ECE 6640 Digital Communications

Transcription

1 ECE 6640 Digital Communications Dr. Bradley J. Bazuin Assistant Professor Department of Electrical and Computer Engineering College of Engineering and Applied Sciences

2 Course/Lecture Overview Syllabus Personal Intro. Textbook/Materials Used Additional Reading ID and Acknowledgment of Policies Textbook Chapter 1 ECE 6640

3 Syllabus Everything useful for this class can be found on Dr. Bazuin s web site! The class web site is at The syllabus ECE

4 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 198 and 1989, respectively. Industrial Experience Digital, ASIC, System Engineering Part-time ARGOSystems, Inc. (purchased by Boeing) Full-time ARGOSystems, Inc Full-time Radix Technologies Academic Experience Electrical and Computer Engineering Term-appointed Faculty, WMU ECE Dept Tenure track Assistant Professor, WMU ECE Dept Tenured Associate Professor, WMU ECE Dept present ECE

5 Research Activities and Interests 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 Center for the Advancement of Printed Electronics (CAPE) Printed electronic device design, fabrication and testing Semiconductor Physics Physical Layer Communication Signal Processing Software Defined Radios (SDR) 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 Xilinx programmable devices Parallel processing, hosts including NVIDIA GPUs with CUDA and multithreaded applications ECE

6 Required Textbook/Materials Bernard Sklar, Digital Communications, Fundamentals and Applications, Prentice Hall PTR, Second Edition, 001. ISBN: SystemView by ELANIX CD with textbook MATLAB, Student Edition MATLAB Signal Processing Toolbox The MATH Works, MATLAB and Signal Processing Toolbox ECE

7 Supplemental Books and Materials John G. Proakis and Masoud Salehi, Digital Communications, 5 th ed., McGraw Hill, Fifth Edition, 008. ISBN: John G. Proakis and Masoud Salehi, Communication Systems Engineering, nd ed., Prentice Hall, 00. ISBN: A. Bruce Carlson, P.B. Crilly, Communication Systems, 5th ed., McGraw-Hill, 010. ISBN: Leon W. Couch II, Digital and Analog Communication Systems, 7th ed., Prentice Hall, 007. ISBN: Stephen G. Wilson, Digital Modulation and Coding, Prentice-Hall, ISBN: Ezio Biglieri, D. Divsalar, P.J. McLane, M.K. Simon, Introduction to Trellis-Coded Modulation with Applications, Macmillan, ISBN: ECE

8 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

9 Course/Text Overview 1. Signals and Spectra. Digital Communication Signal Processing. Classification of Signals. Spectral Density. Autocorrelation. Random Signals. Signal Transmission through Linear Systems. Bandwidth of Digital Data.. Formatting and Baseband Modulation. Baseband Systems. Formatting Textual Data (Character Coding). Messages, Characters, and Symbols. Formatting Analog Information. Sources of Corruption. Pulse Code Modulation. Uniform and Nonuniform Quantization. Baseband Modulation. Correlative Coding. ECE

10 Course/Text Overview () 3. Baseband Demodulation/Detection. Signals and Noise. Detection of Binary Signals in Gaussian Noise. Intersymbol Interference. Equalization. 4. Bandpass Modulation and Demodulation/Detection. Why Modulate? Digital Bandpass Modulation Techniques. Detection of Signals in Gaussian Noise. Coherent Detection. Noncoherent Detection. Complex Envelope. Error Performance for Binary Systems. M-ary Signaling and Performance. Symbol Error Performance for M-ary Systems (M>>). Exam #1 ECE

11 Course/Text Overview (3) 5. Communications Link Analysis. What the System Link Budget Tells the System Engineer. The Channel. Received Signal Power and Noise Power. Link Budget Analysis. Noise Figure, Noise Temperature, and System Temperature. Sample Link Analysis. Satellite Repeaters. System Trade-Offs. ECE

12 Course/Text Overview (4) 6. Channel Coding: Part 1. Waveform Coding. Types of Error Control. Structured Sequences. Linear Block Codes. Error-Detecting and Correcting Capability. Usefulness of the Standard Array. Cyclic Codes. Well-Known Block Codes. 7. Channel Coding: Part. Convolutional Encoding. Convolutional Encoder Representation. Formulation of the Convolutional Decoding Problem. Properties of Convolutional Codes. Other Convolutional Decoding Algorithms. Exam # ECE

13 Course/Text Overview (5) 8. Channel Coding: Part 3. Reed-Solomon Codes. Interleaving and Concatenated Codes. Coding and Interleaving Applied to the Compact Disc Digital Audio System. Turbo Codes. Appendix 8A. The Sum of Log-Likelihood Ratios. 9. Modulation and Coding Trade-Offs. Goals of the Communications System Designer. Error Probability Plane. Nyquist Minimum Bandwidth. Shannon-Hartley Capacity Theorem. Bandwidth Efficiency Plane. Modulation and Coding Trade-Offs. Defining, Designing, and Evaluating Systems. Bandwidth-Efficient Modulations. Modulation and Coding for Bandlimited Channels. Trellis-Coded Modulation. Final Exam ECE

14 Course/Text Overview (6) Advanced Topics (as time permits) 11. Multiplexing and Multiple Access. Allocation of the Communications Resource. Multiple Access Communications System and Architecture. Access Algorithms. Multiple Access Techniques Employed with INTELSAT. Multiple Access Techniques for Local Area Networks. 1. Spread-Spectrum Techniques. Spread-Spectrum Overview. Pseudonoise Sequences. Direct- Sequence Spread-Spectrum Systems. Frequency Hopping Systems. Synchronization. Jamming Considerations. Commercial Applications. Cellular Systems. Final Exam ECE

15 Text Appendices A. A Review of Fourier Techniques. Signals, Spectra, and Linear Systems. Fourier Techniques for Linear System Analysis. Fourier Transform Properties. Useful Functions. Convolution. Tables of Fourier Transforms and Operations. B. Fundamentals of Statistical Decision Theory. Bayes' Theorem. Decision Theory. Signal Detection Example. C. Response of a Correlator To White Noise. D. Often-Used Identities. E. s-domain, z-domain and Digital Filtering. F. List of Symbols. G. SystemView by ELANIX Guide to the CD. ECE

16 Comments from 006 Offering A strong focus on themes and critical results for each chapter covered is needed. The text author provides his own list of critical elements, they can be incorporated into the instructors set. Matlab simulations of all significant concepts should be available. They allow the students to perform theoretical computations and then observe what the computations mean, particularly as it relates to biterror rate performance, digital modulation and coherent and noncoherent demodulation, and channel encoding and decoding. The software that comes with the text provides demonstrations, but it is not user friendly and the software is very out-of-data (no longer supported). ECE

17 Chapter 1 1. Signals and Spectra. 1.1 Digital Communication Signal Processing. 1. Classification of Signals. 1.3 Spectral Density. 1.4 Autocorrelation. 1.5 Random Signals. 1.6 Signal Transmission through Linear Systems. 1.7 Bandwidth of Digital Data. A review of prerequisite material that is critically important when studying digital communication systems. ECE

18 Sklar s Communications System Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 18 Prentice Hall PTR, Second Edition, 001.

19 Simplified Communications System 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 Information Message Format Source Encode Channel Encode Modulation RF Mixing Transmitter Antenna Noise Bits Symbols Signals Interference RF Signal Information Message Reformat Source Decode Channel Decode ECE Demodulation RF Mixing Receiver Antenna

20 Communication Channel Linear Filtering Nonlinear Distortion Attenuation Noise Transmitting Antenna Interference RF Communication Channel Receiving Antenna The channel greatly effects received RF signals 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 ECE

21 Received Signal r t s t h t s t h t s t h t nt c N N 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

22 Why Digital? 1. Noise, Interference, Path Loss, and Channel Impairments (signal environment). Cost 3. Inherent Availability 4. Reliability and Reconfigurability Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, Prentice Hall PTR, Second Edition, 001.

23 Terminology Information Source Textual Message Character Binary Digit (Bit) Bit Stream Symbol Digital Waveform Data Rate Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 3 Prentice Hall PTR, Second Edition, 001.

24 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,

25 Classification of Signals Deterministic and Random Periodic and Non-periodic Analog and Discrete/Digital Energy and Power Signals ECE

26 SKLAR DSP Tutorial The CD that comes with the text includes a Concise DSP Tutorial in pdf format Table of Contents: Frequency Domain Analysis critical importance General Digital Filters important Finite Impulse Response (FIR) Filters critical importance Infinite Impulse Response (IIR) Filters useful but Filter Design Techniques will be discussed and provided Adaptive Filters saved for Dr. Bazuin s ECE6950 course Also see Appendix B: Fundamentals of Statistical Decision Theory Specific material from probability and statistics is required. ECE 6640 (ECE 3800 or ECE580 material) 6

27 Spectral Density Energy Spectral Density E X X x t dt f Xf Xf * Power Spectral Density G X P X 1 T T 0 x 0 T 0 1 T t dt * f lim X f X f T T T ECE

28 Autocorrelation of an Energy Signal R XX x t xt dt Properties: 1. Energy R XX 0 EX X. Symmetry R XX R XX 3. Maximum 4. Transform Pair R R XX R XX XX XX 0 f ECE

29 Autocorrelation of a Power Signal XX lim x t xt T 1 T T T dt Properties: 1. Energy. Symmetry XX 0 XX 1 T T0 x 0 T 0 t XX dt 3. Maximum XX XX 0 4. Transform Pair XX G XX f ECE

30 Random Signals 1 Distribution Functions Probability Distribution Function (PDF) or Cumulative Distribution Function (CDF) [preferred] 0 F X x 1, for x F X 0 and F X 1 F is non-decreasing as x increases X Prx X x F x F x For discrete events For continuous events 1 X X 1 ECE

31 Random Signals. Density Functions Probability Density Function (pdf) f X x 0, for x F X f X x dx 1 x f X u du x dx x X x Pr 1 f X x 1 Functions of random variables dx fy y f X x dy x Probability Mass Function (pmf) f X x 0, for x F X Pr f X x dx 1 x f X u du x X x f x dx 1 x x 1 X ECE

32 Random Signals Mean Values and Moments 1 st, general, nth Moments x X EX x f X x dx or X EX x Pr X x x EgX gx f X x dx or EgX gx Pr X x X n E n n X x f x Central Moments X dx n X X E X X n X X E X X x n n n or X EX x Pr X x n n x X f X x dx n n x X PrX x x Variance and Standard Deviation X X E X X X X E X X x X f X x dx x X PrX x x ECE

33 Random Signals The Gaussian Random Variable f X x 1 x X exp, for x where X is the mean and is the variance F X Unit Normal x v X x exp dv x v 1 1 x X x u u exp x x 1 1 u Qx exp du ECE u x du F X x or F x X 1 x X The Q-function is the complement of the normal function, : (Appendix B)

34 Random Processes 5. Random Processes 5.1. Introduction Ensemble 5.. 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 nd moment based on the probability b) Find the time sample mean and time sample nd moment based on time averaging. c) If the means or nd 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. From: George R. Cooper and Clare D. McGillem, Probabilistic Methods of Signal and ECE 6640 System Analysis, 3rd ed.,oxford University Press Inc., ISBN:

35 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 t X t, X 1, 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 t X t, X 1, 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 35 System Analysis, 3rd ed.,oxford University Press Inc., ISBN:

36 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 36 System Analysis, 3rd ed.,oxford University Press Inc., ISBN:

37 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 37 System Analysis, 3rd ed.,oxford University Press Inc., ISBN:

38 Random Processes: Stationary and Nonstationary () 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 E X t X X and X t X t EX X t t X 0 X R XX for t 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 38 System Analysis, 3rd ed.,oxford University Press Inc., ISBN:

39 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, X n x n f x dx lim T 1 T T T X n t dt Note that ergodicity cannot exist unless the process is stationary! From: George R. Cooper and Clare D. McGillem, Probabilistic Methods of Signal and ECE 6640 System Analysis, 3rd ed.,oxford University Press Inc., ISBN:

40 Random Processes The power spectral density is the Fourier Transform of the autocorrelation: For an ergodic process, w R EX t X t iw S XX XX exp XX lim xt xt dt xt xt T 1 T T T X w X w X w ECE 6640 XX 40 From: George R. Cooper and Clare D. McGillem, Probabilistic Methods of Signal and System Analysis, 3rd ed.,oxford University Press Inc., ISBN: d EX t X t lim x t xt dt exp iw XX 1 T T T lim x t exp iwt xt exp iwt d dt XX T T 1 T T 1 T T T lim x t exp iwt X w dt XX T T 1 T X w lim x t exp i XX T T T w tdt d

41 Binary Sequence, Low Bit Rate Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 41 Prentice Hall PTR, Second Edition, 001.

42 Binary Autocorrelation and PSD Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 4 Prentice Hall PTR, Second Edition, 001.

43 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. Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 43 Prentice Hall PTR, Second Edition, 001.

44 Bandwidth Consideration Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 44 Prentice Hall PTR, Second Edition, 001.

45 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 S0 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 45 System Analysis, 3rd ed.,oxford University Press Inc., ISBN:

46 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 The equivalent noise power is then: S S XX w 0 0 f W W f E W X RXX 0 S0 dw W S 0 For communications, we use ktb. How much noise power, in dbm, would I say that there is in a 1 MHz bandwidth? W ktb dbkt dbb dbm db 114 ECE

47 White Noise in Comm. From the text Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 47 Prentice Hall PTR, Second Edition, 001.

48 Noise as A Gaussian Random Process A Gaussian Random Variable 1 x X f X, x exp for x where X is the mean and is the variance F X x v x exp dv v 1 X What is so special about a Gaussian Distribution? 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 ECE

49 Linear Systems Linear transformation of signals: Convolution Integrals or y y y Y t ht xt s Hs Xs 0 t xt h t t ht x ECE 6640 h t dt 49 d d where for physical realizability and stability constraints we require h t 0 for t 0

50 Transfer Function H f Hf exp j f f tan 1 Im H f Re H f For linear systems: A sinusoidal input results in sinusoidal output modified in magnitude and phase. x t A cos f t 0 y t h t x t y t A Hf cos f t 0 0 f0 ECE

51 ECE Filtering a Random Process The PSD of a filtered response is d h t x d h t x E R YY XX YY R h h d d R exp d iw R h h d d R w S XX YY YY w H H w w S R w S XX YY YY w H w S R w S XX YY YY

52 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 The transfer function is y t K xt t 0 f K Xf exp f H f K exp f A constant gain with a linear phase H f K f f t0 t 0 t 0 ECE

53 Ideal Filter (1) For no distortion, the ideal filter should have the following properties: Hf Hf expj f H f 1, 0, for for f f f f u u f f t0, arbitrary, for for f f f f u u The impulse response is h h f u t 1exp j f t expj f t f f u t expj f t t u 0 fu ECE df df

54 Ideal Filter () Continuing h h h h h f u t expj f t t f u fu exp j f t t0 t j t t0 fu exp j fu t t0 t j t t0 sin fu t t 0 t t t0 t f sinc f t t u The sinc function A non-causal filter u 0 df 0 exp j j fu t t0 t t Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 54 Prentice Hall PTR, Second Edition,

55 Ideal Filters in the Freq. Domain Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 55 Prentice Hall PTR, Second Edition, 001.

56 Realizable Filters, RC Network 1 st order Butterworth Filter Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 56 Prentice Hall PTR, Second Edition, 001.

57 White Noise in an RC Filter The noise PSD has been modified The autocorrelation is spread in time Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 57 Prentice Hall PTR, Second Edition, 001.

58 Signal Filtering in the Real World Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 58 Prentice Hall PTR, Second Edition, 001.

59 Signal Filtering in the Real World () Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 59 Prentice Hall PTR, Second Edition, 001.

60 Bandwidth Considerations, Easy Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 60 Prentice Hall PTR, Second Edition, 001.

61 Bandwidth Considerations, Harder If the spectrum extends to infinity, where do you assume that it can be cut off? Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 61 Prentice Hall PTR, Second Edition, 001.

62 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 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

63 Bandwidths that are Used Notes and figures are based on or taken from materials in the course textbook: ECE 6640 Bernard Sklar, Digital Communications, Fundamentals and Applications, 63 Prentice Hall PTR, Second Edition, 001.

64 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

65 Bandwidth Definitions () (d) Fractional power containment bandwidth. This bandwidth criterion has been adopted by the Federal Communications Commission (FCC Rules and Regulations Section.0) 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

66 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, 66 Prentice Hall PTR, Second Edition, 001.

67 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