: 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:
| 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 ) |
[ = E[s[n]s[n+k]] + E[s[n]w[n+k]] + E[w[n]s[n+k]] + E[w[n]w[n+k]] ] : Always define new symbols the first time they appear
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(DTFT of ( r_xx[k] )) [ S_xx(e^j\omega) = \sum_k=-\infty^\infty r_xx[k] e^-j\omega k = \sum_k r_ss[k] e^-j\omega k + \sigma_w^2 \sum_k \delta[k] e^-j\omega k ] [ \boxed S_xx(e^j\omega) = S_ss(e^j\omega) + \sigma_w^2 ] Find the autocorrelation ( r_xx[k] ) and power
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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] ] So: [ r_xx[k] = r_ss[k] + r_ww[k] =
Chapter 2: Probability Review 2.1 – 2.20 solutions