Mathematical Problems in Engineering

Volume 2018, Article ID 4942938, 14 pages

https://doi.org/10.1155/2018/4942938

## A New Approach for the Characterization of Nonstationary Oscillators Using the Wigner-Ville Distribution

Department of Electrical and Electronic Engineering, Ariel University, Ariel 40700, Israel

Correspondence should be addressed to Chagai Levy; moc.liamg@yveliagahc

Received 28 February 2018; Revised 9 June 2018; Accepted 20 June 2018; Published 11 July 2018

Academic Editor: Akhil Garg

Copyright © 2018 Chagai Levy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Oscillators and clocks are affected by physical mechanisms causing amplitude fluctuations, phase noise, and frequency instabilities. The physical properties of the elements composing the oscillator as well as external environmental conditions play a role in the characteristics of the oscillatory signal produced by the device. Such instabilities demonstrate frequency drifts and modulation and spectrum broadening and are observed to be nonstationary processes in nature. Most of tools which are being used to measure and characterize oscillator stability are based on signal processing techniques, assuming time invariance during a temporal window, during which the signal is assumed to be stationary. This paper proposes a new time-frequency metric for the characterization of frequency sources. Our technique is based on the Wigner-Ville distribution, which extends the spectral measures to consist of the temporal nonstationary behavior of the processes affecting the accuracy of the clock. We demonstrate the use of the technique in the characterization of phase errors, frequency offsets, and frequency drift of oscillators.

#### 1. Introduction

In recent years, due to the high data rates required to be transferred in communication networks, the demand for high accuracy satellite navigation systems and the development of high resolution radars and clocks with very high precision are required. Stable oscillators, including atomic ones like rubidium and even cesium atomic, are often used in communication and navigation facilities [1–6]. In addition to the above applications, many other electronic systems require frequency generators. In such systems, time synchronization and an accurate clock frequency are required [7–12].

Every oscillator has its unique clock frequency consisting of a nominal frequency in addition to a normalized frequency difference of Parts Per Million (PPM) from the nominal frequency multiplied by this nominal frequency. In addition, every oscillator has a random jitter . Therefore, the master and the slave clocks may have different frequencies and thus not be synchronized. For example, suppose that the master’s frequency is () and the slave’s frequency is (); thus, their frequencies are not synchronized. In that case, the clock offset generally keeps increasing and may cause the communication link to fail [13]. Therefore, in such systems, frequency synchronization and an accurate clock frequency are required [14].

For example [13], communication networks such as Long Term Evolution Advanced (LTE-A) based cellular networks require base stations and backhaul equipment to maintain time synchronization in order to provide several new features like synchronized transmissions over frequencies from adjacent base stations and interference coordination. An example of an emerging networked application is smart power grid systems which are characterized by a two-way flow of energy and end-to-end communications. The communications are largely machine-to-machine (M2M) in nature [13, 15] and need to share synchronized information in order to improve efficiency and reliability of power delivery. Therefore, in addition to frequency synchronization, time synchronization is also required [16].

Oscillators and clocks are subjected to environmental conditions, internal malfunctions, and inherent physical phenomena causing instabilities in the oscillatory signal being produced [17]. These instabilities are expressed by amplitude and phase fluctuations causing frequency deviations, drift, and spectrum broadening. Instead of being stationary, single frequency, and temporal coherent, the clock signal is observed to be a stochastic process characterized by statistical and spectral features, which are nonstationary time varying [18–21]. Therefore, the design of a set of tools for time domain analysis of a frequency source is therefore needed and significant to investigate.

Several works [22–25] on the time-frequency (TF) analysis of oscillators and atomic clocks have been carried out recently. In [22, 23], the authors presented methods for the detection and identification of atomic clock anomalies via simulation on experimental data. Those methods are based on derivation of the frequency deviation and then sliding Welchs periodogram on the clock noise data.

Another work [24] presents a time-frequency relationship between the Langevin equation and the harmonic oscillator by deriving relationship between the Wigner distribution of the Green’s function of the Langevin equation and of the harmonic oscillator. Their result paves the way for a simplification of the time-frequency representation of differential equations, as well as for a better understanding and filtering of the interference terms.

Most recently, a study of the important role of time-frequency signal representations was presented [25]. This approach was used for enhancement of the performance of global navigation satellite system (GNSS) receivers by using short-time Fourier transform (STFT) and the Wigner distribution as the TF representations.

In order to measure and characterize the oscillator properties, time-frequency signal processing techniques are required. Most often the Allan variance (or the Allan deviation) is employed [26–32]. It calculates the covariance of two adjacent samples of the signal produced by the under test oscillator. The Allan variance is widely used and recommended in international standards [28]. However, the Allan variance does not give the possibility of identifying and interpreting nonstationary effects.

Recently [18–21], an extended version of the Allan variance, the dynamic Allan variance (DAVAR), was proposed for the characterization of atomic clock stability that can vary with time. However, the DAVAR deals only with characterization of the stability of the frequency source and cannot detect phase and frequency offsets relative to a reference source.

The Allan variance is a fundamental tool for characterizing frequency stability of oscillators since it measures the size of frequency fluctuations at different observation intervals, whereas the DAVAR describes the variations with time of these fluctuations. Please note that in this paper we concentrate in ideal oscillators, where no random frequency fluctuations are considered.

In this paper we propose a time-frequency metric for the characterization of oscillators and frequency generators, considering also nonstationary variations. It is based on the Wigner distribution [33, 34], proposed in 1932 for the characterization of quantum fluctuations. In signal processing, it was used by Ville [35], so that it is often called Wigner-Ville distribution [36]. The applications of Wigner distribution are various: analysis of nonstationary signals [37–40], radar signals [41, 42], biomedical signals [43–45], analysis of time varying filters [46, 47], and image processing [48–50]. However, the Wigner-Ville distribution was not yet used in the characterization of nonstationary oscillators such as detection of the phase and frequency offsets relative to a reference source. In this paper, we will introduce both the discrete and continuous forms of the Wigner-Ville distribution for the characterization of a clock signal (nonstationary oscillator) with respect to a reference source and show how this technique is useful for the detection of the phase and frequency offsets relative to a reference source.

A short review of the oscillator clock model and the Wigner-Ville distribution are introduced in Section 2. The proposed approach for the characterization of nonstationary oscillators is presented in Sections 3-6. In Section 7 numerical simulation results are presented including demonstration of phase and frequency deviations. Section 8 summarizes the paper.

#### 2. Theoretical Background

In this section, we will introduce a short theoretical background of the clock model and the Wigner-Ville distribution. In Section 2.1 we will describe briefly the clock model and in Section 2.2 we will present briefly the Wigner-Ville distribution [33].

##### 2.1. The Clock Model

A model for a real clock which includes variations in the amplitude and phase is given by [17, 26]where is the amplitude and is the frequency of the oscillation. and represent random fluctuations observed in amplitude and phase, respectively.

The amplitude fluctuations can be neglected [17, 26], and thus we can write the approximate model [17]:The phase variations result in an instantaneous frequency which is a time dependent process given by [17]In an ideal clock, the phase is constant and can be set to zero [51]; thus, it generates a sinusoidal oscillation of the form [52, 53]:where is the amplitude of the ideal clock.

Physical signals are real, in which case the Wigner-Ville distribution can be applied to either the (real) signal itself, or a complex version of the signal known as the analytic signal [54].

In the following, we define the analytic form of the clock signal asand for an ideal clock, the analytic signal will be

##### 2.2. The Wigner-Ville Distribution

One of the most representative joint representations of time-frequency analysis is the Wigner distribution (WD), which is a quadratic signal representation introduced by Wigner (1932) [34] and later applied by Ville (1948) [35] to signal analysis. A comprehensive discussion of the WD properties can be found in a series of classical articles by Claasen and Mecklenbruker (1980) [55–57]. Originally, the WD has been applied to continuous variables as follows: consider an arbitrary 1D function , . The WD of is given by [33]where “” denotes complex conjugation. By considering the shifting parameter as a variable, (7) represents the Fourier transform (FT) of the product , where denotes the spatial frequency variable and, hence, the WD can be interpreted as the local spectrum of the signal .

The Wigner distribution satisfies many meaningful mathematical properties that are summarized in Appendix A.

The most powerful property is the “Time-Marginal” property and is given bywhere denotes the norm of the signal . Summing up of the energy distribution for all frequencies at a particular time would give the instantaneous energy of the signal. We will use the “Time-Marginal” property in our development in order to detect phase offset of a frequency source with respect to a reference source.

Modern signal processing uses the discrete form of physical signals. Therefore, the practical computation will use the discrete Wigner-Ville distribution form. There are several presentations in the literature for the discrete Wigner-Ville distribution. In Appendix B we present and explain in detail the discrete form of the Wigner-Ville Distribution and some of the main presentations. In this paper we will use the following discrete Wigner-Ville distribution, similar to the one proposed by Eric Chassande-Mottin and Archana Pai [58]:where , denotes the number of samples, and denotes the index numbers for the time and frequency vectors, respectively.

This equation, designed for discrete time or space signals, shows a great similarity with the original continuous version; therefore, it has been selected for our development.

#### 3. A New Metric for Frequency Source Characterization

The system under consideration is presented in Figure 1.