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Random Processes ECE460 Spring, 2012 Power Spectral Density Generalities : Example: X t , i A cos 2 f 0 t i 2 Example Given a process Yt that takes the values ±1 with equal probabilities: P Yt 1 P Yt 1 1 / 2 P Yt 1 | Yt 2 1 P Yt 1 | Yt 2 1 1 1 , 2T 1 / 2, T T w h ere t 2 t1 Find R Y t1 , t 2 3 Ergodic 1. A wide-sense stationary (wss) random process is ergodic in the mean if the time-average of X(t) converges to the ensemble average: lim t 1 T T /2 T /2 g ( x ( t ; i )) d t E [ g ( X ( t ))] 2. A wide-sense stationary (wss) random process is ergodic in the autocorrelation if the time-average of RX(τ) converges to the ensemble average’s autocorrelation R X x t x t E X t X t R X 3. Difficult to test. For most communication signals, reasonable to assume that random waveforms are ergodic in the mean and in the autocorrelation. 4. For electrical engineering parameters: 1. x x t 2. x 3. R XX 0 x 4. X x 5. X is eq u al to th e rm s valu e o f th e ac co m p o n en t 2 2 is eq u al to th e d c level o f th e sig n a l x t 2 t 2 eq u als th e n o rm alized d c p o w er 2 t is eq u al to th e to tal avg . n o rm alize d p o w er x t 2 is th e avg . n o rm alized ac p o w er 4 Multiple Random Processes h (t ) Filter X (t ) Y (t ) Multiple Random Processes • Defined on the same sample space (e.g., see X(t) and Y(t) above) • For communications, limit to two random processes Independent Random Processes X(t) and Y(t) – If random variables X(t1) and Y(t2) are independent for all t1 and t2 Uncorrelated Random Processes X(t) and Y(t) – If random variables X(t1) and Y(t2) are uncorrelated for all t1 and t2 Jointly wide-sense stationary – If X(t) and Y(t) are both individually wss – The cross-correlation function RXY(t1, t2) depends only on τ = t2 - t1 R XY t , t E X t Y t R X Y 5 Transfer Through a Linear System X t h t Y t Let X(t) be a wss random process and h(t) be the impulse response of a stable filter. Find E{Y(t)} Find the cross-correlation function RXY(t1,t2). 6 Transfer Through a Linear System X t h t Y t For a wss X(t) with autocorrelation RX(τ) and a stable (bibo) filter h(t), X(t) and Y(t) are jointly wide sense stationary: 7 Example X t h t X t A co s 2 f 0 t Y t h t differentiator t w h ere is a ran d o m variab le u n ifo rm ly d istrib u ted o n [0 ,2 ] mY : R X t1 , t 2 : SY f : SXY f : 8 Energy Processes Recall that the energy of signal x(t) was calculated by Ex x 2 t dt if Ex < ∞ then this is an energy signal Define for a random process EX X 2 t dt Then the energy content of this signal can be given by EX E EX 2 E X t dt E X 2 t d t R X t, t dt 9 Power Processes Recall that the power of signal x(t) was calculated by Px lim T T 1 T 2 x T 2 t dt 2 if Px > 0 then this is a power signal Define for a random process P X lim 1 T T T 2 T 2 X t dt 2 Then the power of signal x(t) can be given by PX E P X 1 E lim T T lim T lim T 1 T 1 T T X 2 E X 2 2 T 2 T 2 T t d t 2 T t dt 2 T R X t, t dt 2 10 Example Show that SX(f0) ≈ power of X(t) in [f0, f0+Δf] X t H h t f 1, 0, f f0 , f0 f Y t o th erw ise 11 Thermal Noise R N t Sn f 2K T R W /H z R - valu e o f resister in O h m s T - tem perature of resister in K elvin 23 K - B o ltzm an n 's co n stan t 1 .3 8 1 0 J/K Because of the wide-band of thermal noise, it is usually modeled as white noise: 12 Gaussian Processes A random process X(t) is Gaussian if for every t1, t2, …, tn, and every n, the random variables X(t1), X(t2)…, X(tn) Are jointly Gaussian. The Gaussian random process is completely determined by its mean and autocorrelation functions, i. e., by m X t E X t R X t1 , t 2 E X t 1 X t 2 If a Gaussian process X(t) is passed through a linear filter, the output process is Gaussian If X(t) is a wss Gaussian process with mean mX(t), autocorrelation RX(τ), and an LTI filter with input response h(t), then Y(t) = X(t)* h(t) is a wss Gaussian process with mY t m X t H 0 RY R X h * h 13 Zero-Mean White Gaussian Noise A zero mean white Gaussian noise, W(t), is a random process with 1. E W t 0 t 2. RW 3. SW E W t W t f No No 2 W att/H z 2 4. For any n and any sequence t1, t2, …, tn the random variables W(t1), W(t2), …, W(tn), are jointly Gaussian with zero mean E W t i 0 fo r i 1, 2, ..., n and covariances co v W t i W t j E W t i W t j RW t j t i No 2 t j ti 14 Bandpass Processes Similar to Chapter 2.5 X(t) is a bandpass process R X an d S X is a d eterm in istic b an d p ass sig n al f F R X is n o n -zero ab o u t f 0 Filter X(t) using a Hilbert Transform: h t H f 1 t j sg n f and define X c t X t co s 2 f 0 t X t sin 2 f 0 t X s t X t co s 2 f 0 t X t sin 2 f 0 t If X(t) is a zero-mean stationary bandpass process, then Xc(t) and Xs(t) will be zero-mean jointly stationary processes: E X c t E X s t 0 R X c t , t R X c R X s t , t R X s R X c X s t , t R X c X s 15 Bandpass Processes This results in two key formulas for future use: R X c R X s R X co s 2 f 0 R X sin 2 f 0 R X c X s R X sin 2 f 0 R X co s 2 f 0 Note: Xc(t) and Xs(t) are lowpass processes; i.e., their power spectrum vanishes for |f| ≥ W. Find the power spectrum of the in-phase and quadrature components: 16 Bandpass Example 4.6.1 The white Gaussian noise process N(t) with power spectrum N0/2 passes through an ideal bandpass filter with frequency response H f 1, 0, f fc W O th erw ise where W << fc. The output process is denoted by X(t). Find the power spectrum and the cross-spectral density of the in-phase and quadrature components in the following two cases: 1. f0 is chosen to be equal to fc. 2. f0 is chosen to be equal to fc-W. 17