ECE 521E - Radar Equation Study Guide

Course Alignment

Learning Objectives

By the end of this study guide, the student should be able to:

Explain the physical basis of the radar equation and trace the signal's two-way power budget from transmitter to receiver.
Derive the monostatic radar equation from first principles (free-space spreading) and state every assumption involved.
Identify each parameter in the radar equation and convert among dB, dBW, dBm, and linear units fluently.
Modify the radar equation to account for system losses, atmospheric attenuation, noise figure, and matched-filter processing gain.
Relate received SNR to probability of detection (P_d) and probability of false alarm (P_fa) using the Swerling target-fluctuation models.
Compute maximum detection range, required peak power, and antenna parameters for practical radar problems.
Distinguish the search-radar equation (power-aperture product) and bistatic radar equation from the basic monostatic form.
Apply the radar equation to mini-design problems typical of satellite and terrestrial radar links.

Why This Equation Matters

1. Introduction

The radar equation is the single most important power-budget relationship in radar engineering. It connects what the radar transmits, how the target reflects energy, and what the receiver can detect into one expression. Every radar design — from a police speed gun to a planetary surveillance radar and a spaceborne SAR — begins with the radar equation.

Physically, the equation captures two fundamental effects:

One-way spreading loss (1/R²): as the transmitted wavefront expands, its power density falls with the square of range.
Two-way journey: the wave travels to the target and back, so the round-trip effect enters as 1/R⁴. This steep R⁻⁴ dependence is why doubling radar range requires 16× (12 dB) more power — a fact that dominates radar design economics.

Key insight: The R⁻⁴ law means the radar equation is not an efficiency statement — it is a detectability boundary. Solving it for R gives the maximum range at which a target of a given size can be detected with a specified probability.

1.1 Where the Equation is Used

Design: choosing transmitter power, antenna gain, and frequency for a required range.
Performance prediction: estimating P_d for a given target at a given range.
Radar range instrumentation & satellite ground stations: link-budget verification.
ECM/ECCM studies: quantifying jamming and anti-jam margins.

From First Principles

2. Derivation of the Monostatic Radar Equation

Assume: free space, isotropic-free antenna with gain G, a point target of radar cross-section σ, co-located Tx/Rx (monostatic), and no losses. Track the power in four steps.

1Power density at the target (one-way spreading). An isotropic radiator spreads power P_t over a sphere of area 4πR². A directive antenna concentrates this; the density at range R is:

S₁ = P_t G_t / (4πR²)[W/m²]

2Power intercepted and re-radiated by the target. The target captures S₁ over its effective area σ (the radar cross-section) and scatters it. Treating the target as a lossless isotropically re-radiating scatterer, the scattered power density back at the radar (another 4πR² spreading) is:

S₂ = (S₁ · σ) / (4πR²) = P_t G_t σ / (4πR²)²[W/m²]

3Power captured by the receive antenna. An antenna of effective aperture A_e collects power density S₂. Using the fundamental relation G_r = 4πA_e/λ², the received power is:

P_r = P_t G_t G_r λ² σ / [(4π)³ R⁴]The basic monostatic radar equation [W]

4Same-antenna simplification. For a monostatic radar using one antenna (G_t = G_r = G):

P_r = P_t G² λ² σ / [(4π)³ R⁴]

Equivalently, in terms of effective aperture only:

P_r = P_t A_e² σ / (4π λ² R⁴)(using A_e = Gλ²/4π)

2.1 Alternative Forms in Common Use

Form	Expression	Typical Use
Energy form	P_r = E_t G² λ² σ / [(4π)³ R⁴ τ]	Pulsed radar (E_t = pulse energy)
Average-power form	P_r,avg = P_avg G² λ² σ T_obs / [(4π)³ R⁴]	With coherent integration over dwell T_obs
dB form	10log P_r = 10log P_t + 2(10log G) + 20log λ + 10log σ − 30log(4π) − 40log R	Link-budget bookkeeping

Assumptions to state in exams/designs: free space; target in the far field (R > 2D²/λ); single scatterer; polarization matched; no atmospheric or system losses; antenna gains are peak gains; σ is independent of aspect over the dwell.

Know Every Term

3. Parameters of the Radar Equation

Symbol	Name	Units	Notes
P_t	Transmitted peak power	W (dBW)	Peak of the pulse for pulsed radar
G_t, G_r	Transmit / receive antenna gain	— (dBi)	G = 4πA_e/λ²; related to beamwidth θ_B ≈ 65λ/D degrees
λ	Wavelength (c/f)	m	c = 3×10⁸ m/s
σ	Radar cross-section (RCS)	m² (dBsm)	Ratio of scattered power density at receiver to incident density × 4πR²; see Table 3.1
R	Range to target	m (km)	Slant range; cτ/2 for pulsed (τ = delay)
P_r	Received signal power	W (dBW)	Must exceed the minimum detectable signal

3.1 Typical Radar Cross-Sections (σ)

Target	σ (m²)	σ (dBsm)
Bird / insect	0.01 – 0.1	−20 to −10
Person	~1	0
Small boat / UAV	1 – 10	0 to 10
Fighter aircraft	3 – 10 (head-on), up to 100 (broadside)	5 – 20
Stealth aircraft	0.001 – 0.01	−30 to −20
Large ship	10⁴ – 10⁶	40 – 60

RCS facts for exams: RCS depends on viewing aspect, frequency, and polarization — it is not a physical area. Simple shapes: sphere of radius a (a ≫ λ): σ = πa²; flat plate of area A at normal incidence: σ ≈ 4πA²/λ²; long thin dipole: σ ≈ 0.88λ². Stealth shaping and RAM reduce σ so that R (via R⁻⁴) collapses dramatically.

Reality Check

4. Losses, Noise, and the Modified Radar Equation

The free-space equation is optimistic. Real budgets insert a total system loss factor L (L ≥ 1, i.e. positive dB loss) and a one-way atmospheric attenuation factor (2αR dB, where α is dB/km — it appears twice, once on transmit and once on receive):

P_r = P_t G_t G_r λ² σ / [(4π)³ R⁴ L] · 10^−2αR/10with system loss L (linear) and atmospheric attenuation α (dB per unit range, each way)

4.1 Sources of System Loss L

Hardware losses

Transmitter–antenna feed line loss L_t
Antenna–receiver line loss L_r
Duplexer / T-R switch insertion loss
Antenna pattern loss: beam-shape loss, scan loss (phased arrays off broadside)
Radome loss

Processing & propagation losses

Atmospheric absorption (gaseous, rain — see ITU-R models)
Polarization mismatch loss
Collapsing loss (non-ideal integrators)
Quantization / straddling loss in digital receivers
CFAR loss (adaptive thresholding)

4.2 Receiver Noise and the Minimum Detectable Signal

The radar competes against thermal noise. With receiver bandwidth B and noise figure F (relative to T₀ = 290 K):

N = k T₀ B F = k T_s Bnoise power at receiver output [W]; k = 1.38×10⁻²³ J/K

The minimum detectable signal (MDS) is the signal power that yields the threshold SNR (SNR_min, or S/N required for given P_d, P_fa):

MDS = k T₀ B F · (S/N)_min

Noise figure rule: noise figure F (dB) degrades sensitivity by the same dB amount. A receiver with F = 3 dB needs twice the received power of a noiseless one for equal SNR. For low-noise front-ends (satellite ground stations, passive radars), T_s is quoted directly in kelvin.

Signal-to-Noise Ratio

5. Radar Range Equation in SNR Form

Detection is fundamentally an energy contest, not a power contest. Combining P_r with noise N and processing gain:

SNR = P_r / N = P_t G_t G_r λ² σ n E_p / [(4π)³ R⁴ k T₀ B F L]n = number of coherently integrated pulses; E_p = pulse energy

For a matched filter (B·τ ≈ 1) processing a single pulse, the processing gain is already implicit in the matched filter, giving the compact detectability form:

SNR = P_t τ G² λ² σ / [(4π)³ R⁴ k T₀ F L]P_tτ = pulse energy E_p

5.1 Threshold Detection

A threshold is set at the receiver output. If noise alone exceeds it → false alarm (probability P_fa); if signal-plus-noise exceeds it → detection (probability P_d).

Lowering the threshold raises P_d but also P_fa — the classic trade-off (Neyman–Pearson criterion: maximize P_d subject to a fixed P_fa).
For a square-law detector on Gaussian noise, P_fa ≈ exp(−V_T²/2ψ₀) with integration reducing P_fa dramatically for the same threshold.
Typical operating point: P_fa = 10⁻⁶ per pulse (with integration, effective P_fa falls further).

Required single-pulse SNR (non-fluctuating target, P_fa = 10⁻⁶)	P_d
≈ 10.5 dB	0.50
≈ 13.2 dB	0.90
≈ 16.0 dB	0.99

Albersheim's equation (see summary) gives SNR_min analytically from P_d and P_fa for a non-fluctuating (Swerling 0/5) target in Gaussian noise — a standard exam tool.

Solving for R

6. Maximum Range Equation

Set P_r = MDS (or SNR = SNR_min) and solve for R. Ignoring atmospheric loss:

R_max = [ P_t G_t G_r λ² σ / ((4π)³ k T₀ B F (S/N)_min L) ]^1/4R_max ∝ P_t^1/4, G^1/2, λ^1/2, σ^1/4

The fourth-root law: because R ∝ (power)^1/4, doubling R_max by brute force requires 16× the transmitted power (12 dB). It is usually far cheaper to increase antenna gain or dwell time than transmitter power. Atmospheric loss (2αR) must often be solved iteratively since R appears in the exponent.

6.1 Design Levers Ranked

Increase dwell / integrate more pulses (n): coherent integration gives n-fold SNR gain (∝ n in power; √n in amplitude per non-coherent).
Antenna gain/aperture: G² helps as G in the one-way link; aperture ↑ = beamwidth ↓ = more dwell per beam position.
Lower noise figure / system temperature: every dB of F reduction is a free dB of SNR.
Transmitter power: effective but costly in prime power, cooling, and cost — and only a 1/4-root lever on range.

Target Fluctuation

7. Swerling Target Models

Real targets (aircraft with multiple scatterers) present RCS that varies between looks. Swerling models describe the statistics of σ:

Model	Fluctuation rate	σ distribution	Notes
Swerling 0 / 5	None	Constant (non-fluctuating)	Baseline; Marcum case
Swerling 1	Slow (scan-to-scan)	Exponential, correlated over a dwell	Many independent scatterers of similar size
Swerling 2	Fast (pulse-to-pulse)	Exponential, independent per pulse	Requires frequency agility to decorrelate
Swerling 3	Slow	Chi-square, 4 degrees of freedom	One dominant scatterer + small ones
Swerling 4	Fast	Chi-square, 4 degrees of freedom	Pulse-to-pulse version of Swerling 3

Effect on design: fluctuating targets need more SNR at high P_d than Swerling 0 (e.g., at P_d = 0.9, Swerling 1 needs roughly 6–8 dB more than the non-fluctuating case), but the curves cross at low P_d — fluctuation actually helps weak targets escape detection less often relative to their average. Frequency agility converts Swerling 1 → 2-like behavior, gaining several dB.

Variants

8. Bistatic, Search, and Tracking Forms

8.1 Bistatic Radar Equation

When transmitter and receiver are separated (baseline L), the one-way spreading terms apply to R_tx and R_rx separately:

P_r = P_t G_t G_r λ² σ_b / [(4π)³ R_tx² R_rx² L]σ_b = bistatic RCS; constant-CSP contours are ovals of Cassini

Advantages: covert receiver, no minimum-range blind zone (except near Tx), forward-scatter enhancement (σ can spike when target is near the transmitter–receiver baseline).
Costs: synchronization of time/frequency/phase, complex geometry, bistatic RCS differs from monostatic.

8.2 Search Radar Equation (Power–Aperture Form)

For a search radar covering solid angle Ω in frame time t_f with required P_d, the governing figure of merit is the power–aperture product:

P_avg A_e = (SNR)_min· 4π k T_s L · R⁴ σ-scan / (σ t_f Ω)after substituting dwell = t_fΩ/(4π/n_beams) and beam solid angle ≈ λ²/A_e

Interpretation: to search a fixed volume in fixed time at fixed range, you need a minimum product of average power and aperture — trading small antenna + big power against big antenna + small power. Peak power and pulse width then drop out; dwell and beam scheduling matter.

8.3 Satellite & Deep-Space Link Analogy

For satellite radar/communication links, the two-way equation collapses (one-way) and is written with free-space path loss (4πR/λ)²; rain attenuation per ITU-R, pointing loss, and the receiving-system G/T (gain over system temperature) replace the radar-specific terms. The radar equation's R⁻⁴ vs. the link equation's R⁻² is the essential distinction between active radar and one-way telemetry.

Apply It

9. Worked Examples

Example 1 — Received power from a small aircraft

A monostatic radar has P_t = 100 kW peak, G = 35 dB (≈3162 linear), λ = 0.03 m (X-band), target σ = 2 m² at R = 50 km. Find P_r (ignore losses).

Solution: P_r = P_tG²λ²σ / [(4π)³R⁴]
G = 10^(35/10) = 3162.3, so G² = 10⁷. Numerator: 10⁵ × 10⁷ × (0.03)² × 2 = 10¹² × 9×10⁻⁴ × 2 = 1.8×10⁹.
Denominator: (4π)³ = 1984.4; R⁴ = (5×10⁴)⁴ = 6.25×10¹⁸ → 1.240×10²².
P_r = 1.8×10⁹ / 1.24×10²² ≈ 1.45×10⁻¹³ W ≈ −128.4 dBW.

Example 2 — Maximum range with noise and losses

Radar: P_t = 1 MW, G = 40 dB (10⁴), λ = 0.1 m, σ = 5 m², F = 4 dB (2.51), B = 1 MHz, required SNR = 13 dB (P_d ≈ 0.9, Swerling 0), total system loss L = 6 dB (3.98). Atmospheric loss neglected. Find R_max.

Solution: N = kT₀BF = 1.38×10⁻²³·290·10⁶·2.51 ≈ 1.005×10⁻¹⁴ W.
Required P_r = SNR·N = 10^(13/10)·1.005×10⁻¹⁴ = 19.95×10⁻¹⁴ ≈ 2.0×10⁻¹³ W.
R⁴ = P_tG²λ²σ / [(4π)³·L·P_r] = 10⁶·10⁸·10⁻²·5 / (1984×3.98×2.0×10⁻¹³)
= 5×10¹² / (1.58×10⁻⁹) = 3.16×10²¹ → R = (3.16×10²¹)^0.25 ≈ 237 km.

Example 3 — Effect of 6 dB RCS reduction (stealth)

If the target's σ drops from 5 m² to 5/4 m² (6 dB reduction), what happens to R_max?

Solution: R ∝ σ^¼ → R scales by (1/4)^¼ = 1/√2 ≈ 0.707. Range falls by ~29% (from 237 km to ~168 km). A 20 dB RCS reduction (stealth aircraft vs. conventional) cuts range by (1/100)^¼ = 1/3.16 → range falls to 32% — the strategic impact of low-observable design.

Example 4 — Coherent integration gain

How many pulses must be coherently integrated to drop required single-pulse SNR from 13 dB to 3 dB (10 dB saving)?

Solution: Coherent integration gives n dB for n pulses (power adds). 10 dB saving requires n = 10 pulses. At a PRF of 1 kHz, this needs only 10 ms of dwell — the key to modern pulse-Doppler processing.

Try It Yourself

10. Interactive Radar Equation Calculator

Monostatic Radar Range Calculator

Peak power P_t (kW)

Antenna gain G (dBi)

Frequency f (GHz)

Target RCS σ (m²)

Noise figure F (dB)

Bandwidth B (MHz)

Required SNR (dB)

System loss L (dB)

Pulses integrated n (coherent)

Atm. attenuation α (dB/km, one-way)

Results will appear here…

Note: With α ≠ 0, the equation is solved iteratively (fixed-point on the atmospheric term) — up to 200 iterations.

Check Understanding

11. Self-Test Questions

Q1. Why does the radar equation contain R⁻⁴ rather than R⁻²?

Because radar is a two-way path: the wave spreads over 4πR² on the way to the target and the re-radiated energy spreads over another 4πR² on the return. The product gives R⁴ in the denominator. A one-way communication link has only R⁻².

Q2. State the assumption linking antenna gain and effective aperture, and give the formula.

For an antenna with aperture efficiency η: G = 4πηA/λ², and for the ideal effective aperture A_e = ηA. So G = 4πA_e/λ². This is used to rewrite the radar equation in aperture form.

Q3. A radar must double its maximum range. By how much must P_t increase, all else equal?

R ∝ P_t^1/4 ⇒ P_t,new/P_t,old = 2⁴ = 16 → a 16× increase (12 dB). Often a better alternative is to raise antenna gain (R ∝ G^1/2, so only 4× gain / 6 dB) or integrate more pulses.

Q4. Define radar cross-section and explain why a stealth aircraft has small σ.

σ = 4πR²·(scattered power density at receiver)/(incident power density at target). It measures how detectable a target is, not its physical size. Stealth aircraft use shaping (aligning returns away from the radar), radar-absorbent material, and internal carriage to reduce σ by 20–40 dB.

Q5. Distinguish P_d and P_fa and state the effect of lowering the detection threshold.

P_d = probability of declaring a target present when it is; P_fa = probability of declaring a target when only noise is present. Lowering the threshold raises both P_d and P_fa; the Neyman–Pearson approach maximizes P_d for a fixed acceptable P_fa.

Q6. Write the bistatic radar equation and explain its geometric product term.

P_r = P_tG_tG_rλ²σ_b/[(4π)³R_tx²R_rx²L]. Equal-power contours satisfy R_tx·R_rx = const (ovals of Cassini), collapsing to a single point only in the monostatic limit R_tx = R_rx = R.

Q7. Why does frequency agility improve detection of a Swerling 1 target?

Swerling 1 assumes σ constant over a dwell (correlated scan-to-scan). Changing frequency pulse-to-pulse decorrelates the target echoes, converting slow fluctuation to Swerling-2-like fast fluctuation, whose required SNR at high P_d is several dB lower (diversity gain).

Q8. What is the figure of merit for a search radar and why?

The power–aperture product P_avgA_e. Because search performance is set by the energy placed on each resolution cell within the frame time, and the dwell per cell scales with aperture while power sets energy per dwell — their product is invariant. Tracking radars instead are peak-power/energy limited.

Q9. Convert: P_t = 250 kW, G = 38 dB, σ = 0.5 m² to dBW, linear gain, and dBsm.

P_t = 10log(2.5×10⁵) = 54 dBW; G = 10^(38/10) ≈ 6310; σ = 10log(0.5) ≈ −3 dBsm.

Q10. Atmospheric loss of 0.01 dB/km each way at R = 100 km: what is the total two-way loss in dB, and why can it not simply be lumped into L?

Total = 2αR = 2×0.01×100 = 2 dB. It cannot be lumped into a constant L because it depends on R itself (the exponent 10⁻²αR/10 contains R), so R_max must be found iteratively or graphically.

Revision Sheet

12. Summary

Core equation (monostatic, lossless): P_r = P_tG²λ²σ/[(4π)³R⁴].
With losses: divide by L and apply 10⁻²αR/10 for atmosphere; with noise: N = kT₀BF.
Detectability form: set SNR = SNR_min(P_d, P_fa, Swerling case) and solve for R_max.
Fourth-root scalings: R ∝ P_t^¼σ^¼, R ∝ G^½λ^½, R ∝ (n/BF)^¼ for integration.
Fluctuation: Swerling 1–4 adjust required SNR; frequency agility mitigates slow fluctuation.
Variants: bistatic (R_tx²R_rx²), search (P_avgA_e product), link-budget (R⁻², G/T).
Albersheim's equation (non-fluctuating, n pulses): SNR₁ = A + 0.12AB + 1.7B, where A = ln[0.62/P_fa], B = ln[P_d/(1−P_d)], and total SNR = SNR₁·n^0.85 approximates non-coherent integration.

The Radar Equation