Radar Equation — Study Notes

Section 1

1. Foundations: The Physics of Radar Power Budgets

1.1 What the Radar Equation Is

The radar equation is a power balance between the transmitter and the receiver of a radar system, expressed through the intervening propagation medium and the target. It answers three engineering questions:

1. Direct analysis: Given a radar and a target, what received signal power (and hence what SNR) do we get?
2. Design synthesis: Given a required detection performance, what transmitter power, antenna gain, or dwell time do we need?
3. Range prediction: At what maximum range does a target of given size become detectable with specified probability?

1.2 The Two-Way Path and the R⁻⁴ Law

Electromagnetic energy from a point source spreads uniformly over the surface of an expanding sphere. At distance R the sphere has area 4πR², so power density falls as 1/R². In radar, energy undergoes two such spreading processes:

• Transmit leg: P_t with antenna gain G_t produces density S₁ = P_tG_t/4πR² at the target.
• Receive leg: the target scatters a fraction of that density; the scattered wave spreads again over 4πR² on its way back.

1.3 Assumptions of the Basic Equation

• Free-space propagation (no absorption, refraction, or multipath).
• Target in the antenna far field: R > 2D²/λ (Fraunhofer distance, D = antenna diameter).
• Point target: target extent ≪ radar resolution cell.
• Constant (non-fluctuating) RCS over the observation interval.
• Peak antenna gains aligned with the target; polarization matched.
• Monostatic geometry: transmitter and receiver co-located (or nearly so).
• Matched receiver filtering so that bandwidth B is the noise-defining bandwidth.

1.4 A Brief History

The two-way power law was appreciated in the 1930s during the development of pulsed radar (Watson-Watt's team in the UK, and parallel work in the USA, Germany, and USSR). The modern form with antenna gain and noise figure matured in the 1940s–50s with Skolnik's systematization; the statistical extensions (Swerling 1954, Marcum 1947–50) made the equation a probabilistic design tool rather than a deterministic power balance.

Section 2

2. Full Derivation of the Monostatic Radar Equation

2.1 Important Equivalent Forms

Section 3

3. Radar Cross-Section (σ) in Depth

3.1 Canonical Target RCS

3.2 RCS Dependences

• Aspect angle: aircraft RCS can vary 20–30 dB between nose-on and broadside views (specular flashes from wings/fuselage).
• Frequency: in the optical (high-frequency) region σ is roughly frequency-independent for large smooth bodies; in the Rayleigh region (target ≪ λ) σ falls steeply (∝ f⁻⁴); in the resonance region (target ≈ λ) σ oscillates wildly.
• Polarization: circular polarization discriminates against rain (spherical drops return the opposite-sense circular wave, rejected by the antenna).
• Materials and shaping: radar-absorbent material (RAM) converts incident energy to heat; facet shaping redirects returns away from the threat radar.

3.3 Typical RCS Values

Section 4

4. Decibel Bookkeeping for Radar

Radar engineers work in dB because multiplicative budgets become additive sums. Master these conversions.

4.1 Reference Units

4.2 The Radar Equation in dB Form

where 30log₁₀(4π) = 32.98 dB. Useful building blocks:

• kT₀ = −204 dBW/Hz (thermal noise density at 290 K);
• 20log₁₀λ for λ in metres (e.g., λ = 0.1 m → −20 dB);
• Free-space one-way path loss: L_fs(dB) = 92.45 + 20log₁₀f(GHz) + 20log₁₀R(km) — a handy check for satellite-style links; the radar two-way uses twice this value in dB at the same R.

Section 5

5. Receiver Noise, Noise Figure, and the Minimum Detectable Signal

5.1 Thermal Noise

Any receiver adds noise. The available thermal noise power in bandwidth B at physical temperature T is kTB. Real receivers are worse than the ideal; the noise figure F quantifies the excess:

5.2 Noise Figure and System Temperature

• F = (SNR_in)/(SNR_out) ≥ 1. In dB, NF = 10log₁₀F.
• Cascaded stages (Friis formula): F_total = F₁ + (F₂−1)/G₁ + (F₃−1)/(G₁G₂) + … — the first stage dominates, hence low-noise amplifiers (LNAs) at the antenna.
• T_s = T₀(F−1) + T_ant·… practically T_s = F·T₀ when the antenna contribution is included in F.

5.3 Minimum Detectable Signal (MDS / S_min)

Section 6

6. Detection Statistics: P_d, P_fa, and Required SNR

6.1 The Threshold Detector

The receiver forms a statistic (e.g., envelope or matched-filter output sampled at the expected delay) and compares it with a threshold V_T:

• Noise alone exceeds V_T → false alarm, probability P_fa.
• Signal + noise exceeds V_T → detection, probability P_d.

Both rise as V_T falls — the irreducible trade-off. The Neyman–Pearson criterion fixes P_fa (set by the tolerable false-alarm rate: FAR = P_fa·B per second) and maximizes P_d.

6.2 Single-Pulse, Non-Fluctuating Target (Swerling 0 / Marcum Case)

For a square-law detector, P_d is the Marcum Q-function of threshold and SNR. Representative values (P_fa = 10⁻⁶):

6.3 Albersheim's Equation (Non-Fluctuating, Coherent or Single Pulse)

For coherent integration of n pulses, SNR adds directly in power: SNR_coh = n·SNR₁. For non-coherent integration the effective gain is less than n; a widely used approximation is SNR_total = SNR₁·n^0.8 in power terms, or Albersheim's extended non-coherent integration formula, which adds a 5log₁₀n term with a small correction factor.

6.4 Integration Losses

• Coherent (predetection): voltages add → power SNR × n. Requires stable phase; limited by target motion (range/Doppler walk).
• Non-coherent (post-detection): magnitudes add → SNR ∝ n^0.7–0.9; suffers a small integration loss relative to ideal; robust to phase instability.
• Binary (m-of-n): vote-counting across threshold crossings; simplest, most lossy.

Section 7

7. Maximum Range and Sensitivity

7.1 The Range Equation

Setting P_r = MDS and solving (no atmosphere):

7.2 Scaling Laws (memorize these)

7.3 Range with Atmospheric Attenuation

Atmospheric loss enters twice (two-way): multiply P_r by 10^(−2αR/10), α in dB per unit range. Because R appears in the exponent, R_max no longer has a closed form:

At microwave frequencies, attenuation sources: oxygen absorption (~0.006–0.01 dB/km near 60 GHz resonance), water vapour (~22 GHz resonance), rain (0.01–1+ dB/km depending on rate and frequency), clouds/fog. Radar horizon (refraction) also limits practical range: R_horizon(km) ≈ 4.12(√h_tx + √h_rx) with heights in metres.

7.4 Sensitivity Figures of Merit

• MDS / noise floor: absolute smallest detectable signal (dBm at receiver input).
• Tangential sensitivity (TSS): classical visual criterion, ≈ MDS + a few dB.
• Noise-equivalent flux / detectability factor: SNR needed at the decision point.

Section 8

8. System Losses: The Complete Inventory

Real radars lose signal power and detection efficiency in many places. The total loss L (linear, ≥1) or L(dB) multiplies the required SNR. Typical inventory:

Section 9

9. Target Fluctuation and the Swerling Models

9.1 Why Fluctuation Matters

A complex target (aircraft, ship) comprises many scatterers. The composite return is a vector sum whose amplitude varies with aspect, frequency, and vibration — σ is a random variable. Detection performance must then be averaged over σ's distribution.

9.2 The Five Models

9.3 Detection Consequences

• High P_d region: fluctuating targets need MORE SNR than Swerling 0. At P_d = 0.9, Swerling 1/2 require roughly 6–8 dB above the Marcum value. Design radars are usually sized on Swerling 1 or 2.
• Low P_d region: fluctuation can slightly help (the average return is occasionally large).
• Diversity gain: frequency agility (pulse-to-pulse steps ≥ c/2·target-extent) converts Swerling 1 → 2, recovering several dB at high P_d. Time diversity (multiple scans) similarly decorrelates.

Section 10

10. Search Radar Equation and Tracking Radar Equation

10.1 Search: The Power–Aperture Product

For a search radar covering solid angle Ω in frame time t_f with n beam positions, dwell per cell is T_d = t_f/n, and beam solid angle ≈ λ²/A_e. Substituting into the radar equation with n_p = T_d·PRF pulses integrated:

Interpretation: The invariant figure of merit is the product of average power and effective aperture. A large dish with modest power equals a small dish with heavy power — the choice is driven by cost, mobility, and beam-agility constraints, not detection performance.

10.2 Tracking Radar

A tracking radar stares at one target; energy on target scales with dwell time, and the relevant currency is total transmitted energy E = P_avg·T_obs. Angle precision improves with √(SNR); the radar equation for tracking sets the SNR that meets both detection and the required angle/range accuracy (e.g., monopulse split-error ∝ 1/√(SNR)·θ_B).

10.3 Comparison

Section 11

11. Bistatic and Multistatic Radar Equation

• Ovals of Cassini: constant-(R_tx·R_rx) contours; the useful coverage region is bounded by the product, not the sum, of ranges.
• Forward scatter: when the target lies near the baseline, the bistatic angle β → 180° and σ_b can exceed the monostatic value by 20–40 dB (forward-scatter radar exploits this; e.g., passive radar using broadcast illuminators).
• Practical burdens: precise time/frequency/phase synchronization (GPS-disciplined oscillators), geometry-dependent coverage, data association across receivers, bistatic RCS modeling.
• Passive/bistatic advantage: the receiver emits nothing — valuable in military and spectrum-crowded contexts.

Section 12

12. The Radar Equation in Satellite & Space Contexts

Radar & Satellite Engineering connects the radar equation to the satellite link budget. Key correspondences and differences:

Spaceborne radar (e.g., SAR altimeters, precipitation radars) reintroduces the R⁻⁴ law with very long ranges (600–1200 km), forcing high antenna gain (large deployable apertures), long pulses with pulse compression, and high-duty-cycle transmitters. Rain attenuation at Ku/Ka band and ionospheric scintillation at L band are the dominant propagation corrections.

Section 13

13. Worked Problems (Exam-Style)

Section 14

14. Formula Sheet (Revision)

Section 15

15. Glossary of Key Terms

References

1. M. I. Skolnik, Introduction to Radar Systems, 3rd ed., McGraw-Hill, 2001 — Ch. 2.
2. M. I. Skolnik (ed.), Radar Handbook, 3rd ed., McGraw-Hill, 2008 — Ch. 2.
3. D. K. Barton, Radar Equations for Modern Radar, Artech House, 2013.
4. N. Levanon, Radar Principles, Wiley, 1988.
5. F. E. Nathanson & S. L. Cohen, Radar Design Principles, 2nd ed., 1999.
6. ITU-R Rec. P.676 (gaseous attenuation), P.838 (rain attenuation).