Radar Detection Range Analysis: Monte Carlo Confidence Intervals for Pulsed Systems

The Problem With Single-Point Detection Range Estimates

Every radar textbook gives you the range equation. Plug in your transmit power, antenna gain, noise figure, and target RCS, and out comes a detection range number. Engineers build entire system budgets around that number — and then wonder why the manufactured radar behaves differently from the prediction.

The reason is that the range equation is deterministic but the real world is not. Target RCS fluctuates. Receiver noise figure varies from unit to unit. Transmit power sits at its spec minimum on a cold morning and its spec maximum in a warm rack. Rain attenuation depends on instantaneous rain rate, not an annual average. A single-point estimate hides all of this.

This post walks through using the Radar Detection Monte Carlo simulator to analyze an X-band pulsed radar, showing how Monte Carlo confidence bands give you the information you need to make real design decisions.

The Reference Design

The design is a 10 GHz X-band ground-based radar with the following parameters:

The target is a small UAV or bird — 1 m² mean RCS, slow fluctuation (Swerling I — the RCS is correlated across all integrated pulses in a dwell, changing scan-to-scan).

Setting Up the Nominal Analysis

Enter these values into the Radar Detection Monte Carlo tool. The tool immediately shows:

• R₅₀ = 45.2 km — nominal 50% detection range
• R₉₀ = 28.4 km — 90% detection range (high-confidence)
• Integration gain = 6.3× — from n^0.8 approximation with 10 pulses

The SNR vs range plot shows the post-integration SNR crossing the detection threshold (≈ 12.4 dB above noise floor for Pfa = 10⁻⁶ with 10 pulses) at about 45 km. This is consistent with the classical range equation prediction.

Swerling Model Comparison

Now change the Swerling model from I to 0 (non-fluctuating) and re-run. R₅₀ shifts to 50.1 km — a 10% increase. This seems counterintuitive: shouldn't a fluctuating target be harder to detect?

The answer depends on Pd. At very high Pd (> 0.9), non-fluctuating targets are easier to detect because the RCS never dips to a low value. But at moderate Pd (50%), fluctuating targets (Swerling I) can actually achieve similar or better performance because occasionally the RCS spikes above its mean. The "swerling loss" appears mainly at high Pd requirements.

Switching to Swerling II (fast fluctuation, same chi²(2) RCS distribution) with the same mean RCS gives R₅₀ = 43.8 km — marginally shorter than Swerling I at 50% Pd. The fast fluctuation actually helps when using many integrated pulses because some pulses always see a high-RCS state.

Rain Attenuation Impact

Now add rain: set Rain Rate to 25 mm/hr (heavy tropical rain). Re-run with Swerling I.

The tool applies ITU-R P.838 two-way attenuation:

• At 10 GHz: k = 0.0101, α = 1.276
• Specific attenuation: γ = 0.0101 × 25^1.276 ≈ 0.57 dB/km one-way
• Two-way: 1.14 dB/km

At the nominal 45 km detection range, this totals 51.3 dB of two-way rain loss — catastrophic. R₅₀ drops to 12.3 km. The detection range is now rain-limited, not hardware-limited.

This is why X-band weather radars carry significant margin against their clear-sky detection range. The designer needs to know the R₅₀ under rain, not just nominal conditions.

The MC Confidence Bands

Setting rain back to 0 and looking at the Monte Carlo confidence bands for Swerling I:

• p95 band (best case): R₅₀ = 52.1 km — 15% better than nominal
• p50 band (median): R₅₀ = 45.2 km — matches nominal (expected)
• p5 band (worst case): R₅₀ = 38.7 km — 14% worse than nominal

The asymmetry is small here because the parameter variations (±0.5 dB NF, ±0.3 dB Pt) are modest compared to the Swerling I RCS fluctuation, which dominates the spread.

For a manufacturing specification, the requirement should be written against the p5 curve: the radar must achieve R₅₀ ≥ 38.7 km across all manufactured units, not just on a nominal bench measurement.

ROC Curve Interpretation

The ROC curve shows Pd vs –log₁₀(Pfa) at R₅₀. At the operating point (Pfa = 10⁻⁶, –log₁₀ = 6):

• Pd ≈ 0.50 — by construction (we chose the 50% range)

Sliding Pfa tighter to 10⁻⁸ (–log₁₀ = 8) drops Pd to 0.31. Relaxing Pfa to 10⁻⁴ (–log₁₀ = 4) raises Pd to 0.72. This is the classic detection-vs-false-alarm tradeoff that CFAR processors navigate in real systems.

Key Takeaways for Design

1. Always use the p5 curve for margin allocation. The nominal detection range is an optimistic single-point estimate that only 50% of operating scenarios will meet or exceed.
2. Rain dominates at X-band. In wet environments, the rain-attenuated detection range is the binding constraint, not the clear-sky hardware performance.
3. Swerling model matters at high Pd requirements. At Pd = 0.9, switching from Swerling 0 to Swerling I costs roughly 6–8 dB of SNR (the Swerling loss). This corresponds to roughly a 2× reduction in detection range at 90% Pd.
4. Pulse integration is worth doing. 10 non-coherent pulses provide 6.3× SNR gain, equivalent to increasing peak power by 8 dB or antenna gain by 4 dBi.