Myths concerning Woodward's ambiguity function: analysis and resolution

Baylis, Charles; Cohen, Lawrence S.; Eustice, Dylan; Marks, Robert J.

doi:10.1109/taes.2016.150735

Cited by 7 publications

(2 citation statements)

References 29 publications

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“…[84], Table 2). Woodward is known for first mentioning the sinc function [32,43,100,152,153] and his time-frequency ambiguity function [46,51,56,[154][155][156][157][158][159][160][161][162]. We now let g be the Dirac comb, it leads us to the following lemma [82,142].…”

Section: Definition 1 (Periodization and Discretizationmentioning

confidence: 99%

On the Reversibility of Discretization

Fischer

Stens

2020

Mathematics

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“Discretization” usually denotes the operation of mapping continuous functions to infinite or finite sequences of discrete values. It may also mean to map the operation itself from one that operates on functions to one that operates on infinite or finite sequences. Advantageously, these two meanings coincide within the theory of generalized functions. Discretization moreover reduces to a simple multiplication. It is known, however, that multiplications may fail. In our previous studies, we determined conditions such that multiplications hold in the tempered distributions sense and, hence, corresponding discretizations exist. In this study, we determine, vice versa, conditions such that discretizations can be reversed, i.e., functions can be fully restored from their samples. The classical Whittaker-Kotel’nikov-Shannon (WKS) sampling theorem is just one particular case in one of four interwoven symbolic calculation rules deduced below.

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Section: Definition 1 (Periodization and Discretizationmentioning

confidence: 99%

On the Reversibility of Discretization

Fischer

Stens

2020

Mathematics

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“…The AF is not uniquely defined [8,9], including in the works of Woodward [6]. Further, later works have generalized this definition to handle larger bandwidth signals [10,11], long duration signals [12], volumetric scatterers [13,14], targets with high velocity [15,16], and angular domain [17][18][19].…”

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confidence: 99%

Phase Retrieval for Radar Waveform Design

Pinilla¹,

Mishra²,

Sadler³

et al. 2022

Preprint

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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 can 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 upon 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 even sparsely and randomly sampled AFs with mean-square-error of 1×10 −6 and 9 × 10 −2 for the full noiseless samples and sparse noisy samples, respectively.

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Phase retrieval for radar waveform design

Pinilla,

Mishra,

Sadler

et al. 2024

Information and Inference: A Journal of the IMA

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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.

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Myths concerning Woodward's ambiguity function: analysis and resolution

Cited by 7 publications

References 29 publications

On the Reversibility of Discretization

On the Reversibility of Discretization

Phase Retrieval for Radar Waveform Design

Phase retrieval for radar waveform design

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