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/D
THEORY REFERENCE
ALL KEY RADAR FORMULAS — ORGANIZED BY TOPIC — WITH DERIVATIONS AND CONTEXT
01 — Propagation & Frequency
Fundamental Wavelength Relation
The 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/s
PROPAGATIONFUNDAMENTAL
Radar Band Designations
IEEE 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 cm
BANDSPROPAGATION
Atmospheric Absorption
Water 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)
PROPAGATIONLOSSES
02 — Range Measurement
Range from Echo Delay
Radar 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 m
τ_d — round-trip delay (s) · c — 3×10⁸ m/s
RANGEFUNDAMENTAL
Maximum Unambiguous Range
The 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)
RANGEAMBIGUITY
03 — Range Resolution & Pulse Compression
Pulse Width Resolution
Two 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 m
τ — pulse width (s) · ΔR — minimum resolvable separation (m)
RESOLUTION
Pulse 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 COMPRESSIONLFM
Matched Filter SNR
The 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 FILTERSNR
04 — Doppler & Velocity
Doppler Frequency Shift
A 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 boresight
DOPPLERVELOCITY
Maximum Unambiguous Velocity
The 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_u
v_u — max unambiguous velocity (m/s)
VELOCITYAMBIGUITY
05 — PRF & The Range-Doppler Ambiguity
Ambiguity Product — Fixed by Physics
PRF 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 changes
PRFAMBIGUITYFUNDAMENTAL
Staggered PRF — Resolving Ambiguities
Transmitting 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) = 1
PRF1, PRF2 — staggered pulse rates · p, q — coprime integers
PRFSTAGGERED
06 — Antenna & Beam
Beamwidth
A 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_3dB
D — aperture width (m) · R — range (m) · δ_az — cross-range resolution
ANTENNABEAMWIDTH
Antenna Gain
Gain 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 area
ANTENNAGAIN
Phased Array — Steering & Grating Lobes
A 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/2
BEAMFORMINGPHASED ARRAY
07 — Detection Theory
Hypothesis Testing
Every 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-PEARSON
Rayleigh/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))
DETECTIONRAYLEIGHSWERLING
Coherent Integration
Summing 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% range
INTEGRATIONDETECTION
08 — CFAR — Constant False Alarm Rate
CA-CFAR Threshold
Cell-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 = 16–32
α — CFAR scaling factor · N — number of reference cells
CFARDETECTION
CFAR Variants
CA-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]
CFARDETECTION
09 — System Parameters & Radar Range Equation
Receiver Noise Power
Thermal 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)
NOISERECEIVER
The Radar Range Equation
The 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_dB
Pt — transmit power (W) · G — antenna gain · σ — RCS (m²) · L — losses (linear)
RANGE EQUATIONFUNDAMENTAL
Radar 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]
RCSTARGET
10 — Clutter & MTI
Clutter RCS
Ground 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 dB
CLUTTER
MTI Canceller
Moving 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…
MTICLUTTER
11 — FMCW Radar
Beat Frequency & Range
FMCW 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 sweeps
B — sweep bandwidth (Hz) · T — sweep period (s) · f_b — beat frequency
FMCWCW RADAR
12 — STAP & MIMO
STAP Optimal Weights
Space-Time Adaptive Processing jointly nulls clutter and interference in both angle and Doppler. The optimal weight vector maximizes SINR by whitening the interference covariance before steering.
w_opt = R⁻¹·s / (s^H·R⁻¹·s) R — M×N space-time covariance matrix s — MN×1 space-time steering vector Training requirement: K ≥ 2·M·N (RMB rule)
STAPADAPTIVE
MIMO Virtual Array
MIMO radar transmits orthogonal waveforms from N_t elements and separates them at N_r receive elements, synthesizing N_t×N_r virtual channels. The virtual aperture has N_t times the angular resolution of a conventional phased array.
Virtual elements: N_v = N_t × N_r Virtual aperture: L_v = N_t · N_r · d Beamwidth: θ ≈ λ / L_v DoF gain vs phased array: N_t times more
MIMOVIRTUAL ARRAY
13 — Target Tracking
Kalman Filter
The Kalman filter is the minimum mean-square-error estimator for linear Gaussian systems. Each cycle alternates between prediction (propagating uncertainty forward) and update (correcting with new measurement).
PREDICT: x_{k|k-1} = F·x_{k-1} P_{k|k-1} = F·P·Fᵀ + Q UPDATE: y_k = z_k − H·x_{k|k-1} [innovation] S = H·P·Hᵀ + R [innovation cov] K = P·Hᵀ·S⁻¹ [Kalman gain] x_k = x_{k|k-1} + K·y_k P_k = (I−K·H)·P_{k|k-1}
Q — process noise cov · R — measurement noise cov · F — state transition · H — observation matrix
TRACKINGKALMAN
Tracking Gate & Data Association
An ellipsoidal validation gate selects candidate measurements for each track. The Mahalanobis distance determines whether a measurement falls within the predicted uncertainty ellipsoid.
Gate test: d² = yᵀ·S⁻¹·y ≤ χ²_{n,P_g} S = H·P·Hᵀ + R [innovation covariance] P_g — gate probability (typically 0.95–0.999) χ²_{n,P_g} — chi-squared threshold (n = meas. dim.)
TRACKINGDATA ASSOCIATION
GLOSSARY
RADAR ENGINEERING TERMS — ALPHABETICAL — 57 DEFINITIONS
SEARCH TERMS
A
A-Scope
A radar display that plots received signal amplitude (vertical axis) versus range (horizontal axis). Each target appears as a spike at its corresponding range. The A-scope is the most fundamental radar display and the primary visualization in Modules 02–03.
Ambiguity Function
A 2D function χ(τ, f_d) that describes the matched filter output for a waveform as a function of both delay (range) and Doppler offset (velocity). The ambiguity function fully characterizes the range-Doppler resolution and sidelobe structure of any waveform — an ideal thumbtack (narrow spike at the origin) is the design goal.
|χ(τ,f_d)|² = |∫s(t)·s*(t−τ)·e^(j2πf_d·t)dt|²
Antenna Aperture
The physical collecting area of an antenna, measured in m². Larger aperture means higher gain and narrower beamwidth for a given wavelength. The effective aperture A_eff = η·A accounts for illumination taper and feed losses. The relationship G = 4πA_eff/λ² connects aperture directly to gain.
G = 4π · A_eff / λ²
Atmospheric Attenuation
The absorption and scattering of radar energy by atmospheric gases, primarily water vapor (H₂O) and oxygen (O₂). Expressed in dB/km, it varies strongly with frequency. At X-band (~10 GHz) attenuation is ~0.01 dB/km; at 60 GHz it reaches 15 dB/km due to oxygen absorption. Both one-way and two-way losses must be budgeted in the Radar Range Equation.
B
Bandwidth
The range of frequencies occupied by a radar signal. For a simple pulse, bandwidth ≈ 1/τ. For a chirp, bandwidth B is the frequency sweep extent and directly sets compressed range resolution ΔR = c/(2B). Wider bandwidth gives finer resolution but requires wider receiver filters (more noise). Bandwidth also sets the noise power floor: P_noise = k_B·T·B.
ΔR = c / (2B) P_noise = k_B·T·B
Beamforming
The process of combining signals from multiple antenna elements with appropriate phase shifts (and optionally amplitude weights) to produce a directional beam. Beamforming can be implemented in hardware (analog beamforming), digitally after ADC (digital beamforming), or in hybrid architectures. Digital beamforming enables simultaneous multiple beams from the same array.
Beamwidth
The angular width of the main beam of an antenna pattern, typically measured between the half-power (−3 dB) points. For a uniformly illuminated rectangular aperture: θ_3dB ≈ 0.886λ/D radians. Narrower beamwidth means better angular resolution but requires a larger aperture or higher frequency. The first sidelobe level for a uniform aperture is −13.2 dB below the main lobe peak.
θ_3dB ≈ 0.886 λ/D (rad) ≈ 50.8 λ/D (deg)
Blind Speed
A target radial velocity at which the MTI canceller erroneously cancels the moving target along with stationary clutter. Blind speeds occur when the target's phase change per PRI is a multiple of 2π, making it appear stationary. The first blind speed is v_b = PRF·λ/2. Staggered PRF moves blind speeds to different velocities, effectively eliminating most of them in practice.
v_blind = n · PRF · λ/2, n = 1, 2, 3…
Burn-through Range
The maximum range at which a target's true echo exceeds the jamming noise level, allowing detection despite active noise jamming. Below burn-through range, the radar can detect the target; beyond it, jamming dominates. Lower RCS targets have shorter burn-through ranges. Increasing transmit power, coherent integration, or sidelobe cancellation all extend burn-through range.
C
CA-CFAR Cell-Averaging Constant False Alarm Rate
The most widely used CFAR algorithm. For each range cell under test, it averages the power in N surrounding reference cells and multiplies by a threshold factor α. The threshold rises where noise is high and drops where it is quiet, keeping Pfa constant regardless of noise level. Guard cells adjacent to the CUT prevent the target's own energy from inflating the noise estimate.
T = α · mean(ref cells) α = N·(Pfa^(−1/N) − 1)
CFAR Constant False Alarm Rate
A class of detection algorithms that maintain a constant Pfa regardless of changes in background noise or clutter level, by adaptively setting the detection threshold based on the local environment. The alternative — a fixed threshold — has a Pfa that varies wildly with noise level, producing either excessive false alarms in high-noise regions or missed detections in quiet regions.
Chirp
See LFM Chirp. A waveform whose instantaneous frequency sweeps linearly from f₀ to f₀+B over the pulse duration T. Used in pulse compression to achieve fine range resolution (∝ 1/B) while transmitting a long high-energy pulse (∝ T).
Clutter
Any unwanted radar return that is not the target of interest. Ground clutter (land and sea returns), weather clutter (rain, hail), chaff, and birds all produce clutter. Unlike thermal noise (spectrally flat), clutter has spatial and Doppler structure that radar signal processing can exploit. The signal-to-clutter ratio (SCR) determines detectability in clutter-limited environments.
Coherent Integration
The process of summing multiple pulse returns with phase alignment before detection. Because the target signal adds coherently (amplitudes add) while noise adds incoherently (power adds), coherent integration of N pulses improves SNR by N linear (10 log₁₀ N dB). This is fundamentally different from incoherent integration (√N gain in amplitude) and is the primary tool for extending detection range.
SNR_coh = N · SNR_single
D
Data Association
The problem of determining which radar measurements belong to which tracked targets. In multi-target environments, measurements from different targets can fall within each other's tracking gates, creating ambiguous correspondence. Algorithms range from Nearest Neighbor (simple, brittle) to JPDA (probabilistic, polynomial) to MHT (optimal, exponential worst-case). Correct association is prerequisite to accurate track maintenance.
Detection Probability Pd
The probability that the radar correctly declares a target present when one actually exists. Pd depends on SNR and the detection threshold T: higher SNR means the target distribution is better separated from the noise distribution. For a given SNR, raising the threshold reduces both Pd (more misses) and Pfa (fewer false alarms). The ROC curve traces all Pd–Pfa pairs for a given SNR.
Pd = P(X > T | H₁) Swerling 1: Pd = Pfa^(1/(1+SNR))
Doppler Effect
The apparent change in frequency of a wave caused by relative motion between source and observer. For radar, a target moving radially at velocity v_r shifts the echo frequency by f_d = 2v_r/λ — positive (upshift) if closing, negative (downshift) if opening. The factor of 2 arises from the round-trip: the radar "hears" the shift on both transmission and reception.
f_d = 2·v_r/λ = 2·v_r·f_c/c
DRFM Digital Radio Frequency Memory
An electronic warfare device that captures a radar's transmitted waveform digitally, stores it, and retransmits it with controlled modifications (delay, Doppler shift, amplitude changes). DRFM enables sophisticated deception jamming: false targets at controlled ranges and velocities, range-gate pull-off, velocity gate pull-off, and so on. Modern DRFMs operate with GHz bandwidth and nanosecond timing precision.
Duty Cycle
The fraction of time the radar is transmitting: DC = τ · PRF = τ / PRI. High duty cycle means more average power (better SNR) but less time available to receive echoes (reduced range). CW and FMCW radars have duty cycle = 1 (100%), requiring transmit/receive isolation by physical separation or polarization. Pulsed radars typically have duty cycles of 1–10%.
DC = τ · PRF = Pt_avg / Pt_peak
E
Electronic Warfare EW
All military and security applications of the electromagnetic spectrum for attack, defense, and support. Electronic Attack (EA) includes jamming, deception, and directed energy. Electronic Protection (EP) includes ECCM techniques like frequency agility, sidelobe blanking, LPI waveforms, and spatial nulling. Electronic Support (ES) is passive interception and signals intelligence. Radar and EW systems are in continuous technological competition.
F
False Alarm Pfa
A detection event where the radar declares a target present when no target exists — noise or clutter exceeds the threshold. Expressed as a probability Pfa = P(noise > T). Even Pfa = 10⁻⁶ (one in a million) generates thousands of false alarms per second at typical radar PRFs (10 kHz PRF × 1000 range bins = 10⁷ tests/second). CFAR keeps Pfa constant; a fixed threshold does not.
Pfa = exp(−T²/2σ_n²) [Rayleigh noise]
FMCW Frequency-Modulated Continuous Wave
A radar architecture that transmits a continuous frequency sweep while simultaneously receiving. Mixing transmit and receive signals produces a "beat" frequency directly proportional to target range. FMCW has no minimum range (no T/R switching delay), low peak power (excellent for LPI), compact hardware integration, and simultaneous range-velocity measurement. It is the standard architecture for automotive radar (77 GHz), drone altitude sensors, and industrial level gauges.
f_beat = 2·R·B/(c·T) ΔR = c/(2B)
Frequency Agility
Changing the radar's carrier frequency pseudo-randomly from pulse to pulse across a wide band. Agility provides: (1) ECCM — spot jammers must spread power across the entire band, reducing J/S; (2) RCS decorrelation — independent scintillation samples improve detection of Swerling 1/2 targets; (3) range sidelobe reduction in synthetic aperture processing. The frequency hop band must be wider than the coherent processing bandwidth.
G
Gain (Antenna) G
The ratio of the antenna's peak radiated power density (in its direction of maximum radiation) to that of a lossless isotropic radiator fed with the same total power. Gain is dimensionless but typically expressed in dBi (dB relative to isotropic). Gain appears as G² in the monostatic Radar Range Equation — once for transmit focusing, once for receive collecting area. A 2× aperture area increase gives +3 dBi gain, which improves maximum detection range by ×2^(1/4) ≈ 19%.
G = η·4πA/λ² (aperture antenna) G_dBi = 10·log₁₀(G)
Grating Lobe
Secondary maxima in a phased array beam pattern that appear at the same gain as the main lobe when element spacing d > λ/2. A target at a grating lobe angle is completely indistinguishable from a main-beam target, producing a catastrophic ambiguity. All practical phased arrays use d ≤ λ/2 to push grating lobes to sin(θ) > 1 (outside the visible hemisphere). Widening element spacing to reduce cost or increase bandwidth must be traded against grating lobe appearance.
sin(θ_g) = sin(θ_s) ± n·λ/d, n=1,2,3… No grating lobes when: d ≤ λ/2
H
Hypothesis Testing H₀ / H₁
The statistical framework underlying all radar detection. H₀ (null hypothesis) = no target present; H₁ (alternative) = target present. The Neyman-Pearson lemma proves that the likelihood ratio test is optimal: it maximizes Pd for a given Pfa. In practice the exact likelihood ratio requires knowledge of the target amplitude distribution, which drives the choice of Swerling model and CFAR variant.
I
Improvement Factor IF
For MTI processors, the improvement factor (also called clutter improvement factor or CIF) is the ratio of signal-to-clutter ratio at the output to that at the input, expressed in dB. A single-delay MTI canceller achieves 30–40 dB IF on ideal stationary clutter. IF is degraded by clutter spectral width (wind-induced motion, antenna scanning, platform motion), receiver phase noise, and A/D quantization errors.
IF = SCR_out / SCR_in (linear) IF_dB = SCR_out_dB − SCR_in_dB
J
J/S Ratio Jamming-to-Signal Ratio
The ratio of jamming power to target echo power at the radar receiver. A self-screening jammer (on the target aircraft) sees the signal power fall as 1/R⁴ (two-way propagation) while jamming power falls only as 1/R² (one-way propagation). At long range the jammer wins; at short range (burn-through range) the echo dominates. Stand-off jammers (separate platform) see different geometry.
J/S = (Pj·Gj·R²) / (Pt·G²·σ/(4π)·R²_j) Self-screen: J/S ∝ R² (jammer gains at range)
K
Kalman Filter
A recursive algorithm for optimal state estimation in linear systems with Gaussian noise. Each cycle alternates between prediction (propagating the state estimate and uncertainty forward using the motion model) and measurement update (correcting the prediction using new observations via the Kalman gain). The Kalman gain automatically weights prediction vs measurement based on their respective uncertainties — optimal in the MMSE sense for linear Gaussian systems.
K = P·Hᵀ·(H·P·Hᵀ+R)⁻¹ x←x+K·(z−Hx)
L
LFM Chirp Linear Frequency Modulation
A pulse waveform whose instantaneous frequency sweeps linearly from f₀ to f₀+B over the pulse duration T. The matched filter for an LFM chirp produces a compressed output with width ≈ 1/B — achieving fine resolution independent of pulse length. The time-bandwidth product BT is the pulse compression gain (typically 100–10,000 in modern systems). LFM is the most widely used pulse compression waveform due to its Doppler tolerance and ease of implementation.
ΔR = c/(2B) G_c = B·T
LPI Low Probability of Intercept
A design philosophy and set of waveform/system techniques that minimize the likelihood of an adversary's electronic support (ES) receiver detecting the radar's transmissions. LPI techniques include: FMCW (low peak power), frequency agility (spread signal across wide band), burst mode operation, low sidelobes, and high antenna directivity. LPI radar sacrifices range or revisit rate to reduce detectability.
M
Matched Filter
A linear filter whose impulse response is the time-reversed complex conjugate of the transmitted waveform. The matched filter maximizes the output SNR for any given waveform and is provably optimal under the Neyman-Pearson criterion. Its output is the cross-correlation of the received signal with the transmitted template; peaks in the output correspond to target ranges. The width of the peak equals the inverse signal bandwidth — which is why range resolution equals c/(2B).
h(t) = s*(T−t) SNR_out = 2E/N₀
MIMO Radar Multiple-Input, Multiple-Output
A radar architecture that transmits orthogonal waveforms from multiple antennas simultaneously. Each receive element separates the orthogonal transmit waveforms using matched filters, creating N_t×N_r virtual receive channels. The resulting virtual aperture has N_t times more elements than a conventional phased array of the same physical size, providing superior angular resolution and degrees of freedom for STAP and parameter estimation.
Virtual DoF = N_t × N_r
MTI Moving Target Indication
A signal processing technique that cancels stationary clutter by subtracting consecutive pulse returns. The clutter, nearly identical between pulses, cancels; moving targets change phase each PRI and survive. Single-delay MTI uses H(z)=1−z⁻¹ (notch at DC); double-delay MTI deepens the notch. MTI is the simplest form of Doppler processing and is the predecessor to modern pulse-Doppler and STAP processors.
y[n] = x[n] − x[n−1] → H(e^jω) = 1 − e^{−jω}
N
Noise Figure F or NF
A measure of the excess noise added by a receiver component or chain above the thermal noise floor. Defined as F = SNR_in / SNR_out (linear). A perfect noiseless receiver has F = 1 (0 dB). Real LNAs achieve 0.5–3 dB; system noise figures of 3–10 dB are common. The Friis cascade formula shows that the first element (LNA) contributes most: cooling or improving it gives the largest system benefit.
F_total = F₁ + (F₂−1)/G₁ + (F₃−1)/(G₁G₂) + …
O
Off-Boresight Angle
The angular separation between the radar's boresight (main beam axis) and the direction to a target of interest. Antenna gain falls off as the off-boresight angle increases, following the one-way pattern G(θ). At the 3 dB beamwidth (θ₃dB/2 off boresight), gain drops by 3 dB one-way (6 dB two-way), reducing SNR significantly. Targets at large off-boresight angles may be seen only through sidelobes.
G(θ) ≈ G₀ · sinc²(πDsinθ/λ) (uniform aperture)
Operating Frequency f₀, λ
The carrier frequency at which a radar transmits. Operating frequency determines wavelength (λ = c/f₀), which in turn affects range resolution, Doppler sensitivity, antenna size, and atmospheric propagation losses. Common radar bands: L-band (1–2 GHz, long-range surveillance), S-band (2–4 GHz, weather/ATC), C-band (4–8 GHz, weather), X-band (8–12 GHz, fire control/imaging), Ka-band (27–40 GHz, automotive).
λ = c / f₀ (c ≈ 3×10⁸ m/s)
P
Phased Array
An antenna array in which the phase (and optionally amplitude) of the signal applied to each element is individually controlled to steer and shape the beam electronically, without mechanical movement. Phase steering can redirect the beam in microseconds — versus tens of milliseconds for mechanically scanned antennas. Phased arrays enable simultaneous multiple beams, adaptive nulling, and rapid interleaving of different radar modes.
PPI Scope Plan Position Indicator
A radar display that presents a top-down (azimuth vs range) 2D map of the surveillance area. The radar antenna rotates (or the beam scans electronically), and each range-azimuth cell is mapped to a pixel. Targets appear as bright spots at their true geographic positions. PPI is the standard display for air traffic control, weather radar, and naval surveillance systems.
PRF Pulse Repetition Frequency
The number of pulses transmitted per second (Hz). PRF simultaneously sets maximum unambiguous range (R_u = c/2PRF) and maximum unambiguous velocity (v_u = PRF·λ/4) in opposite directions — increasing PRF extends velocity coverage but reduces range coverage. The PRF ambiguity product R_u·v_u = c²/(8f_c) is fixed by physics. Three PRF regimes: low PRF (range unambiguous), medium PRF (both ambiguous), high PRF (velocity unambiguous).
R_u = c/(2·PRF) v_u = PRF·λ/4
Pulse Compression
A waveform processing technique that transmits a long coded pulse (for energy) but achieves the range resolution of a short pulse (for resolution). The most common form uses an LFM chirp; phase-coded waveforms (Barker, Frank codes) are also used. The time-bandwidth product B·T is the compression gain: the ratio of compressed to uncompressed pulse width. Pulse compression decouples the energy–resolution trade-off that limits simple pulsed radars.
G_c = B·T ΔR = c/(2B) (compressed)
Q
Quadrature Sampling I/Q
A signal representation using two components sampled in phase quadrature (90° apart): In-phase (I) and Quadrature (Q). Together they form a complex-valued signal s(t) = I(t) + jQ(t) that preserves both amplitude and phase of the radar return. I/Q sampling enables coherent processing, Doppler estimation, and unambiguous phase measurement. Modern radars digitize I and Q directly at IF or use digital downconversion from RF samples.
s(t) = I(t) + jQ(t) A = √(I²+Q²) φ = atan2(Q,I)
R
Radar Cross Section RCS, σ
The effective scattering area of a target, defined as the area of an equivalent perfectly reflecting isotropic sphere that would return the same power to the radar. RCS depends on target shape, size, material, orientation (aspect angle), and radar frequency. Typical values: aircraft 1–10 m² (0–10 dBsm), stealth aircraft ~0.001 m² (−30 dBsm), ship 1,000–100,000 m² (30–50 dBsm). A 10 dB reduction in RCS cuts maximum detection range by ~44%.
σ_dBsm = 10·log₁₀(σ) sphere: σ = πr²
Radar Range Equation RRE
The fundamental equation relating radar detection range to system parameters. It expresses the received SNR as a function of transmit power, antenna gain, wavelength, RCS, range, and noise. The R⁴ dependence (two-way propagation squared) is the central challenge of long-range radar — doubling range requires 16× more power or 4× more antenna area. All Phase I–II modules build toward this single equation.
SNR = Pt·G²·λ²·σ / ((4π)³·R⁴·kTBFL)
ROC Curve Receiver Operating Characteristic
A plot of detection probability Pd versus false alarm probability Pfa for all possible threshold values at a given SNR. Every point on the curve represents a different threshold setting. Higher SNR shifts the ROC curve up and to the left (better performance). The ideal ROC curve passes through (Pfa=0, Pd=1) — a perfect detector. The area under the ROC curve (AUC) is a single-number performance metric.
S
SAR Synthetic Aperture Radar
A radar imaging technique that exploits the motion of the platform (aircraft, satellite) to synthesize a virtual aperture far larger than the physical antenna. By coherently processing echoes collected over a long along-track distance L_s, SAR achieves cross-range resolution δ_cr ≈ D/2 — independent of range and wavelength. SAR enables centimeter-resolution imaging from low Earth orbit and is used for terrain mapping, change detection, and reconnaissance.
δ_cr ≈ D/2 (focused SAR, D = physical aperture)
Sidelobes
Secondary maxima of the antenna beam pattern or matched filter output that appear outside the main lobe. Antenna sidelobes allow targets or jammers at off-boresight angles to produce returns that appear to come from the main beam direction. Matched filter range sidelobes create false peaks around a real target. Both are controlled by window functions (tapering) at a cost in main lobe width. The first sidelobe of a uniform aperture/rectangle window is −13.2 dB; Hamming gives −42.7 dB.
SNR Signal-to-Noise Ratio
The ratio of target signal power to noise power at the detector input, expressed linearly or in dB. SNR is the primary determinant of detection performance: higher SNR means better separation of the target and noise distributions, enabling either higher Pd at a given Pfa, or the same Pd at lower Pfa. The SNR required for a given (Pd, Pfa) pair is the detection threshold and is read from ROC curves or Albersheim's equation.
SNR_dB = 10·log₁₀(P_signal / P_noise)
Staggered PRF
A technique using two or more alternating pulse repetition frequencies with incommensurate ratios to extend the unambiguous range and velocity coverage beyond what any single PRF can achieve. Based on the Chinese Remainder Theorem: the extended unambiguous intervals are the least common multiples of the individual intervals. Staggered PRF also moves MTI blind speeds — a target visible at one PRF is likely visible at the other.
STAP Space-Time Adaptive Processing
An adaptive signal processing technique that simultaneously exploits spatial (array element) and temporal (pulse) degrees of freedom to cancel interference. For an airborne radar, STAP places a 2D null in angle-Doppler space along the clutter ridge (where ground clutter appears at all Doppler shifts proportional to platform velocity × angle cosine). STAP generalizes both beamforming (spatial only) and MTI (temporal only) into a unified framework.
w_opt = R⁻¹·s / (sᴴ·R⁻¹·s)
Swerling Models
A set of four statistical models (Sw0–Sw4) describing how target RCS fluctuates over time. Sw0: non-fluctuating (steady RCS). Sw1: exponential RCS distribution, scan-to-scan decorrelation (multiple glints — aircraft body). Sw2: exponential, pulse-to-pulse decorrelation. Sw3: chi-squared 4 DoF, scan-to-scan (dominant scatterer). Sw4: chi-squared 4 DoF, pulse-to-pulse. Fluctuating targets (Sw1–4) require more SNR than non-fluctuating for Pd > 0.5 — the diversity loss.
Sw1: Pd = Pfa^(1/(1+SNR_avg))
System Losses L
All signal power losses that reduce the received SNR below what the Radar Range Equation would predict for an ideal system. Includes: feed and transmission line losses, antenna pointing loss, signal processing losses (range and Doppler straddle losses), A/D quantization loss, matched filter mismatch, propagation losses (rain, atmospheric), and system integration losses. Typically 3–10 dB total. Must be measured and budgeted for each specific system design.
T
Thermal Noise
Random electrical noise generated by the thermal agitation of electrons in any resistive component above absolute zero. It is irreducible — the fundamental detection floor for every radar. The available noise power from a resistor at temperature T over bandwidth B is P = k_B·T·B. At 290 K and 1 MHz bandwidth, this is −114 dBm. Reducing system temperature (cooling the LNA), narrowing bandwidth, or narrowing the noise figure are the only ways to lower the floor.
P_noise = k_B · T · B k_B = 1.38×10⁻²³ J/K
Tracking Gate Validation Gate
An ellipsoidal region in measurement space centered on a track's predicted position, used to determine which measurements could plausibly have originated from that track. Measurements falling inside the gate are candidates for association; those outside are rejected. The gate size is set by the innovation covariance S and a chi-squared threshold corresponding to the desired gate probability P_g. Small gates reduce false association but risk missing the true measurement when prediction is imprecise.
d² = yᵀ·S⁻¹·y ≤ χ²_{n,Pg}
U
Unambiguous Range R_u
The maximum target range at which the radar can correctly identify the echo as belonging to the most recent pulse. Beyond R_u, the echo arrives after the next pulse has been transmitted and is incorrectly attributed to that later pulse, causing the radar to report a "folded" shorter range. R_u is set by the PRF: R_u = c/(2·PRF). Increasing PRF extends Doppler coverage but shrinks R_u.
R_u = c / (2·PRF)
Unambiguous Velocity v_u
The maximum radial velocity the radar can unambiguously measure. Beyond v_u, the target's Doppler phase advance per PRI exceeds π radians (the Nyquist limit), causing the velocity to alias to an incorrect lower value or wrong direction. v_u is set by the PRF and wavelength: v_u = PRF·λ/4. Decreasing PRF (to extend range) shrinks v_u.
v_u = PRF · λ / 4
W
Window Function Taper, Weighting
An amplitude weighting applied to an array aperture (spatial window) or pulse bandwidth (spectral window) to reduce sidelobes at the cost of slightly widened main lobe. Common windows: Rectangular (no taper — highest sidelobes −13.2 dB, narrowest main lobe), Hamming (−42.7 dB sidelobes, 1.46× wider main lobe), Taylor (adjustable −25 to −50 dB sidelobes), Chebyshev (equiripple — minimum main lobe width for given sidelobe level). Used in both antenna pattern shaping and range/Doppler processing.
ABOUT RADAR LABORATORY
A VISUAL COURSE FOR BUILDING PRACTICAL RADAR INTUITION
PURPOSE
Radar Laboratory is an interactive learning environment for engineers who need to understand radar behavior without treating the sensor as a black box.
Each module pairs a visual scene with a plot, live readouts, and a short explanation. The goal is to make the physical behavior visible first, then connect that behavior to the equations.
The material is written for new engineers, systems engineers, analysts, and software developers who work around radar systems but may not have formal RF or electrical-engineering training.
LEARNING PATH
The course starts with the measurements radar makes directly, then builds toward performance limits, detection, digital processing, and advanced phenomenology.
ACCURACY AND SCOPE
The simulations use standard radar relationships and simplified engineering models. They are intended for training, communication, and first-order intuition.
Radar Laboratory is not a replacement for a validated radar performance model, mission-analysis tool, hardware test, or scenario-specific simulation. When a module uses a simplification, the module text states the main assumption and keeps the core physical relationship intact.
HOW TO USE IT
REFERENCE BASIS
The content follows standard radar, antenna, detection, propagation, and DSP references, including Skolnik, Richards, Levanon & Mozeson, Balanis, Van Trees, Nathanson, Ward, Kay, Oppenheim & Schafer, Stimson, and ITU-R recommendations P.676, P.838, and P.530.
Equations are used to support intuition. Engineering decisions should still be checked against authoritative references and validated models.
TECHNOLOGY
Radar Laboratory is a single-file HTML application using HTML5 Canvas, CSS, and vanilla JavaScript. It can be hosted as a static site or opened locally for demonstration and training.
HTML5 CANVAS VANILLA JS SINGLE FILE STATIC SITE READY