通信原理英文版课件：Ch1 Random Processes part1

1、1Communication Systems （Fourth Edition） 2Review of Chapter 0The communication systemSource of informationSpeech, music, pictures, and computer dataTransmitter (digital)Source encoder, channel encoder, modulatorCommunication channelsWireline channels: twisted pair, coaxial cable, and optical fiberWir

2、eless channels: broadcast, mobile radio, and satelliteReceiver (digital)Demodulator, channel decoder, source decoderUser of information3Review of Chapter 0 (Contd)Communication networksThe OSI modelThe InternetB-ISDN and ATMModulationContinuous wave modulationAM, FM, and PMPulse modulationPAM, PDM,

3、and PPMPCMShannons information capacity theory4Review of Chapter 0 (Contd)Some basic conceptsPower limited and bandwidth limitedNoise, Signal-to-Noise Ratio (SNR) and Additive White Gaussian Noise (AWGN) channelCircuit switching and packet switchingSource coding and channel codingBroadcasting and po

4、int-to-point communicationAnalog and digital types of communicationChannel capacityBit Error Rate (BER)Quality of Service (QoS)5CH1：Random ProcessesIntroductionMathematical Definition of a Random ProcessStationary ProcessesMean, Correlation, and Covariance FunctionsErgodic ProcessesTransmission of a

5、 Random Process Through a Linear Time-Invariant FilterPower Spectral DensityGaussian ProcessNoiseNarrowband NoiseRepresentation of Narrowband Noise in Terms of In-phase and Quadrature ComponentsRepresentation of Narrowband Noise in Terms of Envelope and Phase ComponentsSine Wave Plus Narrowband Nois

6、eComputer Experiments: Flat-Fading ChannelSummary and Discussion61.1 IntroductionTwo mathematical modelsDeterministicStochastic (or random)Received signal in a communication system usually consists of:Information-bearing signalRandom interferenceChannel noise Describing the signal using statistical

7、parametersAverage power, power spectral density, 7Random (stochastic) processPropertiesFunction of timeRandomDefinitionEnsemble of time functionsA probability rule1.2 Mathematical Definition of a Random Process81.2 Mathematical Definition of a Random Process (Contd)Figure 1.1 An ensemble of sample f

8、unctions.Some concepts:Sample space SRandom process X(t,S) = X(t)Sample point sjRealization (sample function)xj(t) = X(t, sj)Random variable91.3 Stationary ProcessThe joint distribution function:Strictly stationary： For all time shifts , all k, and all possible choices of observation times t1, , tk,

9、 equation(1) is always true.Two special cases (wide-sense stationary):101.3 Stationary ProcessFigure 1.2 Illustrating the probability of a joint event.11Example 1.1 Three spatial windows located at times t1, t2, and t3, the probability of the joint event: In terms of the joint distribution function,

10、 this probability equals:1.3 Stationary Process121.3 Stationary ProcessFigure 1.3 Illustrating the concept of stationary in Example 1.1.131.4 Mean, Correlation, and Covariance FunctionsMean:Autocorrelation function:Autocovariance function:(Stationary)Cross-correlation function:141.4 Mean, Correlatio

11、n, and Covariance Functions (Contd)The mean and autocorrelation function provide a partial description of a random process.Wide-sense stationaryMean is a constant and autocorrelation function depends only on time difference.Often used in practiceNot necessary strictly stationary, and vise verse.15Pr

12、operties of the Autocorrelation FunctionProperties:Defining autocorrelation function of a stationary process X(t) as:16Properties of the Autocorrelation Function (Contd)Figure 1.4 Illustrating the autocorrelation functions of slowly and rapidly fluctuating random processes.17Example 1.2 Sinusoidal W

13、ave with Random PhaseA and fc are constants, and18Example 1.2 (Contd)The autocorrelation function of X(t) is:19Example 1.2 (Contd) Figure 1.5 Autocorrelation function of a sine wave with random phase.20Example 1.3 Random Binary Wave 21Cross-Correlation FunctionsTwo random processes X(t) and Y(t) wit

14、h autocorrelation functions RX(t,u) and RY(t,u), the two cross-correlation functions of X(t) and Y(t) are defined by:The correlation matrix:A symmetry relationship:22Example 1.4 Quadrature-Modulated Processes 23Example 1.4 (Contd) 241.5 Ergodic ProcessesUsing time averages to approximate ensemble av

15、erages.Considering a sample function x(t) of a stationary process X(t) in an observation window T t T:(The DC value)Time average X(T) represents an unbiased estimate of the ensemble-averaged mean X.251.5 Ergodic Processes (Contd)A process X(t) is ergodic in the mean if two conditions are satisfied:A

16、 process X(t) is ergodic in the autocorrelation function if two conditions are satisfied:For a random process to be ergodic, it has to be stationary; but a stationary random process is not necessarily ergodic.261.6 Transmission of a Random Process Through a Linear Time-Invariant FilterFigure 1.8 tra

17、nsmission of a random process through a linear time-invariant filter.271.6 Transmission Through a Linear Time-Invariant Filter (Contd)281.7 Power Spectral Density (PSD)29Definition of PSDThe power spectral density (or power spectrum) is the Fourier transform of the autocorrelation function.As a resu

18、lt,IfThenand f is small,30Properties of PSDThe PSD and the autocorrelation function form a Fourier-transform pair.The Einstein-Wiener-Khintchine Relations31Properties of PSD (Contd)is a probability density function.32PSD Example 1Sinusoidal wave with random phase33PSD Example 1 (Contd)Figure 1.10 Power spectral density of sine wave with random phase.34PSD Example 2Random binary wave35PSD Example 2 (Contd)Figure 1.11 Power spectral density of random binary wave.36PSDs of Input/O