KR20070108294A - A method for estimating phase angle of time series data by discrete fourier transform - Google Patents

Abstract

A method for estimating a phase angle of time series data by discrete Fourier transform is provided to be reliable and to be very efficient and to reduce calculation time by making a calculation process be simple using a discrete Fourier transformation algorithm. A method for estimating a phase angle of time series data by discrete Fourier transform includes the steps of: calculating a peak frequency and a peak value from a Fourier spectrum; calculating the phase angle of a Fourier spectrum corresponding to the peak value; estimating the phase angle of an important mode in the time series data by replacing the calculated phase angle of the Fourier spectrum with the phase angle of a specific mode in the time series data; and printing the estimated result. A system for implementing the method consists of an input processing unit(10), a discrete time generating unit(40), an operating unit(20) and an outputting unit(30).

Description

A method for estimating phase angle of time series data by discrete Fourier transform}

1 is a schematic block diagram of hardware for carrying out the present invention;

2 is a flowchart briefly illustrating an operation process of an embodiment of the present invention;

3 is the Fourier spectral magnitude of an aperiodic signal,

4 shows the peak frequency and Fourier spectral magnitude for discrete signals,

5 is a peak frequency and Fourier spectral phase for the discrete signal,

6 is a discrete time series signal for an embodiment,

7 is a Fourier spectral magnitude for an example,

8 is a Fourier spectral phase for an embodiment,

9 is a case where the discrete Fourier transform spectrum is asymmetrical,

10 is a case where the resultant discrete Fourier transform spectrum is symmetrical.

<Explanation of symbols for the main parts of the drawings>

10: input processing unit 20: arithmetic unit

30: output unit 40: discrete time generation unit

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a phase estimation method of time series data by Fourier transform and relates to system identification or statistical signal processing for estimating modes or parameters included in discrete time series data.

Until now, the method of estimating a parameter from time series data can be largely divided into a parametric method and a non-parametric method. The parametric method estimates a parameter (mode) included in discrete data by calculating a complex exponential from a linear prediction matrix obtained by converting a signal into an autoregressive moving average (ARMA) model. Way.

The typical parametric method is the Prony method, which requires a lot of computational time because the linear prediction equations must be solved and the solutions of higher-order equations must be solved when the discrete signal is fitted as a complex exponential. You need a computer with a large storage area. In addition, this method has an error in noise and the parameter is sensitively changed according to the interval of signal data and the size of data.

Discrete Fourier transforms are widely used in a variety of industries and are mainly used to calculate frequencies in signals. The nonparametric method using the discrete Fourier transform is a method of estimating a parameter included in the discrete data by repeatedly applying the Fourier transform. The nonparametric method repeatedly estimates the vibration mode corresponding to each frequency by applying the discrete Fourier transform and continuously calculates the magnitude of the spectrum corresponding to each peak and then applies the curve fitting to correspond to the peak frequency. Estimate the mode. However, there are disadvantages in that a number of Fourier transforms are required and a large error occurs when spectral leakage occurs. In particular, a method of estimating a phase corresponding to each mode to be implemented in the present invention has not been developed until now.

The conventional methods include complex numerical algorithms such as linear equations and least squares methods, which require a lot of computation time and require large computer storage.

In order to solve the problems of the prior art, the present invention provides a method for estimating the phase of a direct mode directly from the discrete Fourier spectrum. Since the discrete Fourier transform algorithm is used, it is reliable and the calculation process is very simple. The purpose of the present invention is to provide a method for estimating the phase for each mode included in the time series data as well as greatly reducing the computation time.

[Hardware for Carrying Out the Invention]

FIG. 1 is a schematic block diagram of hardware for carrying out the present invention. As can be seen from the drawings, the hardware for carrying out the present invention includes an input processing unit 10, an operation unit 20, and an output unit ( 30) and the discrete time generator 40 may be configured.

The input processor 10 may analog-to-digital convert time series data of an analog type to be input to the calculator 20.

The calculator 20 performs a discrete Fourier transform on time series data input from the input processor 10 to estimate a mode and estimates a phase of time series data corresponding to each mode from the time series data. .

The output unit 30 outputs the estimation result of the phase estimated by the calculation unit 20.

The discrete time generator 40 generates discrete time data, which is time information necessary for Fourier transform of the calculator, and inputs the discrete time data to the calculator 20.

Hereinafter, an embodiment of the present invention performed through the hardware configured as described above will be described.

The present invention relates to a method for directly obtaining a phase for each mode representing a state of a system in a discrete Fourier transform of time series data, and includes an embodiment of estimating a phase from a Fourier spectrum.

First, the principle applied to the embodiment of the present invention will be described. The Fourier transform equation for the continuous function is shown in Equation 1.

The signal given as the actual time series data is usually the aperiodic signal rather than the periodic signal. The discrete Fourier transform of the signal x [n] defined in the interval 0 ≦ n ≦ N-1 is shown in Equation 2.

In Equation 2, W _N = exp (−j 2π / N), and the spectral coefficient X [k] is defined in the interval 0 ≦ n ≦ N−1. On the other hand, if the signal is aperiodic, the spectrum of the Discrete Fourier Transform is represented continuously. Most of the natural signals are aperiodic, so the spectrum is represented continuously. Therefore, in the Fourier spectrum obtained as a result of the Discrete Fourier Transform, the left and right spectrums centered on the peak spectrum become an envelope form (FIG. 3).

Therefore, in the present invention, the left and right spectrum including the peak spectrum in the Fourier spectrum is assumed as a complex exponential decay function (Equation 3 or Equation 4). Therefore, the frequency and the braking coefficient included in the signal can be estimated from the Fourier spectrum of the complex attenuation function, and the phase corresponding to each mode can be estimated (Fig. 2).

If the continuous function x (t) is an exponential decay cosine function having a braking coefficient α, a magnitude A, and a phase Φ ₁ at a frequency ω ₁ , it can be expressed as Equation 3. In addition, an exponential decay sine function having a phase Φ ₂ can be expressed as Equation (4).

First, in order to consider the exponential decay cosine function, Equation 3 is Fourier transformed.

A phase corresponding to the frequency ω ₁ is called Φ ₁ , and when the braking coefficient is small and ω ₁ ≫α is established, it is expressed by Equation (6).

In summary, this equation is simply expressed as in Equation (7).

Therefore it can be expressed by separating the size X of the spectrum and angle _ω1 ∠X (ω _1), shown in Equation (8), equation (9).

Thus, in an exponential decay cosine function with phase φ ₁ corresponding to frequency ω ₁ , the phase is equal to the phase of the Fourier spectrum.

Meanwhile, the exponential decay sine function shown in Equation 4 is expressed as a cosine function as follows.

Therefore, the phase of the exponential attenuation function in the relationship between the phase of the exponential decay cosine function and the spectral phase represented by Equation 9 can be expressed as shown in Equation 11.

Therefore, the relationship shown in Equation 12 is established between the phase of the exponential attenuation function and the Fourier spectrum phase.

In Equation 12, it can be seen that the phase of the exponential decay function is delayed by 90 ° from the phase of the Fourier spectrum.

Therefore, the phase in the parameters included in the discrete time series data can be estimated directly from the Discrete Fourier Transform for the given time series data. When discrete discrete time series data is discrete Fourier transformed, it can be represented by the magnitude of the Fourier spectrum as shown in FIG. 4 and the Fourier spectral phase as shown in FIG. 5. If the peak frequency ω ₁ is a frequency corresponding to the important mode, the phase becomes φ ₁ so that the phase can be estimated very easily.

The following verify the accuracy and efficiency of the present invention through the examples. FIG. 6 shows data sampled for 30 seconds (N = 1796) with a sampling interval of 1/60 sec for the time function y (t) = 1.0e ^{-0.1 t} cos (4.2t + 30 °). . It can be seen from the figure that it is an exponentially decaying sinusoidal wave. FIG. 7 is a diagram illustrating a power spectrum obtained by applying a discrete Fourier transform to a signal shown in FIG. 6, and has a form similar to that of a Fourier spectrum. This shows that the peak frequency is near 0.67 Hz (4.2 [rad / sec]). In other words, the peak of the power spectrum is formed around 0.67Hz, it can be seen that the frequency of the critical mode is 0.67Hz.

8 illustrates the phase of the Fourier spectrum. It can be seen that the phase for the frequency corresponding to the peak spectrum is formed below about 40 °, and the numerical calculation accurately estimates 32.2 °, resulting in a result close to the initial given phase of 30 °. That is, since the phase corresponding to the important mode is the same as the phase of the spectrum, it can be seen that the phase of the important mode can be estimated by simple complex calculation as shown in FIG. 8, and the efficiency of the present invention can be confirmed.

On the other hand, in the Discrete Fourier Transform, when a given number of data is small, leakage of the spectrum occurs. As a result of the Discrete Fourier Transform, assuming that there is no mode affecting the peaks around significantly, if the spectral leakage is large, the peak left and right spectra become asymmetric about the peak value [Fig. 9]. , The left and right spectrums are symmetrical around the peak [FIG. 10]. That is, when the left and right spectrums of the peak spectrum are asymmetric, as shown in FIG. 9, since the peak frequency is not an accurate value, the phase corresponding to the peak frequency is also not an accurate value. However, if the left and right spectrums of the peak spectrum are close to symmetry as shown in FIG. 10, the peak frequency is an accurate value, and thus the phase calculated using the peak spectrum is an accurate value. Therefore, the accuracy of the phase of the mode estimated from the Fourier spectral phase can be judged from the symmetry of the spectrum. There is shown by the size of the size X ₁ X ₂ and a right spectrum (S220) of the peak left spectrum (S210) to quantify the degree of symmetry in 9 equal to the equation (13). When the magnitude of the spectrum is X ₂ > X ₁ ,

If we define a new variable like

Equation 13 is summarized as a function of spectral deviation as shown in Equation 15.

In Equation 15, the closer the ΔX _idx is to 0%, it may be determined that the phase estimated from the spectrum is an accurate value, and the closer the ΔX _idx is to 100%, the more determined the estimated phase may be incorrect.

As described above, the present invention can estimate a phase representing the characteristics of a dynamic system directly in the Fourier transform of time series data. Unlike conventional methods, since the present invention does not rely on iterative simulation, the Fourier transform is not repeatedly calculated. In one discrete Fourier transform, a phase corresponding to each mode included in the signal may be estimated.

That is, since the present invention greatly shortens the phase estimation time, performance can be improved in a system requiring fast parameter estimation in real time. In particular, since the present invention is a simple and clear algorithm based on a mathematical formula, there is an effect that can be easily applied to any system that applies the Discrete Fourier Transform.

Claims (4)

In the phase estimation method of time series data by the Discrete Fourier Transform, Compute peak and peak values from the Fourier spectrum, Calculate the phase of the Fourier spectrum corresponding to the peak, Estimate the phase for the critical mode included in the time series data by substituting the calculated Fourier spectral phase with the phase for the particular mode included in the time series data, And a step of outputting the estimation result. The phase estimation method of time series data using a Fourier transform. The method of claim 1, wherein the exponential decay cosine function If you simulate with x (t) = Ae ^{-α t} cos (ω ₁ t + φ ₁ ), Phase φ ₁ corresponding to peak frequency ω ₁ is the discrete Fourier transform of the exponential attenuation cosine φ ₁ = ∠X (ω ₁ ) Estimate from the phase of the Fourier spectrum, And a step of outputting the estimation result. The phase estimation method of time series data using a Fourier transform. The exponential decay sign function of claim 1, wherein the time series data If you simulate with x (t) = Ae ^{-α t} sin (ω ₁ t + φ ₂ ), The phase φ ₂ corresponding to the peak frequency ω ₁ is the discrete Fourier transform of the exponential decay sine function. φ ₂ = ∠X (ω ₁ ) + π / 2 Estimate the phase of the Fourier spectrum, And a step of outputting the estimation result. The phase estimation method of time series data using a Fourier transform. In the phase estimation method of time series data by the Discrete Fourier Transform, Compute peak and peak values from the Fourier spectrum, By comparing the left and right Fourier spectral magnitudes of the peak spectrum, In determining the accuracy of the phase for the critical modes included in the estimated time series data, Judgment is made using the relative deviation between the peak and left and right spectrum, And a step of outputting the result of the determination, wherein the phase estimation method for time series data by Fourier transform is included.

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