Chapter Summary

Chapter Summary

Key Points

• 1.

  Van Trees is the Bayesian CRLB. For a random parameter $\theta \sim \pi$ observed through likelihood $f(y\mid\theta)$, every estimator (biased or not) satisfies $\mathbb{E}[(\hat\theta-\theta)^2] \geq 1/I_B$ with $I_B = \mathbb{E}_\pi[I_F(\theta)] + I_P$. Data information and prior information add literally.

• 2.

  Ziv-Zakai captures the threshold effect. The ZZB bounds MSE by an integral of minimum error probabilities between neighbouring parameter values, valley-filled for monotonicity. It asymptotes to the CRLB at high SNR and to the a priori variance at low SNR, tracing the threshold transition that the CRLB misses.

• 3.

  I-MMSE ties MMSE to mutual information. On the canonical channel $Y=\sqrt{\text{SNR}}\,X+N$, $d\,I/d\,\text{SNR}= \tfrac{1}{2}\text{mmse}(\text{SNR})$ for every input distribution. Mutual information is the integral of an MMSE curve.

• 4.

  ISAC replaces either-or with a tradeoff region. The rate-CRB region is traced by SDP-based beampattern design: the non-convex $\mathrm{tr}(\mathbf{J}^{-1})$ minimisation becomes a convex SDP via the Schur complement, and the Pareto boundary morphs continuously between comm-only and sensing-only beams.

• 5.

  Every bound has a proof pattern. Van Trees = Cauchy-Schwarz on the joint score. Ziv-Zakai = pairwise hypothesis testing + valley-filling. I-MMSE = heat equation + Stein's identity. SDP beampattern = Schur complement. Recognising these patterns is how one extends each bound to new settings.