Cross terms vanish: ( E[s[n]w[n+k]] = 0), ( E[w[n]s[n+k]] = 0). So: [ r_xx[k] = r_ss[k] + r_ww[k] = r_ss[k] + \sigma_w^2 \delta[k] ]
| Concept | Recommended notation | |---------|----------------------| | Vectors | bold lower: , y | | Matrices | bold upper: R , A | | Expectation | ( E[\cdot] ) | | Estimate | ( \hatx ) | | Convolution | ( * ) | | Autocorrelation | ( r_xx[k] ) | | Power spectrum | ( S_xx(e^j\omega) ) | | Derivative (gradient) | ( \nabla_\theta ) | Cross terms vanish: ( E[s[n]w[n+k]] = 0), (
: Always define new symbols the first time they appear. 4. Example Solution Entry Problem 3.7 (from “Random Processes” chapter): Let ( x[n] = s[n] + w[n] ), where ( s[n] ) is a zero‑mean WSS signal with autocorrelation ( r_ss[k] ), and ( w[n] ) is white noise with variance ( \sigma_w^2 ), uncorrelated with ( s ). Find the autocorrelation ( r_xx[k] ) and power spectral density ( S_xx(e^j\omega) ). Solution: Example Solution Entry Problem 3
