmax Absolute BW Effective BW
1f 3f 2f 2f
0 3f 3f 3f
0 ? ? 1/X ?
24
Bandwidth of a transmission system

• Is the Range of signal frequencies that are
  adequately passed by the system
• Effectively, the transmission system
  (TX, medium, RX) acts as
  a filter
• Poor transmission media, e.g. twisted pairs, have
  a narrow bandwidth
• This effectively cuts off higher frequency signal
  components
• ? poor signal quality at receive
• ? limit the signal frequencies (Hz) that can be
  used for transmission
• ? this limits the data rates that can be used
  (bps), examples
• Twisted pair 4 KHz BW ? 100 Kbps
• Optical fiber 4 THz BW ? 10 Gbps

25
Limiting Effect of System Bandwidth
Received Waveform
1,3
Better reception requires larger BW
BW 2f
More difficult reception with smaller BW
f
3f
1
1,3,5
BW 4f
5f
f
3f
2
Varying System BW
1,3,5,7
BW 6f
7f
f
5f
3f
3

BW ?
1,3,5,7 ,9,?
?
f
5f
7f
3f
4
Fourier Series for a Square Wave
26
System Bandwidth and Achievable Data Rates

• Any transmission system supports only a limited
  range of frequencies (bandwidth) for satisfactory
  transmission
• For example, this bandwidth is largest for
  expensive optical fibers and smallest for cheap
  twisted pair wires
• So, bandwidth is money ? Economize in its use
• Limited system bandwidth degrades higher
  frequency components of the signal transmitted ?
  poorer received waveforms ? more difficult to
  interpret the signal at the receiver (especially
  with noise) ? Data Errors
• More degradation occurs when higher data rates
  are used (signal will have higher frequency
  components)

27
Maximum Data Rate ( link channel
capacity)Considerations

• Bandwidth of transmission system
• Signal to noise ratio (SNR)
• Receiver type
• Specified acceptable error performance

28
Bandwidth and Data Rates
Data Element Signal Element
Period T 1/f
T/2
Data rate 1/(T/2) (2/T) bits per sec 2f bps
B
0
0
1
1
Data
B 4f
Given a bandwidth B, Data rate 2f B/2
To double the data rate you need to double f Two
ways to do this
1. Double the bandwidth, same received waveform
quality (same RX conditions error rate)
2B 4f
2B
1
1
1
1
0
0
0
0
1
1
1
0
0
1
0
0
New bandwidth 2B, Data rate 2f 2(2f) 4f B
X 2
f
3f
5f
2. Same bandwidth, B, but tolerate poorer
received waveform (needs better receiver, higher
S/N ratio, or tolerating more errors in data)
1
B 2f
1
1
1
0
0
0
0
B
X 2
Bandwidth B, Data rate 2f 2(2f) 4f B
5f
3f
f
29
Bandwidth Data Rates Tradeoffs Compromises

• Increasing the data rate (bps) while keeping BW
  the same (to economize) means working with
  inferior (poorer) waveforms at the receiver,
  which may require
• Ensuring higher signal to noise ratio (SNR) at RX
• Larger transmitted power (may cause interference
  to others!)
• Limited (shorter) link distances
• Use of more en-route repeaters/amplifiers
• Better shielding of cables to reduce noise, etc.
• More sensitive ( costly!) receiver
• Suffering from higher bit error rates
• Tolerate them?
• Add more efficient means for error detection and
  correction- this also increases overhead!.

30
Appendix 3A Decibels and Signal Strength

• The decibel notation (dB) is a logarithmic
  measure of the ratio between two signal power
  levels
• NdB number of decibels
• P1 input power level (Watts)
• P2 output power level (Watts)
• e.g. ? Amplifier gain
• ? Signal loss
• (attenuation) over a link
• Example
• A signal with power level of 10mW is inserted
  into a transmission line
• Measured power some distance away is 5 mW
• Power loss in dBs is expressed as
• NdB 10 log (5/10)10(-0.3) -3 dB
• - ive dBs P2 lt P1 (Loss),
• ive dBs P2 gt P1 (Gain)

P2
P3
P1
Lossy Link
Amplifier
31
Relationship Between dB Values and Power ratio
(P2/P1)
Power Ratio dB Power Ratio dB
1 0
101 10 10-1 -10
102 20 10-2 -20
103 30 10-3 -30
104 40 10-4 -40
105 50 10-5 -50
106 60 10-6 -60
2 3 1/2 -3
32
Decibels and Signal Strength

• Decibel notation is a relative, not absolute,
  measure
• A loss of 3 dB halves the power (could be 100 to
  50, 16 to 8, )
• A gain of 3 dB doubles the power (could be 5 to
  10, 7.5 to 15, )
• Will see shortly how we can handle absolute
  levels
• Advantages of using dBs
• The log allows replacing
• Multiplication with Addition
• C A B
• Log C Log A Log B
• and Division with Subtraction
• A C / B
• Log A Log C - Log B

33
Decibels and Signal Strength

• Example Transmission line with an intermediate
  amplifier

(/10)
(/15.6)
(3162)

• Net power gain over transmission path
• 35 12 10 13 dB ( ive means there is
  net gain)

?

• Received signal power (4 mW) log10-1(13/10) 4
  x 101.3
• 4 x 101.3 mW 79.8 mW

Still we use some multiplication!
34
How to represent absolute power
levels?Decibel-Watt (dBW) and Decibel-mW (dBm)
WK 4

• As a ratio relative to a fixed reference power
  level
• With 1 W used as a reference ? dBW
• With 1 mW used as a reference ? dBm
• Examples
• Power of 1000 W is 30 dBW, 1 W ? dBW
• 10 dBm represents a power of 0.1 mW,
• 1 mW ? dBm X dBW (X ?) dBm

Caution! Must be same units at top and bottom
Caution! Must be same units at top and bottom
35
dBs dBms are added algebraically
P2
P1
G
G is ? Positive for gain ? Negative for
loss (attenuation)
G power ratio
G dBs 10 log10 G
Similarly for dBs dBWs
36
Decibels and Signal Strength

• Example Transmission line with an intermediate
  amplifier

• If all ratios are in dBs and all levels are in
  dBm ? solve by algebraic addition Same for
  dBs and dBWs (No need for
  any multiplication/division)

4 mW
?
Gain 35 dB
Transmitted Signal
Received Signal
Loss 10 dB
Loss 12 dB
Amplifier

• Net power gain over transmission path
• 35 12 10 13 dB ( ive means actual
  net gain)

TX Signal Power in dBm 4 mW 10 log (4/1)
6.02 dBm

• RX signal power (dBm) 6.02 13 19.02 dBm
• Check 19.02 dBm 10 log (RX signal in mW/1 mW)
• ? RX signal log-1 (19.02/10) 79.8 mW

As on Slide 33
37
Decibels Voltage ratios

• Power decibels can also be expressed in terms of
  voltage ratios
• Power P V2/R, assuming same R
• ? Relative
• ? Absolute dBV and dBmV
• Decibel-millivolt (dBmV) is an absolute unit,
  with 0 dBmV being equivalent to 1mV.

dBV is similarly defined
38
Pitfalls with the Decibel Notation

• Wrong to multiply dBs dBs x dBs !
• 35 dBs x 5 dBs (what would be the units of
  the result!)
• Wrong to divide dBs dBs / dBs !
• (caution dBs / (dBs/km) is OK)
• Wrong to mix numerical levels with dBs
• 3.5 W 24 dBs ..
• OK to mix dBs with dBms
• -35 dBm 12 dBs 18 dBs .
• But wrong to use dBm and dBW in the same
  expression
• -35 dBm 12 dBs X dBs 3 dbW

39
Appendix B Fourier Analysis
Signals in Time
Aperiodic
Periodic

Discrete Continuous
Discrete Continuous
DFS
FS
FT
DFT
Use Fourier Series
Use Fourier Transform
FS Fourier Series DFS Discrete Fourier
Series FT Fourier Transform DFT Discrete
Fourier Transform
40
Fourier Series for periodic continuous signals

• Any periodic signal x(t) of period T and
  repetition frequency f0 (f0 1/T) can be
  represented as an infinite sum of sinusoids of
  different frequencies and amplitudes its
  Fourier Series. Expressed in Two forms
• 1. The sine/cosine form

Frequencies are multiples of the
fundamental frequency f0
f0 fundamental frequency 1/T
Where
DC Component
f(n)
Two components at each frequency
All integrals over one period only
If A0 is not 0, x(t) has a DC component
f(n)
41
Fourier Series 2. The Amplitude-Phase form

• Previous form had two components at each
  frequency (sine, cosine i.e. in quadrature) An,
  Bn coefficients
• The equivalent Amplitude-Phase representation has
  only one component at each frequency Cn, qn
• Derived from the previous form using
  trigonometry
• cos (a) cos (b) - sin (a) sin (b) cos a b

Now we have Only one component at each
frequency nf0
Now components have different amplitudes,
frequencies, and phases
The Cs and ?s are obtained from the previous
As and Bs using the equations
42
Fourier Series General Observations
Fourier Series Expansion
Function
Odd Function
Even Function
DC
Function Function Series
No DC A0 0
Even Function x(t) x(-t) Symmetric about Y axis Bn 0 for all n
Odd Function x(t) - x(-t) Symmetric about the origin An 0 for all n
43
Correction
44
Fourier Series Example
x(t)
Note (1) x( t)x(t) ? x(t) is an even
function (2) f0 1 / T ½ Hz
Note A0 by definition is 2 x the DC content
45
Contd
0 for n even
(4/n?) sin (n?/2) for n odd
f0 1/2
a function of n only
? Replace t by t ? Swap limits in the first
integral
- sin(2pnf t)
- dt
Then Bn 0 for all n
x(t), since x(t) is an even function
46
Contd
f0 ½, so 2? f0 ?
A0 0, Bn 0 for all n, An 0 for n
even 2, 4, (4/n?) sin (n?/2) for n
odd 1, 3,
Original x(t) is an even function!
Amplitudes, n odd
Cosine is an even function
2 p 3 (1/2) t
2 p (1/2) t
3rd Harmonic
Fundamental
f0 ½ ?
47
Another Example
Previous Example
x1(t)
1
1
-1
2
-2
-1
T
Note that x1(-t) -x1(t) ? so, x(t) is an odd
function
Also, x1(t)x(t-1/2)
This waveform is the previous waveform shifted
right by 1/2
48
Another Example, Contd
Sine is an odd function
As given before for the square wave on slide 25.
Because
49
Fourier Transform

• For aperiodic (non-periodic) signals in time, the
  spectrum consists of a continuum of frequencies
  (not discrete components)
• This spectrum is defined by the Fourier Transform
• For a signal x(t) and a corresponding spectrum
  X(f), the following relations hold

Imaginary
1
nf0 ? f
T/2 ? ?
Inverse FT (from frequency to time )
Forward FT (from time to frequency)
Real
? Express sin and cos

• X(f) is always complex (Has both real Imaginary
  parts), even for x(t) real.

50
(Continuous in Frequency)
(non-periodic in time)
Sinc function
Sinc2 function
51
Fourier Transform Example
x(t)
A
Sin (x) / x i.e. sinc function
Area of pulse In time domain
52
Fourier Transform Example, contd.
Sin (x) / x sinc function
Lim x?0 (sin x)/x (cos x)x0/1 1
First zero in the Frequency spectrum
sin pft 0 pft p f 1/t
Study the effect of the pulse width ?
53
The narrower a function is in one domain, the
wider its transform is in the other domain
The Extreme Cases
0
54
Power Spectral Density (PSD) Bandwidth

• Absolute bandwidth of any time-limited signal is
  infinite
• But luckily, most of the signal power will be
  concentrated in a finite band of lower
  frequencies
• Power spectral density (PSD) describes the
  distribution of the power content of a signal as
  a function of frequency
• Effective bandwidth is the width of the spectrum
  portion containing most of the total signal power
• We estimate the total signal power in the time
  domain

55
Signal Power in the time domain

• Signal is specified as a function s(t)
  representing signal voltage or current
• Assuming resistance R 1 W,
• Instantaneous signal power (t) v(t)2/1
  i(t)21 s(t)2
• Signal power can be obtained as the average of
  the instantaneous signal power over a given
  interval of time constant
• For periodic signals, this averaging is taken
  over one period, i.e.
• This measure in the time domain gives the total
  signal power
• Effective BW is then determined such that it
  contains a specified portion (percentage) of this
  total signal power

(1)
s
56
Signal Power in the Frequency Domain Periodic
signals

• For periodic signal we have a discrete spectrum
  (the F Series)
• For a DC component, Power Vdc2
• For AC components Power Vrms2 Vpeak 2 (use
  eqn. 1 on prev. slide)
• Power spectral density (PSD) is a discrete
  function of frequency
• Where ?(f) is the Dirac delta function
• Total signal power (watts) up to the j th
  harmonic is

(A function of frequency)
(A quantity, summation of PSD components- not a
function of a frequency)
57
Example

• Consider the following signal
• The PSD is (A function of Frequency)
• The signal power is (A quantity)

(No DC)
58
Signal Power in the Frequency Domain Aperiodic
signals
Watts/Hz

• Continuous (not discrete) frequency spectrum
• PSD (Power spectrum density) function, in
  Watts/Hz, is a continuous function of frequency
  S(f),
• Total signal power contained in the frequency
  band f1lt f lt f2 (in Watts) is given by
• (Integration, instead of summation, over
  frequency)

Components exist in both negative and positive
frequencies
59
Complete Fourier Analysis Example

• Consider the half-wave rectified cosine signal,
  Figure B.1 on page 793
• Write a mathematical expression for s(t) over its
  period T
• Compute the Fourier series for s(t) (Amplitude
  Phase form)
• Get an expression for the power spectral density
  function for s(t)
• Find the total power of s(t) from the time domain
• Find the order of the highest harmonic n such
  that the Fourier series for s(t) contains at
  least 95 of the total signal power
• Determine the corresponding effective bandwidth
  for the signal

60
Example (Cont.)

1. Mathematical expression for s(t)

T/2
Where f0 is the fundamental frequency, f0
(1/T)
61
Example (Cont.)

• 2. Fourier series
• Before we start what to expect?

• DC Component?
• Even or odd function?
• A0 ?
• An ?
• Bn ?

Sine/cosine form of the Fourier Series
To get to the amplitude-phase form of the Fourier
series, we must first obtain the sine-cosine form
62
Example (Cont.)

• Fourier Analysis

f0 (1/T)
DC ?
63
Example (Cont.)
f0 (1/T)

• Fourier Analysis (cont.)

n 1 will be treated Separately later
From integral tables
64
Example (Cont.)

• Fourier Analysis (cont.)
• n ? 1

, for n even
65
Example (Cont.)

• Fourier Analysis (cont.)
• For n 1, A1 is obtained separately

Note cos2q ½(1 cos 2q)
66
Example (Cont.)

• Fourier Analysis (cont.)

-
67
Example (Cont.)

• Fourier Analysis (cont.)
• For n 1, B1 is obtained separately

i.e. Bn 0 for all n (our function is even!)
68
Example (Cont.)

• Fourier Analysis (cont.)

Note qn are not required for PSD and power
calculations
69
Example (Cont.)

• 3. Power Spectral Density function (PSD)

n Even
n 1
n 0 (DC)
For large n, power decays ? (1/n4) Good or bad?
70
Example (Cont.)

• 4. Total Power
• (From the time domain)

Note cos2q ½(1 cos 2q)
0.25 A2
Zero
Half the power of a full sine wave
71
Example (Cont.)

• 5. Finding n such that we get at least 95 of the
  total power

(Only the DC component)
Power
of total power in this component
72
Example (Cont.)

• Finding n such that we get at least 95 of the
  total power, contd.

(DC first harmonic)
Power
of total power in these two components
73
Example (Cont.)

• Finding n such that we get at least 95 of the
  total power, Contd.

(DC first harmonic second harmonic)
Power
OK! ? 95

• n 2, and
• 6. the effective bandwidth is
• Beff fmax fmin
• Beff 2f0 0 2f0

Beff
?
f
2f0
0
f0
3f0
DC
74
Bandwidth about a Center Frequency

• So far we have considered signals in their base
  band form (without modulation)
• Data is often sent as variations in a high
  frequency carrier signal having a frequency fc
  (modulation)
• So, bandwidth (BW) of this signal occupies a
  range of frequencies centered about fc
• The larger fc, the larger the BW obtainable
• Largest BW obtainable for a given center
  frequency fc is 2 fc

Carrier
With Amplitude Modulation, For each component of
the modulating signal
75
Analog and Digital Data Transmission
WK5

• Data
• Entities that convey meaning
• Signals
• Electric or electromagnetic representations of
  data
• Data Transmission
• Communication of data
  through propagation and processing of
  signals that represent them

76
Data types in nature Analog and Digital Data

• Analog Data
• Continuous values within some interval
• Examples audio, video
• Typical bandwidths
• Speech 100Hz to 7kHz
• Voice over telephone 300Hz to 3400Hz
• Video 4MHz
• Digital Data
• Discrete values (not necessarily binary)
• Examples integers, text characters, mixture
• 2347, text, SDR054

77
Analog and Digital Signals

• Means by which data get transmitted over various
  media, e.g. wire, fiber optic, space
• Analog signal
• Continuously variable in time and amplitude
• Digital signal
• Uses a few (two or more) DC levels

78
Analog Signal Example 1 Speech Data

• Frequency range for human hearing 20Hz-20kHz
• Almost fully utilized by music
• Human speech 100Hz-7kHz
• Telephone voice channel Spectrum is further
  limited to 300-3400Hz (why?)
• Mechanical sound waves (data) are easily
  converted into electromagnetic signal for
  processing and transmission
• Mechanical waves (Sound) of varying pitch and
  loudness (Data)
• is represented as
• Electromagnetic signals of different frequencies
  and amplitudes (Signal)

79
Analog Example 1. The Acoustic Spectrum
Dynamic range of the human ear can be as high as
120 dBs!
dBs
Source Data
Hearing Spectrum
Dynamic Range of Signal Power
SPEECH
Frequency Range
-70
Log Scale
80
Conventional Telephony Analog data Analog
Signal

• Telephone mouthpiece converts mechanical voice
  analog data into electromagnetic analog
  electrical signal
• Signal travels on telephone lines
• At receiver, speaker re-converts received
  electrical signal to voice

81
Analog Signal Example 2. Video Data

• Electrical signal proportional to the brightness
  of image spot on a raster-scanned phosphor screen

Interlaced Scan
11 ms
52.5 ms (Active)
Line Scan
Frame Scan
82
Bandwidth of a Black White Video Signal

• USA Specification 525 lines per frame scanned at
  the rate of 30 frames per second
• 525 lines 483 active scan lines 42 lost
  during vertical retrace
• So 525 lines x 30 frames/second 15750 lines per
  second
• Line scan interval 1/15750 63.5?s
• 11?s go for horizontal retrace, so 52.5 ?s for
  active video per line
• Effective vertical resolution 0.7 x 483 338
  lines
• Horizontal resolution 338 x aspect ratio
• 338 x (4/3) 450 dots
• Max frequency is when black and white dots
  alternate
• 450 picture dots correspond to 225 cycles in 52.5
  ?s ? Time period 52.5/225 ?s ? fmax 1/Period
  4.2 MHz
• fmin (DC) 0 ? Bandwidth fmax - fmin 4.2 MHz

83
Digital Signals

• Advantages
• Cheaper and easier to generate No extra
  processing needed
• Less susceptible to noise
• (The threshold effect)
• Disadvantages
• When noise is above threshold ? Total data
  reversal (Bit error) (1? 0, 0 ?1)
• Greater attenuation
• Line capacitances make pulses rounded and smaller
  in amplitude, leading to loss of information
• More so at higher data rates and longer distances
• So, use at low data rates over short distances

84
Attenuation of Digital Signals
1 1 1 1 . . .
0 0 0 0 . . .
Pulse shaping Due to line capacitances Worse
over longer distances
Worse at higher data rates (narrower pulses)
Effect of line capacitances
85
Digital Binary Signal

• Example Between keyboard and computer
• Two bipolar dc levels ( and Why?)
• Bandwidth required depends on the signal
  frequency, which depends on
• The data rate (bps) and
• The actual data sequence transmitted

_
Data
-

• Data rate ?
• - Maximum f ?
• - Minimum f ?

Data element
Signal
86
Data and Signal combinations

• We have seen above (data and signal of same
  type)
• Analog signals carrying analog data Telephony,
  Video
• Digital signals carrying digital data Keyboard
  to PC
• Simple- one only needs a transducer/transceiver
• But we may also have (data and signal of
  different types)
• Analog signal representing digital data Data
  over telephone wires (using a modem)
• Digital signal representing analog data CD
  Audio, PCM (pulse code modulation) (using a
  codec)
• More complex- We Need a converter
• So, all the four data-signal combinations are
  possible!

87
Analog Signals can carry Analog Data or Digital
Data
(Base band)
i.e. in its original form
(Transducer)
(Converter)
We need a converter when the signal type is
different from the data type
88
Digital Signals can carry Analog Data or Digital
Data
Digitized Analog Samples
e.g. using PCM (Pulse Code Modulation)
Coder-Decoder
(Converter)
Transmitter-Receiver
We need a converter when the signal type is
different from the data type
89
Four Data/Signal Combinations
Signal Signal
Analog Digital
Data Analog Two ways Signal has Same spectrum as data (base band) e.g. Telephony to exchange Different spectrum (through modulation) e.g. AM Radio, FDM Use a (converter) codec, e.g. for PCM (pulse code modulation)
Data Digital Use a (converter) modem e.g. the V.90 standard Simple two signal levels e.g. NRZ code Special Encoding e.g. Manchester code (Chapter 5)
90
Two Modes of Transmitting Signals 1. The Analog
Mode (associated with FDM)

• Treats the signal as analog regardless of what
  it represents (Not interested in the data content
  of signal)
• Following attenuation over distance, signal level
  is boosted using amplifiers
• Unfortunately, this also amplifies in-band noise
• With cascaded amplifiers (i.e. one after the
  other at locations along the link), effect on
  noise and distortion is cumulative, i.e. they get
  amplified again and again
• Effect of noise and distortion on analog systems
  may be tolerated, e.g. with telephony you can
  still manage to get it! (Humans are good at
  filling-in gaps!)
• But digital systems are more sensitive to the
  effects of excessive noise and distortion ?
  unacceptable errors
• So Do not transmit digital signals the analog
  way!

91
Two Modes of Transmitting Signals 2. The
Digital Mode (Associated with TDM)

• Concerned with the data content of the signal
• It assumes that the signal carries digital data
• Uses repeaters (not amplifiers), which
• Receive the signal
• Extract the data bit stream from it
• Retransmit a fresh, strong signal representing
  the extracted bit stream
• This way
• Effect of attenuation is overcome
• Noise and distortion are not cumulative

92
Four Signal/Transmission Mode Combinations
Transmission mode Transmission mode
Analog Uses amplifiers Not concerned with what data the signal represents Noise and distortion are cumulative - Associated with FDM Digital Uses repeaters Assumes signal represents digital data, recovers this data and represents it as a new outbound signal This way, noise and distortion are not cumulative Associated with TDM
Signal Analog OK Makes sense only if the analog signal represents digital data! (Ask yourself What data is the repeater going to extract?!)
Signal Digital Avoid OK
Which transmission mode is more versatile and
useful for integrating different signal types?
FDM Frequency Division
Multiplexing TDM Time Division
Multiplexing
93
Advantages of Digital Mode of Transmission

• Use of digital technology
• Lower cost, smaller size, and high speed VLSI
  technology
• Higher data integrity (reliability) as noise
  effects are not cumulative (fresh signal
  restoration en-route)
• Cover longer distances, at higher data rate, at
  low error rates, over lower quality lines
• Easier to implement multiplexing for improved
  utilization of link capacity
• High bandwidth links are now economical (Fiber,
  Satellite)
• To utilize them efficiently we need to do a lot
  of multiplexing
• This is done more efficiently using digital (TDM)
  rather than analog (FDM) (Chapter 8)
• Encryption for data security
• and confidentiality is digital
• Easier to integrate different data types
• Convert analog data to digital signalsand use
  one system to handle all voice, video, and data,
  e.g. one network for all types of traffic

Frequency Division Multiplexing
Time Division Multiplexing
94
Transmission Impairments

• Signal received is often a degraded form of the
  signal transmitted
• Why? What happens en-route?... Impairments
• Attenuation
• Limits the bandwidth of the received signal
• In-band signals arrive weaker
• Attenuation distortion (Attenuation is not
  uniform over bandwidth)
• Delay
• Delay distortion
• Noise and interference (including crosstalk)
• Effect
• On analog data - Some degradation in signal
  quality
• On digital data Fatal bit errors (total bit
  reversals)

95
Attenuation

• Signal strength falls off with distance traveled
• Nature of loss in signal power depends on medium
• Guided (Wires, etc.)
• Exponential drop is signal power with distance
  Pd P0 e-ad
• 10 ln (Pd/P0) -ad
• 10 log (Pd/P0) -ad
• ? Loss a dBs per km (a depends on medium
  type e.g. fiber, twisted pair, cable)
• Unguided (Open space)
• Inverse square law spread with distance P ? P0
  /d2
• ? Loss 6 dBs for each distance doubling
• Absorption, scattering
• May also depend on weather, e.g. rain, sunspots,

Signal power after traveling distance d
96
Effects of Attenuation

• Received signal strength must be
• Sufficiently Large enough to be detected
• Sufficiently higher than noise to be interpreted
  correctly (without error)
• To overcome these problems
• Use amplifiers (analog transmission mode)
• or repeaters (digital transmission mode)
  en-route
• Amplifier gains should not be too large as this
  may cause signal distortion due to saturation
  (nonlinearities)
• Problem with networks distance actually traveled
  (hence attenuation) will depend on actual route
  taken through the network!

97
Attenuation Distortion

• Attenuation usually increases with frequency
• This causes bandwidth limitation (understood)
• Moreover, over the transmitted bandwidth itself
• Different frequency components of the signal get
  attenuated differently ? Signal distortion
• Affects analog signals more
• To overcome this problem
• Use Equalizers that reverse the effect of
  frequency-dependent attenuation distortion
• Passive e.g. loading coils in telephone circuits
• Active Amplifier gain designed specifically for
  this purpose

98
Attenuation Distortion
Equalization To Reduce Attenuation Distortion
Q. What is the signal ?
99
Delay Distortion

• Happens only on guided media
• Wave propagation velocity varies with frequency
• Highest at the center frequency (minimum delay)
• Lower at both ends of the bandwidth (larger
  delay)
• Effect Different frequency components of the
  signal arrive at slightly different times!
  (Dispersion in time)
• Affects digital data more due to bit spill-over
  (timing is more critical here than for analog
  data)
• Again, equalization can help overcome the problem

100
Delay Distortion
Equalization To Reduce Delay Distortion
Without Equalizer
With Equalizer
101
Noise (1)

• Definition Any additional unwanted signal
  inserted between transmitter and receiver
• The most limiting factor in communication systems
• Noise Types
• Thermal Noise
• Inter-modulation Noise
• Crosstalk Noise
• Impulse Noise

102
Noise (2)
PSD

• Thermal (White) Noise
• Due to thermal agitation of electrons
• (Increases with temperature)
• Uniformly distributed over frequency (White
  noise)
• ? Difficult to eliminate
• (exists even in the same bandwidth as your
  signal!)
• Effect is more significant on weak received
  signals, e.g. from satellites

f
103
Thermal Noise, Contd.

• Thermal noise power density in 1 Hz of bandwidth,
  N0 (Constant, Independent of frequency)
• k Boltzmanns constant 1.38?10-23 J/K
• T temperature in degrees Kelvin ( 273 t ?C)
• Thermal noise power in a bandwidth of B Hz

PSD
N0
B
1 Hz
f
Can you see some disadvantage now in having a
larger BW?
10 log k
Example at t 21 ?C (T 294 ?K) and for a
bandwidth of 10 MHz N -228.6 10 log 294
10 log 107 - 133.9 dBW
104
Noise (3)
WK 6

• Inter-modulation Noise
• Signals having the sum and difference (frequency
  mixing) of original frequencies sharing a
  transmission system
• (e.g. in FDM systems)
• f1, f2 ? (f1f2) and (f1-f2)
• Caused by nonlinearities in the medium and
  equipment, e.g. due to overdrive and saturation
  of amplifiers
• Danger Resulting new frequency components may
  fall within valid signal bands, thus causing
  interference

Linear System
A cos q1 B cos q2
A cos q1 B cos q2
Output
K(A cos q1 B cos q2)
A cos q1 B cos q2 f(2q1)f(2q2)f(q1-q2)f(q
1q2)
Non-Linear System
Input
A cos q1 B cos q2
K(A cos q1 B cos q2) K(A cos q1 B cos q2)2
Inter-modulation components
Input
New spurious components can fall within genuine
signal bands causing interference
105
Noise (4)

• Crosstalk Noise
• A signal from one channel picked up by another
  channel in close proximity
• Examples
• Physical proximity coupling between adjacent
  twisted pair channels
• ? Shield cables properly
• Directional proximity antenna pick up from other
  directions ? Use directional antennas
• Spectral proximity leakage between adjacent
  channels in frequency division multiplexing (FDM)
  systems
• ? Use guard bands between adjacent channels

106
Noise (5)

• Impulse Noise
• Pulses (spikes) of irregular shape and high
  amplitude lasting short durations
• Causes External electromagnetic interference due
  to switching large currents, car ignition,
  lightning,
• Minor effect on analog signals (e.g. crackling
  noise in voice channels)
• Major effect on digital signals- Bit reversal
  error!
• More damage at higher data rates
• (a noise pulse of a given width can destroy a
  larger block of bits)

107
Effect of Impulse Noise on a Digital Signal

Impulse

RX
Q What is the effect of the same noise at 10
times the data rate?
108
Channel Capacity

• Channel capacity Maximum data rate usable under
  a given set of communication conditions
• How channel BW (B), signal level, noise and
  impairments, and the amount of data error that
  can be tolerated limit the channel capacity?
• In general, Max possible data rate, C, on a given
  channel
• Function (B, Signal wrt noise, Bit error rate
  allowed)
• Max data rate Max rate at which data can be
  communicated on the channel, bits per second
  (bps)
• Bandwidth BW of the transmitted signal as
  constrained by the transmission system, cycles
  per second (Hz)
• Signal relative to Noise, SNR signal
  power/noise power ratio (Higher SNR ? better
  communication conditions ? higher C)
• Bit error rate (BER) allowed in (bits received
  in error)/(total bits transmitted). Equal to
  the bit error probability.
  e.g. Higher allowed ? higher usable data
  rates ? higher C

109
Channel Capacity, C

• So, in general C bps F(B, SNR, BER)
• Three Formulations under different assumptions

Idealistic
Assumptions Formulation
Ideal Noise-free, Error-free C F(B) Nyquist
Noisy, Error-free C F(B, SNR) Shannon
Practical Noisy, Error C F(B, SNR, BER) Eb/N0 Vs Error Rate
Realistic
110
Bandwidth (or Spectral) Efficiency (BE)

• Measures how well we are utilizing a given
  bandwidth to send data at a high rate.
• Can be greater than 1 (not like engineering
  efficiencies)
• The larger the better

111
1. Nyquist Channel capacity (Noise-free,
Error-free)

• Idealized, theoretical
• Assumes a noise-free ? error-free channel
• Nyquist showed that (without noise, without
  errors) If rate of signal transmission is 2B
  then a signal with frequency components up to B
  Hz is sufficient to carry that signalling rate
• In other words Given bandwidth B, highest
  signalling rate possible is 2B signal elements/s
• How much data rate does this represent?
• (depends on how many bits are represented by
  each signal element!)
• Given a binary signal (1,0), data rate is same as
  signal rate ? Data rate supported by a BW of
  B Hz is 2B bps ? C 2B
• For the same B, data rate can be increased by
  sending one of M different signals (symbols) as
  each signal level now represents log2M bits
• Generalized Nyquist Channel Capacity, C 2B
  log2M bits/s (bps)
• Bandwidth efficiency C/B 2 log2M
• (bits/s)/Hz Dimensionless quantity
  

bits/signal
Signals/s
112
Nyquist Bandwidth Example

• C 2B log2M bits/s
• C Nyquist Channel Capacity
• B Bandwidth
• M Number of discrete signal levels (symbols)
  used
• Data on telephone Channel
  B 3400-300 3100 Hz
• With a binary signal (M 2 symbols, e.g. 2
  amplitudes)
• C 2B log2 2 2B x 1 6200 bps
• With a quadnary signal (M 4 symbols)
• C 2B log2 4 2B x 2 4B 12,400 bps
• Channel capacity increased, but
• disadvantage Larger number of signal levels (M)
  makes it more difficult for the receiver to
  determine data correctly in the presence of noise

Signal Element
1
0
11
10
2 bits/Symbol i.e. 2 bits /signal
element
01
00
113
2. Shannon Capacity Formula (Noisy, Error-Free)

• Highest error-free data rate in the presence of
  noise
• Signal to noise ratio SNR signal / noise levels
• SNRdB 10 log10 (SNR ratio)
• Errors are less likely with lower noise (larger
  SNR ratios). This allows higher error-free data
  rates i.e. larger Shannon channel capacities
• Shannon Capacity C B log2(1SNR)
• Highest data rate transmitted error-free with a
  given noise level
• For a given BW, the larger the SNR the higher the
  data rate I can use without introducing errors
• C/B Spectral (bandwidth) efficiency, BE,
  (bps/Hz) (gt1)
• Larger BEs mean better utilization of a given
  bandwidth B for transmitting data fast.

Caution! Log2 Not Log10
Caution! Ratio- Not dBs
114
Shannon Capacity Formula Comments

• Formula says for data rates ? calculated C, it
  is theoretically possible to find an encoding
  scheme that achieves error-free transmission at
  the given SNR But it does not say how!
• Also
• It is a theoretical approach based on thermal
  (white) noise only. But in practice, we also have
  impulse noise, attenuation and delay distortions,
  etc
• So, maximum error-free data rates measured in
  practice are expected to be lower than the C
  predicted by the Shannon formula due to the
  greater noise
• However, maximum error-free data rates can be
  used to compare practical systems The higher
  that rate the better the system

115
Shannon Capacity Formula Comments Contd.

• Formula suggests that changes in B and SNR can be
  done arbitrarily and independently but
• ? In practice, this may not be the case!
• Higher SNR obtained through excessive
  amplification may also introduce nonlinearities ?
  increased distortion and inter-modulation noise
  which reduces SNR!
• High Bandwidth B opens the system up for more
  thermal noise (kTB), and therefore reduces SNR!

116
Shannon Capacity Formula Example

• Spectrum of communication channel extends from 3
  MHz to 4 MHz
• SNR 24dB
• Then B 4MHz 3MHz 1MHz
• SNRdB 24dB 10 log10 (SNR)
• SNR (ratio) log-110 (24/10) 1024/10 251
• Using Shannons formula C B log2 (1 SNR)
• C 106 log2(1251) 106 8 8 Mbps
• Based on Nyquists formula, determine M that
  gives the above channel capacity
• C 2B log2 M
• 8 106 2 (106) log2 M
• 4 log2 M
• M 16

117
3. Eb/N0 Vs Error Rate Formulation
(Noise and Error are both specified Together)

• Handling both noise and a quantified error rate
  simultaneously
• We introduce Eb/N0 A standard quality measure of
  three channel parameters (B, SNR, R) and can also
  be independently related to the error rate
• R is the data rate. Max value of R is the
  channel capacity C
• It expresses SNR in a manner related to the data
  rate, R
• Eb Signal energy in one bit interval (Joules)
• Signal power (Watts) x bit interval Tb
  (second)
• S x (1/R) S/R
• N0 Noise power (watts) in 1 Hz kT. Two
  formulations

Tb 1/R
SNR/BE
118
Eb/N0 (Cont.)
BER vs Eb/N0 curve for a given encoding scheme
Lower Error Rate larger Eb/N0

• Bit error rate for digital data is a decreasing
  function of Eb/N0 for a given signal encoding
  scheme
• Analysis ? For a given system (SNR, B, R) ?
  (Eb/N0), determine error rate BER
• Design ? Given a desired error rate BER, get
  Eb/N0 to achieve it, then determine other
  parameters from formula, e.g. S, SNR, R, etc.
• Effect of S, R, T on error performance
• Which encoding scheme is better A
  or B?

B
A
Better Encoding
SNR

BE
Max R C, BE C/B
119
Example

• Given
• The effective noise temperature, T, is 290oK
• The data rate, R, is 2400 bps
• Would like to operate with a bit error rate of
  10-4 (e.g. 1 error in 104 bits)
• What is the minimum signal level required for the
  received signal?
• From curve, a minimum Eb/No needed to achieve a
  bit error rate of 10-4 8.4 dB
• 8.4 S(dBW) 10 log 2400 228.6 dBW 10
  log290
• S(dBW) (10)(3.38) 228.6
  (10)(2.46)
• S -161.8 dBW

Design or Analysis?
120
Eb/N0 in terms of BE, assuming Shannon channel
capacity

• From Shannons formula
• C B log2(1SNR)
• We have
• From the Eb/N0 formula
• C/B (bps/Hz) is the spectral (bandwidth)
  efficiency BE based on Shannon channel capacity

121
Example

• Find the minimum Eb/N0 required to achieve a
  Shannon bandwidth efficiency (BECShannon/B)
  of 6 bps/Hz
• Substituting in the equation above
• Eb/N0 (1/6) (26 - 1) 10.5 10.21 dB