COE 341: Data & Computer Communications (T081)Dr. Marwan Abu-Amara
Chapter 3: Data Transmission
COE 341 – Dr. Marwan Abu-Amara 2
Agenda Concepts & Terminology Decibels and Signal Strength Fourier Analysis Analog & Digital Data Transmission Transmission Impairments Channel Capacity
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Terminology (1) Transmitter Receiver Medium
Guided medium e.g. twisted pair, optical fiber
Unguided medium e.g. air, water, vacuum
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Terminology (2) Direct link
No intermediate devices Point-to-point
Direct link Only 2 devices share link
Multi-point More than two devices share the link
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Terminology (3) Simplex
One direction e.g. Television
Half duplex Either direction, but only one way at a time
e.g. police radio
Full duplex Both directions at the same time
e.g. telephone
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Frequency, Spectrum and Bandwidth Time domain concepts
Analog signal Varies in a smooth way over time
Digital signal Maintains a constant level then changes to another
constant level Periodic signal
Pattern repeated over time Aperiodic signal
Pattern not repeated over time
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Analogue & Digital Signals
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PeriodicSignals
S (t+nT) = S (t);Where:t is timeT is the waveform periodn is an integer
T
Temporal Period
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Sine Wave – s(t) = A sin(2ft +) Peak Amplitude (A)
maximum strength of signal unit: volts
Frequency (f) rate of change of signal unit: Hertz (Hz) or cycles per second Period = time for one repetition (T) = 1/f
Phase () relative position in time unit: radians
Angular Frequency () = 2 /T = 2 f unit: radians per second
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Varying Sine Wavess(t) = A sin(2ft +)
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Wavelength () Distance occupied by one cycle Distance between two points of corresponding
phase in two consecutive cycles Assuming signal velocity v
= vT f = v For an electromagnetic wave,
v = speed of light in the medium In free space, v = c = 3*108 m/sec
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Frequency Domain Concepts
Signal usually made up of many frequencies Components are sine waves Can be shown (Fourier analysis) that any
signal is made up of component sine waves Can plot frequency domain functions
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Addition of FrequencyComponents(T=1/f)
Fundamental Frequency
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FrequencyDomainRepresentations
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Spectrum & Bandwidth Spectrum
range of frequencies contained in signal Absolute bandwidth
width of spectrum Effective bandwidth
Often just bandwidth Narrow band of frequencies containing most of
the energy DC Component
Component of zero frequency
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Signal with a DC Component
t
1V DC Level
1V DC Component
t
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Bandwidth for these signals:fmin fmax Absolute
BWEffective
BW
1f 3f 2f 2f
0 3f 3f 3f
0 1/x ?
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Bandwidth and Data Rate Any transmission system supports only a limited band of
frequencies for satisfactory transmission “system” includes: TX, RX, and Medium Limitation is dictated by considerations of cost, number
of channels, etc. This limited bandwidth degrades the transmitted signals,
making it difficult to interpret them at RX For a given bandwidth: Higher data rates More
degradation This limits the data rate that can be used with given
signal and noise levels, receiver type, and error performance
More about this later!!!
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Bandwidth Requirements
)2sin(k
14)(
1 k odd,
kfttsk
f 3f
f 3f 5f
f 3f 5f 7f
f 3f 5f 7f ……
BW = 2f
BW = 4f
BW = 6f
BW =
…
Larger BW
needed for better representation
Mo
re d
iffic
ult
rece
ptio
n w
ith m
ore
lim
ited
BW
1,3
1,3,5
1,3,5,7
1,3,5,7 ,9,…
1
2
3
4Fourier Series for a Square Wave
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Decibels and Signal Strength Decibel is a measure of ratio between two
signal levels NdB = number of decibels
P1 = input power level
P2 = output power level
Example: A signal with power level of 10mW inserted onto a
transmission line Measured power some distance away is 5mW Loss expressed as NdB =10log(5/10)=10(-0.3)=-3 dB
1
210log10
P
PNdB
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Decibels and Signal Strength Decibel is a measure of relative, not absolute, difference
A loss from 1000 mW to 500 mW is a loss of 3dB A loss of 3 dB halves the power A gain of 3 dB doubles the power
Example: Input to transmission system at power level of 4 mW First element is transmission line with a 12 dB loss Second element is amplifier with 35 dB gain Third element is transmission line with 10 dB loss Output power P2
(-12+35-10)=13 dB = 10 log (P2 / 4mW)
P2 = 4 x 101.3 mW = 79.8 mW
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Relationship Between Decibel Values and Powers of 10 Power Power
RatioRatiodBdB Power Power
RatioRatiodBdB
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
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Decibel-Watt (dBW) Absolute level of power in decibels Value of 1 W is a reference defined to be 0 dBW
Example: Power of 1000 W is 30 dBW Power of 1 mW is –30 dBW
W
PowerPower W
dBW 1log10 10
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Decibel & Difference in Voltage Decibel is used to measure difference in
voltage. Power P=V2/R
Decibel-millivolt (dBmV) is an absolute unit with 0 dBmV equivalent to 1mV. Used in cable TV and broadband LAN
1
22
1
22
1
2 log20/
/log10log10
V
V
RV
RV
P
PNdB
mV
VoltageVoltage mV
dBmV 1log20
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Fourier AnalysisSignals
Periodic (fo) Aperiodic
Discrete Continuous Discrete Continuous
DFS FS
DTFT
FT
DFT
Infinite time Finite time
FT : Fourier TransformDFT : Discrete Fourier TransformDTFT : Discrete Time Fourier TransformFS : Fourier SeriesDFS : Discrete Fourier Series
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Fourier Series (Appendix B) Any periodic signal of period T (f0 = 1/T) can be
represented as sum of sinusoids, known as Fourier Series
1
000 )2sin()2cos(
2)(
nnn tnfBtnfA
Atx
T
dttxT
A0
0 )(2
T
n dttnftxT
A0
0 )2cos()(2
T
n dttnftxT
B0
0 )2sin()(2
If A0 is not 0,x(t) has a DC component
DC Component
fundamental frequency
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Fourier Series Amplitude-phase representation
1
00 )2cos(
2)(
nnn tnfC
Ctx
00 AC 22nnn BAC
n
nn A
B1tan
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Fourier Series Representation of Periodic Signals - Example
1
-1
1/2-1/2 1 3/2-3/2 -1 2
T
0111212)(2)(2
2)(
2 1
2/1
2/1
0
1
0
2
00
0 dtdtdttxdttxdttxT
AT
x(t)
Note: (1) x(– t)=x(t) x(t) is an even function(2) f0 = 1 / T = ½
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Fourier Series Representation of Periodic Signals - Example
1
0
0
2/
0
0
0
0 )2cos()(2)2cos()(4
)2cos()(2
dttnftxdttnftxT
dttnftxT
ATT
n
2sin
4)2cos(2)2cos(2
1
2/1
0
2/1
0
0
n
ndttnfdttnf
2/
2/
0
0
0 )2sin()(2
)2sin()(2 T
T
T
n dttnftxT
dttnftxT
B
2/
0
0
0
2/
0 )2sin()(2
)2sin()(2 T
T
dttnftxT
dttnftxT
2/
0
0
2/
0
0 )2sin()(2
)2sin()(2 TT
dttnftxT
dttnftxT
Replacing t by –tin the first integralsin(-2nf t)=- sin(2nf t)
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Fourier Series Representation of Periodic Signals - ExampleSince x(– t)=x(t) as x(t) is an even function, then
Bn = 0 for n=1, 2, 3, …
1
000 )2sin()2cos(
2)(
nnn tnfBtnfA
Atx
tnn
ntx
n
cos2
sin4
)(1
4 4 4 4( ) cos cos3 cos5 cos 7 ...
3 5 7x t t t t t
4 1 1 1( ) cos cos3 cos5 cos 7 ...
3 5 7x t t t t t
Cosine is an even function
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Another Example
1
-1
1-1 2
T
-2
x1(t)
Note that x1(-t)= -x1(t) x(t) is an odd function
Also, x1(t)=x(t-1/2)
4 1 1 1 1 1 1 11( ) cos cos 3 cos 5 cos 7
2 3 2 5 2 7 2x t t t t t
x(t)
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Another Example
4 1 3 1 5 1 71( ) cos cos 3 cos 5 cos 7
2 3 2 5 2 7 2x t t t t t
4 1 1 11( ) in in3 sin 5 in7
3 5 7x t s t s t t s t
tt sin2
cos
tt 3sin
2
33 cos
tt 5sin2
55 cos
tt 7sin
2
77 cos
Sine is an odd function
Where:
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Fourier Transform For a periodic signal, spectrum consists of
discrete frequency components at fundamental frequency & its harmonics.
For an aperiodic signal, spectrum consists of a continuum of frequencies (non-discrete components). Spectrum can be defined by Fourier Transform For a signal x(t) with spectrum X(f), the following
relations hold
dfefXtx ftj 2 )()(
dtetxfX ftj 2 )()(
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Fourier Transform Example
x(t)A
22
dtetxfX ftj 2 )()(
2/
2/
22/
2/
2
2 )(
ftjftj efj
AdteAfX
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Fourier Transform Example
2/2
)2/2sin(
2
2
2
2
22
2 2/22/2
f
ff
f
A
j
ee
f
A fjfj
f
fA
f
fAfX
)sin(
2/2
)2/2sin()(
j
ee jj
2sin
2cos
jj ee
A
1/
f
Study the effect of the pulse width
Sin (x) / x“sinc” function
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The narrower a function is in one domain, the wider its transform is in the other domain
The Extreme Cases
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Power Spectral Density & Bandwidth
Absolute bandwidth of any time-limited signal is infinite
However, most of the signal power will be concentrated in a finite band of frequencies
Effective bandwidth is the width of the spectrum portion containing most of the signal power.
Power spectral density (PSD) describes the distribution of the power content of a signal as a function of frequency
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Signal Power A function x(t) specifies a signal in terms of
either voltage or current Assuming R = 1
Instantaneous signal power = V2 = i2 = |x(t)|2
Instantaneous power of a signal is related to average power of a time-limited signal, and is defined as
For a periodic signal, the averaging is taken over one period to give the total signal power
22
1
1( )
2 1
t
t
x t dtt t
2
0
1( )Total
T
P x t dtT
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Power Spectral Density & Bandwidth For a periodic signal, power spectral density
is
where (f) is
Cn is as defined before on slide 27, and f0 being the fundamental frequency
2
0( ) ( )nn
PSD f C f nf
1 =00 0( ) f
ff
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Power Spectral Density & Bandwidth For a continuous valued function S(f), power
contained in a band of frequencies f1 < f < f2
For a periodic waveform, the power through the first j harmonics is
2
1
)(2f
f
dffSP
j
nnCCP
1
220 2
125.0
COE 341 – Dr. Marwan Abu-Amara 43
Power Spectral Density & Bandwidth - Example Consider the following signal
The signal power is
7in
7
1 5sin
5
1 3in
3
1 in)( tsttststx
watt586.0 49
1
25
1
9
11
2
1
Power
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Fourier Analysis Example Consider the half-wave rectified cosine signal from
Figure B.1 on page 793:1. Write a mathematical expression for s(t)
2. Compute the Fourier series for s(t)
3. Write an expression for the power spectral density function for s(t)
4. Find the total power of s(t) from the time domain
5. Find a value of n such that Fourier series for s(t) contains 95% of the total power in the original signal
6. Determine the corresponding effective bandwidth for the signal
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Example (Cont.)1. Mathematical expression for s(t):
cos(2 ) , -T/4 T/40 , T/4 3T/4( ) oA f t t
ts t
-T/4-3T/4 +3T/4+T/4
Where f0 is the fundamental frequency, f0 = (1/T)
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Example (Cont.)2. Fourier Analysis:
1 )2/sin( where, 2
)2/sin(2)2/sin()2/sin(
)2/sin()2/sin(/2
)/2sin(2
)2cos(2
)(2
4/
4/
4/
4/
4/
4/
0
A
AA
A
T
Tt
T
A
dttfT
Adtts
TA
T
T
T
T
o
T
T
T
dttxT
A0
0 )(2
f0 = (1/T)
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Example (Cont.)2. Fourier Analysis (cont.):
/ 4 / 4
/ 4 / 4
/ 4
/ 4
2 2( )cos(2 ) cos(2 )cos(2 )
sin(2 ( 1) ) sin(2 ( 1) )2 , for 1
4 ( 1) 4 ( 1)
cos( / 2) cos( / 2) , for
( 1) ( 1)
T T
n o o o
T T
T
o o
o o T
AA s t nf t dt f t nf t dt
T T
n f t n f tAn
T n f n f
A n nn
n n
1
2
sin( ) sin( ) cos( )cos( ) , and
2( ) 2( )
sin( ) cos(
Note:
)
ax bx ax bxax bx dx
a b a b
x x
T
n dttnftxT
A0
0 )2cos()(2
f0 = (1/T)
COE 341 – Dr. Marwan Abu-Amara 48
Example (Cont.)2. Fourier Analysis (cont.):
2 2
2 2
2
( ) ( )
( ) ( )
( )
2
0 , for and 1
( 1) ( 1)
( 1) ( 1)
( 1) ( 1) ( 1)( 1) ( 1)
( 1)( 1)
( 1) ( 1) ( 1)( 1)
( 1)
oddn n
n n
n
n
n
A n n
AA
n n
A n n
n n
An n
n
2(1 )
2
2 ( 1) , for
( 1)even
n
An
n
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Example (Cont.)2. Fourier Analysis (cont.):
/ 4 / 4
1
/ 4 / 4
/ 42
/ 4
/ 4
/ 4
2 2( )cos(2 1 ) cos(2 )cos(2 )
2 cos (2 )
sin(4 )2 2 sin( ) sin( )
2 4 2 4 8
2
T T
n o o o
T T
T
o
T
T
o
o oT
AA s t f t dt f t f t dt
T T
Af t dt
T
f tA t A T
T f T f
A
Note: cos2 = ½(1 + cos 2)
COE 341 – Dr. Marwan Abu-Amara 50
Example (Cont.)2. Fourier Analysis (cont.):
/ 4 / 4
/ 4 / 4
/ 4
/ 4
2 2( )sin(2 ) cos(2 )sin(2 )
cos(2 ( 1) ) cos(2 ( 1) )2 , for 1
4 ( 1) 4 ( 1)
0
T T
n o o o
T T
T
o o
o o T
AB s t nf t dt f t nf t dt
T T
n f t n f tAn
T n f n f
, for 1n
cos( ) cos( ) sin( )cos(Note
): )
2( 2( )
ax bx ax bxax bx dx
a b a b
T
n dttnftxT
B0
0 )2sin()(2
COE 341 – Dr. Marwan Abu-Amara 51
Example (Cont.)2. Fourier Analysis (cont.):
/ 4 / 4
1
/ 4 / 4
/ 4
/ 4
/ 4
/ 4
2 2( )sin(2 1 ) cos(2 )sin(2 )
sin(4 )
cos(4 ) cos( ) cos( )4 4
0
T T
n o o o
T T
T
o
T
T
o T
AB s t f t dt f t f t dt
T T
Af t dt
T
A Af t
COE 341 – Dr. Marwan Abu-Amara 52
Example (Cont.)2. Fourier Analysis (cont.):
2
2
1
(1 )
22,4,6,...
o 1
(1 )
2
( ) cos(2 ) sin(2 )2
2 ( 1) cos(2 ) cos(2 )
2 1
2C ,
20 , is odd and 1
2 ( 1) , 2, 4,
( 1)
n
n
on o n o
n
o on
n
n
As t A nf t B nf t
A A Af t nf t
n
A AC
C n n
AC n
n
6,...
00 AC 22nnn BAC
COE 341 – Dr. Marwan Abu-Amara 53
Example (Cont.)3. Power Spectral Density function (PSD):
Or more accurately:
220
1
1
4 2 nn
CPSD C
220
1
1( ) ( )
4 2 n on
CPSD f C f nf
COE 341 – Dr. Marwan Abu-Amara 54
Example (Cont.)3. Power Spectral Density function (PSD):
220
1
2 2 2
2 2 2 22,4,6,...
1( ) ( )
4 2
( )2 ( ) ( )
8 ( 1)
n on
oo
n
CPSD f C f nf
f nfA A Af f f
n
COE 341 – Dr. Marwan Abu-Amara 55
Example (Cont.)4. Total Power:
3 / 4 / 422 2
/ 4 / 4
/ 42
/ 4
2
1( ) cos (2 )
sin(4 )
2 8
4
T T
s o
T T
T
o
o T
AP s t dt f t dt
T T
f tA t
T f t
A
Note: cos2 = ½(1 + cos 2)
COE 341 – Dr. Marwan Abu-Amara 56
Example (Cont.)5. Finding n such that we get 95% of total power:
2 2 220
0 2 2
2
2
For
40.1014
4 4
0.1014% 40.5%
0.25
0
n
C A APSD A
APower
A
n
COE 341 – Dr. Marwan Abu-Amara 57
Example (Cont.)5. Finding n such that we get 95% of total power:
2 2 2 220 1
1 2
2
2
For
0.2264 2 8
0.226% 90.5%
0.
1
25
n
C C A APSD A
APower
n
A
COE 341 – Dr. Marwan Abu-Amara 58
Example (Cont.)5. Finding n such that we get 95% of total power:
6. Effective bandwidth with 95% of total power:
Beff = fmax – fmin
= 2f0 – 0 = 2f0
2 2 2 2 2 220 1 2
2 2 2
2
2
For
20.2485
4 2 2 8 9
0.2485% 99.41
2
2
0. 5%
n
C C C A A APSD A
AP wer
A
n
o
0 f0 2f0 3f0
…
f
Beff
COE 341 – Dr. Marwan Abu-Amara 59
Data Rate and Bandwidth
Any transmission system has a limited band of frequencies
This limits the data rate that can be carried Example on pages 74 – 76 of textbook
COE 341 – Dr. Marwan Abu-Amara 60
Bandwidth and Data Rates
(Bsys = 2B) = (Bsig = 4f)
Bsys = (Bsig = 4f)
f 3f
f 3f 5f
Period T = 1/f
1 0 1 0
Data Element,Signal Element
T/2
Data rate = 1/(T/2) = 2/T bits per sec = 2fGiven a bandwidth B,Data rate = 2f = 2(B/4) = B/2
Two ways to double the data rate… To double the data rate you need to double f
1. Double the transmission system bandwidth, with the same receiver and error rate (same received waveform)
f 3f 5f
New bandwidth: 2B,Data rate = 2f = 2(2B/4) = B
2. Same transmission system bandwidth, B, with a better receiver, higher S/N, or by tolerating more error (poorer received waveform)
(Bsys = B) = (Bsys = 2f)
Bandwidth: B,Data rate = 2f = 2(B/2) = B
1 1 1 10 0 0 0
1 1 1 10 0 0 0
B
2B
B
COE 341 – Dr. Marwan Abu-Amara 61
Bandwidth and Data Rates Increasing the data rate (bps) with the same BW
means working with inferior waveforms at the receiver, which may require: Better signal to noise ratio at RX (larger signal relative to
noise): Shorter link spans Use of more repeaters/amplifiers Better shielding of cables to reduce noise, etc.
More sensitive (& costly!) receiver Dealing with higher error rates
Tolerating them Adding more efficient means for error detection and correction-
this also increases overhead!.
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Bandwidth and Data Rates
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Analog and Digital Data Transmission Data
Entities that convey meaning Signal
Electric or electromagnetic representations of data Transmission
Communication of data by propagation and processing of signals
COE 341 – Dr. Marwan Abu-Amara 64
Analog and Digital Data Transmission Data
Can be either Analog data or Digital data Signal
Can use either Analog signal or Digital signal to convey the data
Transmission Can use either Analog transmission or Digital
transmission to carry the signal
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Analog and Digital Data Analog
Continuous values within some interval e.g. sound, video
Digital Discrete values e.g. text, integers
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Acoustic Spectrum (Analog)
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Analog and Digital Signals Means by which data are propagated Analog
Continuously variable Various media
wire, fiber optic, space Speech bandwidth 100Hz to 7kHz Telephone bandwidth 300Hz to 3400Hz Video bandwidth 4MHz
Digital Use two DC components
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Advantages & Disadvantages of Digital Signals Advantages:
Cheaper Less susceptible to noise
Disadvantages: Greater attenuation
Pulses become rounded and smaller Leads to loss of information
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Attenuation of Digital Signals
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Components of Speech Frequency range (of hearing) 20Hz-20kHz
Speech 100Hz-7kHz Easily converted into electromagnetic signal
for transmission Sound frequencies with varying volume
converted into electromagnetic frequencies with varying voltage
Limit frequency range for voice channel 300-3400Hz
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Conversion of Voice Input into Analog Signal
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Video Components USA - 483 lines scanned per frame at 30 frames
(scans) per second 525 lines but 42 lost during vertical retrace
So 525 lines x 30 frames (scans) = 15750 lines per second 63.5s per line
11s for retrace, so 52.5 s per video line Max frequency if line alternates black and white Horizontal resolution is about 450 lines giving 225
cycles of wave in 52.5 s Max frequency (for black and white video) is 4.2MHz
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Binary Digital Data
From computer terminals etc. Two dc components Bandwidth depends on data rate
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Conversion of PC Input to Digital Signal
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Data and Signals
Usually use digital signals for digital data and analog signals for analog data
Can use analog signal to carry digital data Modem
Can use digital signal to carry analog data Compact Disc audio
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Analog Signals Carrying Analog and Digital Data
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Digital Signals Carrying Analog and Digital Data
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Four Data/Signal Combinations Signal
Analog Digital
Data
Analog
- Same spectrum as data (base band): e.g. Conventional Telephony
- Different spectrum (modulation): e.g. AM, FM Radio
Use a (converter): codec, e.g. for PCM
(pulse code modulation)
Digital Use a (converter): modem e.g. with the
V.90 standard
-Simple two signal levels: e.g. NRZ code-Special Encoding: e.g. Manchester code (Chapter 5)
COE 341 – Dr. Marwan Abu-Amara 79
Analog Transmission
Analog signal transmitted without regard to content
Analog signal may be analog or digital data Attenuated over distance Use amplifiers to boost signal Also amplifies noise
COE 341 – Dr. Marwan Abu-Amara 80
Digital Transmission Concerned with content Integrity endangered by noise, attenuation
etc. Repeaters used
Repeater receives signal Extracts bit pattern Retransmits
Attenuation is overcome by a repeater by reconstructing the signal
Noise is not amplified
COE 341 – Dr. Marwan Abu-Amara 81
Four Signal/Transmission Mode Combinations
Transmission mode
Analog- Uses amplifiers- Not concerned with what data the signal represents
- Noise is cumulative
Digital- Uses repeaters- Assumes signal represents digital data, recovers it and represents it as a new outbound signal
- This way, noise is not cumulative
Signal
Analog
OK
Makes sense only if the analog signal represents digital data
Digital Avoid OK
COE 341 – Dr. Marwan Abu-Amara 82
Advantages of Digital Transmission Digital technology
Low cost LSI/VLSI technology Data integrity
Longer distances over lower quality lines Capacity utilization
High bandwidth links economical High degree of multiplexing easier with digital techniques
Security & Privacy Encryption
Integration Can treat analog and digital data similarly
COE 341 – Dr. Marwan Abu-Amara 83
Transmission Impairments Signal received may differ from signal
transmitted Analog signal - degradation of signal
quality Digital signal - bit errors Caused by
Attenuation and attenuation distortion Delay distortion Noise
COE 341 – Dr. Marwan Abu-Amara 84
Attenuation Signal strength falls off with distance Depends on medium (guided vs. unguided) Attenuation affects received signal strength
received signal strength must be enough to be detected received signal strength must be sufficiently higher than noise to
be received without error signal strength can be achieved by using amplifiers or repeaters
Attenuation is an increasing function of frequency Different frequency components of a signal get attenuated
differently Signal distortion Particularly significant with analog signals
for digital signals, strength of signal falls of rapidly with frequency Can overcome signal distortion using equalizers
COE 341 – Dr. Marwan Abu-Amara 85
Delay Distortion Only in guided media Propagation velocity varies with frequency
Highest at 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)
Badly affects digital data due to bit spill-over (intersymbol interference) major limitation to max bit rate over a transmission channel
Can overcome delay distortion using equalizers
COE 341 – Dr. Marwan Abu-Amara 86
Noise Additional unwanted signals inserted between
transmitter and receiver The most limiting factor in communication
systems Noise categories:
Thermal Intermodulation Crosstalk Impulse
COE 341 – Dr. Marwan Abu-Amara 87
Thermal (White) Noise Due to thermal agitation of electrons Uniformly distributed across the bandwidth Power of thermal noise present in a bandwidth B
(Hz) is given by
T is absolute temperature in kelvin and k is Boltzmann’s constant (k = 1.3810-23 J/K)
0 (watts)
= 228.6 10log 10log (dBw)
N kTB N B
T B
= =
- + +
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
COE 341 – Dr. Marwan Abu-Amara 88
Intermodulation Occurs when signals at different frequencies
share same transmission medium Produces signals that are the sum and/or the
difference of original frequencies sharing the medium f1, f2 (f1+f2) and (f1-f2) Caused by nonlinearities in the medium and
equipment, e.g. due to overdrive and saturation of amplifiers
Resulting frequency components (i.e. f1+f2 and f1-f2) may fall within valid signal bands, thus causing interference
COE 341 – Dr. Marwan Abu-Amara 89
Crosstalk & Impulse Crosstalk
A signal from one channel picked up by another channel e.g. Coupling between twisted pairs, antenna pick up,
leakage between adjacent channels in FDM, etc.
Impulse Irregular pulses or spikes Short duration High amplitude e.g. External electromagnetic interference Minor effect on analog signals but major effect on digital
signals, particularly at high data rates
COE 341 – Dr. Marwan Abu-Amara 90
Channel Capacity Channel capacity: Maximum data rate usable under given
communication conditions How BW, signal level, noise and other impairments, and the amount
of error tolerated limit the channel capacity? Max data rate
= Function (BW, Signal wrt noise, Error rate allowed) Max data rate: Max rate at which data can be communicated, 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) Error rate: bits received in error/total bits transmitted. Equal to the bit
error probability
COE 341 – Dr. Marwan Abu-Amara 91
1. Nyquist Bandwidth: (Noise-free, Error-free) Idealized, theoretical Assumes a noise-free, error-free channel Nyquist: If rate of signal transmission is 2B then a signal with
frequencies no greater than B is sufficient to carry that signalling rate
In other words: Given bandwidth B, highest signalling rate possible is 2B signals/s
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
For the same B, data rate can be increased by sending one of M different signal levels (symbols): as a signal level now represents log2M bits
Generalized Nyquist Channel Capacity, C = 2B log2M bits/s (bps)
Signals/s bits/signal
COE 341 – Dr. Marwan Abu-Amara 92
Nyquist Bandwidth: Examples C = 2B log2M bits/s
C = Nyquist Channel Capacity B = Bandwidth M = Number of discrete signal levels (symbols) used
Telephone Channel: B = 3400-300 = 3100 Hz With a binary signal (M = 2):
C = 2B log2 2 = 2B = 6200 bps With a quandary signal (M = 4):
C = 2B log2 4 = 2B x 2 = 4B = 12,400 bps
Practical limit: larger M makes it difficult for the receiver to operate, particular with noise
0
1
00
01
10
11
COE 341 – Dr. Marwan Abu-Amara 93
2. Shannon Capacity Formula: (Noisy, Error-Free)
Assumes error-free operation with noise Data rate, noise, error: A given noise burst affects more bits at
higher data rates, which increases the error rate So, maximum error-free data rate increases with reduced noise Signal to noise ratio SNR = signal / noise levels
SNRdB= 10 log10 (SNR) dBs
Shannon Capacity C = B log2(1+SNR):
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 errors C/B: Spectral (bandwidth) efficiency, BE, (bps/Hz) (>1) Larger BEs mean better utilizing of a given B for transmitting data
fast.
Caution! Log2 Not Log10
Caution! Ratio- Not log
COE 341 – Dr. Marwan Abu-Amara 94
Shannon Capacity Formula: Comments
Formula says: for data rates calculated C, it is theoretically possible to find an encoding scheme that gives error-free transmission.
But it does not say how… It is a theoretical approach based on thermal (white) noise
only However, in practice, we also have impulse noise and
attenuation and delay distortions So, maximum error-free data rates obtained in practice are
lower than the C predicted by this theoretical formula However, maximum error-free data rates can be used to
compare practical systems: The higher that rate the better the system is
COE 341 – Dr. Marwan Abu-Amara 95
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!
High SNR obtained through excessive amplification may introduce nonlinearities: distortion and intermediation noise!
High Bandwidth B opens the system for more thermal noise (kTB), and therefore reduces SNR!
COE 341 – Dr. Marwan Abu-Amara 96
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 = 251 Using Shannon’s formula: C = B log2 (1+ SNR)
C = 106 * log2(1+251) ~ 106 * 8 = 8 Mbps Based on Nyquist’s formula, determine M that gives the above
channel capacity:
C = 2B log2 M
8 * 106 = 2 * (106) * log2 M
4 = log2 M
M = 16
COE 341 – Dr. Marwan Abu-Amara 97
3. Eb/N0 (Signal Energy per Bit/Noise Power density per Hz) (Noise and Error Together) Handling both noise and error together
Eb/N0: A standard quality measure for digital communication system performance
Eb/N0 Can be independently related to the error rate Expresses SNR in a manner related to the data rate, R Eb = Signal energy per bit (Joules)
= Signal power (Watts) x bit interval Tb (second)
= S x (1/R) = S/R N0 = Noise power (watts) in 1 Hz = kT
0 0
/b bE ST S R S
N N kT kTR
0 0
/b T TE B BS R SSNR
N N N R R
COE 341 – Dr. Marwan Abu-Amara 98
Eb/N0 Example 1:
Given: Eb/No = 8.4 dB (minimum) is needed to achieve a bit error rate of 10-4
Given: The effective noise temperature is 290oK (room temperature) Data rate is 2400 bps
What is the minimum signal level required for the received signal?
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
0
10log 10log 10log
10 log 228.6 10log
bdBW
dB
dBW
ES R k T
N
S R dBW T
COE 341 – Dr. Marwan Abu-Amara 99
Eb/N0 (Cont.) Bit error rate for digital data is a
decreasing function of Eb/N0 for a given signal encoding scheme
Which encoding scheme is better: A or B?
Get Eb/N0 to achieve a desired error rate, then determine other parameters from formula, e.g. S, SNR, R, etc. (Design)
Error performance of a given system (Analysis)
Effect of S, R, T on error performance
0
10log 10log 10log
10log 228.6 10log
bdBW
dB
dBW
ES R k T
N
S R dBW T
Lower E
rror Rate: larger E
b/N0
A B
0 0
/b T TE B BS R SSNR
N N N R R
BetterEncoding
COE 341 – Dr. Marwan Abu-Amara 100
Eb/N0 (Cont.) From Shannon’s formula:
C = B log2(1+SNR)We have:
From the Eb/N0 formula:
With R = C, substituting for SNR we get:
Relates achievable spectral efficiency C/B (bps/Hz) to Eb/N0
COE 341 – Dr. Marwan Abu-Amara 101
Eb/N0 (Cont.) Example 2
Find the minimum Eb/N0 required to achieve a spectral efficiency (C/B) of 6 bps/Hz:
Substituting in the equation above:
Eb/N0 = (1/6) (26 - 1) = 10.5 = 10.21 dB