# Reading — Detection Theory By the end of this lesson you should be able to: 1. Compute the **noise floor** for a given bandwidth and noise figure. 2. Explain the **detection threshold** and the $P_d / P_{fa}$ tradeoff. 3. Read a **receiver operating characteristic (ROC)** curve and identify the SNR that meets a mission requirement. 4. Quantify the SNR gain from **coherent** and **non-coherent** integration. ## Where $S_{\min}$ comes from Back in L3 we wrote $R_{\max}$ in terms of a minimum detectable signal $S_{\min}$, and quietly read it off a data sheet as if it were a constant. It is not. $S_{\min}$ is the end of a chain of choices: how much noise the receiver has, how confident you insist on being that a blip is real, and how long you are willing to dwell. Detection theory is where that chain gets made explicit — and it turns the vague phrase "good radar" into a single number you can engineer: **signal-to-noise ratio at the detector input.** ## The detection problem Every radar return is one signal sample sitting in noise. The radar compares it to a **threshold**: - Above threshold → declare *target*. - Below threshold → declare *noise*. That single comparison has two ways to go wrong: - **Miss** — a target is present but the sample doesn't cross the threshold. Its probability is $1 - P_d$, where $P_d$ is the probability of detection. - **False alarm** — noise alone crosses the threshold. Its probability is $P_{fa}$. :::{admonition} Key Concept :class: key-concept Every detection is a thresholded coin flip. Where you set the threshold trades $P_d$ against $P_{fa}$ — and no threshold gives you both unless the underlying SNR is high enough. Detection is an SNR business. ::: ## The noise floor You cannot see below the noise. Thermal noise sets the floor, and a compact formula gives it in dBm: $$ N_{\text{dBm}} = -174 + 10\log_{10}(B_{\text{Hz}}) + \text{NF}_{\text{dB}}. $$ The three terms each mean something: - $-174$ dBm/Hz is the thermal-noise power spectral density of a cold receiver at 290 K. - $10\log_{10}B$ adds the noise admitted by the receiver **bandwidth** $B$ — wider bandwidth, more noise. - $\text{NF}$ is the **noise figure**, the receiver's own added noise. For example, $B = 10$ MHz and $\text{NF} = 3$ dB give $N = -174 + 70 + 3 = -101$ dBm. Narrow the bandwidth to 1 MHz and the floor drops 10 dB to $-111$ dBm. Bandwidth is one of the cheapest knobs a designer has — but a radar needs enough bandwidth to support its range resolution ($\Delta R = c/2B$), so it cannot be made arbitrarily small. ## SNR and the threshold Define $\text{SNR} = S/N$ at the detector input. Now sweep the threshold: - Raise it: fewer noise spikes survive, so $P_{fa}$ drops — but some genuine returns are also rejected, so $P_d$ drops too. - Lower it: $P_d$ climbs, but so does $P_{fa}$. There is no setting that delivers high $P_d$ *and* low $P_{fa}$ unless the SNR is high enough to separate the signal-plus-noise distribution from the noise-only distribution. A common operating point — $P_d = 0.9$ at $P_{fa} = 10^{-6}$ on a steady (non-fluctuating) target — requires roughly **13 dB** of SNR. A fluctuating target (Swerling I, scan-to-scan) needs considerably more, around 21 dB, because you must protect against the scans where the target happens to fade. ## The ROC curve Plot $P_d$ (vertical) against $P_{fa}$ (horizontal, log scale) as the threshold sweeps, and you get a **receiver operating characteristic**. Each value of SNR produces its own curve: - Higher SNR pushes the curve toward the **top-left corner** — the ideal detector that catches everything and never false-alarms. - Fix a tolerable $P_{fa}$ and read off the achievable $P_d$; or fix a mission $P_d$ and read off the **required SNR**. The ROC is how a commander's preference ("I want to be 90% sure, and I'll tolerate one false alarm in a million") becomes an engineering spec on SNR. ## Integration: spending time for SNR If a single pulse doesn't carry enough SNR, combine $N$ pulses from the same target: - **Coherent integration** sums the complex returns, preserving phase. The SNR gain is $N$ (linear), i.e. $+10\log_{10}N$ dB. One hundred pulses buy **20 dB**. - **Non-coherent integration** sums magnitudes after detection, discarding phase. The gain is roughly $\sqrt{N}$, i.e. about $+5\log_{10}N$ dB. One hundred pulses buy only about **10 dB**. Coherent integration is twice as efficient in dB, but it demands phase coherence across the whole dwell — exactly what a modern AESA is built to maintain. Integration is the radar trading **time** (a longer dwell) for **sensitivity** (more SNR). ## Closing the loop on $R_{\max}$ Now the L3 range equation can be honest about $S_{\min}$: $$ R_{\max} = \left[\frac{P_t\,G_t\,G_r\,\lambda^2\,\sigma}{(4\pi)^3\,S_{\min}}\right]^{1/4}, \qquad S_{\min} = \frac{kTB\cdot\text{NF}\cdot\text{SNR}_{\text{req}}}{G_{\text{int}}}. $$ Reading those together: - A tighter mission ($P_d$ up, $P_{fa}$ down) raises $\text{SNR}_{\text{req}}$, raises $S_{\min}$, and **shortens** $R_{\max}$. - More integration raises $G_{\text{int}}$, lowers $S_{\min}$, and **lengthens** $R_{\max}$. And because $R_{\max}$ scales as $S_{\min}^{-1/4}$, the leverage is the familiar one-quarter: 20 dB of integration gain buys $20/4 = 5$ dB of range, a factor of $10^{5/10} \approx 3.2$ in kilometers. This is precisely how "a longer dwell" turns into "a longer range" — and why an EW technique that forces the radar to integrate longer, or accept a worse $P_{fa}$, directly shrinks the threat's reach. ::::{admonition} Quick Exercise :class: quick-exercise 1. A receiver has $B = 1$ MHz and $\text{NF} = 5$ dB. What is the noise floor in dBm? 2. Your single-pulse $R_{\max}$ at PRF $= 1$ kHz is 80 km. You apply **coherent** integration of 100 pulses. About what is the new $R_{\max}$? 3. A threat radar tightens its false-alarm contract from $P_{fa} = 10^{-6}$ to $10^{-9}$. Which way does its $R_{\max}$ move, and why? :::{admonition} Solution :class: dropdown 1. $N = -174 + 10\log_{10}(10^6) + 5 = -174 + 60 + 5 = -109$ dBm. 2. Coherent gain $= +10\log_{10}100 = 20$ dB, so $S_{\min}$ drops 20 dB and $R_{\max}$ gains $20/4 = 5$ dB: $80\ \text{km}\times10^{5/10} \approx 253$ km. 3. A tighter $P_{fa}$ demands a few extra dB of SNR to hold the same $P_d$, which raises $S_{\min}$ and **shortens** $R_{\max}$. The IADS commander is choosing how false-alarm-tolerant to be — and paying for it in range. ::: :::: ## Wrap-Up Detection is a thresholded comparison against a noise floor of $-174 + 10\log_{10}B + \text{NF}$ dBm, and the threshold trades $P_d$ against $P_{fa}$ at a level set by SNR. The ROC curve turns that tradeoff into a single SNR requirement; integration buys SNR by spending time, coherently at $+10\log N$ dB and non-coherently at $+5\log N$ dB. Finally, $S_{\min}$ is a $P_d/P_{fa}$ contract, not a constant — closing the loop on the range equation from L3. This completes the radar-fundamentals toolkit; **L9** puts L1–L8 to work on the Project 1 B-21 detection-range analysis.