Reading — Detection Theory

Reading — Detection Theory#

By the end of this lesson you should be able to:

Compute the noise floor for a given bandwidth and noise figure.
Explain the detection threshold and the \(P_d / P_{fa}\) tradeoff.
Read a receiver operating characteristic (ROC) curve and identify the SNR that meets a mission requirement.
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 \(P_{fa}\).

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 \(kT_0\), the thermal-noise power spectral density at the standard reference temperature \(T_0 = 290\) K — the noise already present before the receiver adds anything of its own (that is the separate \(\text{NF}\) term).
\(10\log_{10}B\) 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 \(\sqrt{N}\) at low per-pulse SNR — the true gain lies between \(\sqrt{N}\) and \(N\) — 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.

Type-along

Type this in MATLAB. It sets one threshold from a false-alarm contract, then counts.

rng(495); M = 1e6;                    % a million noise-only cells
Pfa_want = 1e-3;                      % the false-alarm contract
T = sqrt(2)*erfcinv(2*Pfa_want);      % threshold that honors it
fprintf('T = %.3f  Pfa = %.2e\n', T, mean(randn(M,1) > T));
for snr_dB = [6 10 13]
    d = sqrt(2*10^(snr_dB/10));       % separation, in noise sigmas
    Pd_count = mean((d + randn(M,1)) > T);
    Pd_calc  = 0.5*erfc((T-d)/sqrt(2));
    fprintf('%2d dB: Pd = %.3f (calc %.3f)\n', snr_dB, Pd_count, Pd_calc);
end

The threshold never moved and \(P_{fa}\) never moved. Only \(P_d\) moved. If you are the B-21, which number in this script can you actually attack, and how?
Why did we contract for \(P_{fa} = 10^{-3}\) instead of the \(10^{-6}\) from the reading?

Solution

Expected output (counted values move a few percent run to run; T and the calc column do not):

T = 3.090  Pfa = 9.51e-04
dB: Pd = 0.394 (calc 0.394)
dB: Pd = 0.916 (calc 0.916)
dB: Pd = 0.999 (calc 0.999)

Not \(T\), and not \(P_{fa}\) — those are the radar’s contract with itself, and it will hold them no matter what you do. The attackable quantity is \(d = \sqrt{2\,\text{SNR}}\), the separation between the two distributions: shrink \(\sigma\) (RCS reduction, so less signal) or raise the noise floor (jamming, so a larger spread). \(P_d\) collapses and the radar never sees its own \(P_{fa}\) change. That is the whole of Block 2 in one line.
At \(P_{fa} = 10^{-6}\) you expect one false alarm in a million trials — Monte Carlo cannot measure it. This is exactly why the closed form exists, and why measuring a real receiver’s \(P_{fa}\) takes hours of collection.

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.

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