Analysis of the Precision of Bispectral Estimation Methods

In order to evaluate the accuracy of bispectral estimation method, signals of cosine function were adopted. Because the cosine signals’ three order moment spectrum and three order cumulant spectrum are zero, Non zero part of the bispectrum estimated by no matter which method is the estimation error. Through the comparison of three kinds of estimation methods: the direct method, indirect method and AR parameter method, errors of various estimation methods were obtained, then changing the values of different parameters in all methods, and observing the bispectral error values changes with the parameters, so as to provide a basis for the various bispectrum estimation method and the selection.

The occurrence of FFT (fast fourier transform) in 1965 produced the periodogram; the fatal weakness of BT and periodogram is the limitation of frequency resolution. In order to overcome this shortcoming, Burg was inspired by the linear prediction method in the study of earthquake application in 1967, and derived the maximum entropy spectrum estimation method. E.p. arzen formally proposed the AR spectrum estimation method in 1968; Since then, many high resolution spectral estimation methods have been developed in the past decade, which is called the modern spectral estimation method, and the BT method and the periodogram are called the traditional spectral estimation method.
Either way, each spectrum estimation technique can be considered a model method. Specifically, based on the prior knowledge of the process, a model of approximate actual process is established. Secondly, the model parameters of the hypothesis are estimated by observation data or autocorrelation function, finally to make a spectrum estimation [1] [2].
The commonly used models include periodogram and BT (sine harmonic summation model), AR (Auto Regression Model), MA (the moving average model), ARMA (the Auto Regression Integrated with Moving), Prony, maximum likelihood, etc. The variation of performance between various estimates is caused by the difference between the hypothetical model and the actual process. Although different models can produce similar results, some models may need fewer parameters.
It is general to estimate the high order cumulant spectrum or high order moment spectra using finite length data. There are two main methods: 1) Conventional (Fourier) method; 2) Parameter methods: AR, MA, ARMA and other methods. This paper discusses the conventional method and the double spectrum estimation effect of AR parameter method. Conventional methods can be divided into the following two categories: 1) Indirect method: based on the definition of Formula (6) or Formula (7) to estimate; 2) Direct method: based on the definition of. Formula (8) to estimate. Due to the inevitable error of various estimation methods, this paper will analyze the error of these methods by the double spectrum estimation of cosine sig-

High Order Cumulant
If {x(n)} is the zero mean k order stationary random process, the k order cumulant ( ) Here, mom() represents the joint moment, the third order cumulant is: The k-order cumulant spectrum is defined as k − 1 dimension Fourier transform of k order cumulant.
k order moment spectrum is defined as k − 1 dimension Fourier transform of k order moment.
The most commonly used high order spectrum is the third-order spectrum (also known as the double spectrum), and the estimation formula as follows: For periodic power signal, the high order moment spectrum is generally estimated, order (7) can be used to calculate the double spectrum by the following formula:

1) AR parameter method
Assuming y 1 (t) is the signal of the actual output signal y(t) of the system, the random vibration signal of the system output is caused by the non-gaussian white noise a(t), whose mean value is equal to zero, and establishes the AR model: is the optimal order of the AR model.
2) The direct method For deterministic signals, the step of the direct method of double spectrum estimation is: Step 1. Divide the data into K segments. Each segment contains M sample points, i.e., N = K * M, and the date of every piece is averaged out. If it's a deterministic signal, then only one record (N = M) is used. If you want to get the length that the FFT algorithm fits, a zero can be added to each piece of data.
Step 2. Suppose Step 3. The final K segment data is averaged to get a double spectrum of the given data.
3) The indirect method For deterministic signals, the two-spectral estimation indirect method is: Step 1. Divide the data into K segments, and each segment has M sample points, i.e., N = K • M; Step 2. the date of every piece is averaged out; Step 3. Suppose Step 4. Double spectrum estimation is performed according to Equation (7).

Two Spectra of Cosine Signal
Supposecosine signal ( ) cos x n t ω = , By third-order cumulants calculation for- can be known, cosine signal of third-order cumulants is 0, so whatever according to type (7) or type (8), the double spectrum are calculated for 0. See literature (5) and (6).

Double Spectrum Estimation Error Analysis of Cosine Signal
It can be seen from the above analysis that the cosine signal is zero for both its third-order cumulant or the third-order moment spectrum. Based on the above three estimation methods. Due to the known cosine signal bispectrum is zero, so no matter what kind of method to estimate the cosine signal power spectrum, as long as there is not zero, it can be regarded as the estimate method of error.
First, the AR parameter method is used to estimate.
This article takes the cosine signal y(t) = cos(2 * PI * f 0 * t), respectively in f 0 1 HZ, 5 HZ and 50 HZ, the sampling frequency is 1000 HZ, this kind of selection is mainly in random, and considering the interval of frequencies simultaneously, the sampling time is 1 second, adopt the AR parameters method, direct method and indirect method for double spectrum estimation, respectively is shown in Figures 1-3. Can be seen from the above image, when f 0 to 1 HZ, estimate the double spectrum distribution minimum non-zero part, but the value is bigger.
Under the condition of the sampling frequency must, therefore, with the increase of signal frequency, whatever estimation method, the double spectrum error estimate its distribution range is increased, but the value is the trend of de-       which indirectly proves that the Frobenius norm value to 0.995 as the rationality of the threshold. As can be seen from the above analysis results, the estimation effect of parametric method is slightly better than that of direct and indirect methods.

Brief Summary
Based on the spectrum and the theoretical value of zero cosine signal bispectrum estimation, three kinds of methods for analysis of the various led to the error of estimation methods, and by changing the parameters, analysis of different estimation methods in different cases, the change of the error, to the precision of estimation methods provide a judgment standard [5] [6].