Egerton University — Department of Electrical & Electronic Engineering

Study Guide: Radar Equation & Radar Performance

Course EEEN 567 – Satellite Engineering • 4th Year BSc. Electrical & Electronic Engineering

Topic 5 • Radar Fundamentals for Satellite & Ground-Station Systems

Learning Objectives

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

  1. Explain the basic principle of pulsed radar operation and distinguish between monostatic and bistatic configurations.
  2. Derive the radar range equation step by step from first principles (power density, antenna gain, RCS and free-space spreading).
  3. Define and compute the radar cross section (RCS, σ) of simple targets such as spheres, cylinders and flat plates.
  4. Account for receiver noise, noise figure and minimum detectable signal (MDS) in radar range prediction.
  5. Calculate the maximum detection range of a radar and analyse its sensitivity to peak power, antenna gain, wavelength and RCS.
  6. Evaluate radar performance limitations: pulse repetition frequency (PRF), unambiguous range, blind speeds, eclipsing and atmospheric attenuation.
  7. Use interactive simulations to verify the radar range equation and PRF constraints.

1. Introduction: The Basic Radar Principle

Radar (Radio Detection and Ranging) is an electromagnetic system used to detect the presence of, and determine the location and velocity of, distant objects (targets). It does so by transmitting electromagnetic energy and processing the echo returned from the target. The distance follows from the measured round-trip delay:

R = c·t / 2 where t is the echo delay (s) and c = 3×108 m/s is the speed of light. The factor 2 accounts for the two-way path.

1.1 The Monostatic Radar

In a monostatic radar, a single antenna (shared through a duplexer) is used for both transmission and reception. This is the most common configuration, used in ground stations, weather radar and many satellite tracking systems.

Monostatic radar block diagram
Fig. 1 — Monostatic radar: transmitter, receiver, shared antenna and display.
Monostatic radar block diagram hand-drawn style
Fig. 2 — Block diagram of a pulsed monostatic radar (transmitter, duplexer, receiver, indicator).

1.2 The Bistatic Radar

In a bistatic radar, the transmitter and receiver are separated by a distance comparable to or greater than the target range. Bistatic geometries are useful in satellite links, passive radar and stealth detection, because the receiver is located away from the transmitter and does not itself radiate.

Bistatic radar diagram
Fig. 3 — Bistatic radar with separate transmit and receive antennas.
Key point: The radar range equation must reflect the geometry. For a monostatic radar, transmit and receive ranges are equal (Rt = Rr = R); for a bistatic radar they differ, giving Rt2·Rr2 spreading losses.

2. Derivation of the Radar Range Equation

The radar range equation relates the received echo power to the transmitted power, antenna parameters, target properties and range. Derive it in four steps.

Step 1 — Power density at the target. An isotropic antenna would spread the peak transmitted power Pt uniformly over a sphere of radius R. A directive antenna of gain G concentrates this power, so the power density at the target is:
Wt = PtG / (4πR2)   [W/m²]
Step 2 — Power intercepted and re-radiated by the target. The target reflects a fraction of the incident power captured by its effective area. By definition this fraction is the radar cross section σ (m²). The target re-radiates the intercepted power PtGσ/(4πR2) isotropically (assumed).
Step 3 — Power density back at the radar. The re-radiated power spreads over a second sphere of radius R, so the returned power density at the receiving antenna is:
Wr = PtGσ / [(4πR2)(4πR2)]
Step 4 — Power captured by the receive antenna. An antenna of gain G has effective aperture Ae = Gλ2/(4π), where λ is the wavelength. The received signal power is therefore:
Pr = PtG2λ2σ / [(4π)3R4]   [W] Monostatic radar range equation (basic form)
Annotated radar maximum range equation
Fig. 4 — The maximum-range form of the radar equation, showing the role of each parameter.
The R−4 law: received power falls off with the fourth power of range — because the energy undergoes two spreading losses (outbound and return). Doubling the range reduces the echo power by a factor of 16 (≈12 dB). This is why radar range improvement is so difficult: to double the detection range you must raise Pt×G2×σ by 16× (≈12 dB).

2.1 Solving for Maximum Range

Setting Pr equal to the minimum detectable signal Smin (see Section 4) and solving for R gives the maximum detection range:

Rmax = [ PtG2λ2σ / ((4π)3Smin) ]1/4

Useful dB form (powers in dBW, G and σ in dB):

Pr(dBW) = Pt + 2G + 10log10σ + 20log10(λ) − 30log10(4π) − 40log10(R)

2.2 Practical (Modified) Range Equation

Real systems include system losses L (feeds, duplexer, atmospheric absorption, beam shape, integration loss) and a pattern propagation factor F:

Pr = PtG2λ2σF4 / [(4π)3R4L]

2.3 Search and Tracking Forms

3. Radar Cross Section (σ)

The radar cross section of a target is the (fictitious) area that would intercept an amount of incident power which, if scattered isotropically, would produce the observed echo power back at the radar:

σ = 4πR2 · (Sr / St)   [m²]  or in dBsm:  σ(dBsm) = 10log10[σ(m²)/1 m²]

σ depends on the target's physical size, shape, material, frequency and aspect angle — it is not simply the physical area.

Concept of radar cross section
Fig. 5 — Concept of RCS: the target behaves as an isotropic re-radiator of the intercepted power.
RCS of common shapes
Fig. 6 — Maximum RCS of simple shapes (sphere, cylinder, flat plate, corner reflectors).
TargetTypical RCSComment
Insect / bird10−5 – 10−2 m²Rayleigh region (a << λ)
Small UAV / missile0.01 – 0.5 m²Stealth shaping reduces this
Fighter aircraft1 – 10 m²Highly aspect-dependent
Large airliner10 – 100 m²Optical region
Ship103 – 105 m²Superstructure dominates
Sphere (radius a, a >> λ)σ = πa2Equals its projected physical area
Satellite link: for satellite tracking and space-debris surveillance, RCS can be below 0.01 m², so extremely large antennas (e.g., Egerton ground-station class dishes) and low noise receivers are required.

4. Receiver Noise, Noise Figure and Minimum Detectable Signal

The weakest echo a radar can detect is set not by the transmitted power but by noise. The dominant noise source is thermal (Johnson–Nyquist) noise from the receiver front end.

4.1 Thermal Noise Power

N = k·T0·Bn k = 1.38×10−23 J/K (Boltzmann), T0 = 290 K standard temperature, Bn = receiver noise bandwidth (Hz)

At T0 = 290 K, the noise spectral density is kT0 = −174 dBm/Hz. For a 1 MHz bandwidth, N ≈ −114 dBm.

4.2 Noise Figure and Noise Temperature

The receiver degrades the signal-to-noise ratio by its noise figure F (or equivalent noise temperature Te = (F−1)T0):

F = (S/N)in / (S/N)out      F(dB) = 10log10F

The total input-referred noise is Ni = kT0BnF. For cascaded stages, use Friis' formula: F = F1 + (F2−1)/G1 + (F3−1)/(G1G2) + ... which is why the first amplifier (LNA) at the antenna must have the lowest possible noise figure.

4.3 Minimum Detectable Signal

The minimum detectable signal (MDS), also called the minimum discernible signal, is the received power for which the output SNR just equals the detectability threshold (SNR)1 required by the detector (typically 10–16 dB, and lower with pulse integration):

Smin = k·T0·Bn·F·(S/N)1

Substituting into the range equation:

Rmax = [ PtG2λ2σ / ((4π)3kT0BnF(S/N)1) ]1/4
Design insight: everything on the bottom line of this expression hurts range. Since range depends only on the 1/4 power of each factor, a 16× reduction in noise (or required SNR) is needed to double Rmax.

5. Factors Affecting Radar Performance

5.1 Pulse Repetition Frequency and Unambiguous Range

A pulsed radar transmits pulses at a rate called the pulse repetition frequency (PRF). The echo from a distant target must return before the next pulse is transmitted, otherwise the measured delay is ambiguous. The maximum unambiguous range is:

Ru = c / (2·PRF)

Low PRF (long unambiguous range) suits range measurement and ground mapping; high PRF suits Doppler/velocity measurement but folds ranges. Medium PRF is a compromise.

Pulse repetition period
Fig. 7 — Pulse repetition period, transmitted pulse and echo pulse.
Multiple PRF ranging
Fig. 8 — Multiple-PRF ranging resolves range ambiguities (Chinese-remainder style true-range resolution).

5.2 Doppler Frequency and Blind Speeds

A target with radial velocity vr produces a Doppler shift fd = 2vr/λ. The PRF also determines the unambiguous Doppler interval (=±PRF/2). Moving-target-indicator (MTI) radars cancel stationary clutter using delay-line cancelers, but targets at blind speeds vb = n·λ·PRF/2 (n = 1, 2, ...) produce zero Doppler after cancellation and are lost.

5.3 Eclipsing and Minimum Range

During the transmitted pulse (width τ), the receiver is blanked; targets closer than Rmin = cτ/2 return echoes that arrive while the radar is still transmitting and are eclipsed. Short pulses improve range resolution (ΔR = cτ/2) but reduce average power.

5.4 Atmospheric Attenuation and Propagation

At radar frequencies, absorption by oxygen and water vapour (peaks near 22 and 60 GHz), rain, and multipath/ducting all reduce performance. The two-way atmospheric loss must be included as a term in L of the modified range equation. This is especially important for satellite-earth paths traversing the whole atmosphere.

5.5 Summary of Performance Trade-offs

Parameter ↑Effect on range/performance
Peak power Pt↑ range (fourth-root law); limited by hardware
Antenna gain GVery strong effect (G appears squared)
Wavelength λLonger λ → more received power, but larger antenna for same gain
Target RCS σLarger σ → longer detection range
Bandwidth Bn↑ B → ↑ noise → ↓ range (but better resolution)
Noise figure F↑ F → ↓ range
PRFSets unambiguous range and Doppler window

6. Animation: Two-Way Radar Propagation & the R−4 Law

Watch a pulse travel from the radar to the target and back. The echo amplitude bar shows how rapidly the returned power falls with range — the physical origin of the 1/R4 dependence in the radar equation.

7. Interactive Simulations

Simulation 7.1 — Radar Range Equation Calculator

Vary the system parameters and observe the received power, SNR and maximum detection range. Use it to verify the worked examples and explore sensitivity (the fourth-root law).

Press Calculate to evaluate the radar equation.

Received SNR vs. Range

Simulation 7.2 — PRF, Unambiguous Range and Blind Speeds

Explore the fundamental PRF trade-off: low PRF gives long unambiguous range; high PRF pushes blind speeds up and widens the unambiguous Doppler window.

8. Self-Check Questions

Attempt each question, then click to reveal the answer.

Q1. Why does the received power vary as 1/R4 rather than 1/R2?
Answer: The transmitted energy undergoes two inverse-square spreading processes — one on the way to the target and one on the return trip — giving (1/R2)×(1/R2) = 1/R4.
Q2. A radar's peak power is increased 16×. By what factor does the maximum range increase (all else constant)?
Answer: Rmax ∝ Pt1/4, so 161/4 = 2×. Only a doubling of range for a 16× power increase — this illustrates why antenna gain (squared term) is so valuable.
Q3. Compute the noise power in dBm for Bn = 2 MHz and F = 3 dB.
Answer: N = −174 dBm/Hz + 10log10(2×106) + F = −174 + 63 + 3 = −108 dBm.
Q4. What is the maximum unambiguous range for PRF = 2 kHz?
Answer: Ru = c/(2×PRF) = 3×108/(4000) = 75 km. Echoes from beyond 75 km appear folded into nearer ranges.
Q5. A 3 GHz radar uses PRF = 1 kHz. Find the first blind speed.
Answer: λ = c/f = 0.1 m. vb1 = λ·PRF/2 = 0.1×1000/2 = 50 m/s (180 km/h).
Q6. Explain why the LNA noise figure dominates overall receiver noise figure.
Answer: By Friis' formula, the first stage's noise figure enters unattenuated while later stages are divided by the preceding gains. Hence a low-NF, high-gain LNA mounted at the antenna sets the system noise figure.
Q7. A sphere of radius 0.5 m operates in the optical region at 10 GHz. Estimate its RCS.
Answer: σ = πa2 = π(0.5)2 ≈ 0.79 m² (−1 dBsm), independent of frequency in the optical region.
Q8. How does atmospheric absorption affect the radar equation for a satellite ground station?
Answer: It adds a two-way loss term Latm (often several dB at microwave bands, peaking near 22 and 60 GHz water-vapour/oxygen lines) to the total loss L in the modified range equation, reducing detectable range.

9. References and Further Reading

  1. M. I. Skolnik, Introduction to Radar Systems, 3rd ed., McGraw-Hill, 2001 (Chapter 2: The Radar Equation).
  2. M. I. Skolnik (ed.), Radar Handbook, 3rd ed., McGraw-Hill, 2008.
  3. C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed., Wiley, 2016 (effective aperture and gain relations).
  4. G. W. Stimson, Introduction to Airborne Radar, 2nd ed., SciTech, 1998.
  5. Agilent / Keysight Application Note 150-9, Radar Measurements.
  6. Electronics Desk, “Radar System — Definition, Block Diagram”, electronicsdesk.com.
  7. RF Cafe, EW & Radar Systems Engineering Handbook — Radar Cross Section, rfcafe.com.
Study tip for EEEN 567: Work through Skolnik's worked examples 2.1–2.4, then reproduce them with Simulation 7.1. Pay special attention to dB arithmetic — most satellite link budgets (EIRP, G/T, and the Friis transmission equation) are direct extensions of the radar range equation.