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For nonstationary time series consisting of multiple time-varying frequency (TVF) components where the frequency of components overlaps in time, classical linear filters fail to extract components. The

The traditional linear filter is defined as

In general, traditional linear filters, such as the Butterworth filter, are time invariant. They are designed to extract components from stationary processes where the frequency behavior of the signal does not change with time. However, for time series data with time-varying frequency behavior (TVF), these filters can produce very poor results because the time-invariant nature of the filters does not properly account for the time-varying frequency behavior of the data. That is, these filters do not properly adjust the cutoff frequency with time according to the frequency behavior of the data. See discussion and examples in Xu et al. [

In order to address the filtering problem for nonstationary data with TVF, Xu et al. [

In Section

In this section, we define and discuss

A process

A process

Some special cases of

when

when

when

when

Jiang et al. [

Given a realization, the frequency change measured by the GIF can be visually represented using time-frequency plots. In this paper we will use the nonparametric Wigner-Ville plots for this purpose. These plots display the time-frequency behavior in the data by computing inverse Fourier transforms of windowed versions of the sample autocovariance function (see [

Xu et al. [

Fit a

Apply traditional time-invariant filtering methods to the dual data to extract components.

Convert filtered dual components back into the original scale.

See Xu et al. [

In Example

We consider the linear chirp process

(a) A realization from the linear chirp model in Example

TVF signal

Wigner-Ville plot for (a)

Low frequency component

Wigner-Ville plot for (c)

High frequency component

Wigner-Ville plot for (e)

To perform Step

(a) The dual data obtained from Figure

Data after time transformation

Wigner-Ville plot for (a)

Low-pass filtered data

Recovered high frequency TVF signal

The low and high frequency dual components are shown in Figures

(a) Recovered low frequency TVF component; (b) recovered high frequency TVF component; (c) Wigner-Ville plot for (a); (d) Wigner-Ville plot for (b).

Recovered low frequency TVF signal

Recovered high frequency TVF signal

Wigner-Ville plot for (a)

Wigner-Ville plot for (b)

It is important to note that we could not have recovered the TVF components by application of the Butterworth filter directly to the data in Figure

For more details concerning the filtering method used in Example

In Example

First, we discuss how the time-deformation technique will affect the spectrum of the data. Suppose that a component has a phase function

For a process consisting of two components, one with a phase function

More generally, it is important to note that regardless of whether

Here, we consider a simulated data set which is the signal consisting of the sum of two quadratic chirps,

(a) A realization of the process in Example

TVF signal

Wigner-Ville plot for (a)

Low frequency TVF component

High frequency TVF component

We apply the

(a) Data on transformed time axis; (b) the Wigner-Ville time-frequency distribution of (a); (c) data in (a) after low-pass filter; (d) data in (a) after high-pass filter.

Data after time transformation

Wigner-Ville plot for (a)

Low-pass filtered data from (a)

High pass filtered data from (a)

(a) Figure

Recovered low frequency TVF signal

Recovered high frequency TVF signal

Wigner-Ville plot for (a)

Wigner-Ville plot for (b)

Note that Example

(a) Sketch of upper and lower Wigner-Ville curves for Example

Dual plot for (a)

Dual plot for (c)

In Figures

In this section we discuss methods for extracting components in more complicated TVF settings in which components may enter and/or exit the overall signal at different times. The procedure will be illustrated in Example

In this example, we simulate a situation similar to that studied by Papandreou-Suppappola and Suppappola [

The overall signal consists of three TVF component signals. We will refer to these as Components 1, 2, and 3. Based on the Wigner-Ville plot we conclude that

component 1 exists for approximately

component 2 exists for approximately

component 3 exists for approximately

Components 1 and 3 have Wigner-Ville curves that appear to be straight lines, suggesting that each of these is a linear chirp signal.

Component 1 exists in the signal by itself for about

The frequencies associated with the components intersect at two places:

frequencies associated with components 1 and 2 intersect at about

frequencies associated with components 2 and 3 intersect at about

Figures

(a) A realization of the process in Example

Data for Example

Wigner-Ville plot for (a)

Component 1

Component 2

Component 3

Since component 1 exists by itself for

the first segment (1–50) which contains only the transformed component 1 signal;

the middle segment (51–100) where the frequencies intersect;

the last segment (101–300) where the frequencies are well separated. In this segment the transformed component 1 signal is consistently higher frequency than the transformed component 2 and component 3 signals. In this range, a constant cutoff frequency (e.g., 0.12) can be used to separate component 1 from the other two components.

In order to extract the stationarized dual signal associated with component 1 we note that no extraction is necessary for the first 50 dual time values (the only signal present is the one associated with the first linear chirp component). As suggested in note (iii) above, extraction of component 1 in the range 101–300 can be accomplished using a high-pass Butterworth filter.

Extraction of the first linear chirp component in the range 51–100 is more difficult. The procedure we have selected is to fit

(a) The dual data (1 : 300) after the first time deformation; (b) the Wigner-Ville time-frequency distribution of the dual; (c) the extracted stationary dual component; (d) the recovered first linear chirp component.

Dual data for Example

Wigner-Ville plot for (a)

Extracted dual component 1

Recovered chirp component 1

In the second step, we will extract component 2. We first subtract the recovered component 1 (shown in Figure

only TVF components 2 and 3 remain;

component 2 exists by itself for

Based on note (ii) above, we apply the method of Jiang et al. [

the second component seems to have been stationarized using this procedure (frequency behavior is represented by a horizontal line at about

the first segment (101–250) contains only the stationarized component 2;

transformed component 2 is consistently lower frequency than transformed component 3 for the range 251–280 and the signals could be separated by a constant cutoff frequency of about 0.075;

the frequencies intersect in the range 281–340;

the transformed component 2 is consistently lower frequency than transformed component 3 for the range 341–400. A constant cutoff frequency of about 0.175 could be used to separate the signals in this range.

Based on these comments we apply a high-pass Butterworth filter to extract component 2 in the range 101–280 and a low-pass Butterworth filter to extract the dual for component 2 for the range 341–400. Forecasting and backcasting are used as in Step

(a) The data (101 : 400) with the first filtered component subtracted; (b) the Wigner-Ville time-frequency distribution of the data; (c) the dual data; (d) the Wigner-Ville time-frequency distribution of the dual; (e) the extracted stationary dual component; (f) the recovered

Data (101–400) with First comp. subtracted

Wigner-Ville plot for (a)

Dual data (101–400)

Wigner-Ville plot for (c)

Extracted dual component 2

Recovered component 2

The third component is obtained by subtracting the two recovered components from the original data; that is,

Figure

(a), (c), and (e): recovered components for Example

Recovered first component (linear chirp)

Wigner-Ville plot for (a)

Recovered second component (

Wigner-Ville plot for (c)

Recovered third component (linear chirp)

Wigner-Ville plot for (e)

For nonstationary time series with multiple time-varying frequency structure, especially where the frequencies of components overlap over time, traditional linear filters are not able to successfully extract components. In order to address this problem, the

In practical situations, most nonstationary data with multiple time-varying frequency structure cannot be transformed to stationary by applying any time deformation. In this paper, we showed that the