Linear Filtering 2.9 2001/08/06 2004/01/07 10:03:35.839 US/Central Behnaam Aazhang aaz@ece.rice.edu Dinesh Rajan dinesh@ece.rice.edu Mohammad Jaber Borran mohammad@ece.rice.edu Roy Ha rha@rice.edu Shawn Stewart mrshawn@alumni.rice.edu Behnaam Aazhang aaz@ece.rice.edu Markov process power spectral density random process stochastic process white process Describes signals that cannot be precisely characterized. Integration Z ω t a b X t ω Linear Processing Y t τ h t τ X τ Differentiation X t t X t Properties Z t a b X t ω t a b μ X t Z 2 t 2 a b X t 2 t 1 a b X t 1 t 2 a b t 1 a b R X t 2 t 1

μ Y t τ h t τ X τ τ h t τ μ X τ If X t is wide sense stationary and the linear system is time invariant μ Y t τ h t τ μ X μ X t h t μ Y R Y X t 2 t 1 Y t 2 X t 1 τ h t 2 τ X τ X t 1 τ h t 2 τ R X τ t 1 R Y X t 2 t 1 τ h t 2 t 1 τ R X τ h R X t 2 t 1 where τ τ t 1 . R Y t 2 t 1 Y t 2 Y t 1 Y t 2 τ h t 1 τ X τ τ h t 1 τ R Y X t 2 τ τ h t 1 τ R Y X t 2 τ R Y t 2 t 1 τ h τ t 2 t 1 R Y X τ R Y t 2 t 1 h ~ R Y X t 2 t 1 where τ t 2 τ and h ~ τ h τ for all τ . Y t is WSS if X t is WSS and the linear system is time-invariant.

X t is a wide sense stationary process with μ X 0 , and R X τ N 0 2 δ τ . Consider the random process going through a filter with impulse response h t a t u t . The output process is denoted by Y t . μ Y t 0 for all t. R Y τ N 0 2 α h α h α τ N 0 2 a τ 2 a X t is called a white process. Y t is a Markov process. Power Spectral Density The power spectral density function of a wide sense stationary (WSS) process X t is defined to be the Fourier transform of the autocorrelation function of X t . S X f τ R X τ 2 f τ if X t is WSS with autocorrelation function R X τ . Properties S X f S X f since R X is even and real. Var X t R X 0 f S X f S X f is real and nonnegative S X f 0 for all f. If Y t τ h t τ X τ then S Y f ℱ R Y τ ℱ h h ~ R X τ H f H ~ f S X f H f 2 S X f since H ~ f t h ~ t 2 f t H f X t is a white process and h t a t u t . H f 1 a 2 f S Y f N 0 2 a 2 4 2 f 2