Phase Retrieval from Modulus Using Homeomorphic Signal Processing and the Complex Cepstrum: An Algorithm for Lightning Protection Systems
In general, the Phase Retrieval from Modulus problem is very difficult. In this report, we solve the difficult, but somewhat more tractable case in which we constrain the solution to a minimum phase reconstruction. We exploit the real-and imaginary part sufficiency properties of the Fourier and Hilbert Transforms of causal sequences to develop an algorithm for reconstructing spectral phase given only spectral modulus. The algorithm uses homeomorphic signal processing methods with the complex cepstrum. The formal problem of interest is: Given measurements of only the modulus {vert_bar}H(k){vert_bar} (no phase) of the Discrete Fourier Transform (DFT) of a real, finite-length, stable, causal time domain signal h(n), compute a minimum phase reconstruction {cflx h}(n) of the signal. Then compute the phase of {cflx h}(n) using a DFT, and exploit the result as an estimate of the phase of h(n). The development of the algorithm is quite involved, but the final algorithm and its implementation are very simple. This work was motivated by a Phase Retrieval from Modulus Problem that arose in LLNL Defense Sciences Engineering Division (DSED) projects in lightning protection for buildings. The measurements are limited to modulus-only spectra from a spectrum analyzer. However, it is desired to perform system identification on the building to compute impulse responses and transfer functions that describe the amount of lightning energy that will be transferred from the outside of the building to the inside. This calculation requires knowledge of the entire signals (both modulus and phase). The algorithm and software described in this report are proposed as an approach to phase retrieval that can be used for programmatic needs. This report presents a brief tutorial description of the mathematical problem and the derivation of the phase retrieval algorithm. The efficacy of the theory is demonstrated using simulated signals that meet the assumptions of the algorithm. We see that for the noiseless case, the reconstructions are extremely accurate. When moderate to heavy simulated white Gaussian noise was added, the algorithm performance remained reasonably robust, especially in the low frequency part of the spectrum, which is the part of most interest for lightning protection. Limitations of the algorithm include the following: (1) It does not account for noise in the given spectral modulus. Fortunately, the lightning protection signals of interest generally have a reasonably high signal-to-noise ratio (SNR). (2) The DFT length N must be even and larger than the length of the nonzero part of the measured signals. These constraints are simple to meet in practice. (3) Regardless of the properties of the actual signal h(n), the phase retrieval results are constrained to have the minimum phase property. In most problems of practical interest, these assumptions are very reasonable and probably valid. They are reasonable assumptions for Lightning Protection applications. Proposed future work includes (a) Evaluating the efficacy of the algorithm with real Lightning Protection signals from programmatic applications, (b) Performing a more rigorous analysis of noise effects, (c) Using the algorithm along with advanced system identification algorithms to estimate impulse responses and transfer functions, (d) Developing algorithms to deal with measured partial (truncated) spectral moduli, and (e) R & D of phase retrieval algorithms that specifically deal with general (not necessarily minimum phase) signals, and noisy spectral moduli.
- Research Organization:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Sponsoring Organization:
- USDOE
- DOE Contract Number:
- W-7405-ENG-48
- OSTI ID:
- 15014444
- Report Number(s):
- UCRL-TR-205254; TRN: US200802%%1231
- Country of Publication:
- United States
- Language:
- English
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