Deterministic DTFT Interpolation for Joint Frequency and Chirp-Rate Estimation: Cell-Uniform Efficiency and Threshold Analysis
TL;DR
A deterministic two-stage estimator that controls both failure modes uniformly over the residual cell and derives a mean-squared-error and threshold characterization over the full SNR range, in closed form except for one calibrated scalar (an effective cell count).
Abstract
Joint frequency and chirp-rate estimation for a noisy chirp signal arises in radar, sonar, and burst satellite communications. Conventional estimators combine a coarse grid search with fine interpolation; accuracy degrades at the edges of the residual cell (the edge effect) and below the breakdown SNR (the threshold effect). We present a deterministic two-stage estimator that controls both failure modes uniformly over the residual cell. The estimator combines a time-centered, zero-padded dechirp-FFT acquisition bank with alternating selectable-$p$ amplitude-interpolation refinements on DTFT samples at fractional bins; in the centered frame, the frequency-chirp-rate cross-term of the Fisher information vanishes. The paper derives a mean-squared-error and threshold characterization over the full SNR range, in closed form except for one calibrated scalar (an effective cell count), to our knowledge the first for the joint problem: the breakdown threshold is governed by the cell count, and its cell-position dependence is dominated by the scalloping loss of the coarse FFT, which the padding bounds at 0.4 dB. An asymptotic uniformity analysis over the cell, including its corners, gives fixed-point variance ratios of $1.003$ and $0.998$, analytically free of the residual. A closed-form bias analysis under a cubic phase mismatch shows the centered chirp-rate estimate is insensitive to first order. Monte Carlo experiments at $N=256$ (validated at $N=32$-$512$) measure frequency- and chirp-rate-axis efficiencies with median $1.03$ and worst case $1.07$ over $144$ cell positions at $-5$ dB. Threshold predictions hold within $1.0$ dB on four configurations not used in the calibration. The dechirp-FFT bank is fully parallel, and each of the four refinement iterations evaluates three DTFT samples per axis; under fixed operating conditions, per-estimate latency is constant at $O(N\log N)$ cost.