MTI Clutter Suppression¶

The clutter model computes how much clutter a geometry produces; it does not say how much of it a radar can remove. That leaves the detection chain broken in the middle. A ground-based L-band radar looking at wooded hills sees roughly 33 dB more clutter than target, which the detection model scores as hopeless — yet such radars work, because they filter clutter in Doppler. This page covers the step that closes the gap:

\[ \text{clutter RCS} \;\rightarrow\; \boxed{\text{MTI improvement factor}} \;\rightarrow\; \text{post-MTI SCNR} \;\rightarrow\; \text{detection} \]

and, through SNR, on into track accuracy.

Clutter spectrum¶

Clutter is modeled as a zero-mean Gaussian Doppler spectrum. The spread is a property of the scatterers — wind-blown foliage, sea-surface motion — and Skolnik notes it is frequency-independent when expressed as a velocity, so the velocity form is the input:

\[ \sigma_c = \frac{2\sigma_v}{\lambda}, \qquad \sigma_\omega = \frac{2\pi\sigma_c}{\mathrm{PRF}} \]

The same hillside therefore produces a wider Doppler spectrum, and is harder to cancel, at higher frequency. Only \(\sigma_\omega\) — the spread relative to the PRF — enters the canceller math.

The normalized autocorrelation of that spectrum (Richards FRSP Eq. 5.53, valid for \(\sigma_\omega \ll \pi\)) is

\[ \rho_c[k] = e^{-(\sigma_\omega k)^2/2} \]

Binomial cancellers¶

The canceller is the \((N-1)\)th-difference FIR filter with weights \(w_k = (-1)^k \binom{N-1}{k}\): \([1,-1]\) for two pulses, \([1,-2,1]\) for three. The weights sum to zero, so stationary clutter is rejected exactly.

This package computes the improvement factor from the general quadratic form

\[ I = \frac{\sum_k w_k^2}{\sum_i \sum_j w_i w_j \, \rho_c[i-j]} \]

rather than from tabulated closed forms. It reduces to them exactly — FRSP Eq. (5.52) \(I = 1/(1-\rho[1])\) for \(N=2\), Eq. (5.54) \(I = 1/(1 - \frac{4}{3}\rho[1] + \frac{1}{3}\rho[2])\) for \(N=3\) — and extends to \(N > 3\) where no closed form is tabulated. It is also better conditioned: the three-pulse closed form differences three terms all near unity, and loses precision as the clutter spectrum narrows.

Improvement factor, not clutter attenuation¶

Three quantities are reported and they are not interchangeable:

Quantity	Meaning
\(G = \sum_k w_k^2\)	average signal gain over Doppler (FRSP p. 246)
\(CA\)	clutter power ratio across the filter (FRSP Eq. 5.43)
\(I = G \cdot CA\)	improvement in signal-to-clutter ratio (Levanon 1988)

\(I\) is what the detection budget consumes, because it accounts for both the filter's gain on the target and its rejection of clutter. Quoting \(CA\) alone understates the benefit by \(G\) — 3.0 dB for a two-pulse canceller, 7.8 dB for a three-pulse.

A target whose Doppler shift is a multiple of the PRF advances in phase by a full cycle between pulses, is indistinguishable from stationary clutter, and is cancelled with it:

\[ v_{blind} = \frac{n \cdot \mathrm{PRF} \cdot \lambda}{2} \]

A single-PRF MTI is only usable where the anticipated target Doppler band sits clear of these nulls.

Worked case: ARSR-3¶

An L-band air traffic control radar — 1.3 GHz, PRF 400 Hz, 2 µs pulse, 1.25° azimuth beam, wooded-hill clutter at \(\sigma^0 = -20\) dB, clutter velocity spread 1.16 km/hr, 2 m² target at 30 nmi, 15 dB required S/C.

Step	Value
Unambiguous range \(c/2\mathrm{PRF}\)	375 km
First blind speed	46.1 m/s (400 Hz, clear of the 50–350 Hz target band)
Clutter cell area \(R\theta_{az}(c\tau/2)\)	3.64 × 10⁵ m²
Clutter RCS	3637 m² = 35.6 dBsm
Required attenuation	15 − (3 − 35.6) = 47.6 dB
\(\sigma_c\), \(\sigma_\omega\)	2.79 Hz, 0.0438 rad
Two-pulse: \(I\), \(G\), \(CA\)	30.2 dB, 3.0 dB, 27.2 dB — insufficient
Three-pulse: \(I\), \(G\), \(CA\)	57.3 dB, 7.8 dB, 49.5 dB — sufficient

The three-pulse canceller is the shortest binomial canceller that meets the requirement. Every figure in this table is asserted in tests/test_radar_mti_oracles.py, which also gives the clutter model its first end-to-end worked case.

Running the same geometry through evaluate_case shows what the missing step was worth: with no MTI the target is undetectable (\(P_d = 0\)), with a two-pulse canceller it is detected.

References¶

Richards, Fundamentals of Radar Signal Processing: Eq. (5.43), p. 246, Eq. (5.53), Eqs. (5.52)/(5.54) p. 247.
Skolnik, Introduction to Radar Systems, ch. 15.
Levanon, Radar Principles, Wiley, 1988.