Radar Performance

A Comprehensive Study Guide — Detection, Noise, Losses, Accuracy, and Jamming

ECE 521E • 5th Year BSc. Electrical & Communication Engineering • Masinde Muliro University of Science and Technology

Unit Overview

This study guide supports the Radar Performance unit of ECE 521E. A radar may have the finest antenna and the most sophisticated signal processor, but its worth is expressed through a short list of performance questions: At what range will it detect the target? How often will it lie? How accurately will it measure? Will it survive jamming? This unit supplies the quantitative machinery to answer all four.

Learning Outcomes

After studying this unit and completing the tutorials you should be able to:

  • Derive and apply the radar range equation in its power and energy forms.
  • Compute receiver noise power, noise figure, and cascaded noise temperature.
  • Explain and compute the Pd/Pfa operating point of a detector.
  • Quantify the benefit of coherent and non-coherent pulse integration (Albersheim).
  • Build and defend a complete system loss budget.
  • Balance search dwell against coverage rate; compute tracking accuracy bounds.
  • Predict propagation effects: horizon, multipath nulls, rain attenuation, ducting.
  • Analyse jamming scenarios using the burn-through concept.

Indicative Study Time

ActivityTime
Guided reading (Sections 1–8)5 hours
Figures re-plotted / examples reworked by hand2 hours
Tutorial problems (Section 9)3 hours
Revision of key equations & glossary1 hour
Prerequisite knowledge: radar fundamentals (PRF, PRI, duty cycle, unambiguous range), logarithms and dB arithmetic, basic probability (PDFs, Gaussian and exponential distributions), Fourier concepts for bandwidth–resolution relations.

Section 1 — The Radar Range Equation

1.1 Derivation

Power density at range R from an antenna of gain Gt: S = PtGt/4πR². The target intercepts PtGtσ/4πR² and scatters it isotropically; the returning density is that value over 4πR². The receiving aperture Ae = Grλ²/4π collects the echo:

Pr = Pt G² λ² σ / [(4π)³ R⁴ L]

Assumptions: monostatic radar (Gt = Gr = G), matched polarisation, point target, free-space propagation, all losses lumped into L.

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Fig. B — Echo power in dBm versus range for four target sizes. Detection range is read where each curve crosses the required-power line: σ = 100 m² ≈ 300 km here, while the small drone (σ = 0.1 m²) is seen only to ≈ 95 km. The 40 dB/decade slope is the R⁴ law.

1.2 The Energy Form (for System Design)

Rmax = [ Pav G² λ² σ ts n / ((4π)³ k Ts (S/N)req L) ]1/4

Detection is fundamentally about energy: average power × time-on-target × number of integrated pulses. Peak power alone is meaningless without duty cycle.

1.3 Sensitivity to Parameters

Parameter ↑ 4×Range becomesPractical lever
Pt (power)41/4 = 1.41×Expensive, thermal limits
G (antenna gain)(4²)1/4 = 2×Bigger aperture — best leverage
σ (target size)1.41×Not under designer's control
λ (wavelength)41/4 = 1.41×Conflicts with resolution, antenna size
1/L (loss reduction)41/4 = 1.41×Engineering discipline (Section 5)
Example 1.1 — Compute Pr for Pt = 100 kW, G = 35 dB, λ = 0.1 m, σ = 10 m², R = 200 km, L = 5 dB.
G = 3162, L = 3.16. Pr = (10⁵)(3162²)(0.01)(10)/[1984×(2×10⁵)⁴×3.16] ≈ 2×10⁻¹⁴ W — a twentieth of a picowatt. If the noise floor is −110 dBm and required SNR 13 dB, required Pr = −97 dBm = 2×10⁻¹³ W, so this target is undetectable at 200 km with one pulse — integration or more power is needed.

Check Your Understanding 1

  1. Why does the received power fall as R⁻⁴ and not R⁻²?
  2. Using Fig. B, estimate the detection range of the σ = 1 m² target.
  3. Show that doubling antenna gain doubles detection range (range ∝ G1/2).
Answers
  1. Two-way spherical spreading: 1/R² outbound, 1/R² inbound → 1/R⁴.
  2. σ = 1 m² curve crosses −100.8 dBm at ≈ 170 km.
  3. Rmax = […G²…]1/4 → G → 2G gives (2²)1/4 = √2… wait: (2G)² = 4G², 41/4 = 1.41. Careful: doubling gain multiplies range by 22/4 = 21/2 = 1.41 — doubling range requires 4× gain.

Section 2 — Receiver Noise

2.1 The Thermal Floor

N = k T0 B  →  −114 dBm per MHz at T0 = 290 K

2.2 Noise Figure and Effective Temperature

F = (S/N)in/(S/N)out = 1 + Te/T0;  Ts = Ta + (F−1)T0

2.3 Friis Cascade Formula

Ftot = F₁ + (F₂−1)/G₁ + (F₃−1)/(G₁G₂) + …
Iron rule: any passive loss (cable, duplexer, rotary joint) placed before the LNA adds its dB value directly to the system noise figure. A 2 dB cable before the LNA costs 2 dB of sensitivity ≈ 6% of detection range; after a 25 dB LNA it costs 2/316 ≈ 0.006 dB.
Example 2.1 — Chain: 1.5 dB cable (F₁ = 1.41, G₁ = 0.71), LNA F₂ = 2 dB, G₂ = 25 dB; mixer/IF F₃ = 10 dB.
Ftot = 1.41 + 0.585/0.71 + 9/(0.71×316) = 1.41 + 0.82 + 0.04 = 2.27 → 3.6 dB. Relocating the LNA before the cable yields ≈ 1.6 dB — a 2 dB sensitivity recovery worth ~12% range.

Check Your Understanding 2

  1. A receiver has B = 5 MHz, F = 5 dB. Compute the input-referred noise power in dBm.
  2. Why is antenna temperature Ta not always 290 K?
Answers
  1. kT0B = −107 dBm; +5 dB → −102 dBm.
  2. Ta is the brightness temperature of what the antenna sees: ≈ 30–50 K for cold sky, ≈ 290 K for warm ground — hence low-sidelobe, high-mounted antennas lower Ts.

Section 3 — Detection Statistics: Pd and Pfa

3.1 Hypotheses and Errors

Declare ↓ / Truth →Noise only (H₀)Target present (H₁)
"Target"False alarm, PfaDetection, Pd
"No target"Correct rejectionMiss, 1 − Pd
Pfa = exp(−T²/2σ²);  mean false-alarm time Tfa = 1/(Pfa · Ncells · rate)
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Fig. A — Pd versus single-pulse SNR for a non-fluctuating target at four false-alarm rates. Two lessons: detection probability is a steep function of SNR (the "S-curve"), and lowering Pfa by 100× costs only ~1–2 dB of extra SNR at Pd = 0.9.

3.2 The Operating Point

Designers quote a dual specification, e.g. "Pd = 0.9 at Pfa = 10⁻⁶". For a non-fluctuating target with no integration this demands SNR ≈ 13.2 dB. The radar must meet the spec at the end of the loss budget — a fact often forgotten.

3.3 Fluctuation Models

ModelRCS in timePDFExample target
Marcum (case 0)Constant—Sphere, corner reflector
Swerling IConstant within scan, varies scan-to-scanExponential (2 dof)Complex aircraft
Swerling IIVaries pulse-to-pulseExponentialAircraft with rapid aspect change
Swerling III/IVAs I/IIChi-square, 4 dofOne dominant + small scatterers
Fluctuation loss: at Pd = 0.9, a Swerling I target can need several dB more single-pulse SNR than a Marcum target of the same average σ, because detection must survive deep fades. With integration, pulse-to-pulse fluctuation helps (diversity) while scan-to-scan fluctuation hurts.

Check Your Understanding 3

  1. A radar processes 2×10⁶ cells/s and must average one false alarm per second. What Pfa is needed?
  2. Reading Fig. A, what SNR gives Pd = 0.5 at Pfa = 10⁻⁸?
Answers
  1. Pfa = 1/(2×10⁶) = 5×10⁻⁷.
  2. ≈ 10.5 dB.

Section 4 — Integration and Detection Curves

4.1 Coherent versus Non-Coherent

Coherent (pre-detection)Non-coherent (post-detection)
Summed quantityComplex I/Q phasorsDetected magnitudes
SNR gain, n pulsesn (10 log n)≈ n with a 1–2 dB integration loss
RequirementPhase coherence, Doppler alignmentAmplitude only

4.2 Albersheim's Approximation

SNR₁ = −5 log₁₀n + (6.2 + 4.54/√(n+0.44)) · log₁₀(A + 0.12AB + 1.7B)
A = ln(0.62/Pfa); B = ln[Pd/(1−Pd)]
Example 4.1 — Pd = 0.9, Pfa = 10⁻⁶, n = 20 pulses.
A = 13.34, B = 2.197. SNR₁ = −5(1.301) + (6.2+4.54/√20.44)·log₁₀(13.34+0.12×13.34×2.197+1.7×2.197) = −6.5 + 8.20×log₁₀(19.8) = 4.4 dB/pulse. Twenty pulses at 4.4 dB replace one pulse at 13.2 dB — an 8.8 dB saving bought with dwell time.

4.3 Cumulative Detection on Scan

Pd,cum(m scans) = 1 − (1 − p)m

Search radars legitimately accept per-scan Pd of 0.5–0.7: over 5–10 scans the cumulative probability approaches unity, which is what operators actually experience.

Check Your Understanding 4

  1. Why is coherent integration gain exactly 10 log n while non-coherent falls short?
  2. Per-scan Pd = 0.6: what is the probability of at least one detection in 5 scans?
Answers
  1. Signal phasors add in-phase (power ∝ n²) while noise adds in power (∝n): net SNR ∝ n. The envelope detector's nonlinearity discards phase, costing the "integration loss".
  2. 1 − 0.4⁵ = 0.9898.

Section 5 — The System Loss Budget

Losses enter the range equation as L in the denominator. Building the budget is a discipline: every dB must be owned by a named mechanism.

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Fig. C — An example 8-line loss budget totalling ≈ 8 dB. Consequence: Rmax is multiplied by 10−8/40 ≈ 0.71 — the paper radar sees 30% farther than the real one.
LossMechanismTypical
Beam-shapeTarget not at beam peak during scan1–3 dB
ScanBeam motion during echo transit0.5–1.5 dB
Feed & duplexerRF plumbing1–2 dB
AtmosphericGas & rain absorption0.1–several dB
CFAREstimated-noise threshold margin1–3 dB
Filter mismatchNon-matched receiver filtering0.5–1 dB
IntegrationNon-coherent combining1–2 dB
QuantizationADC bits, phase shifter steps0.5–1 dB

Check Your Understanding 5

  1. A designer discovers a forgotten 6 dB loss. By what factor does detection range fall?
  2. Which single loss entry would you attack first in a cost-constrained upgrade, and why?
Answers
  1. 10−6/40 ≈ 0.71 — a 29% range loss (three quarters of the power-aperture product wasted).
  2. Typically feed/duplexer or beam-shape: they are often reducible with better hardware or scan scheduling at modest cost, unlike atmospheric loss which physics fixes.

Section 6 — Search and Tracking Performance

6.1 The Search Triangle

nbeam = Ω/ΩB;  tframe = nbeam · tdwell;  npulses = tdwell·fr

Dwell buys integration (Section 4) but frame time costs coverage rate. The energy form of the range equation ties it together: performance per target competes with volume coverage per second. Phased arrays escape the rigid triangle by adaptively scheduling dwells per track-quality need.

6.2 Accuracy Bounds

σR ≈ c/(2B√(2·SNR));  σθ ≈ θB/(km√(2·SNR));  σv ≈ λ/(2T√(2·SNR))
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Fig. D — Theoretical monopulse angle error for θB = 2°. Every 6 dB of SNR halves σθ: 10 dB → 0.26°, 16 dB → 0.13°, 22 dB → 0.066°.

6.3 Tracker Architectures

ArchitectureMechanismTrade-off
Single-target track (STT)Beam locked on one targetBest accuracy; ignores all else
Track-while-scan (TWS)Search scan feeds many α–β/Kalman filtersMany tracks, moderate accuracy
Phased-array adaptiveDwell scheduled by track covarianceOptimal but complex scheduling

Check Your Understanding 6

  1. A designer doubles dwell time, antenna and PRF unchanged. State all consequences.
  2. σR for B = 10 MHz, SNR = 13 dB?
Answers
  1. +3 dB integrated SNR (coherent) or better Pd/range; frame time doubles; coverage rate halves; track update latency increases.
  2. c/(2B√(2·20)) = 3×10⁸/(2×10⁷×6.32) ≈ 2.4 m.

Section 7 — Propagation Effects

EffectCausePerformance impactMitigation
Standard refractionRefractive index gradient bends raysHorizon extended: Rh ≈ 4.12(√h₁+√h₂) kmUse 4/3-Earth model in siting
Multipath nullsDirect + ground-reflected interference20–30 dB fades below ≈0.5θB elevationFrequency agility, site elevation
Rain attenuationScattering/absorption by drops, ∝ f²–f³ above ~3 GHzSevere at X/Ku-band in heavy rainS/C-band choice, power margin
DuctingTrapping inversion layersAnomalous over-horizon detections and clutterPropagation monitoring, adaptive maps
Example 7.1 — Radar at 40 m mast, target at 2000 m: Rh = 4.12(√40 + √2000) = 4.12(6.32+44.7) ≈ 210 km. At X-band in heavy rain of 2 dB/km one-way, a 210 km path loses ≈ 840 dB two-way? No — realistic heavy rain at 3 cm is ~0.5–1 dB/km one-way maximum in extreme cells; even 1 dB/km two-way over 50 km through a cell is 100 dB — impossible, so targets behind intense cells are simply lost; S-band (~0.05 dB/km) survives. This is why weather-penetration radars use long wavelengths.

Check Your Understanding 7

  1. Why are low-flying aircraft disproportionately affected by multipath, even within the radar horizon?
  2. Estimate the horizon for a 25 m antenna against a sea-skimming missile at 5 m altitude.
Answers
  1. They dwell continuously inside the interference lobe (grazing geometry persists with range), whereas high fliers cross it quickly.
  2. 4.12(5 + 2.24) ≈ 29.8 km.

Section 8 — Performance Under Jamming (ECCM)

8.1 The Jamming Equation

(S/J) = PtGtσ Rj² Bj / (4π PjGj R⁴ L)  (self-screening jammer)

The jammer's noise enters through the antenna sidelobes (Gj replaced by Gsl when off-axis), so sidelobe level appears in the equation too.

8.2 Burn-Through Range

Rbt: the range where (S/J)(Rbt) = (S/J)req

Because echo ∝ R⁻⁴ and jammer noise ∝ R⁻², the radar wins at short range: S/J improves 6 dB per octave of closing range. Burn-through is where detection becomes possible again.

8.3 ECCM Measures and Their Costs

ThreatECCMCost
Noise jammingFrequency agility, high ERP, sidelobe blanking/cancellationDwell per frequency ↓
Range-gate stealingPulse-to-pulse agility, leading-edge trackingProcessing complexity
ChaffDoppler/MTI discriminationAlready in design
ARM (anti-radiation missile)Low sidelobes, emission control (EMCON)Coverage sacrificed

Check Your Understanding 8

  1. Why does S/J improve 6 dB per octave of range closure against a self-screening jammer?
  2. State two ways to increase burn-through range.
Answers
  1. S ∝ R⁻⁴, J ∝ R⁻² → S/J ∝ R⁻² → each halving of distance multiplies S/J by 4 = 6 dB.
  2. More ERP (PtGt); narrower beam / lower sidelobes reducing jammer entry; frequency agility forcing the jammer to spread power.

Section 9 — Tutorial Problems

Attempt all problems before opening the solutions. Constants: c = 3×10⁸ m/s, k = 1.38×10⁻²³ J/K, T₀ = 290 K, (4π)³ ≈ 1984.

  1. Derive the radar range equation, stating each assumption. Hence explain why aperture is the most efficient range lever.
  2. A radar with Pt = 250 kW, G = 33 dB, λ = 0.15 m observes a σ = 3 m² target at R = 150 km with total loss L = 7 dB. Compute Pr in dBm.
  3. Receiver: B = 4 MHz, F = 4.5 dB. Compute noise power in dBm. If required SNR is 12 dB, what is the minimum detectable signal?
  4. Chain: antenna → 2 dB cable → LNA (F = 2.5 dB, G = 28 dB) → mixer (F = 9 dB). Compute system noise figure. Repeat with the LNA placed before the cable and comment.
  5. Use Albersheim to find the per-pulse SNR for Pd = 0.95, Pfa = 10⁻⁵, n = 8 pulses.
  6. Per-scan Pd = 0.55. Find the probability of ≥1 detection in 3 scans and in 8 scans.
  7. Construct a loss budget totalling no more than 8 dB for a ground-based S-band surveillance radar, justifying each entry.
  8. Monopulse radar, θB = 1.2°, SNR = 18 dB: compute σθ. What SNR quarters the error relative to this value?
  9. Antenna at 60 m: compute radar horizon for targets at 100 m, 1000 m, and 10 000 m altitude. Comment on low-altitude surveillance.
  10. A self-screening jammer delivers J such that S/J = −10 dB at 100 km. Required (S/J)req = 13 dB. Estimate burn-through range.
  11. Explain quantitatively why heavy rain at X-band can remove 30% of detection range, and state the design responses.
  12. Fig. B shows the σ = 1 m² target detected to ≈170 km with one pulse at 13.2 dB SNR. How many coherently integrated pulses (same per-pulse power) extend this to 250 km for the same target? (Hint: required extra SNR from R⁻⁴ scaling.)
Selected solutions
  1. As Section 1.1; aperture enters as G² (or Ae²) in the bracket raised to 1/4 → range ∝ √Ae: doubling aperture ≈ 41% more range for no extra power — unmatched efficiency.
  2. G = 1995, L = 5.01. Pr = 2.5×10⁵×(1995)²×0.0225×3/[1984×(1.5×10⁵)⁴×5.01] ≈ 5.36×10⁻¹⁴ W ≈ −102.7 dBm.
  3. kT₀B = 1.38e−23×290×4e6 = 1.6e−14 W ≈ −108 dBm; +4.5 → −103.5 dBm. MDS = −103.5 + 12 = −91.5 dBm.
  4. Cable first: F = 1.585 + (1.78−1)/0.63 + (7.94−1)/(0.63×631) = 1.585+1.24+0.017 = 2.84 → 4.5 dB. LNA first: 1.78 + 0.585/631 + 6.94/(631×0.63)... = 1.78+0.001+0.017 ≈ 1.80 → 2.5 dB. Lesson: 2 dB of cable before the LNA = 2 dB system NF.
  5. A = ln(0.62/1e−5) = 11.13; B = ln(19) = 2.944. SNR₁ ≈ −5log8 + (6.2+4.54/√8.44)·log₁₀(11.13+0.12×11.13×2.944+1.7×2.944) ≈ −4.5+8.15×log₁₀(17.3) ≈ 5.4 dB.
  6. 1−0.45³ = 0.909; 1−0.45⁸ = 0.9984.
  7. Example: beam-shape 1.5, scan 0.6, feed 1.2, atmospheric 0.8 (S-band, humid), CFAR 1.2, mismatch 0.6, integration 1.0, quantization 0.5 → 7.4 dB.
  8. σθ = 1.2/(1.7√(2·63.1)) = 1.2/19.1 ≈ 0.063°. Error ∝ 1/√SNR → quartering needs 16× SNR = +12 dB → 30 dB.
  9. 4.12(√60+√100) = 73 km; 4.12(7.75+31.6) = 162 km; 4.12(7.75+100) = 444 km. Low-altitude targets are horizon-limited regardless of radar power.
  10. S/J ∝ R⁻²: need +23 dB → R factor 10^(−23/40)... (S/J) gain 23 dB → distance ratio 10^(23/20)? No: S/J ∝ 1/R² → +23 dB needs R² to shrink 10^2.3 → Rbt = 100/10^(23/20) ≈ 7.1 km.
  11. 30% range loss = 12 dB total loss (two-way) → e.g. 0.6 dB/km two-way over 20 km path through cells at X-band — realistic in tropical rain. Responses: S/C-band, power/aperture margin, adaptive waveform, fade-tolerant CFAR.
  12. Extra range factor 250/170 = 1.47 needs Pr margin (1.47)⁴ = 4.7 = 6.7 dB; coherent n pulses give 10 log n ≥ 6.7 → n ≥ 4.7 → 5 pulses.

Section 10 — Glossary of Key Terms

Burn-through range
The range at which S/J equals the required value so detection against a jammer becomes possible again.
CFAR
Constant False-Alarm Rate detector: adaptive threshold set from local noise estimates; costs 1–3 dB.
Clutter attenuation / improvement factor
Reduction of clutter power (MTI); ratio of output to input S/C (see MTI unit).
Coherent / non-coherent integration
Summing I/Q phasors (gain n) versus summing detected magnitudes (gain ≈ n with loss).
False-alarm time
Mean time between false alarms: 1/(Pfa·cells·rate).
Fluctuation loss
Extra SNR required for a fluctuating (Swerling) target relative to a non-fluctuating one of equal mean RCS.
Marcum / Swerling models
Target RCS fluctuation models used in detection-curve computation.
MIC / MDS
Minimum detectable / discriminable signal: threshold-level echo power for the specified Pd, Pfa.
Noise figure (F)
SNR degradation factor of a device; F = 1 + Te/T0.
Radar horizon
Maximum line-of-sight range set by Earth curvature and refraction: 4.12(√h₁+√h₂) km.
Subclutter visibility
How far below clutter (dB) a target may lie and still be detected (MTI unit).
Volume coverage rate
Search volume scanned per unit time at full performance — the currency of search-radar design.

Section 11 — References and Further Reading

  1. M. I. Skolnik, Introduction to Radar Systems, 3rd ed., McGraw-Hill, 2001.
  2. M. I. Skolnik (ed.), Radar Handbook, 3rd ed., McGraw-Hill, 2008.
  3. D. K. Barton, Radar Equations for Modern Radar, Artech House, 2013 — definitive loss-budget treatment.
  4. N. Levanon, Radar Principles, Wiley-Interscience, 1988.
  5. B. R. Mahafza, Radar Systems Analysis and Design Using MATLAB, 3rd ed., CRC Press, 2013.
Exam guidance: When quoting a required SNR, always state Pd, Pfa, fluctuation model, and integration count — a bare number is ambiguous and earns no marks.