# METHODS — E4-H160

## 1. Geometry

Uniform linear array, pitch `d`, N elements, first grating order. Near a design angle θ₀ with
`λ₀ = d sin θ₀`, the steering relation is linearised as

    sin θ(λ) = sin θ₀ + (λ − λ₀)/d ,      dθ/dλ = 1/(d cos θ₀) .

The main-lobe half-width (Rayleigh) is `Δθ = λ₀/(N d cos θ₀)`.

**Element count versus aperture.** `D = N d`. The two are not independent at fixed pitch; every
statement below is made at fixed pitch, and the N-dependence is reported explicitly.

**Angular cosine factors.** `dθ/dλ = 1/(d cos θ₀)` — the cos θ₀ factor is retained in the derivation.
In `B_eff` it cancels against the half-width, which is why the normal-incidence form `c/(Nλ₀)` is
recovered at θ₀ = 20°. This cancellation was checked numerically, not assumed.

**Grating order.** First order throughout. Higher orders change `dθ/dλ` multiplicatively and do not
change the structure of the result.

## 2. Illuminated bandwidth

A point target at angle θ₀ is illuminated while `|θ(f) − θ₀| < Δθ`. Converting through
`dθ/dλ` and `dλ/df = λ²/c`:

    B_eff = Δθ · c/(λ₀² |dθ/dλ|) = c/(N λ₀) .

## 3. Signal model

The received spectrum for a single target at delay `τ = 2R/c` is

    y(f) = A(f) e^{−2πi f τ} + n(f) ,

with `A(f)` the illumination envelope (Gaussian main lobe for a point target; flat for an angularly
extended target), `n` complex white Gaussian noise, and `σ² = E|n|²`.

## 4. Information bound

For a known envelope and unknown delay, the Fisher information is

    J(τ) = (2π)² Σ_f f² |A(f)|² / σ² ,

so the Cramér–Rao bound is `στ = 1/√J`. This is the quantity tabulated in T2. It depends on the
**RMS bandwidth of the illumination-weighted spectrum**, not on the beam width per se — the two
coincide only for a point target with a Gaussian envelope.

## 5. Estimators

- **Matched**: peak of `|Σ_f y(f) A(f) e^{+2πi f τ}|` over a zero-padded inverse FFT of length 2²⁰,
  refined by three-point parabolic interpolation.
- **Unweighted**: identical, with `A` replaced by 1 — i.e. no envelope knowledge. This isolates how
  much of the performance is due to processing rather than to information.

## 6. Numerical checks performed

- Target range placed inside the unambiguous range `c/(2Δf)` at every bandwidth tested. The first run
  used 1 km and aliased; that run's output was discarded and is not reported.
- Delay-axis indexing verified against `fftshift` convention after an initial off-by-half-array error.
- Sample count held fixed across bandwidth sweeps, so `Δf` changes with B; this is stated because it
  changes the unambiguous range and the noise-per-sample.
- Bandwidth sweep stops at 100 GHz for the Monte-Carlo because of delay-grid wrap; the CRB table (T2)
  extends to 2000 GHz because it has no delay grid.

## 7. What was deliberately not done

- No deconvolution experiment. The CRB already bounds *any* estimator, so a deconvolution study could
  not beat it; the question "can processing help?" is answered by the unweighted comparison instead.
- No phase-steered comparison (claim C7) — it requires a second architecture and does not affect the
  bound argument.
- No empirical data.