Phase Retrieval for Radar Waveform Design

TL;DR

This work treats radar waveform design as a phase retrieval problem by recovering a transmit signal from the magnitude of its ambiguity function, $A[p,k]$, and provides theoretical uniqueness guarantees for time- and band-limited signals: a $B$-band-limited signal is recoverable from at least $3B-1$ AF measurements (or $2B-1$ with known power spectrum), and an $S$-time-limited signal from at least $3S-1$. It introduces a non-convex, smoothing-based optimization framework with a trust-region gradient method, complemented by a spectral-like initialization to robustly converge to a correct solution, even with incomplete or noisy AF data. The algorithm scales via block stochastic gradients and leverages Wirtinger derivatives to handle complex-valued signals. Numerical experiments demonstrate accurate recovery of both time- and band-limited signals under complete, incomplete, and noisy AF, including LFM/NLFM-modulated waveforms, highlighting the method’s practicality for AF-constrained radar design.

Abstract

The ability of a radar to discriminate in both range and Doppler velocity is completely characterized by the ambiguity function (AF) of its transmit waveform. Mathematically, it is obtained by correlating the waveform with its Doppler-shifted and delayed replicas. We consider the inverse problem of designing a radar transmit waveform that satisfies the specified AF magnitude. This process may be viewed as a signal reconstruction with some variation of phase retrieval methods. We provide a trust-region algorithm that minimizes a smoothed non-convex least-squares objective function to iteratively recover the underlying signal-of-interest for either time- or band-limited support. The method first approximates the signal using an iterative spectral algorithm and then refines the attained initialization based on a sequence of gradient iterations. Our theoretical analysis shows that unique signal reconstruction is possible using signal samples no more than thrice the number of signal frequencies or time samples. Numerical experiments demonstrate that our method recovers both time- and band-limited signals from sparsely and randomly sampled, noisy, and noiseless AFs.

Figures (13)

• Figure 1: Illustration of multiple signals with trivial ambiguities mapping to the same AF. On the left, two different signals on top and bottom are correlated with their Doppler- and delay-shifted replicas followed by the magnitude-squared operation.
• Figure 2: (a) The cost function $h(\textbf{z},\mu)$ in the optimization problem \ref{['eq:auxproblem']}, when evaluated at $\mu = 0$ and $\mathbf{A}$ computed for $\textbf{x}=[1;0]$. (b) Side-view of the cost function $h(\textbf{z},0)$ from $\mathbf{z}[1]$ dimension. The only local minima are also global minima, located at $\textbf{z}=\pm \textbf{x}$ and $\textbf{z}=\pm \hat{\textbf{x}}$ with $\hat{\textbf{x}}[n] = \textbf{x}[n-1]$, corresponding to the ambiguities T1-T4. There are four saddle points near to $\pm [0.5;0.5]$, and $\pm [-0.5;0.5]$. (c) The top view of the same function $h(\textbf{z},0)$ plotted with respect to the values along each dimension of the 2-D variable $\textbf{z}$.
• Figure 3: Reconstructed time- and band-limited signals with their AFs in the absence of noise. In both cases, the attained relative error as defined by \ref{['eq:distance']} is $1\times 10^{-6}$. For time-limited [band-limited] signal, (a) [(c)] and (b) [(d)] show the original and recovered AFs, respectively; (e) [(g)] and (f) [(h)] are 1-D slices of the AFs at zero delay and Doppler, respectively; (i) [(k)] and (j) [(l)] are the, respectively, magnitude and phase of recovered (red) signal juxtaposed with the original (blue).
• Figure 4: As in Fig. \ref{['fig:complete_sinnoiseresults']} but in the presence of noise with SNR = $20$ dB. The attained relative error is $5\times 10^{-2}$ for both time- and band-limited signals.
• Figure 5: As in Fig. \ref{['fig:complete_sinnoiseresults']} but in the presence of noise with SNR = $20$ dB and with 50% of the delays uniformly removed from the AF. The attained relative error is $5\times 10^{-2}$ for both time- and band-limited signals. Additionally, For time-limited [band-limited] signal, (a) [(d)], (b) [(e)], and (c) [(f)] show the original, undersampled, and recovered AFs, respectively; (g) [(i)] and (h) [(j)] are 1-D slices of the AFs at zero delay and Doppler, respectively; (k) [(m)] and (l) [(n)] are the, respectively, magnitude and phase of recovered (red) signal juxtaposed with the original (blue).

Theorems & Definitions (23)

• Definition 1: $B$-band-limitedness
• Lemma 1
• Lemma 2
• Proposition 1
• proof
• Definition 2
• Definition 3: $S$-time-limitedness
• Corollary 1
• proof
• Theorem 1