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Probability and Random Signals for Electrical Engineers, Neon Garcia References: Digital Communications, Fourth Edition, J.G. Proakis, McGraw Hill, 2000. Course BooksSlide 3: Course Outline Review of Probability Signal and Spectra (Chapter 1) Formatting and Base band Modulation (Chapter 2) Base band Demodulation/Detection (Chapter 3) Channel Coding (Chapter 6, 7 and 8) Band pass Modulation and Demod./Detect. (Chapter 4) Spread Spectrum Techniques (Chapter 12) Synchronization (Chapter 10) Source Coding (Chapter 13) Fading Channels (Chapter 15)Today’s Goal: Today’s Goal Review of Basic Probability Digital Communication BasicCommunication: Communication Main purpose of communication is to transfer information from a source to a recipient via a channel or medium. Basic block diagram of a communication system: Source Transmitter Receiver Recipient ChannelBrief Description: Brief Description Source: analog or digital Transmitter: transducer, amplifier, modulator, oscillator, power amp., antenna Channel: e.g. cable, optical fibre, free space Receiver: antenna, amplifier, demodulator, oscillator, power amplifier, transducer Recipient: e.g. person, (loud) speaker, computerSlide 7: Types of information Voice, data, video, music, email etc. Types of communication systems Public Switched Telephone Network (voice,fax,modem) Satellite systems Radio,TV broadcasting Cellular phones Computer networks (LANs, WANs, WLANs)Information Representation: Information Representation Communication system converts information into electrical electromagnetic/optical signals appropriate for the transmission medium. Analog systems convert analog message into signals that can propagate through the channel. Digital systems convert bits(digits, symbols) into signals Computers naturally generate information as characters/bits Most information can be converted into bits Analog signals converted to bits by sampling and quantizing (A/D conversion)Why digital?: Why digital? Digital techniques need to distinguish between discrete symbols allowing regeneration versus amplification Good processing techniques are available for digital signals, such as medium. Data compression (or source coding) Error Correction (or channel coding)(A/D conversion) Equalization Security Easy to mix signals and data using digital techniquesSlide 13: Basic Digital Communication Transformations Formatting/Source Coding Transforms source info into digital symbols (digitization) Selects compatible waveforms (matching function) Introduces redundancy which facilitates accurate decoding despite errors It is essential for reliable communication Modulation/Demodulation Modulation is the process of modifying the info signal to facilitate transmission Demodulation reverses the process of modulation. It involves the detection and retrieval of the info signal Types Coherent: Requires a reference info for detection Noncoherent: Does not require reference phase informationBasic Digital Communication Transformations : Basic Digital Communication Transformations Coding/Decoding Translating info bits to transmitter data symbols Techniques used to enhance info signal so that they are less vulnerable to channel impairment (e.g. noise, fading, jamming, interference) Two Categories Waveform Coding Produces new waveforms with better performance Structured Sequences Involves the use of redundant bits to determine the occurrence of error (and sometimes correct it) Multiplexing/Multiple Access Is synonymous with resource sharing with other users Frequency Division Multiplexing/Multiple Access (FDM/FDMAPerformance Metrics: Performance Metrics Analog Communication Systems Metric is fidelity: want SNR typically used as performance metric Digital Communication Systems Metrics are data rate (R bps) and probability of bit error Symbols already known at the receiver Without noise/distortion/sync. problem, we will never make bit errorsMain Points: Main Points Transmitters modulate analog messages or bits in case of a DCS for transmission over a channel. Receivers recreate signals or bits from received signal (mitigate channel effects) Performance metric for analog systems is fidelity, for digital it is the bit rate and error probability.Why Digital Communications? : Why Digital Communications? Easy to regenerate the distorted signal Regenerative repeaters along the transmission path can detect a digital signal and retransmit a new, clean (noise free) signal These repeaters prevent accumulation of noise along the path This is not possible with analog communication systems Two-state signal representation The input to a digital system is in the form of a sequence of bits (binary or M_ary) Immunity to distortion and interference Digital communication is rugged in the sense that it is more immune to channel noise and distortionWhy Digital Communications? : Why Digital Communications? Hardware is more flexible Digital hardware implementation is flexible and permits the use of microprocessors, mini-processors, digital switching and VLSI Shorter design and production cycle Low cost The use of LSI and VLSI in the design of components and systems have resulted in lower cost Easier and more efficient to multiplex several digital signals Digital multiplexing techniques – Time & Code Division Multiple Access - are easier to implement than analog techniques such as Frequency Division Multiple AccessWhy Digital Communications? : Why Digital Communications? Can combine different signal types – data, voice, text, etc. Data communication in computers is digital in nature whereas voice communication between people is analog in nature The two types of communication are difficult to combine over the same medium in the analog domain. Using digital techniques, it is possible to combine both format for transmission through a common medium Encryption and privacy techniques are easier to implement Better overall performance Digital communication is inherently more efficient than analog in realizing the exchange of SNR for bandwidth Digital signals can be coded to yield extremely low rates and high fidelity as well as privacyWhy Digital Communications? : Why Digital Communications? Disadvantages Requires reliable “synchronization” Requires A/D conversions at high rate Requires larger bandwidth Nongraceful degradation Performance Criteria Probability of error or Bit Error RateGoals in Communication System Design : Goals in Communication System Design To maximize transmission rate, R To maximize system utilization, U To minimize bit error rate, P e To minimize required systems bandwidth, W To minimize system complexity, C x To minimize required power, E b /N oSlide 23: Comparative Analysis of Analog and Digital CommunicationDigital Signal Nomenclature : Digital Signal Nomenclature Information Source Discrete output values e.g. Keyboard Analog signal source e.g. output of a microphone Character Member of an alphanumeric/symbol (A to Z, 0 to 9) Characters can be mapped into a sequence of binary digits using one of the standardized codes such as ASCII: American Standard Code for Information Interchange EBCDIC: Extended Binary Coded Decimal Interchange CodeDigital Signal Nomenclature: Digital Signal Nomenclature Digital Message Messages constructed from a finite number of symbols; e.g., printed language consists of 26 letters, 10 numbers, “space” and several punctuation marks. Hence a text is a digital message constructed from about 50 symbols Morse-coded telegraph message is a digital message constructed from two symbols “ Mark ” and “ Space ” M - ary A digital message constructed with M symbols Digital Waveform Current or voltage waveform that represents a digital symbol Bit Rate Actual rate at which information is transmitted per secondDigital Signal Nomenclature: Digital Signal Nomenclature Baud Rate Refers to the rate at which the signaling elements are transmitted, i.e. number of signaling elements per second. Bit Error Rate The probability that one of the bits is in error or simply the probability of error1.2 Classification Of Signals 1. Deterministic and Random Signals : 1.2 Classification Of Signals 1. Deterministic and Random Signals A signal is deterministic means that there is no uncertainty with respect to its value at any time. Deterministic waveforms are modeled by explicit mathematical expressions, example: A signal is random means that there is some degree of uncertainty before the signal actually occurs. Random waveforms/ Random processes when examined over a long period may exhibit certain regularities that can be described in terms of probabilities and statistical averages.2. Periodic and Non-periodic Signals: 2. Periodic and Non-periodic Signals A signal x( t ) is called periodic in time if there exists a constant T 0 > 0 such that (1.2) t denotes time T 0 is the period of x ( t ) .3. Analog and Discrete Signals : 3. Analog and Discrete Signals An analog signal x ( t ) is a continuous function of time; that is, x ( t ) is uniquely defined for all t A discrete signal x ( kT ) is one that exists only at discrete times; it is characterized by a sequence of numbers defined for each time, kT, where k is an integer T is a fixed time interval.4. Energy and Power Signals : 4. Energy and Power Signals The performance of a communication system depends on the received signal energy; higher energy signals are detected more reliably (with fewer errors) than are lower energy signals x ( t ) is classified as an energy signal if, and only if, it has nonzero but finite energy (0 < E x < ∞) for all time, where: (1.7) An energy signal has finite energy but zero average power. Signals that are both deterministic and non-periodic are classified as energy signals4. Energy and Power Signals : Power is the rate at which energy is delivered. A signal is defined as a power signal if, and only if, it has finite but nonzero power (0 < P x < ∞) for all time, where (1.8) Power signal has finite average power but infinite energy. As a general rule, periodic signals and random signals are classified as power signals 4. Energy and Power Signals5. The Unit Impulse Function: Dirac delta function δ ( t ) or impulse function is an abstraction—an infinitely large amplitude pulse, with zero pulse width, and unity weight (area under the pulse), concentrated at the point where its argument is zero. (1.9) (1.10) (1.11) Sifting or Sampling Property (1.12) 5. The Unit Impulse Function1.3 Spectral Density : 1.3 Spectral Density The spectral density of a signal characterizes the distribution of the signal’s energy or power in the frequency domain. This concept is particularly important when considering filtering in communication systems while evaluating the signal and noise at the filter output. The energy spectral density (ESD) or the power spectral density (PSD) is used in the evaluation.1. Energy Spectral Density (ESD) : 1. Energy Spectral Density (ESD) Energy spectral density describes the signal energy per unit bandwidth measured in joules/hertz. Represented as ψ x (f), the squared magnitude spectrum (1.14) According to Parseval’s theorem, the energy of x(t): (1.13) Therefore: (1.15) The Energy spectral density is symmetrical in frequency about origin and total energy of the signal x(t) can be expressed as: (1.16)2. Power Spectral Density (PSD): 2. Power Spectral Density (PSD) The power spectral density (PSD) function G x ( f ) of the periodic signal x ( t ) is a real, even, and nonnegative function of frequency that gives the distribution of the power of x ( t ) in the frequency domain. PSD is represented as: (1.18) Whereas the average power of a periodic signal x(t) is represented as: (1.17) Using PSD, the average normalized power of a real-valued signal is represented as: (1.19)1.4 Autocorrelation 1. Autocorrelation of an Energy Signal: 1.4 Autocorrelation 1. Autocorrelation of an Energy Signal Correlation is a matching process; autocorrelation refers to the matching of a signal with a delayed version of itself. Autocorrelation function of a real-valued energy signal x ( t ) is defined as: (1.21) The autocorrelation function R x ( τ ) provides a measure of how closely the signal matches a copy of itself as the copy is shifted τ units in time. R x ( τ ) is not a function of time; it is only a function of the time difference τ between the waveform and its shifted copy.1. Autocorrelation of an Energy Signal: 1. Autocorrelation of an Energy Signal The autocorrelation function of a real-valued energy signal has the following properties: symmetrical in about zero maximum value occurs at the origin autocorrelation and ESD form a Fourier transform pair, as designated by the double-headed arrows value at the origin is equal to the energy of the signal2. Autocorrelation of a Power Signal: 2. Autocorrelation of a Power Signal Autocorrelation function of a real-valued power signal x ( t ) is defined as: (1.22) When the power signal x ( t ) is periodic with period T 0 , the autocorrelation function can be expressed as (1.23)2. Autocorrelation of a Power Signal: 2. Autocorrelation of a Power Signal The autocorrelation function of a real-valued periodic signal has the following properties similar to those of an energy signal: symmetrical in about zero maximum value occurs at the origin autocorrelation and PSD form a Fourier transform pair value at the origin is equal to the average power of the signal1.5 Random Signals 1. Random Variables: 1.5 Random Signals 1. Random Variables All useful message signals appear random; that is, the receiver does not know, a priori, which of the possible waveform have been sent. Let a random variable X ( A ) represent the functional relationship between a random event A and a real number. The (cumulative) distribution function F X ( x ) of the random variable X is given by (1.24) Another useful function relating to the random variable X is the probability density function (pdf) (1.25)1.1 Ensemble Averages: 1.1 Ensemble Averages The first moment of a probability distribution of a random variable X is called mean value m X , or expected value of a random variable X The second moment of a probability distribution is the mean-square value of X Central moments are the moments of the difference between X and m X and the second central moment is the variance of X Variance is equal to the difference between the mean-square value and the square of the mean2. Random Processes : 2. Random Processes A random process X ( A, t ) can be viewed as a function of two variables: an event A and time.220.127.116.11 Statistical Averages of a Random Process : 18.104.22.168 Statistical Averages of a Random Process A random process whose distribution functions are continuous can be described statistically with a probability density function (pdf). A partial description consisting of the mean and autocorrelation function are often adequate for the needs of communication systems. Mean of the random process X ( t ) : (1.30) Autocorrelation function of the random process X ( t ) (1.31)22.214.171.124 Stationarity: 126.96.36.199 Stationarity A random process X ( t ) is said to be stationary in the strict sense if none of its statistics are affected by a shift in the time origin. A random process is said to be wide-sense stationary (WSS) if two of its statistics, its mean and autocorrelation function, do not vary with a shift in the time origin. (1.32) (1.33)188.8.131.52 Autocorrelation of a Wide-Sense Stationary Random Process : 184.108.40.206 Autocorrelation of a Wide-Sense Stationary Random Process For a wide-sense stationary process, the autocorrelation function is only a function of the time difference τ = t 1 – t 2 ; (1.34) Properties of the autocorrelation function of a real-valued wide-sense stationary process are S ymmetrical in τ about zero M aximum value occurs at the origin A utocorrelation and power spectral density form a Fourier transform pair V alue at the origin is equal to the average power of the signal1.5.3. Time Averaging and Ergodicity : 1.5.3. Time Averaging and Ergodicity When a random process belongs to a special class, known as an ergodic process, its time averages equal its ensemble averages. The statistical properties of such processes can be determined by time averaging over a single sample function of the process. A random process is ergodic in the mean if (1.35) It is ergodic in the autocorrelation function if (1.36)1.5.4. Power Spectral Density and Autocorrelation: 1.5.4. Power Spectral Density and Autocorrelation A random process X ( t ) can generally be classified as a power signal having a power spectral density (PSD) G X ( f ) Principal features of PSD functions A nd is always real valued f or X(t) real-valued P SD and autocorrelation form a Fourier transform pair R elationship between average normalized power and PSD1.5.5. Noise in Communication Systems : 1.5.5. Noise in Communication Systems The term noise refers to unwanted electrical signals that are always present in electrical systems; e.g spark-plug ignition noise, switching transients, and other radiating electromagnetic signals. Can describe thermal noise as a zero-mean Gaussian random process. A Gaussian process n ( t ) is a random function whose amplitude at any arbitrary time t is statistically characterized by the Gaussian probability density function (1.40)Noise in Communication Systems : Noise in Communication Systems The normalized or standardized Gaussian density function of a zero-mean process is obtained by assuming unit variance.220.127.116.11 White Noise : 18.104.22.168 White Noise The primary spectral characteristic of thermal noise is that its power spectral density is the same for all frequencies of interest in most communication systems Power spectral density G n ( f ) (1.42) Autocorrelation function of white noise is (1.43) The average power P n of white noise is infinite (1.44)Slide 54: The effect on the detection process of a channel with additive white Gaussian noise (AWGN) is that the noise affects each transmitted symbol independently. Such a channel is called a memoryless channel. The term “additive” means that the noise is simply superimposed or added to the signal1.6 Signal Transmission through Linear Systems: 1.6 Signal Transmission through Linear Systems A system can be characterized equally well in the time domain or the frequency domain, techniques will be developed in both domains The system is assumed to be linear and time invariant. It is also assumed that there is no stored energy in the system at the time the input is applied1.6.1. Impulse Response: 1.6.1. Impulse Response The linear time invariant system or network is characterized in the time domain by an impulse response h ( t ),to an input unit impulse ( t) (1.45) The response of the network to an arbitrary input signal x ( t )is found by the convolution of x ( t )with h ( t ) (1.46) The system is assumed to be causal, which means that there can be no output prior to the time, t =0,when the input is applied. The convolution integral can be expressed as: (1.47a)1.6.2. Frequency Transfer Function: 1.6.2. Frequency Transfer Function The frequency-domain output signal Y ( f )is obtained by taking the Fourier transform (1.48) Frequency transfer function or the frequency response is defined as: (1.49) (1.50) The phase response is defined as: (1.51)22.214.171.124. Random Processes and Linear Systems: 126.96.36.199. Random Processes and Linear Systems If a random process forms the input to a time-invariant linear system,the output will also be a random process. The input power spectral density G X ( f )and the output power spectral density G Y ( f )are related as: (1.53)1.6.3. Distortionless Transmission What is the required behavior of an ideal transmission line? : 1.6.3. Distortionless Transmission What is the required behavior of an ideal transmission line? The output signal from an ideal transmission line may have some time delay and different amplitude than the input It must have no distortion—it must have the same shape as the input. For ideal distortionless transmission: (1.54) (1.55) (1.56) Output signal in time domain Output signal in frequency domain System Transfer FunctionWhat is the required behavior of an ideal transmission line? : What is the required behavior of an ideal transmission line? The overall system response must have a constant magnitude response The phase shift must be linear with frequency All of the signal’s frequency components must also arrive with identical time delay in order to add up correctly Time delay t 0 is related to the phase shift and the radian frequency = 2 f by: t 0 (seconds) = (radians) / 2 f (radians/seconds ) (1.57a) Another characteristic often used to measure delay distortion of a signal is called envelope delay or group delay: (1.57b)188.8.131.52. Ideal Filters: 184.108.40.206. Ideal Filters For the ideal low-pass filter transfer function with bandwidth W f = f u hertz can be written as: Figure1.11 (b) Ideal low-pass filter (1.58) Where (1.59) (1.60)Ideal Filters: Ideal Filters The impulse response of the ideal low-pass filter :Ideal Filters: Ideal Filters For the ideal band-pass filter transfer function For the ideal high-pass filter transfer function Figure1.11 (a) Ideal band-pass filter Figure1.11 (c) Ideal high-pass filter220.127.116.11. Realizable Filters: 18.104.22.168. Realizable Filters The simplest example of a realizable low-pass filter; an R C filter 1.63) Figure 1.13Realizable Filters: Realizable Filters Phase characteristic of RC filter Figure 1.13Realizable Filters: Realizable Filters There are several useful approximations to the ideal low-pass filter characteristic and one of these is the Butterworth filter (1.65) Butterworth filters are popular because they are the best approximation to the ideal, in the sense of maximal flatness in the filter passband.1.7. Bandwidth Of Digital Data 1.7.1 Baseband versus Bandpass: An easy way to translate the spectrum of a low-pass or baseband signal x(t) to a higher frequency is to multiply or heterodyne the baseband signal with a carrier wave cos 2 f c t x c (t) is called a double-sideband (DSB) modulated signal x c (t) = x(t) cos 2 f c t (1.70) From the frequency shifting theorem X c (f) = 1/2 [X(f-f c ) + X(f+f c ) ] (1.71) Generally the carrier wave frequency is much higher than the bandwidth of the baseband signal f c >> f m and therefore W DSB = 2f m 1.7. Bandwidth Of Digital Data 1.7 .1 Baseband versus Bandpass1.7.2 Bandwidth Dilemma : Theorems of communication and information theory are based on the assumption of strictly bandlimited channels The mathematical description of a real signal does not permit the signal to be strictly duration limited and strictly bandlimited. 1.7.2 Bandwidth Dilemma1.7.2 Bandwidth Dilemma : 1.7.2 Bandwidth Dilemma All bandwidth criteria have in common the attempt to specify a measure of the width, W, of a nonnegative real-valued spectral density defined for all frequencies f < ∞ The single-sided power spectral density for a single heterodyned pulse x c ( t ) takes the analytical form: (1.73)Different Bandwidth Criteria : Different Bandwidth Criteria (a) Half-power bandwidth. (b) Equivalent rectangular or noise equivalent bandwidth. (c) Null-to-null bandwidth. (d) Fractional power containment bandwidth. (e) Bounded power spectral density. (f) Absolute bandwidth. 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