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COE 342: Data & Computer Communications (T042)Dr. Marwan Abu-Amara
Chapter 3: Data Transmission
COE 342 (T042) – Dr. Marwan Abu-
Amara 2
Agenda Concepts & Terminology Decibels and Signal Strength Fourier Analysis Analog & Digital Data Transmission Transmission Impairments Channel Capacity
COE 342 (T042) – Dr. Marwan Abu-
Amara 3
Terminology (1) Transmitter Receiver Medium
Guided medium e.g. twisted pair, optical fiber
Unguided medium e.g. air, water, vacuum
COE 342 (T042) – Dr. Marwan Abu-
Amara 4
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 5
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 6
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 7
Analogue & Digital Signals
COE 342 (T042) – Dr. Marwan Abu-
Amara 8
PeriodicSignals
COE 342 (T042) – Dr. Marwan Abu-
Amara 9
Sine Wave Peak Amplitude (A)
maximum strength of signal volts
Frequency (f) Rate of change of signal Hertz (Hz) or cycles per second Period = time for one repetition (T) T = 1/f
Phase () Relative position in time
COE 342 (T042) – Dr. Marwan Abu-
Amara 10
Varying Sine Wavess(t) = A sin(2ft +)
COE 342 (T042) – Dr. Marwan Abu-
Amara 11
Wavelength Distance occupied by one cycle Distance between two points of
corresponding phase in two consecutive cycles
Assuming signal velocity v
= vT f = v c = 3*108 m/sec (speed of light in free space)
COE 342 (T042) – Dr. Marwan Abu-
Amara 12
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 13
Addition of FrequencyComponents(T=1/f)
COE 342 (T042) – Dr. Marwan Abu-
Amara 14
FrequencyDomainRepresentations
COE 342 (T042) – Dr. Marwan Abu-
Amara 15
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 16
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 17
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 18
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 19
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 20
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 21
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 22
Fourier Series Any periodic signal 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
COE 342 (T042) – Dr. Marwan Abu-
Amara 23
Fourier Series Amplitude-phase representation
1
00 )2cos(
2)(
nnn tnfC
Ctx
00 AC 22nnn BAC
n
nn A
B1tan
COE 342 (T042) – Dr. Marwan Abu-
Amara 24
COE 342 (T042) – Dr. Marwan Abu-
Amara 25
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 26
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)
COE 342 (T042) – Dr. Marwan Abu-
Amara 27
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 28
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)
2
17 cos
7
1
2
15 cos
5
1
2
13 cos
3
1
2
1 cos
4)(1 tttttx
COE 342 (T042) – Dr. Marwan Abu-
Amara 29
Another Example
2
77 cos
7
1
2
55 cos
5
1
2
33 cos
3
1
2 cos
4)(1
tttttx
7in
7
1 5sin
5
1 3in
3
1 in
4)(1 tsttststx
tt sin2
cos
tt 3sin
2
33 cos
tt 5sin2
55 cos
tt 7sin
2
77 cos
COE 342 (T042) – Dr. Marwan Abu-
Amara 30
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. 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 )()(
COE 342 (T042) – Dr. Marwan Abu-
Amara 31
COE 342 (T042) – Dr. Marwan Abu-
Amara 32
Fourier Transform Example
x(t)A
22
dtetxfX ftj 2 )()(
2/
2/
22/
2/
2
2 )(
ftjftj efj
AdteAfX
COE 342 (T042) – Dr. Marwan Abu-
Amara 33
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 34
Signal Power A function x(t) specifies a signal in terms of
either voltage or current Instantaneous power of a signal is related to
average power of a time-limited signal, and is defined as
For a periodic signal, the average power in one period is
2)(tx2
2
1
1( )
2 1
t
t
x t dtt t
T
dttxT
0
2)(
1
COE 342 (T042) – Dr. Marwan Abu-
Amara 35
Power Spectral Density & Bandwidth Absolute bandwidth of any time-limited signal is
infinite. Most power in a signal is concentrated in finite
band. Effective bandwidth is the spectrum portion
containing most of the power. Power spectral density (PSD) describes power
content of a signal as a function of frequency
COE 342 (T042) – Dr. Marwan Abu-
Amara 36
Power Spectral Density & Bandwidth For a periodic signal, power spectral density
is
where (f) is
2
0( ) ( )nn
PSD f C f nf
1 =00 0( ) f
ff
COE 342 (T042) – Dr. Marwan Abu-
Amara 37
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 342 (T042) – Dr. Marwan Abu-
Amara 38
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 39
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. Find the total power of s(t)
4. Find a value of n such that Fourier series for s(t) contains 95% of the total power in the original signal
5. Write an expression for the power spectral density function for s(t)
COE 342 (T042) – Dr. Marwan Abu-
Amara 40
Example (Cont.)1. Mathematical expression for s(t):
cos(2 ) , -T/4 T/40 , T/4 3T/4( ) oA f t t
ts t
COE 342 (T042) – Dr. Marwan Abu-
Amara 41
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 42
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 43
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 44
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 45
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 46
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 342 (T042) – Dr. Marwan Abu-
Amara 47
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,...
COE 342 (T042) – Dr. Marwan Abu-
Amara 48
Example (Cont.)3. 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
COE 342 (T042) – Dr. Marwan Abu-
Amara 49
Example (Cont.)4. 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 342 (T042) – Dr. Marwan Abu-
Amara 50
Example (Cont.)4. 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 342 (T042) – Dr. Marwan Abu-
Amara 51
Example (Cont.)4. Finding n such that we get 95% of total power:
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 52
Example (Cont.)5. 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 342 (T042) – Dr. Marwan Abu-
Amara 53
Example (Cont.)5. 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 342 (T042) – Dr. Marwan Abu-
Amara 54
Signal with DC Component
COE 342 (T042) – Dr. Marwan Abu-
Amara 55
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 65 & 66
COE 342 (T042) – Dr. Marwan Abu-
Amara 56
Analog and Digital Data Transmission Data
Entities that convey meaning Signals
Electric or electromagnetic representations of data Transmission
Communication of data by propagation and processing of signals
COE 342 (T042) – Dr. Marwan Abu-
Amara 57
Analog and Digital Data Analog
Continuous values within some interval e.g. sound, video
Digital Discrete values e.g. text, integers
COE 342 (T042) – Dr. Marwan Abu-
Amara 58
Acoustic Spectrum (Analog)
COE 342 (T042) – Dr. Marwan Abu-
Amara 59
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 60
Advantages & Disadvantages of Digital Cheaper Less susceptible to noise Greater attenuation
Pulses become rounded and smaller Leads to loss of information
COE 342 (T042) – Dr. Marwan Abu-
Amara 61
Attenuation of Digital Signals
COE 342 (T042) – Dr. Marwan Abu-
Amara 62
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 63
Conversion of Voice Input into Analog Signal
COE 342 (T042) – Dr. Marwan Abu-
Amara 64
Video Components USA - 483 lines scanned per frame at 30 frames per
second 525 lines but 42 lost during vertical retrace
So 525 lines x 30 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 of 4.2MHz
COE 342 (T042) – Dr. Marwan Abu-
Amara 65
Binary Digital Data
From computer terminals etc. Two dc components Bandwidth depends on data rate
COE 342 (T042) – Dr. Marwan Abu-
Amara 66
Conversion of PC Input to Digital Signal
COE 342 (T042) – Dr. Marwan Abu-
Amara 67
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
COE 342 (T042) – Dr. Marwan Abu-
Amara 68
Analog Signals Carrying Analog and Digital Data
COE 342 (T042) – Dr. Marwan Abu-
Amara 69
Digital Signals Carrying Analog and Digital Data
COE 342 (T042) – Dr. Marwan Abu-
Amara 70
Analog Transmission
Analog signal transmitted without regard to content
May be analog or digital data Attenuated over distance Use amplifiers to boost signal Also amplifies noise
COE 342 (T042) – Dr. Marwan Abu-
Amara 71
Digital Transmission Concerned with content Integrity endangered by noise, attenuation
etc. Repeaters used Repeater receives signal Extracts bit pattern Retransmits Attenuation is overcome Noise is not amplified
COE 342 (T042) – Dr. Marwan Abu-
Amara 72
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 342 (T042) – Dr. Marwan Abu-
Amara 73
Transmission Impairments Signal received may differ from signal
transmitted Analog - degradation of signal quality Digital - bit errors Caused by
Attenuation and attenuation distortion Delay distortion Noise
COE 342 (T042) – Dr. Marwan Abu-
Amara 74
Attenuation Signal strength falls off with distance Depends on medium Received signal strength:
must be enough to be detected must be sufficiently higher than noise to be
received without error Attenuation is an increasing function of
frequency
COE 342 (T042) – Dr. Marwan Abu-
Amara 75
Delay Distortion
Only in guided media Propagation velocity varies with frequency
COE 342 (T042) – Dr. Marwan Abu-
Amara 76
Noise (1) Additional signals inserted between
transmitter and receiver Thermal
Due to thermal agitation of electrons Uniformly distributed White noise
Intermodulation Signals that are the sum and difference of original
frequencies sharing a medium
COE 342 (T042) – Dr. Marwan Abu-
Amara 77
Noise (2) Crosstalk
A signal from one line is picked up by another Impulse
Irregular pulses or spikes e.g. External electromagnetic interference Short duration High amplitude
COE 342 (T042) – Dr. Marwan Abu-
Amara 78
More on Thermal (White) Noise 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
= =
- + +
COE 342 (T042) – Dr. Marwan Abu-
Amara 79
Channel Capacity Data rate
In bits per second Rate at which data can be communicated
Bandwidth In cycles per second of Hertz Constrained by transmitter and medium
COE 342 (T042) – Dr. Marwan Abu-
Amara 80
Nyquist Bandwidth If rate of signal transmission is 2B then signal
with frequencies no greater than B is sufficient to carry signal rate
Given bandwidth B, highest signal rate is 2B Given binary signal, data rate supported by B
Hz is 2B bps Can be increased by using M signal levels C= 2B log2M
COE 342 (T042) – Dr. Marwan Abu-
Amara 81
Shannon Capacity Formula Consider data rate,noise and error rate Faster data rate shortens each bit so burst of
noise affects more bits At given noise level, high data rate means higher
error rate Signal to noise ratio (in decibels) SNRdB
=10 log10 (signal/noise)
Capacity C=B log2(1+SNR) This is error free capacity
COE 342 (T042) – Dr. Marwan Abu-
Amara 82
Eb/N0 Determines digital data rates and error rates Standard quality measure for digital
communication system performance Ratio of signal energy per bit to noise power
density per Hertz Eb = energy per bit in a signal (Joules) = STb,
where S = signal power (Watts), Tb = time required to send 1 bit (seconds) R = bit rate = 1/ Tb
0 0
/b bE ST S R S
N N kT kTR
COE 342 (T042) – Dr. Marwan Abu-
Amara 83
Eb/N0 (Cont.)
Bit error rate for digital data is a decreasing function of Eb/N0
Given Eb/N0 to achieve a desired error rate, parameters in formula above may be selected
Eb/N0 does not depend on bandwidth (vs. SNR)
N = N0BT
0
10log 10log 10log
10 log 228.6 10log
bdBW
dB
dBW
ES R k T
N
S R dBW T
0 0
/b T TE B BS R SSNR
N N N R R
COE 342 (T042) – Dr. Marwan Abu-
Amara 84
Eb/N0 (Cont.) Shannon’s result can be rewritten as:
Relates achievable spectral efficiency C/B to Eb/N0
0
0
2 1
Setting and in
2 1
C B
b TT
C Bb
SSNR
NE B
B B R C SNRN R
E B
N C