Lesson 8 Flashcards

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1. What are the two error modes in detection?

A miss (target present, threshold not crossed; probability \(1-P_d\)) and a false alarm (noise alone crosses the threshold; probability \(P_{fa}\)).

2. Give the noise-floor formula.

\(N_{\text{dBm}} = -174 + 10\log_{10}(B_{\text{Hz}}) + \text{NF}_{\text{dB}}\): cold-receiver density at 290 K, plus bandwidth, plus receiver noise figure.

3. Compute the noise floor for \(B=10\) MHz, NF \(=3\) dB.

\(N = -174 + 70 + 3 = -101\) dBm. Narrowing to 1 MHz drops it 10 dB to \(-111\) dBm.

4. What happens to \(P_d\) and \(P_{fa}\) as you raise the threshold?

Both go down: fewer noise spikes survive (lower \(P_{fa}\)), but some real returns are rejected too (lower \(P_d\)). You can't lower one without the other unless SNR rises.

5. What SNR does a common operating point require?

\(P_d=0.9\) at \(P_{fa}=10^{-6}\) on a steady target needs about 13 dB. A fluctuating (Swerling I) target needs much more, around 21 dB.

6. What does a ROC curve plot, and what does higher SNR do to it?

\(P_d\) versus \(P_{fa}\) (log axis) as the threshold sweeps. Higher SNR pushes the curve toward the top-left corner — the ideal detector.

7. Why is the ROC curve useful?

It converts a commander's preference (a tolerable \(P_{fa}\) and required \(P_d\)) into a single engineering spec: the required SNR at the detector input.

8. Coherent vs non-coherent integration gain for \(N\) pulses?

Coherent: gain \(=N\), i.e. \(+10\log_{10}N\) dB (100 pulses → 20 dB). Non-coherent: gain \(\approx\sqrt{N}\), i.e. \(\approx +5\log_{10}N\) dB (100 pulses → ~10 dB).

9. What does coherent integration require?

Phase coherence across the whole dwell — summing complex returns with phase preserved. Modern AESAs are built to maintain it.

10. Write \(S_{\min}\) in terms of the detection contract.

\(S_{\min} = kTB\cdot\text{NF}\cdot\text{SNR}_{\text{req}}/G_{\text{int}}\). It is a \(P_d/P_{fa}\) contract, not a constant.

11. How does 20 dB of integration gain change \(R_{\max}\)?

\(S_{\min}\) drops 20 dB; since \(R_{\max}\propto S_{\min}^{-1/4}\), range gains \(20/4=5\) dB — a factor of \(\approx 3.2\times\) in km. A longer dwell becomes a longer range.

12. A radar tightens \(P_{fa}\) from \(10^{-6}\) to \(10^{-9}\). Effect on \(R_{\max}\)?

It shortens. A tighter false-alarm contract needs a few more dB of SNR to hold the same \(P_d\), raising \(S_{\min}\) and shrinking range.