Radar Performance Quiz — ECE 521E Radar & Satellite Engineering

Instructions to Candidates

This quiz contains 20 multiple-choice questions covering the first four levels of Bloom's taxonomy (Remember, Understand, Apply, Analyse).
Select the best answer for each question, then press "Check Answers" for instant marking with explanations, or consult the Post-Test Answer Key at the end.
Where calculations are required: c = 3×10⁸ m/s, k = 1.38×10⁻²³ J/K, T₀ = 290 K, (4π)³ ≈ 1984.
Total time: 45 minutes (indicative).

Level 1
Remember (Q1–5)

Level 2
Understand (Q6–10)

Level 3
Apply (Q11–15)

Level 4
Analyse (Q16–20)

Level 1 — Remember (Q1–Q5)

In the radar range equation, the received echo power varies with range as:Remember

(a) 1/R² (b) 1/R⁴ (c) 1/R (d) 1/R³

Explanation: Spherical spreading on the way out (1/R²) and again on the return (1/R²) gives the famous R⁴ law. Doubling the detection range therefore requires 16× (12 dB) more received power — the "tyranny" of radar range.

The available thermal noise power at T₀ = 290 K in a 1 MHz bandwidth is approximately:Remember

(a) −114 dBm (b) −104 dBm (c) −124 dBm (d) −141 dBm

Explanation: N = kT₀B = 1.38×10⁻²³ × 290 × 10⁶ ≈ 4×10⁻¹⁵ W ≈ −114 dBm. This is the absolute physical floor per MHz — every radar receiver starts here.

The probability of detection P_d is:Remember

(a) The probability that noise alone crosses the detection threshold (b) The mean time between false alarms (c) The probability that a target echo plus noise crosses the detection threshold when a target is present (d) The probability that the radar horizon exceeds 4.12√h km

Explanation: P_d is defined under hypothesis H₁ (target present). Option (a) defines the probability of false alarm P_fa, and (b) defines false-alarm time. The pair (P_d, P_fa) completely characterises a radar's detection performance.

The noise figure F of a receiver is defined as:Remember

(a) F = (S/N)_out / (S/N)_in (b) F = (S/N)_in / (S/N)_out (c) F = kT₀B (d) F = P_out / P_in

Explanation: F is the degradation of signal-to-noise ratio through the device, so F = (S/N)_in/(S/N)_out ≥ 1. Equivalently F = 1 + T_e/T₀, linking noise figure to effective noise temperature.

A Swerling I target is one whose radar cross-section:Remember

(a) Is constant from pulse to pulse and scan to scan (b) Fluctuates independently on every pulse (c) Follows a chi-square distribution with 4 degrees of freedom, constant within a scan (d) Is constant during a single scan but varies independently from scan to scan (exponential PDF of RCS)

Explanation: Swerling I = scan-to-scan fluctuation with exponential RCS (many equal scatterers). Option (a) is the Marcum (non-fluctuating) case, (b) is Swerling II, and (c) describes Swerling III. The fluctuation model chosen directly changes the required SNR for a given (P_d, P_fa).

Level 2 — Understand (Q6–Q10)

Pulse integration improves detection probability fundamentally because:Understand

(a) It increases the transmitter peak power (b) It narrows the receiver bandwidth (c) Target echoes add coherently (in phase) while noise adds only in power, so SNR grows approximately with the number of pulses (d) It eliminates the noise figure of the receiver

Explanation: Coherent integration: signal amplitudes add (voltage ×n, power ×n²) while independent noise powers add (×n), so SNR ×n. Non-coherent integration achieves slightly less due to the nonlinear detector. Peak power, bandwidth, and noise figure are unaffected by integration — it trades time for SNR.

In the Friis cascade formula, the noise figure of the FIRST stage dominates because:Understand

(a) The first stage always has the highest gain (b) Later-stage noise contributions are divided by the accumulated gain of the preceding stages (c) The formula only works for two stages (d) Later stages operate at higher temperatures

Explanation: F_total = F₁ + (F₂−1)/G₁ + (F₃−1)/(G₁G₂) + … — each subsequent term is suppressed by the gain already accumulated. Hence any loss (cable, duplexer) placed before the LNA adds dB-for-dB to the system noise figure, which is why the LNA sits at the antenna feed.

A fluctuating target (Swerling I) requires a HIGHER single-pulse SNR than a non-fluctuating target of the same average RCS for high P_d, because:Understand

(a) The exponential RCS distribution means deep fades occur, and detection must be sized for the fade margin (b) Fluctuation increases the thermal noise temperature (c) The radar equation no longer applies to fluctuating targets (d) Fluctuating targets always travel faster

Explanation: With scan-to-scan fluctuation, a single scan may catch the target in a deep fade (RCS well below average). Achieving P_d = 0.9 reliably demands extra SNR to survive the fades — the classic "fluctuation loss". Counter-intuitively, pulse-to-pulse fluctuation with integration can help, because diversity means at least one pulse likely catches a peak.

A CFAR detector raises its threshold near the edge of a strong clutter region in order to:Understand

(a) Increase the probability of detection in that region (b) Reduce the receiver bandwidth (c) Compensate for the R⁴ range loss (d) Hold the false-alarm rate constant when the local noise/clutter level rises

Explanation: CFAR estimates the local interference level from surrounding cells and sets the threshold a fixed margin above it, keeping P_fa (and hence false-alarm time) constant across heterogeneous scenes. The price is the "CFAR loss" — a slightly higher SNR requirement (1–3 dB) because the threshold must sit above an estimated, not known, noise level.

Low-altitude aircraft are hardest to detect at long range primarily because:Understand

(a) Their RCS decreases with altitude (b) The atmosphere absorbs more energy at low elevation (c) The direct ray and the ground-reflected ray interfere, producing deep multipath nulls at low grazing angles (d) The radar horizon blocks all low-altitude targets

Explanation: Near grazing incidence the direct and surface-reflected paths are nearly equal in length but opposite in phase → destructive interference, with fades of 20–30 dB. This interference region extends to roughly half a beamwidth above the horizon, exactly where low fliers live. The horizon formula limits absolute range, but the multipath null is what erodes detection within that range.

Level 3 — Apply (Q11–Q15)

A radar transmits P_t = 200 kW with G = 30 dB at λ = 0.1 m. The received power from a σ = 2 m² target at R = 100 km is approximately:Apply

(a) 2×10⁻⁹ W (b) 2×10⁻¹⁴ W (c) 2×10⁻¹⁷ W (d) 2×10⁻¹¹ W

Explanation: P_r = P_tG²λ²σ/[(4π)³R⁴] = (2×10⁵)(10³)²(0.01)(2) / [1984 × (10⁵)⁴] = 4×10⁹ / 1.984×10²³ ≈ 2×10⁻¹⁴ W. A hundredth of a picowatt — radar echoes are tiny, which is why noise figure matters so much.

A receiver has bandwidth B = 2 MHz and noise figure F = 3 dB. The total input-referred noise power is:Apply

(a) −111 dBm (b) −105 dBm (c) −108 dBm (d) −114 dBm

Explanation: kT₀B = 1.38×10⁻²³×290×2×10⁶ ≈ 8×10⁻¹⁵ W ≈ −111 dBm; adding F = 3 dB: N = −108 dBm. (Option (a) is the thermal floor without the noise figure.)

Using Albersheim's approximation, the required single-pulse SNR for P_d = 0.9, P_fa = 10⁻⁶ with n = 10 non-coherently integrated pulses is approximately:Apply

(a) 13.2 dB (b) 10.0 dB (c) 7.8 dB (d) 4.9 dB

Explanation: A = ln(0.62/10⁻⁶) = 13.34, B = ln(0.9/0.1) = 2.197. SNR₁ = −5log₁₀(10) + (6.2 + 4.54/√10.44)·log₁₀(13.34 + 0.12·13.34·2.197 + 1.7·2.197) ≈ −5 + 7.61·log₁₀(19.8) ≈ 4.9 dB per pulse. Integration recovers ≈ 8 dB relative to the single-pulse requirement (13.2 dB, option (a)).

A search radar achieves P_d = 0.5 per scan on a target. The probability the target is detected at least once in 4 consecutive scans is:Apply

(a) 0.9375 (b) 0.5000 (c) 0.8750 (d) 0.0625

Explanation: P_d,cum = 1 − (1 − p)^m = 1 − (0.5)⁴ = 1 − 0.0625 = 0.9375. Option (d) is the miss-all probability; (c) wrongly adds 4×0.125. Cumulative detection is why search radars tolerate modest per-scan P_d.

An antenna is mounted 50 m above the ground. Under the standard 4/3-Earth model, its radar horizon against a target at the same height is approximately:Apply

(a) 14.6 km (b) 29.1 km (c) 58.2 km (d) 206 km

Explanation: R_h = 4.12√h = 4.12√50 = 4.12 × 7.07 ≈ 29.1 km. Option (a) forgets the square root (dividing by √50 instead); (c) doubles it; (d) uses h = 2500 m. For two unequal heights, add 4.12(√h₁ + √h₂).

Level 4 — Analyse (Q16–Q20)

Flight trials show a new surveillance radar detects a test aircraft well at 3° elevation but loses it repeatedly below 1°, while a beacon on the aircraft confirms continuous presence. The most technically sound diagnosis is:Analyse

(a) The transmitter peak power is insufficient below 1° (b) The unambiguous range has been exceeded (c) Direct-path/ground-reflected multipath interference creates deep nulls in the low-elevation interference lobe, possibly combined with clutter residue (d) The receiver noise figure increases at low elevation angles

Explanation: The elevation-dependent, intermittent fading (target present per beacon, absent per radar) is the signature of multipath nulls — the interference lobe structure extends to ≈ 0.5θ_B above the horizon. Options (a), (b), and (d) would produce range- or time-dependent losses, not an elevation-banded fade. Mitigations: frequency agility (fills nulls at other frequencies), taller antenna siting, or MTI plus elevation diversity.

During a tropical downpour, an X-band (λ = 3 cm) coastal radar's detection range on ships collapses far more than an adjacent S-band (λ = 10 cm) radar's. The correct analysis and remedy is:Analyse

(a) X-band noise figures are inherently worse — recalibrate the receiver (b) Rain attenuation rises steeply with frequency (∝ f²–f³ above ~3 GHz), so the X-band link loses several dB/km — remedy: design power/aperture margin, or prefer S/C-band for all-weather long-range surveillance (c) Rain changes the radar horizon — raise the antenna mast (d) The S-band radar benefits from the R⁴ law; X-band cannot

Explanation: The differential, weather-correlated degradation between two co-sited radars isolates propagation — not receiver — loss. Because one-way attenuation in heavy rain at X-band is several dB/km (two-way on radar), it directly consumes link margin. Since range ∝ (1/L)^1/4, even 12 dB of two-way rain loss cuts range ~30%. The horizon and R⁴ arguments are frequency-generic and cannot explain the band-selective effect.

A search radar designer doubles the dwell time per beam position (same antenna, same PRF, same power). Analysing ALL consequences, she should conclude:Analyse

(a) P_d improves and coverage rate is unchanged (b) Frame time halves and nothing else changes (c) Unambiguous range doubles (d) Twice the hits are integrated (better P_d/longer range), but frame time doubles — slower track update and longer target acquisition, so coverage rate per unit time is halved

Explanation: Doubling dwell doubles n = t_dwellf_r, buying ≈3 dB of integration (coherent) for P_d or range — but with fixed beam count, t_frame = n_beam·t_dwell doubles. The coverage trade is fundamental: energy per target vs volume coverage rate. Option (c) is wrong because R_u depends only on PRF. Professional designs resolve this with phased-array adaptive dwell rather than a uniform change.

After a maintenance crew reroutes a 4 m coaxial run (2.5 dB loss) BETWEEN the antenna feed and the LNA, bench tests show system noise figure jumped from 2.1 dB to about 4.6 dB. The analysis is:Analyse

(a) Pre-LNA losses add directly to system noise figure (Friis: the lossy cable becomes F₁ with gain < 1, so 2.5 dB of cable ≈ 2.5 dB of added NF); the measured jump confirms the cabling must be moved after the LNA (b) The LNA has failed and must be replaced (c) Cable loss cannot affect noise figure, only gain (d) The increase is due to atmospheric absorption inside the cable

Explanation: A passive loss L at temperature T₀ has F = L and G = 1/L. Placed first, Friis gives F_sys ≈ L·F_LNA — the 2.5 dB cable converts directly into ~2.5 dB extra noise figure (2.1 + 2.5 ≈ 4.6 dB ✓). This is the classic field mistake: every dB before the LNA is a dB of sensitivity lost, roughly 6% of detection range per dB.

A shipborne radar is jammed by a noise jammer on an attacking aircraft. The radar designer's correct analysis of "burn-through range" is:Analyse

(a) The range beyond which the jammer's noise no longer reaches the radar (b) The range at which the radar echo power equals the jammer noise power at the receiver — inside this range S/J exceeds the required (S/J)_req and detection is again possible, since echo power falls as R⁻⁴ while jammer noise falls only as R⁻² (c) The maximum unambiguous range of the radar under jamming (d) The range at which the radar's own transmitter overheats from continuous operation

Explanation: S/J ∝ R⁻⁴/R⁻² = R⁻² — the radar wins the exchange at short range because its echo suffers two-way spreading while the jammer's noise suffers only one-way spreading. Burn-through range R_bt satisfies (S/J)(R_bt) = (S/J)_req; increasing radar ERP, narrowing beam (mainlobe vs sidelobe geometry), or frequency agility extends it. The R⁻² vs R⁻⁴ asymmetry is the heart of the jamming equation.

Post-Test Answer Key & Summary

Q	Key	Bloom Level	Concept Tested
1	b	Remember	R⁴ range law
2	a	Remember	Thermal noise floor (−114 dBm/MHz)
3	c	Remember	Definition of P_d
4	b	Remember	Noise figure definition
5	d	Remember	Swerling I fluctuation model
6	c	Understand	Why integration improves SNR
7	b	Understand	Friis cascade — first-stage dominance
8	a	Understand	Fluctuation loss mechanism
9	d	Understand	CFAR purpose and cost
10	c	Understand	Low-elevation multipath nulls
11	b	Apply	Range-equation computation (2×10⁻¹⁴ W)
12	c	Apply	Noise power with NF (−108 dBm)
13	d	Apply	Albersheim with n = 10 (4.9 dB)
14	a	Apply	Cumulative detection (0.9375)
15	b	Apply	Radar horizon (29.1 km)
16	c	Analyse	Elevation-banded multipath fade diagnosis
17	b	Analyse	Rain attenuation frequency dependence
18	d	Analyse	Dwell-time vs coverage trade-off
19	a	Analyse	Pre-LNA cable loss in Friis formula
20	b	Analyse	Burn-through range, R⁻⁴ vs R⁻²

Grading guide: 18–20 = Excellent (design-ready understanding of radar performance); 14–17 = Good; 10–13 = Fair — revisit the noise figure and range equation sections; <10 = Review the Radar Performance study notes before re-attempting.