Novel parameter estimation of high-order polynomial phase signals using group delay
Introduction
The parameter estimation of polynomial phase signals (PPSs) is an important issue in numerous fields of science, particularly in radar, sonar, biomedical, and geophysical signal processing [1], [2], [3], [4], [5]. A large number of estimators for PPSs have been studied and implemented.
Maximum likelihood techniques exhibit good performance in terms of mean square error (MSE). However, their computational load is high because they perform multidimensional search [6]. Several suboptimal parametric methods have been proposed to reduce computational load. Among these, the high-order ambiguity function (HAF) [7], [8], [9] is the most widely used technique. The HAF employs phase differentiation (PD) to reduce the order of the phase polynomial to 1 (complex sinusoidal). Then, it estimates the highest-order coefficient using the maximisation of the Fourier transform. The other coefficients are estimated by repeating the same procedure on the signal with removed previously estimated highest-order phase coefficients [10]. However, this strategy has two major drawbacks. First, each PD shortens the signal and amplifies the effective amount of noise, resulting in an increased signal-to-noise-ratio (SNR) threshold [10,11]. The SNR threshold increases by approximately 6 dB with each degree of the polynomial in the signal phase [12]. Second, the estimation error in higher-order parameters affects the estimation of lower-order parameters (error propagation) [10]. These two problems are particularly apparent for higher-order PPSs. The cubic phase function (CPF) [13,14] is another method for the parameter estimation of PPSs. This method was originally proposed for the parameter estimation of a third-order PPS. Subsequently, numerous CPF extensions, such as the high-order phase function [15,16] and hybrid CPF-HAF [17,18], were developed for the parameter estimation of higher-order PPSs. Estimators based on the CPF improve the HAF in terms of estimation performance; however, they are still limited because they use phase differencing [19]. An estimator based on phase unwrapping can also be used as an effective approach for the parameter estimation of PPSs. In [20], a -order polynomial phase is unwrapped by the integration of the -order difference of the principal value of the phase [19]. However, the use of phase differencing results in an extremely high SNR threshold [21,22]. Recent advances in the field, such as Bayesian unwrapping [19] and least squares phase unwrapping [22,23], have improved the SNR threshold. However, these techniques have high computational requirements.
In recent years, Igor Djurović et al. proposed the quasi-maximum likelihood (QML) estimator [12,24,25] as an effective technique for the parameter estimation of PPSs. Simulation results show that the performance of the QML estimator is better than those of all existing approaches [26]. However, it is considerably computationally demanding because the short-time Fourier transform (STFT) process is repeated for various window widths. As a result, it is not feasible to apply the QML estimator to large data samples. Numerous researchers have attempted to reduce the computation complexity of the QML estimator. In [27], the finite difference algorithm utilises the framework of the QML estimator but uses the finite difference instead of the STFT. In [2], a method based on the filtering and phase unwrapping of a transform function is proposed to obtain a coarse estimation. These methods reduce computational complexity but adversely impact noise performance. Simulation results show that their performance is worse than that of the QML estimator in the aspect of the SNR threshold. Certain researchers have attempted to further improve the SNR threshold. In [26], the QML estimator is combined with the random sample consensus (RANSAC) algorithm to propose the QML-RANSAC estimator. The SNR threshold obtained using the estimator is better by approximately 3 dB compared to that obtained using the standard QML estimator [24]. The modified QML estimator is proposed in [28]. This approach is based on a two-dimensional search over various window widths in the STFT and various IF estimates in polynomial regression. Computational complexity increases considerably because of two-dimensional search, and there is a negligible improvement in the SNR threshold.
In this paper, we propose a novel phase-based estimator for PPSs. As the discrete Fourier transform (DFT) enhances the output SNR of signals, the phase in the frequency domain, namely, the spectrum phase (SP), is significantly more robust to the influence of noise compared to the instantaneous phase (IP) of signals. Therefore, unlike existing phase-based methods, we first propose using the SP instead of the IP to estimate parameters. We expect that this strategy can obtain a high SNR threshold with a low computational load. However, it is considerably difficult or even impossible to directly estimate polynomial coefficients from the SP because the theoretical expression of the SP for general PPSs cannot be expressed in an explicit form. However, the first derivative of the SP, namely, the well-known group delay (GD), is the inverse function of the instantaneous frequency (IF) for PPSs with monotonic IF laws [29]. This property states that the IF and GD provide the same information about the time-frequency characterisation of a signal. Thus, we can estimate parameters by performing polynomial regression on the GD instead of the IF. The proposed procedure first unwraps the SP and then performs a difference operation on the unwrapped SP to obtain the GD. Then, polynomial regression is used on the GD instead of the IF to estimate phase parameters. However, the proposed method can be applied only to PPSs with monotonic IF laws, which we refer to as strictly monotonic PPSs (SMPPSs). An extremely simple phase compensation function (PCF), which can transform general PPSs into SMPPSs, is developed. The proposed strategy can be easily extended to general PPSs using the PCF. The obtained estimates are robust to the influence of noise. However, they are biased because the GD approximately approaches the IF in practice. This bias can be reduced by employing the O'Shea refinement technique [30]. In this manner, we obtain the final estimation that achieves the Cramér–Rao lower bound (CRLB).
In comparison with the HAF and CPF-HAF, the proposed method avoids multiple PD and effectively improves the SNR threshold. Compared to the QML, state-of-the-art technology, the proposed algorithm uses DFT instead of STFT to reduce computational complexity effectively. Moreover, for SMPPSs, the proposed method has the same SNR threshold as the QML; for general PPSs, the SNR threshold of the proposed method is only worse than that of the QML by 4-6 dB.
The remainder of this paper is organised as follows: The proposed estimator for SMPPSs is introduced in Section 2. In Section 3, we introduce the PCF and extend the proposed method to general PPSs. In Section 4, we consider the proposed estimator for multi-component PPSs. In Section 5, we investigate the computational complexity of the proposed method. Section 6 describes the results of Monte Carlo simulations, which compare the proposed estimator with the HAF, hybrid CPF-HAF, and QML estimators. The conclusions are presented in Section 7.
Section snippets
Signal model
The following model can describe PPSs:where is the signal amplitude, is the signal duration, is the IP function, is the order of the polynomial phase, and the phase coefficients are unknown real parameters. Sampled discrete-time PPSs are modelled aswhere is the sampling interval and is the number of discrete samples.
First, we divide PPSs into two classes according to the
Method for transforming non-SMPPSs into SMPPSs
The proposed estimator is only applicable to SMPPSs. However, most PPSs are non-SMPPSs. As the IF of non-SMPPSs is not monotonic, Eq. (5) has multiple solutions and Eq. (11) becomes inapplicable. Thus, Algorithm 1 cannot be utilised for non-SMPPSs. However, the proposed method can be used if non-SMPPSs can be transformed into SMPPSs. A method for transforming non-SMPPSs into SMPPSs is developed.
For a general PPS modelled by (1), the second derivative of can be expressed as
Multi-component PPS
The proposed method is a systematic tool for handling mono-component PPSs. However, situation is not bright for multi-component PPSs. In general, there are numerous techniques [26,[34], [35], [36], [37] for estimation of the multi-component PPS, but this topic is still wide open [10]. In this section, we investigate the proposed method for multi-component PPSs. The following model can describe multi-component PPSs:where and are the
Calculation complexity
A key strength of the proposed algorithm is its computational complexity. Algorithm 1 mainly consists of spectrum calculation, an SP unwrapping procedure, polynomial regression, and a refinement step. The fast Fourier transform (FFT) can be used to calculate the spectrum of PPSs, and the calculation complexity of the FFT is . The phase unwrapping procedure first compares the spectral amplitude to find the frequency interval, which only requires real additions. Then, it requires
Simulation results
In this section, we describe the performance evaluation of the proposed approach on SMPPSs and non-SMPPSs using various PPS estimators. The MSEs of estimated parameters are used as a performance measure, which is calculated aswhere is the estimate of at the iteration, is the true parameter value, and is the number of Monte Carlo simulations.
The proposed estimator, referred to as the SP unwrapping (SPU) method, is compared
Conclusion
We propose a method for the parameter estimation of high-order PPSs at low SNRs. The estimator first obtains the GD by unwrapping the SP and then performs polynomial regression on the GD instead of the conventional IF to estimate the parameters of PPSs. Finally, the obtained result is refined to reduce bias. However, this method is unsuitable for non-SMPPSs. To solve this problem, we develop a PCF for transforming non-SMPPSs into SMPPSs. The proposed method can be applied to general PPSs using
CRediT authorship contribution statement
Xiaodong Jiang: Conceptualization, Methodology, Formal analysis, Software, Writing - original draft, Writing - review & editing. Siliang Wu: Conceptualization, Writing - original draft. Yuan Chen: Software.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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