LINEAR SYSTEMS

$$y(t)=x(t)\ast h(t)$$

$$y(t)=\int h(t-\tau)x(\tau)d\tau$$

What can we say about y(t), y(t₁), x(t₂) etc. when x(t) is stationary.

Thm.

1.

$$m_y=m_x\int_{-\infty}^a h(\tau)d\tau$$

2.

$$R_{XY}(\tau)=R_X(\tau)\ast h(-\tau)$$

3.

$$R_Y(\tau)=R_X(\tau)\ast h(\tau)\ast h(-\tau)$$

Def.

1.

$$E(Y(t))=\int_{-a}^{\infty}h(t-\tau)E(x(\tau))d\tau=m_x\int_{-\infty}^{\infty}h(\tau)d\tau$$

2.

$$\begin{eqnarray*}R_{XY}(t_1,t_2)&=&E\left[\int_{-\infty}^{\infty}x(t_1)x(t_2-\si... ...infty}R_X(\tau-\sigma)h(-\sigma)d\sigma\\ &=&R_X(\tau)*h(-\tau) \end{eqnarray*}$$

3.

$$\begin{eqnarray*}R_Y(t_1,t_2)&=&\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} E... ...o)h(\sigma)h(\rho)d\rho d\sigma\\ &=&R_X(\tau)*h(\tau)*h(-\tau) \end{eqnarray*}$$

Punchline $\rightarrow$ WSS in $\rightarrow$ WSS out
Also true that SSS in $\rightarrow$ SSS out, but we did not prove it

Power spectra
Def.
Define x_T(t)=x(t)(u(t+T/2)-u(t-T/2))

X_T(f)=F[x_T(t)]

$$S_X(f)=E\lim_{T\rightarrow\infty} \frac{\vert X_T(f)\vert^2}{T} =\lim_{T\rightarrow\infty} \frac{\vert X(f)\vert^2}{T}$$

Now need to relate R_X(t₁,t₂) to S_X(f)

Thm. (Weiner Khinchin)

If $R_X(t+\tau,t)$ is integrable on any interval of length $\vert\tau\vert$ with $\vert\tau\vert<\infty$, i.e.

$$\left\vert\int_a^{a+\tau} R_X(t+\tau ,t)dt\right\vert<\infty \qquad \forall a\in R$$

then

$$S_X(f)=F\left[<R_X(t+\tau,t)>\right]$$

where

$$<R_X(t+\tau,t)>=\lim_{T\rightarrow\infty}\frac{1}{T} \int_{-T/2}^{T/2} R_X(t+\tau,t)dt$$

Please look but don't touch

Corollaries

1.

X(t) stationary and $\tau R_X(\tau)$ finite for $\vert\tau\vert<\infty$

$$S_X(f)=F(R_X(\tau))$$

Pf.
W-K $\rightarrow$

$$\lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/2} R_X(\tau)d\tau=R_X(\tau)$$

because $\tau R_X(\tau)$ finite, says

$$\int_a^{a+\tau}R_X(t+\tau,t)dt=\int_a^{a+\tau}R_X(t)dt=\tau R_X(\tau)<\infty$$

2.

X(f) cyclostationary w/ period

$$\left\vert\int_0^{T_0} R_X(t+\tau,t)dt\right\vert<\infty$$

then

$$S_X(f)=F(\bar{R_X}(\tau))$$

$$R_X(\tau)=\frac{1}{T_0}\int_{-T_0/2}^{T_0/2} R_X(t+\tau,t)dt$$

pf. conditions satisfied for W-K if satisfied over a single period and

$$\begin{eqnarray*}\lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/2}R_X(t+\tau... ...tau,t)dt\\ &=&\frac{1}{T_0}\int_{-T_0/2}^{T_0/2}R_X(t+\tau,t)dt \end{eqnarray*}$$

Useful Stuff

1.

$$P_X=\left.\int_{-\infty}^{\infty}S_X(f)df=<R_X(t+\tau,t)>\right\vert _{\tau=0}$$

2.

PSD's of each sample function are equal for an ergodic process

3.

Turns out that S_X(f) is nonnegative and even

LTI systems again
Remember

$$m_Y=m_X \int_{-\infty}^{\infty}h(\tau)d\tau$$

$$R_{XY}(\tau)=R_X(\tau)*h(-\tau)$$

$$R_Y(\tau)=R_X(\tau)*h(-\tau)*h(\tau)$$

So

$$m_Y=m_X H(0)=\left.\int_{-\infty}^{\infty}e^{-j2\pi f\tau}h(\tau)d\tau\right\vert _{f=0}$$

S_XY(f)=S_X(f)H^*(f)

S_YX(f)=S_X(f)H(f)

S_Y(f)=S_X(f)|H(f)|²

NICE PICTURE

Sums

z(t)=x(t)+y(t)

m_Z=m_Y+m_X

$$R_Z(\tau)=R_X(\tau)+R_Y(\tau)+R_{XY}(\tau)+R_{YX}(\tau)$$

S_Z(f)=S_X(f)+S_Y(f)+2Re[S_XY(f)]

Uncorrelated?

$$R_{XY}(\tau)=m_X m_Y$$

and if m_Y or m_X is zero

S_Z(f)=S_X(f)+S_Y(f)