Relaxed Projection Phase Retrieval for Radar Waveforms
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Solution Overview
Problem
Existing methods for reconstructing constant envelope signals in adaptive radar systems face challenges such as slow convergence, convergence stagnation, permutation ambiguities, and sensitivity to initial conditions, particularly in phase retrieval processes involving Fourier transform operations.
Innovation Solution
An iterative process using relaxed projections in both time and spectral domains, where estimates are computed as weighted sums of projections onto constraint sets, with relaxation parameters σ and μ chosen between 0.7 and 0.9, to iteratively refine the signal x(t) until predefined criteria are met.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If alternating projection iterative algorithms (such as Gerchberg-Saxton, Error Reduction Algorithm, steepest-descent/conjugate-gradient, BIO, or HIO algorithms) are used for phase retrieval, then the problem of reconstructing constant envelope signals with prescribed Fourier transform can be solved, but the algorithms suffer from slow convergence, convergence stagnation, permutation and scaling ambiguities, and sensitivity to initial seed
Solution Approach 1:
The patent applies preliminary action by using stationary phase approximation to generate an initial seed signal before starting the iterative phase retrieval process. This pre-computed initial signal with known phase characteristics provides a better starting point for iteration, avoiding the sensitivity to random initial seeds and reducing the number of iterations required for convergence.
Solution Approach 2:
The patent transforms the phase retrieval problem into a frequency warping parameter estimation problem. By changing the parameter space from direct phase retrieval to frequency warping parameters, the algorithm achieves faster convergence and avoids the stagnation issues that plague traditional iterative methods while maintaining retrieval accuracy.
2Use of energy by moving object
If power amplifier is operated at saturation for maximum efficiency, then energy efficiency is improved, but the system requires a temporal signal with constant envelope which constrains the waveform design
Solution Approach 1:
The patent segments the signal design process into two independent parts: first designing the optimal adaptive waveform in the frequency domain without envelope constraints, then separately retrieving the phase information to enforce constant envelope. This segmentation allows the amplitude and phase to be designed independently, maintaining waveform optimization while satisfying power amplifier constraints.
Solution Approach 2:
The patent moves the design freedom to the phase dimension while constraining the amplitude dimension. By working primarily in the frequency domain and using phase retrieval techniques, the system maintains adaptability in spectral shaping while enforcing constant envelope in the time domain, effectively trading time-domain flexibility for frequency-domain flexibility.
3Reliability
If traditional iterative algorithms are used for phase retrieval, then constant envelope signals can be reconstructed, but the algorithms exhibit permutation and scaling ambiguities that affect solution uniqueness
Solution Approach 1:
The patent incorporates feedback mechanisms in the iterative phase retrieval process by continuously monitoring the constant envelope constraint satisfaction and adjusting the phase estimation accordingly. The algorithm uses the error between the current estimate and the desired constant envelope property to guide subsequent iterations, reducing ambiguity and improving convergence to the correct solution.
Data Source
AI summary
The invention is an iterative process for performing iteratively the phase retrieval of an adaptive signal x(t) matching two sets of constraint both concerning the time envelope ue(t) of signal x(t) and magnitude distribution Um(f) of its spectral representation. At each iteration k the process computes an estimate {tilde over (x)}k(t) of signal x(t) which is obtained from a first projection PA on a first set of constraint in time domain of a computed value xk(t) of x(t), xk(t) deriving from an estimate {tilde over (X)}k−1(f) of the spectrum of signal x(t), said estimate {tilde over (X)}k−1(f) being itself obtained from a second projection PB on a second set of constraints in spectral domain of the Fourier transform Xk(f) of the estimate {tilde over (x)}k−1(t) of x(t) computed at iteration k−1. Iterative computation of estimate {tilde over (x)}k(t) is repeated until {tilde over (x)}k(t) meets a predefined criterion which indicates that estimate {tilde over (x)}k(t) is close enough to expected signal x(t).