Chapter Summary

Chapter 4 Summary: Sequential Detection and CFAR

Key Points

• 1.

  The SPRT accumulates log-likelihood ratios $S_n = \sum_{i=1}^n \ell(y_i)$ and stops the first time $S_n$ exits the interval $[B, A]$ with $A \approx \log\frac{1-\beta}{\alpha}$, $B \approx \log\frac{\beta}{1-\alpha}$.

• 2.

  Wald's identity $\mathbb{E}[S_N] = \mathbb{E}[N] \cdot \mathbb{E}[\ell]$ yields the average sample number: the SPRT uses roughly $2\log(1/\alpha) / D(p_1 \| p_0)$ samples on average, versus $n \sim \log(1/\alpha) / D$ for a fixed-sample LRT — a 2x saving.

• 3.

  The Wald-Wolfowitz theorem: among all tests (sequential or fixed) with the same $P_f$ and $P_d$, the SPRT minimizes $\mathbb{E}_0[N]$ AND $\mathbb{E}_1[N]$.

• 4.

  CUSUM detects a change from $p_0$ to $p_1$ via $W_n = \max(0, W_{n-1} + \ell(y_n))$; stop at the first $n$ with $W_n \geq h$. Lorden's minimax framework: CUSUM asymptotically minimizes the worst-case detection delay for a given mean-time-between-false-alarms $\gamma$.

• 5.

  For CUSUM with threshold $h$: $\mathrm{ARL}_0 \approx e^h / D(p_0 \| p_1)$ and worst-case detection delay $\approx h / D(p_1 \| p_0)$. Delay grows logarithmically with the false-alarm rate.

• 6.

  CFAR detectors keep $P_f$ constant even when the noise level is unknown. CA-CFAR estimates noise from $N$ adjacent cells and scales the threshold as $T = \alpha_{CA} \hat\sigma^2$; the multiplier $\alpha_{CA} = N(P_f^{-1/N} - 1)$ follows from the F-distribution of the test statistic.

• 7.

  CA-CFAR degrades at clutter edges and near interfering targets; OS-CFAR (kth-order statistic) is more robust to interference; GO-CFAR/SO-CFAR trade clutter-edge performance against multi-target masking.

• 8.

  Applications: early-stopping in iterative decoding (SPRT on posterior LLRs), handover triggering (CUSUM on RSS), radar target detection (CFAR on range-Doppler maps), and spectrum sensing for cognitive radio (sequential energy detection).