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

Authors: Chagai Levy; Monika Pinchas; Yosef Pinhasi
Journal: Mathematical Problems in Engineering
 (2018)

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## 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.


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## Body

## 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 frequencyν0 in addition to a normalized frequency difference of x Parts Per Million (PPM) from the nominal frequency multiplied by this nominal frequency. In addition, every oscillator has a random jitter ϕ. 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 (ν0+xν0) and the slave’s frequency is (ν0-xν0); 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 Section2. 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 Section2.1 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](1)rt=A+ϵtcos⁡2πν0t+φtwhere A is the amplitude and ν0 is the frequency of the oscillation. ϵ(t) and φ(t) 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]:(2)rt≃Acos⁡2πν0t+φtThe phase variations φ(t) result in an instantaneous frequency which is a time dependent process given by [17](3)νt=ν0+12πdφtdtIn 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]:(4)st=A0cos⁡2πν0twhere A0 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 as(5)r~t≃Aei2πν0t+φtand for an ideal clock, the analytic signal will be(6)s~t=A0ei2πν0t


### 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 r~(t)∈C, t∈R. The WD of r~(t) is given by [33](7)Wr~t,f=∫-∞∞r~t+τ2r~∗t-τ2e-i2πfτdτwhere “∗” denotes complex conjugation. By considering the shifting parameter τ as a variable, (7) represents the Fourier transform (FT) of the product r~(t+τ/2)r~∗(t-τ/2), where f denotes the spatial frequency variable and, hence, the WD can be interpreted as the local spectrum of the signal r~(t).The Wigner distribution satisfies many meaningful mathematical properties that are summarized in AppendixA.The most powerful property is the “Time-Marginal” property and is given by(8)∫-∞∞Wr~t,fdf=r~t2where r~(t) denotes the norm of the signal r~(t). 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 AppendixB 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]:(9)wr~n,m=ts∑k=-knknr~n+k2r~∗n-k2e-i2πmk/2Nwhere kn=min[2n,2N-1-2n], N denotes the number of samples, and n,m 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.


## 2.1. The Clock Model

A model for a real clock which includes variations in the amplitude and phase is given by [17, 26](1)rt=A+ϵtcos⁡2πν0t+φtwhere A is the amplitude and ν0 is the frequency of the oscillation. ϵ(t) and φ(t) 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]:(2)rt≃Acos⁡2πν0t+φtThe phase variations φ(t) result in an instantaneous frequency which is a time dependent process given by [17](3)νt=ν0+12πdφtdtIn 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]:(4)st=A0cos⁡2πν0twhere A0 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 as(5)r~t≃Aei2πν0t+φtand for an ideal clock, the analytic signal will be(6)s~t=A0ei2πν0t


## 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 r~(t)∈C, t∈R. The WD of r~(t) is given by [33](7)Wr~t,f=∫-∞∞r~t+τ2r~∗t-τ2e-i2πfτdτwhere “∗” denotes complex conjugation. By considering the shifting parameter τ as a variable, (7) represents the Fourier transform (FT) of the product r~(t+τ/2)r~∗(t-τ/2), where f denotes the spatial frequency variable and, hence, the WD can be interpreted as the local spectrum of the signal r~(t).The Wigner distribution satisfies many meaningful mathematical properties that are summarized in AppendixA.The most powerful property is the “Time-Marginal” property and is given by(8)∫-∞∞Wr~t,fdf=r~t2where r~(t) denotes the norm of the signal r~(t). 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 AppendixB 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]:(9)wr~n,m=ts∑k=-knknr~n+k2r~∗n-k2e-i2πmk/2Nwhere kn=min[2n,2N-1-2n], N denotes the number of samples, and n,m 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 Figure1.Figure 1
Block diagram of the proposed system. First we subtract the reference signals~(t) from the received one r~(t), and then the Wigner-Ville distribution function is performed on the difference r~(t)-s~(t).At first we will derive a general expression forWr~-s~(t,f) as a function of φ(t); then we will develop simplified and closed expression for the Time Marginal of Wr~-s~(t,f) for three different cases.Case A.
In Section4, we derive the analytic continuous and discrete forms for the Wigner-Ville distribution and its Time-Marginal expression for a constant phase offset case.Case B.
In Section5, we derive the analytic expressions for the continuous Wigner-Ville distribution and its Time-Marginal expression for the case of constant phase and frequency offsets.Case C.
In Section6, we derive the analytic expressions for the continuous Wigner-Ville distribution and its Time-Marginal expression for the case of frequency drift.
In this paper, we make use of an analytic signal which is the complex form of the real clock signal. Our choice is based on [54] because of the following reasons. All of the transformations in Cohens class of distributions (of which the Wigner-Ville is one) produce better results when applied to a modified version of the waveform termed the analytic signal, a complex version of the real signal. While the real signal can be used, the analytic signal has several advantages. The most important advantage is due to the fact that the analytic signal does not contain negative frequencies, so its use will reduce the number of cross products. If the real signal is used, then both the positive and negative spectral terms produce cross products.Theorem 1.
For the following assumptions:(1)
without loss of generality,A=A0=1,(2)
ν0 is the nominal frequency,(3)
s~(t) denotes the continuous complex form of an ideal clock signal,(4)
r~(t) denotes the continuous complex form of the real clock signal.The Wigner-Ville distribution of the differencer~(t)-s~(t) is defined as(10)Wr~-s~t,f=δf-ν0+∫-∞∞eiφt+τ/2e-iφt-τ/2e-i2πf-ν0τdτ-2Re∫-∞∞ei2πν0τ+φt+τ/2e-i2πfτdτProof of Theorem1.
The Wigner-Ville distribution (WD) ofr~-s~ is given by(11)Wr~-s~t,f=WDei2πν0t+φt-ei2πν0tBy using the “Summation Property” given in Appendix A (A.2) we obtain(12)Wr~-s~t,f=Wr~t,f+W-s~t,f-2ReWr~,s~t,fNote that W-s~(t,f)=Ws~(t,f) for s~(t)∈C. Then, by using (12) we have(13)Wr~-s~t,f=Wr~t,f+Ws~t,f-2ReWr~,s~t,fLet us get first a closed form expression for the second component in the right side of (13)(14)Ws~t,f=∫-∞∞s~t+τ2s~∗t-τ2e-i2πfτdτ=∫-∞∞ei2πν0t+τ/2e-i2πν0t-τ/2e-i2πfτdτ=∫-∞∞e-i2πf-ν0τdτ=δf-ν0In order to get a closed form expression for the first component on the right side of (13) we will use the next definition:(15)r~t=ei2πν0t+φt=C~tD~twhere C~(t)=ei(2πν0t) and D~(t)=eiφ(t).
By using the “Convolution Property” given in AppendixA we obtain(16)Wr~t,f=∫-∞∞WC~t,f′WD~t,f-f′df′where WC~(t,f) and WD~(t,f) are the WDs of C~(t) and D~(t), respectively.
Note that the closed form expression forWC~(t,f′)=Ws~(t,f′), where Ws~(t,f′) is given in (14) for f′=f.
By substituting (14) into (16) we obtain(17)Wr~t,f=∫-∞∞δf′-ν0WDt,f-f′df′=WDt,f-ν0=∫-∞∞eiφt+τ/2e-iφt-τ/2e-i2πf-ν0τdτLet us define for simplicity the variable Δφ(τ) as(18)Δφτ=φt-τ2-φt+τ2Thus, (17) can be written as(19)Wr~t,f=WDt,f-ν0=∫-∞∞e-iΔφτe-i2πf-ν0τdτNow we turn to the “cross-term” component of Wr~-s~(t,f) which is the third component on the right side of (13):(20)Wr~,s~t,f=∫-∞∞ei2πν0t+τ/2+φt+τ/2e-i2πν0t-τ/2e-i2πfτdτ=∫-∞∞ei2πν0τ+φt+τ/2e-i2πfτdτUsing (13), (17), and (20), the WD of (12) is given by(21)Wr~-s~t,f=δf-ν0+∫-∞∞eiφt+τ/2e-iφt-τ/2e-i2πf-ν0τdτ-2Re∫-∞∞ei2πν0τ+φt+τ/2e-i2πfτdτThis completes our proof of Theorem 1.


## 4. Constant Phase Offset

As we all know, modern signal processing uses the discrete form of physical signals. Therefore, the practical computation will use the discrete Wigner-Ville distribution form as is introduced later in this subsection. Due to the simplicity of the mathematical development of using the continuous form of the WD upon its discrete one, we will use henceforth in our development process only the continuous form. In order to do so, we need to verify that there is a complete match between the analytic continuous form and the discrete one. Hence, in this subsection we will derive the analytic discrete form for the Wigner-Ville distribution and its Time-Marginal expression for a constant phase offset. In the end of the discrete form development, we will compare the discrete expression against the continuous expression introduced in the last subsection in order to verify a complete match and hence use only the continuous form.


### 4.1. Continuous Form

In this subsection, we will apply the analytic continuous form of the Wigner-Ville distribution onr~-s~ and then we will derive its Time-Marginal expression for a constant phase offset case.Theorem 2.
For the following assumptions:(1)
without loss of generality,A=A0=1,(2)
x0 is the constant phase offset from a reference clock signal,(3)
ν0 is the nominal frequency,(4)
s~(t) denotes the continuous complex form of an ideal clock signal,(5)
r~1(t) denotes the continuous complex form of the real clock signal for the case of constant phase offset.The Time-Marginal expression of the Wigner-Ville distribution forr~1(t)-s~(t) is defined as (22)r~1t-s~t2=21-cos⁡2πν0x0Proof of Theorem2.
For simplicity in the following we denoter~(t) as r~1(t).
The complex form of a clock signal with constant phase offset is given by(23)r~1t=ei2πν0t+φ1twhere the phase variation considering the constant phase is defined by(24)φ1t=2πν0x0The terms presented in (18) are given by(25)φ1t+τ2=φ1t-τ2=2πν0x0By substituting (25) into (17) we obtain(26)Wr~1t,f=∫-∞∞e-i2πf-ν0τdτ=δf-ν0By substituting (25) into (20) we obtain(27)Wr~1,s~t,f=∫-∞∞ei2πν0τ+φ1t+τ/2e-i2πfτdτ=∫-∞∞ei2πν0τ+2πν0x0e-i2πfτdτ=ei2πν0x0δf-ν0Then, the real part of Wr~1,s~(t,f) is(28)ReWr~1,s~t,f=cos⁡2πν0x0δf-ν0By substituting (14), (26), and (28) into (13) we obtain(29)Wr~1-s~t,f=Wr~1t,f+Ws~t,f-2ReWr~1,s~t,f=2δf-ν01-cos⁡2πν0x0Let us derive a closed form expression for the Time Marginal of (29)(30)r~1t-s~t2=∫-∞∞Wr~1-s~t,fdf=21-cos⁡2πν0x0∫-∞∞δf-ν0df=21-cos⁡2πν0x0This completes our proof of Theorem 2.


### 4.2. Discrete Form

Theorem 3.
For the following assumptions:(1)
without loss of generality,A=A0=1,(2)
x0 is the constant phase offset from a reference clock signal,(3)
ν0 is the nominal frequency,(4)
s~[n] denotes the discrete complex form of an ideal clock signal,(5)
r~1[n] denotes the discrete complex form of the real clock signal for the case of constant phase offset.The discrete Time-Marginal expression of the Wigner-Ville distribution forr~1[n]-s~[n] is defined as(31)r~1n-s~n2=21-cos⁡2πν0x0Proof of Theorem3.
For discrete time and frequency variables the Wigner distribution is defined as [58](32)wr~n,m=ts∑k=-knknr~n+k2r~∗n-k2e-i2πmk/2Nwhere kn=min[2n,2N-1-2n], N denotes the number of samples, and n,m denotes the index numbers for the time and frequency vectors, respectively.
Let us derive a closed form expression for the discrete Wigner-Ville distribution (DWD) ofr~-s~:(33)wr~-s~n,m=DWDei2πν0n+φn-ei2πν0nBy using the “Summation Property” given in Appendix A (A.2) we obtain(34)wr~-s~n,m=wr~n,m+ws~n,m-2Rewr~,s~n,mLet us get first a closed form expression for the second component on the right side of (34)(35)ws~n,m=ts∑k=-knkns~n+k2s~∗n-k2e-i2πmk/2N=ts∑k=-knknei2πν0n+k/2e-i2πν0n-k/2e-i2πmk/2N=ts∑k=-knkne-i2πm-2Nν0k/2N=tsδm-2Nν0Let us derive the first component on the right side of (34)(36)wr~1n,m=ts∑k=-knknr~2n+k2r~2n-k2e-i2πmk/2N=ts∑k=-knknei2πν0n+k/2+φ1n+k/2e-i2πν0n-k/2+φ1n-k/2e-i2πmk/2N=ts∑k=-knkne-i2πm-2Nν0k/2N=tsδm-2Nν0Now we turn to the “cross-term” component of wr~-s~(n,m) which is the third component on the right side of (34).
The complex form of a discrete clock signal with constant phase offset is given by(37)r~1n=ei2πν0n+φnwhere the constant phase is defined by(38)φ1n=2πν0x0The term presented in (18) is given by(39)φ1n+k2=2πν0x0Then, the cross-term derivation is given by(40)wr~,s~n,m=ts∑k=-knkne-i2πmk/2Nei2πν0n+k/2+φn+k/2e-i2πν0n-k/2=ts∑k=-knknei2πν0k+φn+k/2e-i2πmk/2NBy substituting (39) into (40) we obtain(41)wr~1,s~n,m=ts∑k=-knknei2πν0k+φn+k/2e-i2πmk/2N=ts∑k=-knknei2πν0k+2πν0x0e-i2πmk/2N=ei2πν0x0tsδm-2Nν0Then, the real part of wr~1,s~(n,m) is(42)Rewr~1,s~n,m=cos⁡2πν0x0tsδm-2Nν0By substituting (35), (36), and (42) into (34) we obtain(43)wr~1-s~n,m=2tsδm-2Nν01-cos⁡2πν0x0Finally, let us drive the closed form expression for the Time Marginal of (43)(44)r~1n-s~n2=fs2N∑m=02N-1wr~1-s~n,m=fs2N2ts1-cos⁡2πν0x0∑m=02N-1δm-2Nν0=fs2N2ts1-cos⁡2πν0x02N=21-cos⁡2πν0x0This completes our proof of Theorem 3.As can be seen clearly, the discrete expression is the same as the continuous expression (22) introduced in the last subsection. This means that henceforth we can use only the continuous form and it will describe correctly the practical discrete signal processing.


## 4.1. Continuous Form

In this subsection, we will apply the analytic continuous form of the Wigner-Ville distribution onr~-s~ and then we will derive its Time-Marginal expression for a constant phase offset case.Theorem 2.
For the following assumptions:(1)
without loss of generality,A=A0=1,(2)
x0 is the constant phase offset from a reference clock signal,(3)
ν0 is the nominal frequency,(4)
s~(t) denotes the continuous complex form of an ideal clock signal,(5)
r~1(t) denotes the continuous complex form of the real clock signal for the case of constant phase offset.The Time-Marginal expression of the Wigner-Ville distribution forr~1(t)-s~(t) is defined as (22)r~1t-s~t2=21-cos⁡2πν0x0Proof of Theorem2.
For simplicity in the following we denoter~(t) as r~1(t).
The complex form of a clock signal with constant phase offset is given by(23)r~1t=ei2πν0t+φ1twhere the phase variation considering the constant phase is defined by(24)φ1t=2πν0x0The terms presented in (18) are given by(25)φ1t+τ2=φ1t-τ2=2πν0x0By substituting (25) into (17) we obtain(26)Wr~1t,f=∫-∞∞e-i2πf-ν0τdτ=δf-ν0By substituting (25) into (20) we obtain(27)Wr~1,s~t,f=∫-∞∞ei2πν0τ+φ1t+τ/2e-i2πfτdτ=∫-∞∞ei2πν0τ+2πν0x0e-i2πfτdτ=ei2πν0x0δf-ν0Then, the real part of Wr~1,s~(t,f) is(28)ReWr~1,s~t,f=cos⁡2πν0x0δf-ν0By substituting (14), (26), and (28) into (13) we obtain(29)Wr~1-s~t,f=Wr~1t,f+Ws~t,f-2ReWr~1,s~t,f=2δf-ν01-cos⁡2πν0x0Let us derive a closed form expression for the Time Marginal of (29)(30)r~1t-s~t2=∫-∞∞Wr~1-s~t,fdf=21-cos⁡2πν0x0∫-∞∞δf-ν0df=21-cos⁡2πν0x0This completes our proof of Theorem 2.


## 4.2. Discrete Form

Theorem 3.
For the following assumptions:(1)
without loss of generality,A=A0=1,(2)
x0 is the constant phase offset from a reference clock signal,(3)
ν0 is the nominal frequency,(4)
s~[n] denotes the discrete complex form of an ideal clock signal,(5)
r~1[n] denotes the discrete complex form of the real clock signal for the case of constant phase offset.The discrete Time-Marginal expression of the Wigner-Ville distribution forr~1[n]-s~[n] is defined as(31)r~1n-s~n2=21-cos⁡2πν0x0Proof of Theorem3.
For discrete time and frequency variables the Wigner distribution is defined as [58](32)wr~n,m=ts∑k=-knknr~n+k2r~∗n-k2e-i2πmk/2Nwhere kn=min[2n,2N-1-2n], N denotes the number of samples, and n,m denotes the index numbers for the time and frequency vectors, respectively.
Let us derive a closed form expression for the discrete Wigner-Ville distribution (DWD) ofr~-s~:(33)wr~-s~n,m=DWDei2πν0n+φn-ei2πν0nBy using the “Summation Property” given in Appendix A (A.2) we obtain(34)wr~-s~n,m=wr~n,m+ws~n,m-2Rewr~,s~n,mLet us get first a closed form expression for the second component on the right side of (34)(35)ws~n,m=ts∑k=-knkns~n+k2s~∗n-k2e-i2πmk/2N=ts∑k=-knknei2πν0n+k/2e-i2πν0n-k/2e-i2πmk/2N=ts∑k=-knkne-i2πm-2Nν0k/2N=tsδm-2Nν0Let us derive the first component on the right side of (34)(36)wr~1n,m=ts∑k=-knknr~2n+k2r~2n-k2e-i2πmk/2N=ts∑k=-knknei2πν0n+k/2+φ1n+k/2e-i2πν0n-k/2+φ1n-k/2e-i2πmk/2N=ts∑k=-knkne-i2πm-2Nν0k/2N=tsδm-2Nν0Now we turn to the “cross-term” component of wr~-s~(n,m) which is the third component on the right side of (34).
The complex form of a discrete clock signal with constant phase offset is given by(37)r~1n=ei2πν0n+φnwhere the constant phase is defined by(38)φ1n=2πν0x0The term presented in (18) is given by(39)φ1n+k2=2πν0x0Then, the cross-term derivation is given by(40)wr~,s~n,m=ts∑k=-knkne-i2πmk/2Nei2πν0n+k/2+φn+k/2e-i2πν0n-k/2=ts∑k=-knknei2πν0k+φn+k/2e-i2πmk/2NBy substituting (39) into (40) we obtain(41)wr~1,s~n,m=ts∑k=-knknei2πν0k+φn+k/2e-i2πmk/2N=ts∑k=-knknei2πν0k+2πν0x0e-i2πmk/2N=ei2πν0x0tsδm-2Nν0Then, the real part of wr~1,s~(n,m) is(42)Rewr~1,s~n,m=cos⁡2πν0x0tsδm-2Nν0By substituting (35), (36), and (42) into (34) we obtain(43)wr~1-s~n,m=2tsδm-2Nν01-cos⁡2πν0x0Finally, let us drive the closed form expression for the Time Marginal of (43)(44)r~1n-s~n2=fs2N∑m=02N-1wr~1-s~n,m=fs2N2ts1-cos⁡2πν0x0∑m=02N-1δm-2Nν0=fs2N2ts1-cos⁡2πν0x02N=21-cos⁡2πν0x0This completes our proof of Theorem 3.As can be seen clearly, the discrete expression is the same as the continuous expression (22) introduced in the last subsection. This means that henceforth we can use only the continuous form and it will describe correctly the practical discrete signal processing.


## 5. Constant Frequency Offset

In this section, we will derive the analytic continuous form for the Wigner-Ville distribution and its Time-Marginal expression for the frequency offset case in addition to the phase offset case.Theorem 4.
For the following assumptions:(1)
without loss of generality,A=A0=1,(2)
x0 is the constant phase offset from a reference clock signal,(3)
y0 is the constant fractional frequency offset from a reference clock signal,(4)
ν0 is the nominal frequency,(5)
s~(t) denotes the continuous complex form of an ideal clock signal,(6)
r~2(t) denotes the continuous complex form of the real clock signal for the case of constant phase and frequency offsets.The Time-Marginal expression of the Wigner-Ville distribution forr~2(t)-s~(t) is defined as (45)r~2t-s~t2=21-cos⁡2πν0x0+y0tProof of Theorem4.
For simplicity in the following we denoter~(t) as r~2(t).
The complex form of a clock signal with constant phase and frequency offsets is given by(46)r~2t=ei2πν0t+φ2twhere the phase variation considering the constant phase and frequency offsets is defined by(47)φ2t=2πν0x0+y0twhere we may define(48)φ2t±τ2=2πν0x0+y0t±τ2By substituting (48) into (18) we obtain(49)Δφ2τ=φ2t-τ2-φ2t+τ2=2πν0-y0τIn order to carry out the first term in the right side of (12), the expression in (49) is substituted into (19):(50)Wr~2t,f=∫-∞∞e-iΔφ2τ-i2πf-ν0τdτ=∫-∞∞e-i2πν0-y0τ-i2πf-ν0τdτ=δf-ν0y0+1Next, in order to carry out the third term in the right side of (12), the expression in (48) is substituted into (20):(51)Wr~2,s~t,f=∫-∞∞ei2πν0τ+φ2t+τ/2e-i2πfτdτ=∫-∞∞ei2πν0τ+2πν0x0+y0t+τ/2e-i2πfτdτ=ei2πν0x0+y0tδf-ν0y02+1Then, the real part of (51) is(52)ReWr~2,s~t,f=cos⁡2πν0x0+y0tδf-ν0y02+1By substituting (14), (50), and (52) into (13) we obtain(53)Wr~2-s~t,f=δf-ν0y0+1+δf-ν0-2cos⁡2πν0x0+y0tδf-ν0y02+1Now we turn to derive the Time-Marginal expression of (53)(54)r~2t-s~t2=∫-∞∞Wr~2-s~t,fdf=∫-∞∞δf-ν0y0+1df+∫-∞∞δf-ν0df-∫-∞∞2cos⁡2πν0x0+y0tδf-ν0y02+1df=21-cos⁡2πν0x0+y0tThis completes our proof of Theorem 4.As we can see the result that we get is a periodic signal with a linear period of1/[ν0(x0+y0t)]. In Section 7, we will examine the analytic expression versus the numerical simulation results in order to verify our expression.


## 6. Frequency Drift

Now we derive the analytic continuous form for the Wigner-Ville distribution and its Time-Marginal expression for the case of frequency drift along with constant phase and frequency offsets.Theorem 5.
For the following assumptions:(1)
without loss of generality,A=A0=1,(2)
x0 is the constant phase offset from a reference clock signal,(3)
y0 is the constant fractional frequency offset from a reference clock signal,(4)
ν0 is the nominal frequency,(5)
a is the linear frequency drift coefficient (a≠0),(6)
s~(t) denotes the continuous complex form of an ideal clock signal,(7)
r~3(t) denotes the continuous complex form of the real clock signal for the case of frequency drift.The Time-Marginal expression of the Wigner-Ville distribution for[r~3(t)-s~(t)] is defined as (55)Fora<0:r~3t-s~t2=2-24ν0a∫-∞∞cos⁡4πν0af-ν0a2t+y02+12-π4+2πν0x0+y0t+at22dfFora>0:r~3t-s~t2=2-24ν0a∫-∞∞cos⁡-4πν0af-ν0a2t+y02+12-π4+2πν0x0+y0t+at22dfProof of Theorem5.
For simplicity in the following we denoter~(t) as r~3(t).
The complex form of a clock signal with frequency drift along with constant phase and frequency offsets is given by(56)r~3t=ei2πν0t+φ3twhere the phase φ3(t) is defined by(57)φ3t=ω0x0+y0t+at22=2πν0x0+y0t+at22Based on (57), φ3(t±τ/2) can be written as(58)φ3t±τ2=2πν0x0+y0t±τ2+a2t±τ22Now, by substituting (58) into (18) we get(59)Δφ3τ=φ3t-τ2-φ3t+τ2=2πν0-y0τ-atτNext, by substituting (59) into (17) we obtain(60)Wr~3t,f=∫-∞∞e-iΔφ3τ-i2πf-ν0τdτ=∫-∞∞e-i2πν0-y0τ-atτ-i2πf-ν0τdτ=δf-ν0y0+at+1Substituting (58) into (20) we obtain(61)Wr~3,s~t,f=-∫-∞∞eiω0τ+φ3t+τ/2e-i2πfτdτ=-∫-∞∞eiω0τ+2πν0x0+y0t+τ/2+at+τ/22/2e-i2πfτdτ=-ei2πν0x0+y0t+at2/2Feiω0τ+2πν0y0τ/2+atτ/2ei2πν0aτ2/8where F{A} denotes the Fourier transform of A.
In the following we derive the outcome ofF{ei{ω0τ+2πν0(y0τ/2+atτ/2)}ei2πν0aτ2/8}. For that purpose we define(62)gτ=exp⁡iω0τ+2πν0y0τ2+atτ2(63)hτ=exp⁡i2πν0aτ28The Fourier transform of a product is defined by [59](64)Fgτhτ=Gf∗Hfwhere ∗ denotes convolution.
Based on (62), G(f) can be written as(65)Gf=∫-∞∞eiω0τ+2πν0y0τ/2+atτ/2e-i2πfτdτ=∫-∞∞e-i2πf+ν0-y0/2-at/2-1τdτ=δf-ν0y02+at2+1Based on (63), H(f) can be defined as(66)Hf=∫-∞∞ei2πν0aτ2/8e-i2πfτdτThe Fourier transform of a complex Gaussian is given by [59](67)Fork>0:Fe-ikτ2=πkexp⁡iπ2f2k-π4Fork<0:Fe-ikτ2=πkexp⁡-iπ2f2k-π4Fork=0:Fe-ikτ2=δfBased on (66) and (67) the expression for k is given by k=-πν0a/4.
Thus, based on (66) and (67) and that k=-πν0a/4 we have(68)Hf=∫-∞∞ei2πν0aτ2/8e-i2πfτdτ=Fora<0:Hf=4-ν0aexp⁡i4πf2-ν0a-π4Fora>0:Hf=π-πν0a/4exp⁡-iπ2f2-πν0a/4-π4By substituting (65) and (68) into (64) we obtain(69)Fora<0:Fgτhτ=δf-ν0y02+at2+1∗4ν0aexp⁡i4πf2ν0a-π4Fora>0:Fgτhτ=δf-ν0y02+at2+1∗4ν0aexp⁡-i4πf2ν0a-π4We use the following property [59]:(70)Gf∗Hf=∫-∞∞GρHf-ρdρNext, by substituting (69) into (70) we obtain(71)Fora<0:Gf∗Hf=∫-∞∞δρ-ν0y02+at2+14ν0aexp⁡⁡i4πν0af-ρ2-π4dρFora>0:Gf∗Hf=∫-∞∞δρ-ν0y02+at2+14ν0aexp⁡-i4πν0af-ρ2-π4dρAfter carrying out the integration in (71) we have(72)Fora<0:Gf∗Hf=4ν0aexp⁡i4πν0af-ν0y02+at2+12-π4Fora>0:Gf∗Hf=4ν0aexp⁡-i4πν0af-ν0y02+at2+12-π4By substituting (72) into (61) we obtain(73)Fora<0:Wr~3,s~t,f=-ei2πν0x0+y0t+at2/24ν0aexp⁡i4πν0af-ν0y02+at2+12-π4Fora>0:Wr~3,s~t,f=-ei2πν0x0+y0t+at2/24ν0aexp⁡-i4πν0af-ν0y02+at2+12-π4Then, the real part of Wr~3,s~(t,f) is(74)Fora<0:ReWr~3,s~t,f=-4ν0acos⁡4πν0af-ν0a2t+y02+12-π4+2πν0x0+y0t+at22Fora>0:ReWr~3,s~t,f=-4ν0acos⁡-4πν0af-ν0a2t+y02+12-π4+2πν0x0+y0t+at22By substituting (14), (60), and (74) into (13) we obtain(75)Fora<0:Wr~3-s~t,f=δf-ν0y0+at+1+δf-ν0-24ν0acos⁡4πν0af-ν0a2t+y02+12-π4+2πν0x0+y0t+at22Fora>0:Wr~3-s~t,f=δf-ν0y0+at+1+δf-ν0-24ν0acos⁡-4πν0af-ν0a2t+y02+12-π4+2πν0x0+y0t+at22Based on (8) and (75) we obtain(76)Fora<0:r~3t-s~t2=2-24ν0a∫-∞∞cos⁡4πν0af-ν0a2t+y02+12-π4+2πν0x0+y0t+at22dfFora>0:r~3t-s~t2=2-24ν0a∫-∞∞cos⁡-4πν0af-ν0a2t+y02+12-π4+2πν0x0+y0t+at22dfThis completes our proof of Theorem 5.


## 7. Numerical Simulations

In this section, we will show the simulation results for the continuous and discrete Time-Marginal Wigner-Ville distribution of the differences [r~1(t)-s~(t)] and [r~1[n]-s~[n]] compared to our analytical proposed expressions (22) and (31), respectively. In order to demonstrate different nominal frequencies, we simulate the Time-Marginal Wigner-Ville distribution of the differences [r~1(t)-s~(t)] for three different nominal frequencies.Figure2 shows the contour plot of the Wigner-Ville distribution of a reference 10MHz signal s~(t), where the sampling frequency is 40MHz.Figure 2
The contour plot of a reference 10MHz Wigner-Ville distribution signals~(t). The sampling frequency is 40MHz.Figures3–5 show for three different values of ν0 the comparison between the outcomes of the following expressions, (22) and (31), which were derived for the continuous and discrete cases, respectively.Figure 3
The Time-Marginal Wigner-Ville distributionr~1(t)-s~(t) versus the constant phase offset φ1(t)=2πν0x0. The nominal frequency ν0 is 10MHz and the sampling frequency is 40MHz.Figure 4
The Time-Marginal Wigner-Ville distributionr~1(t)-s~(t) versus the constant phase offset φ1(t)=2πν0x0. The nominal frequency ν0 is 5MHz and the sampling frequency is 20MHz.Figure 5
The Time-Marginal Wigner-Ville distributionr~1(t)-s~(t) versus the constant phase offset φ1(t)=2πν0x0. The nominal frequency ν0 is 20MHz and the sampling frequency is 80MHz.According to Figures3–5, there are high correlation between the outcomes of (22) and (31).Figures6 and 7 show the contour plot of (53), where the upper, lower, and the middle frequencies lines match the first, second, and the third terms of (53), respectively.Figure 6
The contour of the Wigner-Ville distribution of the differencer~2(t)-s~(t), where y0=0.2. The nominal frequency ν0 is 10MHz and the sampling frequency is 40MHz.Figure 7
The contour of the Wigner-Ville distribution of the differencer~2(t)-s~(t), where y0=0.1. The nominal frequency ν0 is 10MHz and the sampling frequency is 40MHz.Figure8 shows the contour plot of (75), for a=-12KHz, y0=0.15, ν0=10MHz, and sampling frequency of 40MHz. We can see clearly that the linear chirp line follows the first term of (75). The constant frequency line matches the second term of (75) and the cross-term frequencies (in the middle) correspond to the third term of (75).Figure 8
The contour of the Wigner-Ville distribution of the differencer~3(t)-s~(t), where a=-12KHz, y0=0.15, and x0=0. The nominal frequency ν0 is 10MHz and the sampling frequency is 40MHz.


## 8. Summary and Conclusion

In this paper, we have presented a new tool for frequency source characterization based on the Wigner-Ville distribution, whose objective is to represent the variation of clock noise when no stationarity assumption is made. This means that the new tool can be used for a nonstationary phase variation. We presented the analytic terms for the Wigner-Ville distribution and its Time Marginal for constant phase and frequency offsets and frequency drift from a reference source. Simulation results indicate that our derivations hold.



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*Source: 4942938-2018-07-11.xml*