Radar Laboratory — Interactive Radar Phenomenology

RADAR LABORATORY QUICK REF · λ=c/f · R=cτ_d/2 · ΔR=cτ/2 · f_d=2v_r/λ · v_u=PRF·λ/4 · θ=0.886λ/DTHEORY REFERENCEALL KEY RADAR FORMULAS — ORGANIZED BY TOPIC — WITH DERIVATIONS AND CONTEXT01 — Propagation & FrequencyFundamental Wavelength RelationThe wavelength λ of an electromagnetic wave is inversely proportional to frequency. Every radar formula contains λ — choosing the operating frequency is the first and most consequential design decision.λ = c / f c = 3×10⁸ m/s (speed of light) f = carrier frequency (Hz)λ — wavelength (m) · f — frequency (Hz) · c — 3×10⁸ m/sPROPAGATIONFUNDAMENTALRadar Band DesignationsIEEE letter-band designations define standard operating ranges. Band choice determines resolution, attenuation, target interaction, and hardware constraints.L-band: 1–2 GHz λ ≈ 15–30 cm S-band: 2–4 GHz λ ≈ 7.5–15 cm C-band: 4–8 GHz λ ≈ 3.75–7.5 cm X-band: 8–12 GHz λ ≈ 2.5–3.75 cm Ku-band: 12–18 GHz λ ≈ 1.7–2.5 cm Ka-band: 26–40 GHz λ ≈ 0.75–1.15 cmBANDSPROPAGATIONAtmospheric AbsorptionWater vapor (H₂O) peaks at 22 GHz (~0.18 dB/km) and 183 GHz. Oxygen (O₂) dominates at 60 GHz (~15 dB/km) and 119 GHz. Atmospheric windows at 35, 77, and 94 GHz are exploited by automotive and military radars.

L_atm (dB) = α(f) × R_km Two-way loss = 2 × α × R_km α: dB/km (frequency-dependent)α — specific attenuation (dB/km) · R_km — one-way range (km)PROPAGATIONLOSSES02 — Range MeasurementRange from Echo DelayRadar times the two-way travel of a pulse. The round-trip delay τ_d gives range exactly. Electromagnetic waves travel at c = 3×10⁸ m/s ≈ 150 m/μs (one-way).R = c · τ_d / 2 1 μs delay → R = 150 mτ_d — round-trip delay (s) · c — 3×10⁸ m/sRANGEFUNDAMENTALMaximum Unambiguous RangeThe radar must receive the previous pulse's echo before firing again. If the PRI is too short, a distant echo arrives after the next transmission and is reported at a false closer range.R_u = c / (2 · PRF) PRI = 1 / PRF (pulse repetition interval) R_app = R_true mod R_u (folded range)PRF — pulse repetition frequency (Hz) · PRI — 1/PRF (s)RANGEAMBIGUITY03 — Range Resolution & Pulse CompressionPulse Width ResolutionTwo targets closer than ΔR cannot be separated — their echoes overlap in the receiver. The matched filter output width equals cτ/2, which is why resolution and pulse duration are the same formula.ΔR = c · τ / 2 τ = 1 μs → ΔR = 150 m τ = 10 ns → ΔR = 1.5 mτ — pulse width (s) · ΔR — minimum resolvable separation (m)RESOLUTIONPulse Compression (LFM Chirp)A chirp sweeps frequency across bandwidth B during pulse duration T. The matched filter compresses the pulse to width 1/B, independent of T. This breaks the energy–resolution trade-off.

ΔR_compressed = c / (2B) Compression gain: G_c = B·T Peak sidelobes: −13.2 dB (rect window) −42.7 dB (Hamming window)B — chirp bandwidth (Hz) · T — pulse duration (s)PULSE COMPRESSIONLFMMatched Filter SNRThe matched filter is optimal — it maximizes SNR for any given waveform. The output SNR depends only on the signal energy E and noise spectral density N₀, not on pulse shape.SNR_out = 2E / N₀ E = Pt · τ (pulse energy) N₀ = k_B · T_sys · F (noise density)E — signal energy (J) · N₀ — noise spectral density (W/Hz)MATCHED FILTERSNR04 — Doppler & VelocityDoppler Frequency ShiftA moving target compresses (approaching) or stretches (receding) the reflected wavefront, shifting the echo frequency by f_d. Positive Doppler = closing, negative = opening.f_d = 2 · v_r / λ = 2 · v_r · f_c / c v_r = f_d · λ / 2 (velocity from Doppler) v_r = v · cos(θ) (radial component)v_r — radial velocity (m/s) · θ — angle from boresightDOPPLERVELOCITYMaximum Unambiguous VelocityThe radar samples echo phase once per PRI. The Nyquist limit for phase sampling is π per sample — a target exceeding v_u aliased to a wrong (lower) apparent velocity. This is the Doppler counterpart of range ambiguity.v_u = PRF · λ / 4 Phase advance per PRI: Δφ = π · v_r / v_u At v_r = v_u: Δφ = π (Nyquist limit) Aliased velocity: v_app = v_r mod v_uv_u — max unambiguous velocity (m/s)VELOCITYAMBIGUITY05 — PRF & The Range-Doppler AmbiguityAmbiguity Product — Fixed by PhysicsPRF simultaneously sets both R_u and v_u in opposite directions. Their product is fixed by the carrier frequency alone — independent of PRF.

No single PRF can simultaneously maximize both.R_u · v_u = c² / (8 · f_c) = λ · c / 8 (in terms of wavelength) Product is CONSTANT for a given frequency.f_c — carrier frequency (Hz) · invariant under PRF changesPRFAMBIGUITYFUNDAMENTALStaggered PRF — Resolving AmbiguitiesTransmitting alternating PRFs with ratio p:q (p, q coprime) moves blind speeds and ghosted ranges. The Chinese Remainder Theorem extends unambiguous intervals to lcm(R_u1, R_u2) in range and lcm(v_u1, v_u2) in velocity.R_u_stag = lcm(R_u1, R_u2) v_u_stag = lcm(v_u1, v_u2) Choose PRF ratio p/q where gcd(p,q) = 1PRF1, PRF2 — staggered pulse rates · p, q — coprime integersPRFSTAGGERED06 — Antenna & BeamBeamwidthA uniformly illuminated aperture of width D produces a sinc² beam pattern. The 3 dB (half-power) beamwidth in radians is 0.886λ/D. It is always the ratio λ/D that matters — not D or λ independently.θ_3dB ≈ 0.886 · λ / D (radians) θ_3dB ≈ 50.8 · λ / D (degrees) Angular resolution: δ_az = R · θ_3dBD — aperture width (m) · R — range (m) · δ_az — cross-range resolutionANTENNABEAMWIDTHAntenna GainGain G is the ratio of peak radiated intensity to that of an isotropic radiator at the same total power. For a uniformly illuminated aperture, G is proportional to A/λ².

Aperture efficiency η accounts for non-uniform illumination (typically 0.6–0.8).G = η · 4π · A / λ² G = 4π · A_eff / λ² (A_eff = η·A) G_dBi = 10·log₁₀(G)A — physical aperture area (m²) · η — aperture efficiency · A_eff — effective areaANTENNAGAINPhased Array — Steering & Grating LobesA progressive phase shift φ_n steers the main beam to angle θ_s. Element spacing d must satisfy d ≤ λ/2 to push grating lobes outside the visible hemisphere. Violating this creates ambiguous returns at grating lobe angles.Steering phase: φ_n = n·2π(d/λ)·sin(θ_s) Array factor: |AF|² = sin²(Nψ/2)/sin²(ψ/2) ψ = 2π(d/λ)(sinθ − sinθ_s) Grating lobe: sin(θ_g) = sin(θ_s) ± nλ/d Condition for no grating lobe: d ≤ λ/2BEAMFORMINGPHASED ARRAY07 — Detection TheoryHypothesis TestingEvery range cell is tested against two hypotheses: H₀ (noise only) vs H₁ (target + noise). The threshold T sets the trade-off between false alarm probability Pfa and detection probability Pd. No threshold can eliminate both errors simultaneously — the distributions always overlap.H₀: p(x) = N(0, σ_n²) [Gaussian model] H₁: p(x) = N(A_s, σ_n²) Pfa = P(x > T | H₀) = Q((T)/σ_n) Pd = P(x > T | H₁) = Q((T-A_s)/σ_n)DETECTIONNEYMAN-PEARSONRayleigh/Rice Model (Envelope Detection)Real radar receivers use envelope detection, making the noise Rayleigh-distributed (not Gaussian). The Marcum Q₁ function gives Pd for a non-fluctuating target. This is a better model for envelope-detected radar returns. The right model still depends on where in the receiver chain you place the detector and test statistic.

H₀ (noise only): Rayleigh(σ_n) Pfa = exp(−T²/2σ_n²) H₁ (target+noise): Rice(A_s, σ_n) Pd = Q₁(A_s/σ_n, T/σ_n) [Marcum Q] Swerling 1: Pd = Pfa^(1/(1+SNR))DETECTIONRAYLEIGHSWERLINGCoherent IntegrationSumming N pulses coherently (phase-aligned) improves SNR by exactly N (linear), or 10 log₁₀(N) dB. This is the fundamental lever for extending detection range without increasing transmit power.SNR_coh = N · SNR_single SNR_improvement = 10·log₁₀(N) dB Range extension ∝ N^(1/4) Example: N=16 → +12 dB → +88% rangeINTEGRATIONDETECTION08 — CFAR — Constant False Alarm RateCA-CFAR ThresholdCell-Averaging CFAR estimates local noise power from N reference cells surrounding each cell under test (CUT). The threshold scales with the noise estimate, keeping Pfa constant as noise level changes. Guard cells prevent target energy from contaminating the noise estimate.T = α · mean(reference cell powers) α = N · (Pfa^(−1/N) − 1) Guard cells: typically 2–4 each side Reference cells: typically N = 16–32α — CFAR scaling factor · N — number of reference cellsCFARDETECTIONCFAR VariantsCA-CFAR fails at clutter edges and in target-rich environments. Variants address specific failure modes at a cost in detection performance.CA-CFAR: mean of all reference cells GO-CFAR: max(left mean, right mean) [clutter edges] SO-CFAR: min(left mean, right mean) [multiple targets] OS-CFAR: k-th order statistic [non-Rayleigh clutter]CFARDETECTION09 — System Parameters & Radar Range EquationReceiver Noise PowerThermal noise sets the absolute detection floor. The noise figure F quantifies the excess noise added by the receiver chain above the thermal minimum. The first amplifier (LNA) dominates the cascade.

P_noise = k_B · T_sys · B k_B = 1.38×10⁻²³ J/K (Boltzmann) T_sys = T₀(F−1) + T_ant [system temp] T₀ = 290 K (standard reference) F_cascade = F₁ + (F₂−1)/G₁ + ... [Friis]k_B — Boltzmann constant · T_sys — system noise temperature (K) · B — bandwidth (Hz)NOISERECEIVERThe Radar Range EquationThe central equation of radar design. Every parameter in the RRE has been covered in the curriculum. The R⁴ dependence means doubling range requires 16× more power, or 4× more antenna gain.SNR = (Pt · G² · λ² · σ) / ((4π)³ · R⁴ · k_B · T_sys · B · F · L) R_max = [ Pt·G²·λ²·σ / ((4π)³·SNR_min·kTBFL) ]^(1/4) In dB: SNR_dB = Pt_dBW + 2G_dBi + 20log(λ) + σ_dBsm − 30·log(4π) − 40log(R_m) − 10log(kTBF) − L_dBPt — transmit power (W) · G — antenna gain · σ — RCS (m²) · L — losses (linear)RANGE EQUATIONFUNDAMENTALRadar Cross Section (RCS)RCS is the effective scattering area of a target — the area of an equivalent isotropic reflector producing the same power density back at the radar. Highly aspect-angle and frequency dependent.σ = lim(R→∞) 4πR² · |E_s|²/|E_i|² Metallic sphere (optical): σ = πr² Flat plate (normal incidence): σ = 4πA²/λ² σ_dBsm = 10·log₁₀(σ) [dB sq.

meters]RCSTARGET10 — Clutter & MTIClutter RCSGround and sea clutter are distributed targets characterized by the normalized clutter cross-section σ⁰ (sigma-naught) in dB. The total clutter RCS in one range-azimuth cell depends on the cell geometry.σ_c = σ⁰ · A_c A_c = (c·τ/2) · R · θ_az [range-azimuth cell] SCR = σ_target / σ_c [signal-to-clutter] Typical σ⁰: farmland −25 dB, urban −10 dBCLUTTERMTI CancellerMoving Target Indication subtracts consecutive pulse returns. Ground clutter (Doppler ≈ 0) cancels; moving targets survive. The improvement factor (IF) measures cancellation quality.Single delay: y[n] = x[n] − x[n−1] → H(z) = 1−z⁻¹ Double delay: y[n] = x[n] − 2x[n−1] + x[n−2] Improvement Factor: IF = SCR_out / SCR_in Blind speeds: v_b = n·PRF·λ/2, n = 1, 2, 3…MTICLUTTER11 — FMCW RadarBeat Frequency & RangeFMCW mixes transmit and receive signals to produce a constant beat frequency proportional to target range. A Doppler component also appears as a frequency offset between up-sweep and down-sweep measurements.Beat frequency: f_b = 2·R·B / (c·T) Range from beat: R = f_b·c·T / (2·B) Range resolution: ΔR = c/(2B) Velocity: v_r from phase difference between sweepsB — sweep bandwidth (Hz) · T — sweep period (s) · f_b — beat frequencyFMCWCW RADAR12 — STAP & MIMOSTAP Optimal WeightsSpace-Time Adaptive Processing jointly nulls clutter and interference in both angle and Doppler.