# 3.2 Linear filtering

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Describes signals that cannot be precisely characterized.

## Integration

$Z()=\int_{a}^{b} {X}_{t}()\,d t$

## Linear processing

${Y}_{t}=\int_{()} \,d$ h t X

## Differentiation

${X}_{t}^\prime =\frac{d {X}_{t}}{d t}}$

## Properties

• $\langle Z\rangle =\langle \int_{a}^{b} {X}_{t}()\,d t\rangle =\int_{a}^{b} {}_{X}(t)\,d t$
• $\langle Z^{2}\rangle =\langle \int_{a}^{b} {X}_{{t}_{2}}()\,d {t}_{2}\int_{a}^{b} \overline{{X}_{{t}_{1}}}\,d {t}_{1}\rangle =\int_{a}^{b} \int_{a}^{b} {R}_{X}({t}_{2}, {t}_{1})\,d {t}_{1}\,d {t}_{2}$

${}_{Y}(t)=\langle \int_{()} \,d \rangle$ h t X h t X
If ${X}_{t}$ is wide sense stationary and the linear system is time invariant
${}_{Y}(t)=\int_{()} \,d$ h t X X t h t Y
${R}_{YX}({t}_{2}, {t}_{1})=\langle {Y}_{{t}_{2}}\overline{{X}_{{t}_{1}}}\rangle =\langle \int_{()} \,d \rangle$ h t 2 X X t 1 h t 2 R X t 1
${R}_{YX}({t}_{2}, {t}_{1})=\int_{()} \,d ^\prime$ h t 2 t 1 R X h R X t 2 t 1
where $^\prime =-{t}_{1}$ .
${R}_{Y}({t}_{2}, {t}_{1})=\langle {Y}_{{t}_{2}}\overline{{Y}_{{t}_{1}}}\rangle =\langle {Y}_{{t}_{2}}\int_{()} \,d \rangle$ h t 1 X h t 1 R Y X t 2 h t 1 R Y X t 2
${R}_{Y}({t}_{2}, {t}_{1})=\int_{()} \,d ^\prime$ h t 2 t 1 R Y X R Y t 2 t 1 h ~ R Y X t 2 t 1
where $^\prime ={t}_{2}-$ and $\stackrel{~}{h}()=h(-)$ for all $\in \mathbb{R}$ . ${Y}_{t}$ is WSS if ${X}_{t}$ is WSS and the linear system is time-invariant.

${X}_{t}$ is a wide sense stationary process with ${}_{X}=0$ , and ${R}_{X}()=\frac{{N}_{0}}{2}()$ .Consider the random process going through a filter with impulse response $h(t)=e^{-(at)}u(t)$ .The output process is denoted by ${Y}_{t}$ . ${}_{Y}(t)=0$ for all $t$ .

${R}_{Y}()=\frac{{N}_{0}}{2}\int_{()} \,d$ h h N 0 2 a 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 functionof ${X}_{t}$ .
${S}_{X}(f)=\int_{()} \,d$ R X 2 f
if ${X}_{t}$ is WSS with autocorrelation function ${R}_{X}()$ .

## Properties

• ${S}_{X}(f)={S}_{X}(-f)$ since ${R}_{X}$ is even and real.
• $\mathrm{Var}({X}_{t})={R}_{X}(0)=\int_{()} \,d f$ S X f
• ${S}_{X}(f)$ is real and nonnegative ${S}_{X}(f)\ge 0$ for all $f$ .

If ${Y}_{t}=\int_{()} \,d$ h t X then

${S}_{Y}(f)=({R}_{Y}())=((h, \stackrel{~}{h}, {R}_{X}()))=H(f)\stackrel{~}{H}(f){S}_{X}(f)=\left|H(f)\right|^{2}{S}_{X}(f)$
since $\stackrel{~}{H}(f)=\int_{()} \,d t$ h ~ t 2 f t H f

${X}_{t}$ is a white process and $h(t)=e^{-(at)}u(t)$ .

$H(f)=\frac{1}{a+i\times 2\pi f}$
${S}_{Y}(f)=\frac{\frac{{N}_{0}}{2}}{a^{2}+4\pi ^{2}f^{2}}$

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