WaveMax: Radar Waveform Design via Convex Maximization of FrFT Phase Retrieval

TL;DR

WaveMax addresses the inverse problem of recovering a radar waveform from a FrFT-based ambiguity function magnitude by formulating a convex, low-rank matrix recovery problem. The key idea exploits the AF rotation property under FrFT to construct a convex program that recovers $\boldsymbol{x}$ (up to trivial ambiguities) from AF measurements via a lifted variable $\boldsymbol{Z}=\boldsymbol{x}\boldsymbol{x}^{H}$ and a data-driven initialization $\boldsymbol{x}^{(0)}$. The authors prove uniqueness guarantees for band-limited signals when the AF is sufficiently sampled, provide a data-efficient spectral initialization, and demonstrate global convergence and practical performance in noiseless and noisy, complete and sparse AF scenarios. Numerically, WaveMax recovers band- and time-limited signals with missing AF entries (e.g., 75% removal) and outperforms non-convex baselines in initialization robustness, offering a tractable, provable alternative for AF-based radar PR. This approach advances radar waveform design by enabling exact recovery from FrFT-based AF magnitudes and suggesting avenues for extending to other time-frequency kernels.

Abstract

The ambiguity function (AF) is a critical tool in radar waveform design, representing the two-dimensional correlation between a transmitted signal and its time-delayed, frequency-shifted version. Obtaining a radar signal to match a specified AF magnitude is a bi-variate variant of the well-known phase retrieval problem. Prior approaches to this problem were either limited to a few classes of waveforms or lacked a computable procedure to estimate the signal. Our recent work provided a framework for solving this problem for both band- and time-limited signals using non-convex optimization. In this paper, we introduce a novel approach WaveMax that formulates waveform recovery as a convex optimization problem by relying on the fractional Fourier transform (FrFT)-based AF. We exploit the fact that AF of the FrFT of the original signal is equivalent to a rotation of the original AF. In particular, we reconstruct the radar signal by solving a low-rank minimization problem, which approximates the waveform using the leading eigenvector of a matrix derived from the AF. Our theoretical analysis shows that unique waveform reconstruction is achievable with a sample size no more than three times the signal frequencies or time samples. Numerical experiments validate the efficacy of WaveMax in recovering signals from noiseless and noisy AF, including scenarios with randomly and uniformly sampled sparse data.

Figures (6)

• Figure 1: An illustration of the relationship between the FrFT-based AF $A_{X_{\alpha}}(\omega,s)$ and the conventional AF $A_{x}(\tau,\nu)$ of the signal $x(t)$. (a) Signal $x$ represented in the time-frequency $t$-$f$ plane. (b) Representation of the AF of the same signal in the delay-Doppler $\tau$-$\nu$ plane obtained using the operator $\mathcal{A}$ on $x(t)$. (c) Illustration of rotation of the signal in the $t$-$f$ plane by applying the FrFT operator $\mathcal{F}_{\alpha,\zeta}$. (d) Representation of the AF of the FrFT $X_{\alpha}$ shows that $A_{X_{\alpha}}$ is the rotation of $A_x$ using the coordinate transformation matrix $\boldsymbol{R}$ to the new $\omega$-$s$ plane. (e) Example of a signal $x(t)$. (f) AF $A_x(\tau,\nu)$ in the normalized delay-Doppler plane. (g) FrFT $X_{\alpha}(\zeta)$ plotted for various values of $\alpha$ from $-\pi/2$ to $\pi/2$ in increments of approximately $\pi/128$. (h) The FrFT-based AF $A_{X_{\alpha}}(\omega,s)$ is given by $A_{X_{\alpha}}(\omega,s)=A_{x}(\tau \cos \alpha-\nu \sin \alpha, \tau \sin \alpha+ \nu \cos \alpha)$.
• Figure 2: Ordered values of $\boldsymbol{Y}[\alpha,\ell]$ as defined in \ref{['eq:system2']} for $N=128$ in log-scale. The variable $\alpha$ (angle of FrFT) was uniformly sampled from $[\frac{\pi}{2},\frac{3\pi}{2}]$ with number of samples varying from $\lceil N/64 \rceil$ to $\lceil N/19 \rceil$. Observe that all values of $\boldsymbol{Y}[\alpha,\ell]$ are smaller than $10^{-1}$, which implies that $\boldsymbol{x}$ is nearly orthogonal to a large number of $\boldsymbol{u}_{n,\alpha}$.
• Figure 3: Reconstructed band- and time-limited signals when 75% of the samples of their AFs are uniformly removed. The sparse AFs were corrupted by noise such that SNR = $20$ dB. The attained relative error as in \ref{['eq:distance']} was $5\times 10^{-2}$ for both signals. For the band-limited [time-limited] signal, (a) [(d)],(b) [(e)], and (c) [(f)] are the original, sub-sampled, and recovered AFs, respectively; (g) [(i)], and (h) [(j)] are 1-D slices of AFs in the angle and Doppler dimensions, respectively; (k) [(m)] and (l) [(n)] are, respectively, magnitude and phase of recovered (red) signal juxtaposed over the original (blue).
• Figure 4: Reconstructed LFM and NLFM signals when 75% of the samples of their AFs are uniformly removed. The sparse AFs were corrupted by noise such that SNR = $20$ dB. The attained relative error as in \ref{['eq:distance']} was $5\times 10^{-2}$ for both signals. For the LFM [NLFM] signal, (a) [(d)],(b) [(e)], and (c) [(f)] are the original, sub-sampled, and recovered AFs, respectively; (g) [(i)], and (h) [(j)] are 1-D slices of AFs in the angle and Doppler dimensions, respectively; (k) [(m)] and (l) [(n)] are, respectively, magnitude and phase of recovered (red) signal juxtaposed over the original (blue).
• Figure 5: Empirical success rate of solving \ref{['eq:wavemax']} for complete and sparse AF samples in the absence of noise using the proposed Algorithm \ref{['alg:algorithm']} for FrFT-based AF and the non-convex BanRaW method of pinilla2024phase for the conventional AF.

Theorems & Definitions (16)

• Definition 1: $B$-band-limitedness
• Proposition 1
• Definition 2
• Lemma 1
• Remark 1
• Theorem 1
• Remark 2
• Lemma 2
• Theorem 2
• Lemma 3