Nonstationary Wave Field Analysis using new Harmonic Phase Tracking Technique
Authors: Palo Paul 
Published: October 2008

Abstract
This paper introduces a new signal processing technique called Hannonic Phase Tracking (HPT), and uses it to investigate physically-meaningful properties of random seas. The fundamental improvement over FFTs is that HPT adaptively solves for the rank and best component frequencies along with amplitudes and phases. HPT can be applied to wideband ocean waves to recover time-varying information associated with nonstationary conditions. Analyses suggest that random seas are in fact "quasi-detenninistic", with wave packets at discrete frequencies, with continuity for long periods of time (e.g., one hour) and large spatial extents (several wavelengths).

Introduction
The keynote paper in this session reviews analysis issues associated with wave directionality (Borgman1). That paper argues that most commonly-used analysis techniques are limited by their inherent requirement for stationarity via the use of ensemble averaged expected values. This paper directly addresses that analysis limitation by introducing a new signal processing technique called Harmonic Phase Tracking (HPT) that does not require stationarity of the ignal. This technique is based on a very simple premise that yields a surprisingly useful tool. The paper objective is to introduce this new technique and its potential application to ocean wave fields. The paper is divided into four numbered sections that:review some of the limitations of existing techniques;present the fundamentals of HPT,illustrate and contrast its capabilities versus other techniques, andillustrate its application to wave problems such as spatial coherence and directionality.
While this content is unquestionably ambitious for one paper, it is necessary to provide a sufficient understanding of this new approach and its potential impact on modeling wave fields.

1. Limitations of Existing Signal Processing Techniques
It is instructive to preface the description of HPT in the nextsection by briefly reviewing some of the inherent limitations of existing signal processing techniques, which are grouped into two broad categories:spectral analysis techniques, andtime-frequency distributions.
This emphasis on limitations is done with full recognition of the many advantages and historical importance of these techniques to engineering and science. All comments are directed towards stochastic signals. The spectral techniques are the most commonly used, with two subcategories of either low or high resolution techniques. Low resolution techniques are exemplified by the FFf, Blackman-Tukey, ARMA, and Maximum Entropy and Likelihood based approaches. These techniques trade a degradation of information for the convenience of universal applicability to all signals and minimal computational effort. For example, the familiar Fast Fourier Transform (FFT) uses a frequency domain mapping onto a basis vector set of orthogonal sinusoids defined as integer superharmonics of the segment length. The inherent consequence of this is that all FFT-derived functions exhibit large variances due to interfrequency leakage that can only be reduced by assuming ergodicity and applying expected value operations. In other words, by choosing a FFf-based approach the analyst accepts the fact that each individual (raw) FFT estimate is statistically unreliable and instead settles for identifying long-term average behavior. This, in turn, is valid only when the signal is truly stationary (which is difficult to rigorously quantify for ocean wave signals).