ECE 6640
Digital Communications
Dr. Bradley J. Bazuin
Assistant Professor
Department of Electrical and Computer Engineering
College of Engineering and Applied Sciences
Course/Lecture Overview
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Syllabus
Personal Intro.
Textbook/Materials Used
Additional Reading
ID and Acknowledgment of Policies
• Textbook
• Chap. 1
ECE 6640
2
Syllabus
•
Everything useful for this class can be found on Dr. Bazuin’s web site!
– http://homepages.wmich.edu/~bazuinb/
•
The class web site is at
– http://homepages.wmich.edu/~bazuinb/ECE6640/ECE6640_Sp16.html
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The syllabus …
– http://homepages.wmich.edu/~bazuinb/ECE6640/Syl_6640.pdf
ECE 6640
3
Who am I?
• Dr. Bradley J. Bazuin
– Born and raised in Grand Rapids Michigan
– Undergraduate BS in Engineering and Applied Sciences, Extensive
Electrical Engineering from Yale University in 1980
– Graduate MS and PhD in Electrical Engineering from Stanford
University in 1982 and 1989, respectively.
– Industrial Experience – Digital, ASIC, System Engineering
• Part-time ARGOSystems, Inc. (purchased by Boeing) 1981-1989
• Full-time ARGOSystems, Inc. 1989-1991
• Full-time Radix Technologies 1991-2000
– Academic Experience – Electrical and Computer Engineering
ECE 6640
• Term-appointed Faculty, WMU ECE Dept. 2000-2001
• Tenure track Assistant Professor, WMU ECE Dept. 2001-2007
• Tenured Associate Professor, WMU ECE Dept. 2007- present
4
Research Activities and Interests
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Sunseeker
– Adviser to solar car team
– Electrical Systems: Li battery protection system, Controller Area Network (CAN)
based sensors and controllers, Solar Array Energy Collection and Conversion
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Center for the Advancement of Printed Electronics (CAPE)
– Printed electronic device design, fabrication and testing
– Semiconductor Physics
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Physical Layer Communication Signal Processing
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Software Defined Radios (SDR) – USRP and GNURadio.org
Mulitrate Signal Processing (digital channel bank analysis and synthesis, filter-decimation and
interpolation-filter design methods)
Adaptive Filtering and Systems (channel equalization, smart-antenna spatial beamforming)
Communication-based Digital Signal Processing Algorithm Implementation
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–
ECE 6640
Xilinx programmable devices
Parallel processing, hosts including NVIDIA GPUs with CUDA and multithreaded applications
5
Required Textbook/Materials
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Digital Communications, 5th ed., John G. Proakis and
Masoud Salehi, McGraw-Hill Higher Education, 2008.
ISBN: 978-0-07-295716-7.
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MATLAB, Student Edition
MATLAB Signal Processing Toolbox
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The MATH Works,
MATLAB and Signal Processing Toolbox
http://www.mathworks.com/
• Recommend: Communication Systems Toolbox
ECE 6640
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Supplemental Books and Materials
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ECE 6640
Bernard Sklar, “Digital Communications, Fundamentals and Applications,”
Prentice Hall PTR, Second Edition, 2001.
ISBN: 0-13-084788-716-7.
Michael Rice, Digital Communications: A Discrete Approach, Pearson
Prentice Hall, 2009. ISBN: 978-0-13-030497-1.
John G. Proakis and Masoud Salehi, “Communication Systems Engineering,
2nd ed.”, Prentice Hall, 2002. ISBN: 0-13-061793-8.
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGrawHill, 2010. ISBN: 978-0-07-338040-7.
Leon W. Couch II, “Digital and Analog Communication Systems, 7th ed.”,
Prentice Hall, 2007. ISBN: 0-13-142492-0.
Stephen G. Wilson, “Digital Modulation and Coding, ” Prentice-Hall, 1996.
ISBN: 0-13-210071-1.
Ezio Biglieri, D. Divsalar, P.J. McLane, M.K. Simon, “Introduction to
Trellis-Coded Modulation with Applications”, Macmillan, 1991. ISBN: 0-02309965-8.
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Identification and Acknowledgement
• Identification for Grade Posting,
Course and University Policies, and
Acknowledgement
• Please read, provide unique identification, sign and date,
and return to Dr. Bazuin.
ECE 6640
8
Course/Text Overview
Chap. 1: Introduction
Chap. 2: Deterministic and Random
Signal Analysis
Chap. 3: Digital Modulation Schemes
Chap. 4: Optimum Receivers for
AWGN Channels
Chap. 5: Carrier and Symbol
Synchronization
Chap. 6: An Introduction to
Information Theory
Chap. 7: Linear Block Codes
Chap. 8: Trellis and Graph Based
Codes
ECE 6640
Chap. 9: Digital Communication
Through Band-Limited
Channels
Chap. 10: Adaptive Equalization
Chap. 11: Multichannel and
Multicarrier Systems
Chap. 12: Spread Spectrum Signals
for Digital Communications
Chap. 13: Fading Channels I:
Characterization and
Signaling
Chap. 14: Fading Channels II:
Capacity and Coding
Chap. 15: Multiple-Antenna Systems
Chap. 16: Multiuser Communications
9
Text Appendices
A. Matrices.
Eigenvalues and Eigenvectors of a Matrix. Singular-Value
Decomposition. Matrix Norm and Condition Number. MoorePenrose-PseudoInverse.
B. Error Probability for Multichannel Binary Signals.
C. Error Probabilities for Adaptive Reception of M-Phase Signals
D. Square Root Factorization.
ECE 6640
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Chap. 1
1. Introduction.
1.1 elements of Digital Communication Systems.
1.2 Communication Channels and Their
Characteristics.
1.3 Mathematical Models for Communication
Channels.
1.4 A Historical Perspective in the Development of
Digital Communications.
1.5 Overview of the Book
1.6 Bibliographical Notes and References
ECE 6640
11
Sklar’s Communications System
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
12
Sklar Signal Processing Functions
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
13
Simplified Communications System
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Information
Message
Format: making the message compatible with digital processing
Source Coding: efficient descriptions of information sources
Channel Coding: signal transformation enabling improved reception
performance after expected channel impairments
Modulation: formation of the baseband waveform
RF Mixing: frequency domain translation of baseband signal
Transmit/Receive: RF Amplifiers and Filters
Source
Encode
Format
Channel
Encode
Modulation
RF Mixing
Transmitter
Antenna
RF Signal
Noise
Bits
Symbols
Signals
Interference
Information
Message
ECE 6640
Reformat
Source
Decode
Channel
Decode
Demodulation
RF Mixing
Receiver
Antenna
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Communication Channel
Linear
Filtering
Nonlinear
Distortion
Attenuation
Noise
Interference
Transmitting
Antenna
Receiving
Antenna
RF Communication Channel
• The channel greatly effects received RF signals
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–
–
ECE 6640
–
Frequencey, Bandwidth, Transmitted Signal Power, RF Propagation
Attenuation, Nonlinear Distortion, Multipath, Range, Direction
Signal-to-Noise Ratio (SNR) and Signal-to-Interference Ratio (SIR)
Minimum Detectable Signal Level (MDS), Noise Floor
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Guided Wire Channels
• Switching rates for
“connected”
communications
systems
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
16
Wireless Electromagnetic Channels
• Wireless Frequency
Allocation Chart
• See next page or:
http://www.ntia.doc.g
ov/files/ntia/publicatio
ns/2003-allochrt.pdf
ECE 6640
Notes and figures are based on or taken from
materials in the course textbook:
J.G. Proakis and M.Salehi, Digital
Communications, 5th ed., McGraw-Hill, 2008.
17
Frequency Bands
The Electromagnetic Spectrum
ECE 4600
18
http://www.ntia.doc.gov/files/ntia/publications/2003-allochrt.pdf
Common Frequencies
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AM Radio:
FM Radio:
ISM:
ISM:
ISM:
Cell Phone:
PCS:
AWS:
BRS/EBS:
535-1705 kHz
88-108 MHz
433.05-434.79 and 902-928 MHz
2.4-2.5 GHz (wireless Ethernet & Bluetooth)
5.725-5.875 GHz (wireless Ethernet)
824-849 and 869-894 MHz
1850-1910 and 1930-1990 MHz
1710-1755 and 2110-2155 MHz
2.496–2.690 GHz
•
More Frequencies
– https://en.wikipedia.org/wiki/Cellular_frequencies
– https://en.wikipedia.org/wiki/ISM_band
ECE 4600
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Channel Considerations
Not Shown:
Point-to-point Communications – with
considerations for multipath.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
20
Models for Channels (1)
• Basic channels models for studying communications
r t   s t   hc t   s2 t   h2 t     s N t   hN t  nt 
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
21
Models for Channels (2)
• Significantly more complex models
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ECE 6640
Wide-bandwidth
Multipath
Weather
Most “real”, “commercial” systems you care about!
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
22
Received Signal
r t   s t   hc t   s2 t   h2 t     s N t   hN t  nt 
• The receiver must extract the original message as best
possible!
• Multiple signals with similar channel characteristics may be
present
• The RF channel(s) must be allocated and efficiently utilized.
– Frequency band assignments and regulations (power, direction, etc.)
– Signal modulation structures have different characteristics
ECE 6640
23
Why Digital?
1.
2.
3.
4.
ECE 6640
Noise, Interference, Path Loss, and Channel Impairments
(signal environment)
Cost
Inherent Availability
Reliability and Reconfigurability
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
24
Shannon Capacity
• A capacity limit defining the communication that is
possible in a channels with a defined bandwidth and
involving the signal-to-noise ratio.
• See information on the Shannon–Hartley Theorem
https://en.wikipedia.org/wiki/Shannon%E2%80%93Hartley_theorem

PS
C (bits / sec)  W  log 2 1 
 N o W
ECE 6640



25
Textbook Overview (1)
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Chapter 2 presents a review of deterministic and random signal analysis. Our
primary objectives in this chapter are to review basic notions in the theory of
probability and random variables and to establish some necessary notation.
Chapters 3 through 5 treat the geometric representation of various digital
modulation signals, their demodulation, their error rate performance in
additive, white Gaussian noise (AWGN) channels, and methods for
synchronizing the receiver to the received signal waveforms.
Chapters 6 to 8 treat the topics of source coding, channel coding and decoding,
and basic information theoretic limits on channel capacity, source information
rates, and channel coding rates.
Chapter 11 is focused on multichannel and multicarrier communication
systems, their efficient implementation, and their performance in AWGN
channels.
Chapter 12 presents an introduction to direct sequence and frequency hopped
spread spectrum signals and systems and an evaluation of their performance
under worst-case interference conditions.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
26
Textbook Overview (2)
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•
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The design of efficient modulators and demodulators for linear filter channels
with distortion is treated in Chapters 9 and 10. Channel equalization methods
are described for mitigating the effects of channel distortion.
The design of signals and coding techniques for digital communication
through fading multipath channels is the focus of Chapters 13 and 14. This
material is especially relevant to the design and development of wireless
communication systems.
Chapter 15 treats the use of multiple transmit and receive antennas for
improving the performance of wireless communication systems through signal
diversity and increasing the data rate via spatial multiplexing. The capacity of
multiple antenna systems is evaluated and space-time codes are described for
use in multiple antenna communication systems.
Chapter 16 of this book presents an introduction to multiuser communication
systems and multiple access methods.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
27
PREREQUISITE CONCEPTS
AND MATERIAL
ECE 6640
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Classification of Signals
• Deterministic and Random
• Periodic and Non-periodic
• Analog and Discrete/Digital
• Energy and Power Signals
ECE 6640
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Signal Processing “Tools”
• Communications involves:
– Frequency Domain Analysis – critical importance
• Continuous and Discrete
– Trig. and Complex Numbers – and all related math
– Analog and Digital Filters – important
• Finite Impulse Response (FIR) Filters – critical importance
• Filter Design Techniques – will be discussed and provided
– Adaptive Filters – saved for Dr. Bazuin’s ECE6950 course
– Probability and statistics is required (see Chap. 2)
(ECE 3800 or ECE5820 material)
• Random variables and processes, correlation, etc.
• Detection and estimation theory – will be discussed and provided
– Simulation of concepts – MATLAB (or similar software tools)
ECE 6640
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Spectral Density
• Energy Spectral Density

EX 
 x t   dt
2

X f   X f   X f 
*
• Power Spectral Density
T0
1
PX 
T0
2
 x t   dt
2

T0
2
1
*
G X f   lim  X T f   X T f  
T  T


ECE 6640
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Autocorrelation
• of an Energy Signal
R XX   

 xt  x t   dt

•
Properties:
1. Energy
ECE 6640
 
R XX 0   E X 2  X 2
2. Symmetry
R XX    R XX   
3. Maximum
R XX    R XX 0
4. Transform Pair
R XX   XX f 
32
Autocorrelation
• of a Power Signal
T
1 2
 XX    lim  x t   x t     dt
T  T
T
2
•
Properties:
1. Energy
ECE 6640
T0
1
 XX 0  
T0
2
 x t 
2

T0
 dt
2
2. Symmetry
XX   XX  
3. Maximum
XX   XX 0
4. Transform Pair
XX   G XX f 
33
Random Signals
1 Distribution Functions
Probability Distribution Function (PDF) or
Cumulative Distribution Function (CDF) [preferred]
 0  FX  x   1, for    x  
 FX     0 and FX    1
 FX is non-decreasing as x increases
 Pr  x1  X  x 2   FX  x 2   FX  x1 
For discrete events
For continuous events
ECE 6640
34
Random Signals
2. Density Functions
Probability Density Function (pdf)
 f X x   0, for    x  
Probability Mass Function (pmf)
 f X  x   0, for    x  



 f x   dx  1
X



 f X u   du
FX 


Pr  x1  X  x 2  
 f x   dx
X
 f X u   du

Pr x1  X  x 2  
x2
x1

Functions of random variables
dx
f Y  y   f X x  
dy
ECE 6640
X

x
x
FX 
 f x   dx  1

x2
 f x   dx
X
x1
35
Random Signals
Mean Values and Moments
1st, general, nth Moments

X  EX  
x f
X
or X  E  X  
x   dx

 gX  f
X
or E g  X  
x   dx
X
n
 E X   x

n
 f X  x   dx or X
n

 g  X   Pr X  x
x  


 x  Pr X  x
x  

E g  X  


n
 E X    x
n
n
 Pr  X  x 
x  

Central Moments
X  X 
n
X  X 
n

    x  X 

E XX

E XX
n
 f X  x   dx
n


    x  X 
n
n
 Pr  X  x 
x  
Variance and Standard Deviation


2
  X X
2

2  X X

2

E X X

E XX
    x  X 

2
 f X  x   dx
2


    x  X 
2
2
 Pr  X  x 
x  
ECE 6640
36
Random Signals
The Gaussian Random Variable
 x X 2 
1
, for    x  
f X x  
 exp
2


2  
 2 

where
X is the mean and  is the variance




 v X 2 
  dv
 exp
FX x  


2
2  
2 


v  
Unit Normal
x

x  
1
x
 u2 
  du

exp 

2 
2

u  

1
 x   1  x 
x X 
 x X 

 or FX  x   1  
FX  x   

  





The Q-function is the complement of the normal function, :
(Appendix B)
Q x  
ECE 6640

 u2 
  du

exp 

2 
2

ux
1

37
Random Processes
5. Random Processes
5.1. Introduction
Ensemble
5.2. Continuous and Discrete Random Processes
5.3. Deterministic and Nondeterministic Random Processes
5.4. Stationary and Nonstationary Random Processes
5.5. Ergodic and Nonergodic Random Processes
A Process for Determining Stationarity and Ergodicity
a) Find the mean and the 2nd moment based on the probability
b) Find the time sample mean and time sample 2nd moment based on time
averaging.
c) If the means or 2nd moments are functions of time … non-stationary
d) If the time average mean and moments are not equal to the probabilistic mean
and moments or if it is not stationary, then it is non ergodic.
ECE 6640
From: George R. Cooper and Clare D. McGillem, Probabilistic Methods of Signal and
System Analysis, 3rd ed.,Oxford University Press Inc., 1999. ISBN: 0-19-512354-9
38
Random Processes: Continuous,
Discrete and Mixed
Continuous and Discrete Random Processes
A continuous random process is one in which the random variables, such as X t1 , X t 2 ,  X t n  ,
can assume any value within the specified range of possible values. A more precise definition for a
continuous random process also requires that the cumulative distribution function be continuous.
A discrete random process is one in which the random variables, such as X t1 , X t 2 ,  X t n  ,
can assume any certain values (though possibly an infinite number of values). A more precise
definition for a discrete random process also requires that the cumulative distribution function
consist of numerous discontinuities or steps. Alternately, the probability density function is better
defined as a probability mass function … the pdf is composed of delta functions.
A mixed random process consists of both continuous and discrete components. The probability
distribution function consists of both continuous regions and steps. The pdf has both continuous
regions and delta functions.
ECE 6640 From: George R. Cooper and Clare D. McGillem, Probabilistic Methods of Signal and
System Analysis, 3rd ed.,Oxford University Press Inc., 1999. ISBN: 0-19-512354-9
39
Random Processes: Deterministic and
Nondeterministic
Deterministic and Nondeterministic Random Processes
A nondeterministic random process is one where future values of the ensemble cannot be predicted
from previously observed values.
A deterministic random process is one where one or more observed samples allow all future values
of the sample function to be predicted (or pre-determined). For these processes, a single random
variable may exist for the entire ensemble. Once it is determined (one or more measurements) the
sample function is known for all t.
ECE 6640 From: George R. Cooper and Clare D. McGillem, Probabilistic Methods of Signal and
System Analysis, 3rd ed.,Oxford University Press Inc., 1999. ISBN: 0-19-512354-9
40
Random Processes: Stationary and
Nonstationary (1)
Stationary and Nonstationary Random Processes
The probability density function for random variables in time as been discussed, but what is the
dependence of the density function on the value of time, t, when it is taken?
If all marginal and joint density functions of a process do not depend upon the choice of the time
origin, the process is said to be stationary (that is it doesn’t change with time). All the mean values
and moments are constants and not functions of time!
For nonstationary processes, the probability density functions change based on the time origin or in
time. For these processes, the mean values and moments are functions of time.
In general, we always attempt to deal with stationary processes … or approximate stationary by
assuming that the process probability distribution, means and moments do not change significantly
during the period of interest.
ECE 6640
From: George R. Cooper and Clare D. McGillem, Probabilistic Methods of Signal and
System Analysis, 3rd ed.,Oxford University Press Inc., 1999. ISBN: 0-19-512354-9
41
Random Processes: Stationary and
Nonstationary (2)
Stationary and Nonstationary Random Processes
The requirement that all marginal and joint density functions be independent of the choice of time
origin is frequently more stringent (tighter) than is necessary for system analysis.
A more relaxed requirement is called stationary in the wide sense: where the mean value of any
random variable is independent of the choice of time, t, and that the correlation of two random
variables depends only upon the time difference between them.
That is
E X t   X   X and
E X t1   X t 2   E X 0   X t 2  t1   X 0   X    R XX   for   t 2  t1
You will typically deal with Wide-Sense Stationary Signals.
ECE 6640
From: George R. Cooper and Clare D. McGillem, Probabilistic Methods of Signal and
System Analysis, 3rd ed.,Oxford University Press Inc., 1999. ISBN: 0-19-512354-9
42
Random Processes: Ergodicity
Ergodic and Nonergodic Random Processes
Ergodicity deals with the problem of determining the statistics of an ensemble based on
measurements from a sample function of the ensemble.
For ergodic processes, all the statistics can be determined from a single function of the process.
This may also be stated based on the time averages. For an ergodic process, the time averages
(expected values) equal the ensemble averages (expected values).
That is to say,

Xn 

1
x n  f  x   dx  lim
T   2T

T
X
n
t   dt
T
Note that ergodicity cannot exist unless the process is stationary!
ECE 6640
From: George R. Cooper and Clare D. McGillem, Probabilistic Methods of Signal and
System Analysis, 3rd ed.,Oxford University Press Inc., 1999. ISBN: 0-19-512354-9
43
Random Processes
The power spectral density is the Fourier Transform of the autocorrelation:

S XX w  R XX   
 EX t   X t    exp iw   d

For an ergodic process,
1
 XX    lim
T   2T
T
 xt   xt     dt 
T
 
xt   xt   
T

1

lim
 XX    E X t   X t    
xt   xt     dt   exp iw   d
T   2T

 
T

T


1

 XX    lim
xt   exp iwt   xt     exp iwt     d   dt
T   2T
T
 




1
 XX    lim
T   2T

T
 xt   exp iwt X w  dt
T
1
 XX    X w  lim
T   2T
ECE 6640
T
 xt   exp i wt  dt
T
 XX    X w  X  w  X w
2
From: George R. Cooper and Clare D. McGillem, Probabilistic Methods of Signal and System Analysis, 3rd ed.,Oxford
University Press Inc., 1999. ISBN: 0-19-512354-9
44
Binary Sequence, Low Bit Rate
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
45
Binary Autocorrelation and PSD
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
46
Bandwidth Consideration
• The first spectral null occurs are 1/T. Therefore one
measure of bandwidth could be the null.
• Are there others bandwidth measures?
– 3dB bandwidth
– 99% Power
– If it were a rectangle with Gx(0) given, how wide would it be
(Noise Equivalent Bandwidth)
– Etc.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
47
Bandwidth Consideration
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
48
White Noise
Noise is inherently defined as a random process.
You may be familiar with “thermal” noise, based on the energy of an atom and the mean-free path
that it can travel.
As a random process, whenever “white noise” is measured, the values are uncorrelated with each
other, not matter how close together the samples are taken in time.
Further, we envision “white noise” as containing all spectral content, with no explicit peaks or
valleys in the power spectral density.
As a result, we define “White Noise” as
R XX    S 0   t 
S XX w   S 0
This is an approximation or simplification because the area of the power spectral density is infinite!
ECE 6640
From: George R. Cooper and Clare D. McGillem, Probabilistic Methods of Signal and
System Analysis, 3rd ed.,Oxford University Press Inc., 1999. ISBN: 0-19-512354-9
49
Band Limited White Noise
Thermal noise at the input of a receiver is defined in terms of kT, Boltzmann’s constant times
absolute temperature, in terms of Watts/Hz. Thus there is kT Watts of noise power in every Hz of
bandwidth.
For communications, this is equivalent to –174 dBm/Hz or –144 dBW/Hz.
For typical applications, we are interested in Band-Limited White Noise where
 S 0
S XX w   
0
f W
W  f
The equivalent noise power is then:
 
E X 2  R XX 0  
W
 S 0  dw  2  W  S 0
W
For communications, we use kTB.
How much noise power, in dBm, would I say that there is in a 1 MHz bandwidth?
dBkTB   dBkT   dBB   174  60  114 dBm
ECE 6640
50
White Noise in Comm.
• From the text
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
51
Noise as A Gaussian Random Process
A Gaussian Random Variable
f X x  
where


 x X 2 
, for    x  
 exp
2
 2 

2  


X is the mean and  is the variance
1


 v X 2 
  dv
FX x  
 exp
 2  2 
2  


v  
x

•
1
What is so special about a Gaussian Distribution?
–
–
–
–
ECE 6640
Result of summing a large number of random variables
Linear systems produce Gaussian Outputs
Well know/studied characteristics
Used to define the characteristics of numerous natural, real-world signals
52
Linear Systems
Linear transformation of signals:
y t   ht   xt 
Ys   Hs   Xs 
Convolution Integrals
y t  

 xt     h   d
0
or
y t  
t
 ht     x   d

where for physical realizability and stability constraints we require
ht   0
for t  0

ECE 6640
 ht   dt  

53
Transfer Function
Hf   Hf   exp j  f 
 ImHf 
f   tan 1 

 ReHf 
• For linear systems: A sinusoidal input results in sinusoidal
output modified in magnitude and phase.
x t   A  cos2  f 0  t 
yt   h t   x t 
yt   A  Hf 0   cos2  f 0  t  f 0 
ECE 6640
54
Filtering a Random Process
• The PSD of a filtered response is
 
 

 


RYY    E  xt  1   h1   d1    xt    2   h2   d2 

 

 0
 0








RYY    d1  d2  h1   h2   R XX   1  2 


0
0









SYY w  RYY    d1  d2  h1   h2  
R XX   1  2   exp iw   d 



0
 

0





SYY w   R YY   SXX w   Hw   H w 
ECE 6640
SYY w  RYY    S XX w  H w
2
55
Distortionless Transmission and
the Ideal Filter
• To receive a signal without distortion, only changes in the
magnitude and/or a time delay is allowed.
yt   K  x t  t 0 
Yf   K  Xf   exp 2  f  t 0 
• The transfer function is
Hf   K  exp 2  f  t 0 
• A constant gain with a linear phase
Hf   K
ECE 6640
f   2  f  t 0
56
Ideal Filter (1)
• For no distortion, the ideal filter should have the following
properties:
Hf   Hf   exp j  f 
1,
Hf   
0,
for f  f u
for f  f u
2  f  t 0 ,
f   
arbitrary,
for f  f u
for f  f u
• The impulse response is
fu
h t    1 exp j2  f  t 0  exp j2  f  t  df
f u
fu
h t    exp j2  f  t  t 0  df
ECE 6640
f u
57
Ideal Filter (2)
• Continuing
fu
h t    exp j2  f  t  t 0  df
f u
exp j2  f  t  t 0 
h t  
j2  t  t 0 
f
fu
h t  
h t  
u
exp j2  f u  t  t 0  exp j2   f u   t  t 0 

j2  t  t 0 
j2  t  t 0 
2  sin2  f u  t  t 0 
2   t  t 0 
h t   2  f u  sinc2  f u  t  t 0 
• The sinc function
– A non-causal filter
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
58
Ideal Filters in the Freq. Domain
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
59
Realizable Filters, RC Network
1st order
Butterworth
Filter
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
60
White Noise in an RC Filter
• The noise PSD has been modified
• The autocorrelation is spread in time
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
61
Signal Filtering in the Real World
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
62
Signal Filtering in the Real World (2)
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
63
Bandwidth Considerations, Easy
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
64
Bandwidth Considerations, Harder
• If the spectrum extends to infinity, where do you assume
that it can be cut off?
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
65
Bandwidth Considerations
• Note 1 that as soon as time is limited, the signal has been
multiplied by a rect function in the time domain.
– A rect in the time domain creates an infinite sinc convolution in the
frequency domain!
• Note 2 that a bandlimited frequency domain signal can be
generated by multiplying by a rect function in the
frequency domain.
– A rect in the frequency domain results in a non-causal, infinite
time convolution in the time domain!
• For mathematicians, a real signal can not be both time
limited and frequency band limited?!
ECE 6640
66
Bandwidths that are Used
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
67
Bandwidth Definitions
(a) Half-power bandwidth. This is the interval between frequencies at
which Gx(f ) has dropped to half-power, or 3 dB below the peak value.
(b) Equivalent rectangular or noise equivalent bandwidth. The noise
equivalent bandwidth was originally conceived to permit rapid
computation of output noise power from an amplifier with a wideband
noise input; the concept can similarly be applied to a signal bandwidth.
The noise equivalent bandwidth WN of a signal is defined by the
relationship WN = Px/Gx(fc), where Px is the total signal power over
all frequencies and Gx(fc) is the value of Gx(f ) at the band center
(assumed to be the maximum value over all frequencies).
(c) Null-to-null bandwidth. The most popular measure of bandwidth for
digital communications is the width of the main spectral lobe, where
most of the signal power is contained. This criterion lacks complete
generality since some modulation formats lack well-defined lobes.
ECE 6640
68
Bandwidth Definitions (2)
(d) Fractional power containment bandwidth. This bandwidth criterion
has been adopted by the Federal Communications Commission (FCC
Rules and Regulations Section 2.202) and states that the occupied
bandwidth is the band that leaves exactly 0.5% of the signal power
above the upper band limit and exactly 0.5% of the signal power below
the lower band limit. Thus 99% of the signal power is inside the
occupied band.
(e) Bounded power spectral density. A popular method of specifying
bandwidth is to state that everywhere outside the specified band, Gx(f )
must have fallen at least to a certain stated level below that found at
the band center. Typical attenuation levels might be 35 or 50 dB.
(f) Absolute bandwidth. This is the interval between frequencies, outside
of which the spectrum is zero. This is a useful abstraction. However,
for all realizable waveforms, the absolute bandwidth is infinite.
ECE 6640
69
Spectrum and Time Domain of a
Band-limited Bandpass Signal
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.
70
Summary
• Communication must consider a number of aspects
–
–
–
–
Time and Frequency Domain Signals
Discrete and Continuous Time Signal Constructs
Deterministic and Random Signal Properties
Models of Signal Propagation
• Simple time and amplitude changes
• Complex channel impairments
– Models of Other Signals in the Environment
• Noise (white, Gaussian, or more complex)
• Interference
• Multipath
• To successfully model and analyze modern communication
systems, there is a lot of prerequisite knowledge required.
ECE 6640
71