3.2 Linear filtering

Properties

• $\langle Z\rangle =\langle \int_a^b X_t()\,d t\rangle =\int_a^b {}_X(t)\,d t$
• $\langle Z^2\rangle =\langle \int_a^b X_{t_2}()\,d t_2\int_a^b \overline{X_{t_1}}\,d t_1\rangle =\int_a^b \int_a^b R_X(t_2, t_1)\,d t_1\,d t_2$

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