Henry Stark and John W. Woods, Probability, Statistics, and Random Variables for Engineers, 4th ed., Pearson Education Inc., 2012. ISBN: 978-0-13-231123-6 Chapter 9 Random Processes Sections 9.1 Basic Definitions 544 9.2 Some Important Random Processes 548 9.3 Continuous-Time Linear Systems with Random Inputs 572 White Noise 577 9.4 Some Useful Classifications of Random Processes 578 Stationarity 579 9.5 Wide-Sense Stationary Processes and LSI Systems 581 Wide-Sense Stationary Case 582 Power Spectral Density 584 An Interpretation of the Power Spectral Density 586 More on White Noise 590 Stationary Processes and Differential Equations 596 9.6 Periodic and Cyclostationary Processes 600 9.7 Vector Processes and State Equations 606 State Equations 608 Summary 611 Problems 611 References 633 Notes and figures are based on or taken from materials in the course textbook: Probability, Statistics and Random Processes for Engineers, 4th ed., Henry Stark and John W. Woods, Pearson Education, Inc., 2012. B.J. Bazuin, Fall 2016 1 of 78 ECE 3800 9.1 Basic Concepts A random process is a collection of time functions and an associated probability description. When a continuous or discrete or mixed process in time/space can be describe mathematically as a function containing one or more random variables. A sinusoidal waveform with a random amplitude. A sinusoidal waveform with a random phase. A sequence of digital symbols, each taking on a random value for a defined time period (e.g. amplitude, phase, frequency). A random walk (2-D or 3-D movement of a particle) The entire collection of possible time functions is an ensemble, designated as xt , where one particular member of the ensemble, designated as xt, is a sample function of the ensemble. In general only one sample function of a random process can be observed! Think of: X t Asinwt , 0 2 where A and w are known constants. Note that once a sample has been observed … x1 t1 Asinwt1 the function is known for all time, t. Note that, xt2 is a second time sample of the same random process and does not provide any “new information” about the value of the random variable. x1 t2 Asinwt2 There are many similar ensembles in engineering, where the sample function, once known, provides a continuing solution. In many cases, an entire system design approach is based on either assuming that randomness remains or is removed once actual measurements are taken! For example, in communications there is a significant difference between coherent (phase and frequency) demodulation and non-coherent (i.e. unknown starting phase) demodulation. On the other hand, another measurement in a different environment might measure x2 t1 A2 sinwt1 2 In this “space” the random variables could take on other values within the defined ranges. Thus an entire “ensemble” of possibilities may exist based on the random variables defined in the random process. Notes and figures are based on or taken from materials in the course textbook: Probability, Statistics and Random Processes for Engineers, 4th ed., Henry Stark and John W. Woods, Pearson Education, Inc., 2012. B.J. Bazuin, Fall 2016 2 of 78 ECE 3800 For example, assume that there is a known AM signal transmitted: st 1 b Atsinwt at an undetermined distance the signal is received as yt 1 b At sinwt , 0 2 The received signal is mixed and low pass filtered … xt h t y t cos wt ht 1 b Atsinwt coswt,0 2 xt h t y t cos wt ht 1 b At 0.5sin2 wt sin ,0 2 If the filter removes the 2wt term, we have 1 b At xt h t y t cos w t sin , 0 2 2 Notice that based on the value of the random variable, the output can change significantly! From producing no output signal, ( 0, ), to having the output be positive or negative ( 0to or to 2 ). P.S. This is not how you perform non-coherent AM demodulation. To perform coherent AM demodulation, all I need to do is measured the value of the random variable and use it to insure that the output is a maximum (i.e. mix with coswt m , where. m t1 Note: the phase is a function of frequency, time, and distance from the transmitter. Notes and figures are based on or taken from materials in the course textbook: Probability, Statistics and Random Processes for Engineers, 4th ed., Henry Stark and John W. Woods, Pearson Education, Inc., 2012. B.J. Bazuin, Fall 2016 3 of 78 ECE 3800 From our textbook Random Stochastic Sequence Definition 8.1-1. Let ,,P be a probability space. Let . Let Xn,be a mapping of the sample space into a space of complex-valued sequences on some index set Z. If, for each fixed integer n Z , Xn, is a random variable, then Xn, is a ransom (stochastic) sequence. The index set Z is all integers, n , padded with zeros if necessary, Definition 9.1-1. Let ,,P be a probability space. then define a mapping of X from the sample space to a space of continuous time functions. The elements in this space will be called sample functions. This mapping is called a random process if at each fixed time the mapping is a random variable, that is, Xt, for each fixed t on the real line t . Example sets of random sequence. Figure 8.1-1 Illustration of the concept of random sequence X(n,ζ), where the ζ domain (i.e., the sample space Ω) consists of just ten values. (Samples connected only for plot.) Example sets of random process. Figure 9.1-1 A random process for a continuous sample space Ω = [0,10]. Notes and figures are based on or taken from materials in the course textbook: Probability, Statistics and Random Processes for Engineers, 4th ed., Henry Stark and John W. Woods, Pearson Education, Inc., 2012. B.J. Bazuin, Fall 2016 4 of 78 ECE 3800 Example 9.1-2 Separable random process may be constructed by combining a deterministic sequence with one or more random variables. The classic example already shown is a sinusoid with random amplitude and phase: X t, A sin2 f 0 t Where the amplitude and phase are R.V. defined based on the probability space selected. Example 9.1‐3 A random process used to model a continuous sequence of random communication symbols. X t An pt Tn n In a communication class, Dr. Bazuin would typically use the following X t An p t n T , for pt non zero for 0 t k T n Here An is the amplitude and phase of a complex communication symbol and p(t) is the deterministic time function, the simplest of which is a rectangular pulse in time. This can be used to describe a wide range of digital communication systems, including; Phase- Shift Keyed (PSK) or Quadrature Amplitude Modulation (QAM) communication signals. Notes and figures are based on or taken from materials in the course textbook: Probability, Statistics and Random Processes for Engineers, 4th ed., Henry Stark and John W. Woods, Pearson Education, Inc., 2012. B.J. Bazuin, Fall 2016 5 of 78 ECE 3800 The application of the Expected Value Operator Moments play an important role and, for Ergodic Processes, they can be estimated from a single process in time of the infinite number that may be possible. Therefore, X t EX t and the correlation functions (auto- and cross-correlation) * RXX t1,t2 EX t1 X t2 * RXY t1,t2 EX t1 Yt2 and the covariance functions (auto- and cross-correlation) * K XX t1,t2 EX t1 X t1 X t2 X t2 * K XY t1,t2 EX t1 X t1 Yt2 Y t2 with * K XX t1,t2 RXX t1,t2 X t1 X t2 Note that the variance can be computed from the auto-covariance as * 2 K XX t,t EX t X t X t X t X t and the “power” function can be computed from the auto-correlation * 2 RXX t,t EX t X t EX t 2 2 2 For real X(t) RXX t,t EX t X t X t Notes and figures are based on or taken from materials in the course textbook: Probability, Statistics and Random Processes for Engineers, 4th ed., Henry Stark and John W. Woods, Pearson Education, Inc., 2012. B.J. Bazuin, Fall 2016 6 of 78 ECE 3800 Example 9.1‐5 Auto‐correlation of a sinusoid with random phase Think of: X t Asinwt , where A and w are known constants. And theta is a uniform pdf covering the unit circle. The mean is computed as X t EX t EAsinwt X t EX t A Esinwt 1 t E X t A sin wt d X 2 A t EX t coswt X 2 A A t EX t coswt cos wt 0 0 X 2 2 ( What would happen if 0 instead? ) The auto-correlation is computed as * * RXX t1,t2 EX t1 X t2 EAsinwt1 Asinwt2 R t ,t E X t X t * A2 E 1 cos w t t 1 cos w t t 2 XX 1 2 1 2 2 1 2 2 1 2 A2 A2 R t ,t cosw t t Ecosw t t 2 XX 1 2 2 1 2 2 1 2 A2 A2 R t ,t cosw t t 0 cosw t t XX 1 2 2 1 2 2 1 2 ( This works if 0 instead. ) Note that if A was a random variable (independent of phase) we would have … EA2 EA2 R t ,t cosw t t R cosw XX 1 2 2 1 2 XX 2 and we would still have EA t EX t 0 0 X 2 Note: this Random Process is Wide-Sense stationary (mean and variance not a function of time) Notes and figures are based on or taken from materials in the course textbook: Probability, Statistics and Random Processes for Engineers, 4th ed., Henry Stark and John W.