Cubic phase function: A simple solution to polynomial phase signal analysis

Abstract

This article provides an overview of the cubic phase function (CPF) as a tool proposed for both parametric and nonparametric estimation of the frequency modulated (FM) and in particular polynomial phase signals (PPS). This simple tool motivated small revolution in this field with numerous extensions and applications. We are describing the CPF and compare some of its extensions for both one-dimensional and two-dimensional signals. The comparisons are performed in terms of accuracy (measured with signal-to-noise (SNR) threshold and mean-squared error (MSE)) and computational complexity. Also, we review the CPF and related transforms applications.

Introduction

Engineers in many fields often encounter non-stationary signals including biological, speech and music signals, radio signals in wireless communications and radars, and dispersive seismic signals [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57]. The conventional Fourier transform (FT), a popular tool to bridge between time and frequency, is considered to be inadequate to analyze such real-life signals [1], [2], [4]. In contrast, joint time-frequency (TF) analysis is an efficient way to reveal frequency contents of signals evolving over time, alternatively known as the instantaneous frequency (IF).

One particularly interesting model of non-stationary signals is the polynomial phase signal (PPS) model. The last 25 years have witnessed tremendous developments in the area of PPS parameter estimation, driven by applications originated in radars, sonars, biomedicine, machine engine testing, etc. [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87], [88], [89], [90], [91], [92], [93], [94], [95]. The maximum likelihood (ML) estimator has limited application due to a required multi-dimensional search over the parameter space. Early developments for the PPS parameter estimation are based on high-order ambiguity function (HAF) and its product form (PHAF) [91], [96]. The HAF-based estimation procedure consists of phase order decrementing by the process known as the phase differentiation (PD) until obtained signal is a sinusoid (the PPS of the first order). Then, the highest-order phase parameter is estimated using the fast algorithm, i.e., by an one-dimensional (1-D) search over the parameter space. This strategy is efficient but with numerous shortcomings. Firstly, in each stage of the procedure, the PD (performed by the auto-correlation function) reduces signal length and increases the number of noise-related terms in the resulted signal. These effects increase the signal-to-noise-ratio (SNR) threshold and estimation mean squared error (MSE). The auto-correlation also introduces cross-terms when multicomponent signals are considered. Finally, after estimation of the highest-order phase parameter, the same procedure is performed on the dechirped signal. Dechirping procedure causes error propagation from higher- to lower-order phase parameters. Some negative effects of the HAF, in particular cross-terms, are mitigated using the PHAF obtained as the multiplication of several HAFs calculated with different lag sets sharing the same product. The alternative technique is the integrated generalized ambiguity function (IGAF) [97]. It is accurate but, unfortunately, with heavy computations. Specifically this technique requires integrations over multi-dimensional lag space. Large number of integrals (or sums), i.e., calculation complexity, limits application of the IGAF to lower-order PPSs. Nevertheless, the IGAF is known to enhance the signal term and suppress the oscillating cross-terms and noise from the (coherent) lag integrations. In addition, two highest-order parameters are estimated at once meaning that the effect of error propagation is also reduced.

Aside from the IF, instantaneous frequency rate (IFR) or chirp-rate (CR) provides additional insights into the signal's frequency changing rate [1], [2], [98], [99], [100] and has received significant attention after O'Shea's seminal paper of [88]. At first, O'Shea proposed the original cubic phase function (CPF) for the parameter estimation of a third-order PPS, i.e., a cubic phase (CP) signal [88], [89]. The CPF-based procedure requires only one PD resulting in significantly better performances with respect to the (P)HAF-based alternatives for the CP signal. Later, numerous researchers strive for its extensions to higher-order PPSs [69], [70], [101], [102], [103], [104]. Besides, the CPF maps a signal to a 2-D joint time-chirp (frequency) rate domain. The time-CR domain and representations are still not well understood compared to the TF domain and representations. Meanwhile, there exist considerable interests in generalization of this transform to a 2-D PPS [67].

The aim of this paper is to show how this, at the first glance, simple modification of the CPF can motivate significant developments in the field of the PPS estimation and in more general nonstationary signal analysis. These developments resulted in significant improvement in the PPS estimation performance with respect to the state-of-the-art techniques. This overview article is devoted to the CPF since this transform and related approaches still reverberate in the community working with both theoretical and practical developments in the PPS estimation and with general nonstationary phase signals. The remaining of this paper is organized as follows. Theoretical background on the CPF is given in Section 2 with basic performance analysis of this technique both in parametric and nonparametric estimation. Various extensions of the CPF are presented and compared in Section 3. 2-D PPSs are considered in Section 4, while Section 5 brings literature overview of some practical applications where the CPF is used or where CP signals appear.