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##### Lenses

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# Linear Filtering

Module by: Behnaam Aazhang. E-mail the author

Summary: Describes signals that cannot be precisely characterized.

## Integration

Zω=ab X t ωd t Z ω t a b X t ω
(1)

## Linear Processing

Y t =htτ X τ d τ Y t τ h t τ X τ
(2)

## Differentiation

X t =dd t X t X t t X t
(3)

## Properties

1. Z-=ab X t ωd t -=ab μ X td t 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 -=abab 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

μ Y t=htτ X τ d τ -=htτ μ X τd τ 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=htτ μ X d τ = μ X htd 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 d τ X t 1 ¯-=h t 2 τ R X τ t 1 d τ 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 τd 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 ¯d τ -=h t 1 τ R Y X t 2 τd τ =h t 1 τ R Y X t 2 τd τ 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 =hτ( t 2 t 1 ) R Y X τd 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=hτ h ~ τ h τ for all τR τ . Y t Y t is WSS if X t X t is WSS and the linear system is time-invariant.

## 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(at)ut 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 2hαhατd α = 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(i2πfτ)d τ 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 fd f Var X t R X 0 f S X f
3. S X f S X f is real and nonnegative S X f0 S X f 0 for all ff.

If Y t =htτ X τ d τ Y t τ h t τ X τ then

S Y f= R Y τ=h* h ~ * R X τ=Hf H ~ f S X f=|Hf|2 S X f S Y f R Y τ h h ~ R X τ H f H ~ f S X f H f 2 S X f
(12)
since H ~ f= h ~ te(i2πft)d t =Hf¯ H ~ f t h ~ t 2 f t H f

## Example 2

X t X t is a white process and ht=e(at)ut 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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#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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