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Author: Dr. Nir Regev

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1. Introduction and Motivation

Why Polynomial Phase Estimation?

In radar systems, moving targets create signals with polynomial phase characteristics due to their motion dynamics. When a target moves with:

• Constant velocity → Linear phase term
• Constant acceleration → Quadratic phase term
• Constant Jerk (changing acceleration) → Cubic phase term

The received radar signal has the general form:

$$ y(t) = b_0 \exp \{ j\sum_{m=0}^M a_m t^m\}, $$

where:

• $b_0$ = complex amplitude
• $a_m$ = polynomial coefficients related to motion parameters
• $M$ = polynomial order (degree)

The Challenge

Traditional Fourier analysis fails for polynomial phase signals because:

1. Time-varying instantaneous frequency: $f_{inst}(t) = \frac{1}{2\pi}\frac{d\phi}{dt} = \frac{1}{2\pi}\sum_{m=1}^M m \cdot a_m t^{m-1}$
2. Spectral spreading: Energy spreads across frequencies instead of concentrating at a single peak
3. Loss of resolution: Cannot accurately estimate motion parameters