Monopulse Radar Tracking

Σ = A + B   (Sum beam — used for range and Doppler, broad peak at boresight)
Δ = A − B   (Difference beam — deep null at boresight, sensitive to off-axis error)

The monopulse ratio (angle error signal):
ε = Re(Δ / Σ)   — normalised error voltage, proportional to angular error θ_err

Key property: ε is independent of target RCS (σ), range (R), and instantaneous amplitude fluctuations.
The real part is used to preserve the sign convention (positive = right of boresight, negative = left).

Input signals: A and B (RF signals from antenna halves)

Sum port: Σ = (A + B) / √2
Difference port: Δ = (A − B) / √2

The √2 factor is the signal splitting loss — 3 dB per port. This is fundamental; there is no way to avoid it.

Note: The comparator must be extremely well matched. Any amplitude or phase imbalance between the two signal paths introduces a false boresight offset (boresight error). Path length difference of λ/360 (1°) causes measurable boresight shift.

A(θ) = G₀ · exp[−2.77·(θ + θ_sq)² / θ₃dB²]
B(θ) = G₀ · exp[−2.77·(θ − θ_sq)² / θ₃dB²]

For small angular errors (θ ≪ θ₃dB), Taylor-expanding to first order:

Σ(θ) ≈ 2G₀·exp[−2.77·θ_sq²/θ₃dB²]   (≈ constant near boresight)
Δ(θ) ≈ 2G₀·exp[−2.77·θ_sq²/θ₃dB²]·(11.08·θ·θ_sq / θ₃dB²)

Therefore the normalised error voltage:
ε = Δ/Σ = k_m · θ   (linear in angle for small errors!)

where the monopulse slope (discriminant) is:
k_m = 11.08 · θ_sq / θ₃dB²   (rad⁻¹, or deg⁻¹ if θ in degrees)

Maximum slope occurs at θ_sq,opt = θ₃dB / √(2 × 11.08) ≈ θ₃dB / 4.71 ≈ 0.212 × θ₃dB

Δφ = 2π · d · sin(θ) / λ   (exact expression)
Δφ ≈ 2π · d · θ / λ   (small-angle approximation, θ in radians)

Solving for angle:
θ = arcsin(λ · Δφ / (2π · d))

Sensitivity (phase difference per unit angle):
dΔφ/dθ = 2π·d·cos(θ)/λ   ≈   2π·d/λ   (near boresight)

Larger baseline d = finer angular sensitivity, but phase ambiguity when d > λ/2:
If d > λ/2: |Δφ| can exceed π → angle ambiguity (multiple solutions)
No ambiguity: d ≤ λ/2    (unambiguous range ±90°)
Best accuracy: d ≈ 2–5λ with phase ambiguity resolution

For a Gaussian-shaped beam with 3dB beamwidth θ₃dB:

Optimal squint: θ_sq,opt = θ₃dB / √(2 ln 2 × 2) ≈ 0.212 × θ₃dB
At this squint angle: crossover loss = −3 dB relative to peak gain
Maximum slope: k_m,max = √(2e) / θ₃dB ≈ 2.33 / θ₃dB

Practical ranges:
θ_sq = 0.15–0.20 × θ₃dB → higher sum gain, less slope (range-limited applications)
θ_sq = 0.21 × θ₃dB → maximum discriminant slope (tracking-optimised)
θ_sq = 0.25–0.30 × θ₃dB → wider linear range, lower gain (wide-angle acquisition)

Do NOT exceed θ_sq > 0.35 × θ₃dB — the gain crossover drops below −6 dB and the sum beam develops a split (double peak), destroying the single-target tracking assumption.

If the null depth is D (as a power ratio, e.g. D = 10⁻⁴ for −40 dB), the residual boresight error voltage is:

ε_offset = √D   (as a voltage ratio)

Equivalent angular boresight offset: θ_offset = √D / k_m

Example: Null depth = −40 dB (D = 10⁻⁴), k_m = 2 deg⁻¹, θ₃dB = 1°
Boresight error = 10⁻² / 2 = 0.005° = 0.09 mrad

Rule of thumb: Null depth should be at least 20 dB deeper than the required boresight accuracy (in dB).

k_m = d(Δ/Σ)/dθ |_{θ=0} = 11.08 · θ_sq / θ₃dB²   (Gaussian beams, θ in radians)

Equivalently, in terms of normalised squint angle d_n = θ_sq/θ₃dB:
k_m = 11.08 · d_n / θ₃dB   (= 2 ln 4 × d_n / θ₃dB)

At optimal squint (d_n = 0.212):
k_m,max = 11.08 × 0.212 / θ₃dB = 2.35 / θ₃dB

Typical measured values for well-designed reflector/phased array monopulse:
k_m ≈ 1.3 to 1.8 per beamwidth (=1.3/θ₃dB to 1.8/θ₃dB)

σ_θ = θ₃dB / (k_m · √(2 · SNR_Σ))

where SNR_Σ is the signal-to-noise ratio in the sum channel.

Expanded form:
σ_θ = θ₃dB / (k_m · √(2 · SNR_Σ))
     ≈ 0.5 × θ₃dB / √SNR_Σ   (using k_m ≈ 1/θ₃dB as a rough rule)

Example: θ₃dB = 2°, k_m = 1.2 deg⁻¹, SNR_Σ = 20 dB (100 linear)
σ_θ = 2° / (1.2 × √200) = 2° / 16.97 = 0.12° = 2.1 mrad

Key takeaways:
— Accuracy improves as 1/√SNR (10 dB more SNR → 3× better accuracy)
— Narrower beam → better accuracy (proportional to θ₃dB)
— Steeper discriminant k_m helps (but is bounded by the optimum squint angle)
— Pulse-to-pulse integration of N pulses improves SNR by N, accuracy by √N

Noise bandwidth B_L — the loop filter bandwidth. Controls the trade-off between:
  • Narrower B_L → less noise on track estimate, but slower response to target manoeuvres
  • Wider B_L → faster response, but more noise, more glint error

Tracking RMS error (noise-dominated):
σ_θ = θ₃dB · √(B_L / (k_m² · PRF · SNR_pulse))

Typical tracking bandwidths:
Aircraft tracking: B_L = 1–10 Hz (aircraft manoeuvre rate ≈ 1–3 Hz)
Missile guidance: B_L = 10–100 Hz (fast-manoeuvring target)
Satellite tracking: B_L = 0.01–1 Hz (predictable slow motion)

Loop order: 2nd-order loop tracks constant velocity with no lag. 3rd-order tracks constant acceleration. Most fire-control radars use a 2nd or 3rd order loop.

If path A and path B have amplitude ratio r = 10^(ΔA_dB/20) and phase imbalance Δφ rad:

|Δ|² / |Σ|² = (1 + r² − 2r·cos(Δφ)) / (1 + r² + 2r·cos(Δφ))

For small imbalance (r ≈ 1, Δφ ≈ 0):
Null depth ≈ [(ΔA/2)² + (Δφ/2)²]   (linear amplitude imbalance + phase)

To achieve −40 dB null depth: need ΔA < 0.2 dB AND Δφ < 1.1°
To achieve −50 dB null depth: need ΔA < 0.06 dB AND Δφ < 0.36°