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Linear Filtering

Module by: Behnaam Aazhang

Summary: Describes signals that cannot be precisely characterized.

Integration Zω=∫ab X t ωdt Z ω t a b X t ω (1)
Linear Processing Y t =∫-∞∞htτ X τ dτ Y t τ h t τ X τ (2)
Differentiation X t ′=ddt X t X t t X t (3)
    Properties
  1. Z¯=∫ab X t ωdt¯=∫ab μ X tdt Z t a b X t ω t a b μ X t
  2. Z2¯=∫ab X t 2 d t 2 ∫ab X t 1 ¯d t 1 ¯=∫ab∫ab R X t 2 t 1 d t 1 d t 2 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
Figure3-12.png
Figure 1
μ Y t=∫-∞∞htτ X τ dτ¯=∫-∞∞htτ μ X τdτ μ Y t τ h t τ X τ τ h t τ μ X τ (4)
If X t X t is wide sense stationary and the linear system is time invariant
μ Y t=∫-∞∞ht-τ μ X dτ= μ X ∫-∞∞h t ′d t ′= μ Y μ Y t τ h t τ μ X μ X t h t μ Y (5)
R Y X t 2 t 1 = Y t 2 X t 1 ¯¯=∫-∞∞h t 2 -τ X τ dτ X t 1 ¯¯=∫-∞∞h t 2 -τ R X τ- t 1 dτ 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 (6)
R Y X t 2 t 1 =∫-∞∞h t 2 - t 1 - τ ′ R X τ ′d τ ′=h* R X t 2 - 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 (7)
where τ ′=τ- t 1 τ τ t 1 .
R Y t 2 t 1 = Y t 2 Y t 1 ¯¯= Y t 2 ∫-∞∞h t 1 τ X τ ¯dτ¯=∫-∞∞h t 1 τ R Y X t 2 τdτ=∫-∞∞h t 1 -τ R Y X t 2 -τdτ 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 (8)
R Y t 2 t 1 =∫-∞∞h τ ′- t 2 - t 1 R Y X τ ′d τ ′= R Y t 2 - t 1 = h ~ * R Y X t 2 t 1 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 (9)
where τ ′= t 2 -τ τ t 2 τ and h ~ τ=h-τ h ~ τ h τ for all τ∈ℝ τ . Y t Y t is WSS if X t X t is WSS and the linear system is time-invariant.
Figure3-13.png
Figure 2
Example 1 
X t X t is a wide sense stationary process with μ X =0 μ X 0 , and R X τ= N 0 2δτ R X τ N 0 2 δ τ . Consider the random process going through a filter with impulse response ht=ⅇ-atut h t a t u t . The output process is denoted by Y t Y t . μ Y t=0 μ Y t 0 for all tt.
R Y τ= N 0 2∫-∞∞hαhα-τdα= N 0 2ⅇ-a|τ|2a R Y τ N 0 2 α h α h α τ N 0 2 a τ 2 a (10)
X t X t is called a white process. Y t Y t is a Markov process.
Definition 1: Power Spectral Density
The power spectral density function of a wide sense stationary (WSS) process X t X t is defined to be the Fourier transform of the autocorrelation function of X t X t .
S X f=∫-∞∞ R X τⅇ-ⅈ2πfτdτ S X f τ R X τ 2 f τ (11)
if X t X t is WSS with autocorrelation function R X τ R X τ .
    Properties
  1. S X f= S X -f S X f S X f since R X R X is even and real.
  2. Var X t = R X 0=∫-∞∞ S X fdf Var X t R X 0 f S X f
  3. S X f S X f is real and nonnegative S X f≥0 S X f 0 for all ff.
If Y t =∫-∞∞ht-τ X τ dτ Y t τ h t τ X τ then
S Y f=ℱ R Y τ=ℱh* h ~ * R X τ=Hf H ~ f S X f=|Hf|2 S X f 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 (12)
since H ~ f=∫-∞∞ h ~ tⅇ-ⅈ2πftdt=Hf¯ H ~ f t h ~ t 2 f t H f
Example 2 
X t X t is a white process and ht=ⅇ-atut h t a t u t .
Hf=1a+ⅈ2πf H f 1 a 2 f (13)
S Y f= N 0 2a2+4π2f2 S Y f N 0 2 a 2 4 2 f 2 (14)

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