Chapter Summary

Chapter 15 Summary: Linear Systems with Random Inputs

Key Points

• 1.

  WSS through LTI: If $X(t)$ is WSS and the LTI system is BIBO-stable, the output $Y(t)$ is also WSS with $\mu_Y = \mu_X \check{h}(0)$, $r_{yy}(\tau) = h * h^*(-\cdot) * r_{xx}(\tau)$, and $P_y(f) = |\check{h}(f)|^2 P_x(f)$.

• 2.

  Matched filter: The filter $h(t) = s^*(t_0 - t)$ maximizes the output SNR for a known signal $s(t)$ in AWGN. The maximum SNR is $\text{SNR}_{\max} = 2E_s/N_0$, depending only on signal energy and noise PSD — not signal shape.

• 3.

  Wiener filter: The non-causal LMMSE filter for extracting a WSS signal from uncorrelated noise has $\check{h}_{\text{opt}}(f) = P_x(f)/(P_x(f) + P_N(f))$, weighting each frequency by the local signal fraction.

• 4.

  Noise bandwidth: $B_N$ is the width of the equivalent ideal filter passing the same noise power. For white noise, $\mathcal{P}_N = N_0 B_N$. Real filters have $B_N > f_{3\text{dB}}$.

• 5.

  Noise figure: $F = \text{SNR}_{\text{in}}/\text{SNR}_{\text{out}}$ quantifies SNR degradation. Friis's cascade formula shows the first stage dominates: $F_{\text{total}} \approx F_1$ when $G_1$ is large.

• 6.

  Unifying theme: All results in this chapter are consequences of the PSD relation $P_y(f) = |\check{h}(f)|^2 P_x(f)$. The matched filter, Wiener filter, and noise bandwidth are three different applications of the same fundamental identity.