1. Random Processes ECE460 Spring, 2012

2. Power Spectral Density Generalities : Example: 2

3. Example Given a process Y t that takes the values ±1 with equal probabilities: Find 3

4. 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: 2.A wide-sense stationary (wss) random process is ergodic in the autocorrelation if the time-average of R X (τ) converges to the ensemble average’s autocorrelation 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: 4

5. 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 ( t 1 ) and Y ( t 2 ) are independent for all t 1 and t 2 Uncorrelated Random Processes X ( t ) and Y ( t ) – If random variables X ( t 1 ) and Y ( t 2 ) are uncorrelated for all t 1 and t 2 Jointly wide-sense stationary – If X ( t ) and Y ( t ) are both individually wss – The cross-correlation function R XY ( t 1, t 2 ) depends only on τ = t 2 - t 1 5 Filter

6. Transfer Through a Linear System 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 R XY ( t 1,t 2 ). 6

7. Transfer Through a Linear System For a wss X ( t ) with autocorrelation R X ( τ ) and a stable (bibo) filter h ( t ), X ( t ) and Y ( t ) are jointly wide sense stationary: 7

8. Example 8

9. Energy Processes Recall that the energy of signal x(t) was calculated by if E x < ∞ then this is an energy signal Define for a random process Then the energy content of this signal can be given by 9

10. Power Processes Recall that the power of signal x(t) was calculated by if P x > 0 then this is a power signal Define for a random process Then the power of signal x(t) can be given by 10

11. Example Show that S X (f 0 ) ≈ power of X ( t ) in [ f 0, f 0 +Δf] 11

12. Thermal Noise Because of the wide-band of thermal noise, it is usually modeled as white noise: 12

13. Gaussian Processes A random process X ( t ) is Gaussian if for every t 1, t 2, …, t n, and every n, the random variables X ( t 1 ), X ( t 2 )…, X ( t n ) Are jointly Gaussian. The Gaussian random process is completely determined by its mean and autocorrelation functions, i. e., by 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 m X ( t ), autocorrelation R X (τ), and an LTI filter with input response h ( t ), then Y ( t ) = X ( t )* h ( t ) is a wss Gaussian process with 13

14. Zero-Mean White Gaussian Noise A zero mean white Gaussian noise, W ( t ), is a random process with 4.For any n and any sequence t 1, t 2, …, t n the random variables W (t 1 ), W( t 2 ), …, W( t n ), are jointly Gaussian with zero mean and covariances 14

15. Bandpass Processes Similar to Chapter 2.5 X ( t ) is a bandpass process Filter X ( t ) using a Hilbert Transform: and define If X ( t ) is a zero-mean stationary bandpass process, then X c ( t ) and X s ( t ) will be zero-mean jointly stationary processes: 15

16. Bandpass Processes This results in two key formulas for future use: Note: X c ( t ) and X s ( 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

17. Bandpass Example 4.6.1 The white Gaussian noise process N(t) with power spectrum N 0 /2 passes through an ideal bandpass filter with frequency response where W << f c. 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.f 0 is chosen to be equal to f c. 2.f 0 is chosen to be equal to f c - W. 17