1/f spectrum estimation based on α-stable distribution in colored Gaussian noise environments

Abstract

The main goal of this paper is to suppress the effect of unavoidable colored Gaussian noise on declining accuracy of transistor 1/f spectrum estimation. Transistor noises are measured by a nondestructive cross-spectrum measurement method, which is first to amplify the voltage signals through ultra-low noise amplifiers, then input the weak signals into data acquisition card. The data acquisition card collects the voltage signals and outputs the amplified noise for further analysis. According to our studies, the output 1/f noise can be characterized more accurately as non-Gaussian α-stable distribution rather than Gaussian distribution. Therefore, by utilizing the properties of α-stable distribution, we propose a cross-spectrum method effective in noisy environments based on samples normalized cross-correlation function. Simulation results and diodes output noise spectrum estimation results confirm the effectiveness of our method.

Introduction

Spectrum estimation of inherent low-frequency 1/f noise has been generally expected to be qualified for indicating the reliability of diverse transistors. Therefore, as an important indicator of reliability, spectrum estimation of transistors’ output noise with its dominant 1/f noise has aroused great attention and a great deal of research has been carried out by Vandamme, ¹ Jevtic, ² and Dai. ³ The principal feature of 1/f noise spectrum is that it increases monotonically with decreasing frequency. The low-frequency noise of transistor is manifested as voltage fluctuation at both ends of the transistor. Traditional method of accelerating electrical aging technique has caused a certain degree of internal damage to transistors and affects the reliability. Therefore, the noise analysis method is proposed to measure the noise of the transistor under regular working conditions. This method possesses advantages of high efficiency and is nondestructive to the test samples, thus it becomes the universal noise measurement method. The measurement technique of noise analysis is to filter the voltage signals through the capacitor in ultra-low noise amplifier to filter the DC voltage, and then the remaining weak AC voltage signal is amplified and input into DAQ. The DAQ collects the voltage signal and outputs the amplified signal to computer for further analysis. Amplifier will introduce background colored Gaussian noise (CGN) into the output signal, which obviously leads to the decline of 1/f noise spectrum estimation accuracy in noisy environments.

Previous studies have assumed that 1/f stochastic process can be described by Gaussian distribution. Gaussian assumption makes it possible to use second-order moments and higher order cumulants as signal processing tools, which greatly facilitates estimating the spectrum of transistor noise. Unfortunately, according to our studies, in the case of zener diode, its output 1/f noise can be characterized more accurately as non-Gauss α-stable process ⁴ rather than Gaussian process. Thus, Gaussian assumption is proved to be inadequate for spectrum estimation. The diode itself does not have the function of amplification, so it is more difficult to measure and estimate 1/f noise spectrum than other kinds of transistors. In this paper, once we break through the problem of spectrum estimation of zener diode noise, then the spectrum estimation problem of other kinds of transistor noises can be naturally solved. α-stable distribution is one of the classes, which meet the generalized central limit theorem, and Gaussian distribution is its special case (while $\alpha =2$ ). Thus, α-stable distribution has more extensive applicability and more universality than Gaussian distribution. In this paper, we refer in particular to non-Gaussian α-stable noise as α-stable noise except the case when α equals 2. Relationships between 1/f noise and α-stable distribution have been established in Yin and Pethpulu⁵ and Takayasu. ⁶ They proved that 1/f noise belongs to α-stable distribution, and the characteristic index of α-stable distribution is a simple function of the exponent of 1/f spectrum. The spectrum estimation method in Wang ⁴ and Takayasu ⁶ give no considerations of the problem of 1/f spectrum estimation in Gaussian noisy environments.

Since α-stable process is assumed to be generated by exciting an autoregressive (AR) process^7–9 with independent and identically distributed (i.i.d.) symmetric stable noise ( $S\alpha S$ ),^10–12 we deal specifically with the problem of 1/f noise spectrum estimation based on AR $S\alpha S$ model unavoidably under additive colored and/or white Gaussian noise background. It should be pointed out that similar problems exist widely in radar, ¹³ sonar, wireless communication,⁵ geological exploration and medical science, ¹⁴ seismology, ¹⁵ signal processing, and so on. Therefore, research of this problem has certain universality.

For an α-stable process, due to its absence of fractional moments of order $p$ with $p\succ \alpha$ , where $\alpha$ stands for the characteristic exponent, existing signal processing methods based on second-order and higher order statistics become invalid. Fortunately, the fractional lower order moments (FLOM) ¹⁶ exist in α-stable process, thus available parameter estimation methods of α-stable process are basically based on FLOM and its derived fractional lower order statistics ¹⁷ (FLOS), e.g. fractional lower order covariance (FLOC) and covariation. Correspondingly, the spectrum representation of an α-stable process consists of FLOC spectrum, ¹⁸ covariation spectrum, ¹⁹ and α-spectrum.¹¹ Among them, α-spectrum method does not directly employ Fourier transform to estimate spectrum. This method regards α-stable process as a rational system generated by i.i.d. $S\alpha S$ process to produce.^20–23 α-spectrum method avoids the disadvantages of low spectrum resolution, spectrum window leakage, and frequency mixing in Fourier transform as its inherent defects. Therefore, we adopt α-spectrum method to estimate 1/f noise spectrum.

Although FLOS-based methods can effectively solve the parameter estimation problem of α-stable process, there exists a common problem: such methods can only solve the parameter estimation problem under noise-free environments, thus it cannot be used to solve the parameter estimation problem of α-stable process in colored and/or white Gaussian noise environments pointed out in this paper. This common problem is due to the inherent nonlinear property of FLOS itself, so FLOS-based methods cannot be adopted to solve the parameter estimation problem of α-stable in noisy environments.

To solve the above problem, a samples normalized cross-correlation (SNCC)-based parameter estimation algorithm of AR model in CGN is developed, and based on that, we propose a cross-spectrum method effective in noisy environments based on SNCC function, and comprehensive studies will be carried out on the diodes noise measurement circuit, the definition of SNCC function, the AR model parameter estimation method of diodes output noise, and its performance. The simulation results and diodes output noise spectrum estimation confirm the effectiveness of the proposed method.

The structure of this paper is organized as follows. First, the observed data model is given. Then, the definition of SNCC, its operation properties, and the suppression of CGN is analyzed. Also, an SNCC-based parameter estimation algorithm of AR model in CGN is developed, and spectrum estimation method is given on that basis. Then, experiments are carried out on both simulation data and diodes noise. Finally, some conclusions are drawn in the last section.

Observed data model

The observed data are output diodes signal with additive system noise. Output diodes signal is the measured weak voltage, reflected as the level of the AC voltage at both ends of a diode, including 1/f noise and white noise. Noise measurement technique adopted by us is nondestructive noise analysis method, which is to filter the voltage signals through ultra-low noise amplifier, and then the weak noise is amplified and input into data acquisition card (DAQ). The DAQ collects the voltage signal from sensors and outputs the amplified signal for further analysis. Amplifier will introduce its own background noise into the output signal, which obviously leads to the decline of spectrum estimation accuracy. In order to solve this problem, a method of cross-spectrum measurement technique was proposed by Dai ³ to effectively eliminate the influence of noise produced by the amplifiers on the spectrum estimation accuracy. The specific experimental platform is shown in a later section. The main facilities we adopted are test sample circuit and shielding box, two amplifiers (which lead to a single-channel input and two-channel output linear time invariant system) and a DAQ, as depicted in Figure 1.

Figure 1.

Noise measurement platform.

From Figure 1, we can find that the dual-channel output diodes signals, which pass through different amplifiers, can be expressed by α-stable noise and additive white Gaussian noise (WGN) through the same linear system, and both signals are corrupted by different additive colored Gaussian noise (CGN) introduced respectively by the two amplifiers. The two different CGN of amplifiers are independent, because the amplifiers are powered by independent electric pools in this cross-spectrum measurement technique. This measurement method can improve the accuracy of spectrum estimation by effectively eliminating the influence of zero-mean colored and/or white Gaussian noise on it.

The output 1/f noise can be modeled as α-stable process. Yin and Pethpulu⁵ and Takayasu ⁶ have shown that a l/f process can be expressed as a discretized fractional integral of white noise, and follows an α-stable model. According to our studies, in the case of zener diodes, the experimental results show that there holds pulse characteristic in the time domain of the diode 1/f noise. Through the examinations of infinite variance test, the consistency of the model parameter estimation results, the statistical characteristics of 1/f noise signal, and the combination of hypothesis test and actual experiment, 1/f noise signal is verified to satisfy the α-stable distribution. ⁴

Since α-stable process can be expressed by an AR process driven by $S\alpha S$ , the single-channel observed time series is modeled as the output of an AR system excited by an unobservable $S\alpha S$ input, and corrupted by additive CGN (the outcome of adding up both colored and white Gaussian noise). The AR process of $X\left(n\right)$ is described by ²⁴

$$X\left(n\right)=-{\displaystyle \sum _{k=1}^{p}a\left(k\right)}X\left(n-k\right)+U\left(n\right)$$

(1)

where input $U\left(n\right)$ is i.i.d. $S\alpha S$ with characteristic exponent $\alpha \left(1\prec \alpha \prec 2\right)$ , $a\left(k\right)$ is the AR coefficients, $p$ s the order of AR model. By adding CGN, then

$$Y\left(n\right)=X\left(n\right)+N_1\left(n\right)$$

(2)

where $N_1\left(n\right)$ is the i.i.d. CGN, with variance $\sigma 12$ .

By passing through dual amplifiers, two-channel observed data are yielded. We address the problem of estimating AR coefficients of processes $Y\left(n\right)$ and $Z\left(n\right)$

$$Y\left(n\right)=X\left(n\right)+N_1\left(n\right)=-{\displaystyle \sum _{k=1}^{p}a\left(k\right)X\left(n-k\right)}+U\left(n\right)+N_1\left(n\right)$$

(3)

$$Z\left(n\right)=X\left(n\right)+N_2\left(n\right)=-{\displaystyle \sum _{k=1}^{p}a\left(k\right)X\left(n-k\right)}+U\left(n\right)+N_2\left(n\right)$$

(4)

given only the two-channel observed data $Y\left(n\right)$ and $Z\left(n\right)$ , where $X\left(n\right)$ is the output diode noise, $N_1\left(n\right)$ and $N_2\left(n\right)$ are additive CGN. The transfer function of this AR system is: $H\left(z\right)=\frac{1}{1+a\left(1\right)z^{-1}+\cdots +a\left(p\right)z^{-p}}$ , $p$ is the order of AR model.

AR model assumptions are as follows:

[AS1] The order ( $p$ ) of AR model is determined;

[AS2] The system is causal, linear time invariant, exponentially stable, and minimum phase, i.e. the roots of the $A\left(z\right)$ polynomial lie within the unit circle; in the complex Z domain, the transfer function $H\left(z\right)$ of the model conforms to pole-zero cancellations, which means all poles lie inside of the unit circle.

[AS3] $a\left(0\right)=1$ . $a\left(1\right),\cdots ,a\left(p\right)$ are nonzero.

[AS4] The input $U\left(n\right)$ is considered to be unobservable, zero-mean, i.i.d. non-Gaussian, and α-stable process with characteristic exponent $\alpha \left(1\prec \alpha \prec 2\right)$ .

[AS5] The 1/f signal $X\left(n\right)$ is considered to be unobservable, zero-mean, i.i.d. non-Gaussian, and α-stable process, with characteristic exponent $\alpha \left(1\prec \alpha \prec 2\right)$ , which is the same as $U\left(n\right)$ .

[AS6] The additive noise $N_1\left(n\right)$ and $N_2\left(n\right)$ are unobservable, zero-mean colored Gaussian processes respectively with variance ${\sigma }_1^2$ and ${\sigma }_2^2$ , and they are respectively independent of the input $U\left(n\right)$ .

[AS7] The noisy output $Y\left(n\right)$ and $Z\left(n\right)$ are considered to be observable, zero-mean, i.i.d. non-Gaussian and α-stable processes, with characteristic exponent ${\alpha }_Y\left(1\prec {\alpha }_Y\prec 2\right)$ and ${\alpha }_Z\left(1\prec {\alpha }_Z\prec 2\right)$ , which is different from $X\left(n\right)$ and $U\left(n\right)$ .

The dual-channel observed output data $Y\left(n\right)$ and $Z\left(n\right)$ are measured by cross-spectrum measurement technique, which can improve the accuracy of spectrum estimation through a method effectively eliminating the influence of zero-mean CGN on it, and the specific method will be elaborated in the following section.

Samples normalized cross-correlation and its operation properties

Definitions of SNSC and SNCC

From above, we can conclude that the accuracy of spectrum estimation can be improved by a method utilizing cross-spectrum measurement technique, while the method can achieve such a goal by means of SNCC function.

The definition of SNCC is analog to that of samples normalized self-correlation (SNSC). According to literature,^20,24 SNSC $R_{xx}$ of $X\left(n\right)$ converges in probability to certain constants as $n\to \infty$

$$R_{xx}=\frac{\frac{1}{N}{\displaystyle {\sum }_{n=1}^N\left[X\left(n\right)X\left(n+m\right)\right]}}{\frac{1}{N}{\displaystyle {\sum }_{n=1}^N\left[X^2\left(n\right)\right]}}=\frac{{\displaystyle {\sum }_{n=1}^N\left[X\left(n\right)X\left(n+m\right)\right]}}{{\displaystyle {\sum }_{n=1}^N\left[X^2\left(n\right)\right]}}$$

(5)

where $N$ is sample number, and $\left|m\right|$ is far smaller than N

The following identities can be used to establish the existence and boundedness of SNSC

$${\displaystyle \sum _{n=1}^N{\left[X\left(n\right)\pm X\left(n+m\right)\right]}^2}\ge 0$$

(6)

By expanding (6)

$${\displaystyle \sum _{n=1}^N{\left[X\left(n\right)\pm X\left(n+m\right)\right]}^2}={\displaystyle \sum _{n=1}^N\left[X^2\left(n\right)\right]}+{\displaystyle \sum _{n=1}^N\left[X^2\left(n+m\right)\right]}\pm 2{\displaystyle \sum _{n=1}^N\left[X\left(n\right)X\left(n+m\right)\right]}$$

(7)

In virtue of $X\left(n\right)$ satisfying α-stable distribution, it has a stable ergodicity, ²⁵ when $N\to \infty$ , it shows that

$${\displaystyle \sum _{n=1}^N\left[X^2\left(n\right)\right]}={\displaystyle \sum _{n=1}^N\left[X^2\left(n+m\right)\right]}$$

(8)

In order to ensure the existence of $E\left[X\left(n\right)\right]$ , the bounds for $\alpha$ is: $1\le \alpha \prec 2$ . Substituting equation (8) into equation (6)

$${\displaystyle \sum _{n=1}^N{\left[X\left(n\right)\pm X\left(n+m\right)\right]}^2}=2{\displaystyle \sum _{n=1}^N\left[X^2\left(n\right)\right]}\pm 2{\displaystyle \sum _{n=1}^N\left[X\left(n\right)X\left(n+m\right)\right]}\ge 0$$

(9)

$$\text{ and }{\displaystyle \sum _{n=1}^N\left[X^2\left(n\right)\right]}\succ 0,\text{then}\pm {\displaystyle \sum _{n=1}^N\left[X\left(n\right)X\left(n+m\right)\right]}\ge -{\displaystyle \sum _{n=1}^N\left[X^2\left(n\right)\right]}$$

(10)

when $N\to \infty$ , $\left|\frac{{\displaystyle {\sum }_{n=1}^N\left[X\left(n\right)X\left(n+m\right)\right]}}{{\displaystyle {\sum }_{n=1}^N\left[X^2\left(n\right)\right]}}\right|\le 1$ .

$$\underset{N\to \infty }{\lim }{R}_{XX}=\frac{\underset{N\to \infty }{\lim }\frac{1}{N}{\displaystyle {\sum }_{n=1}^N\left[X\left(n\right)X\left(n+m\right)\right]}}{\underset{N\to \infty }{\lim }\frac{1}{N}{\displaystyle {\sum }_{n=1}^N\left[X^2\left(n\right)\right]}}=\frac{E\left[X\left(n\right)X\left(n+m\right)\right]}{E\left[X^2\left(n\right)\right]}$$

(11)

Then

$$\left|\underset{N\to \infty }{\lim }{R}_{XX}\right|=\left|\frac{E\left[X\left(n\right)X\left(n+m\right)\right]}{E\left[X^2\left(n\right)\right]}\right|\le 1$$

(12)

Since $X\left(n\right)$ , $N_1\left(n\right)$ , and $N_2\left(n\right)$ are independent, $N_1\left(n\right)$ and $N_2\left(n\right)$ are zero-mean CGN, according to equations (3) and (4). Let

$$g_{XX}=\frac{E\left[X\left(n\right)X\left(n+m\right)\right]}{E\left[X^2\left(n\right)\right]},g_{YZ}\left(m\right)=\frac{E\left[Y\left(n\right)Z\left(n+m\right)\right]}{E\left[Y\left(n\right)Z\left(n\right)\right]},g_{ZY}\left(m\right)=\frac{E\left[Z\left(n\right)Y\left(n+m\right)\right]}{E\left[Z\left(n\right)Y\left(n\right)\right]}$$

Then

$$\begin{array}{l}{g}_{YZ}\left(m\right)=\frac{E\left[Y\left(n\right)Z\left(n+m\right)\right]}{E\left[Y\left(n\right)Z\left(n\right)\right]}=\frac{E\left\{\left[X\left(n\right)+N_1\left(n\right)\right]\left[X\left(n+m\right)+N_2\left(n+m\right)\right]\right\}}{E\left\{\left[X\left(n\right)+N_1\left(n\right)\right]\left[X\left(n\right)+N_2\left(n\right)\right]\right\}}\\ \hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{1em}=\frac{\left\{\begin{array}{l}E\left[X\left(n\right)X\left(n+m\right)\right]+E\left[X\left(n\right)N_2\left(n+m\right)\right]\\ +E\left[X\left(n+m\right)N_1\left(n\right)\right]+E\left[N_1\left(n\right)N_2\left(n+m\right)\right]\end{array}\right\}}{E\left[X^2\left(n\right)\right]+E\left[X\left(n\right)N_2\left(n\right)\right]+E\left[X\left(n\right)N_1\left(n\right)\right]+E\left[N_1\left(n\right)N_2\left(n\right)\right]}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{1em}{\to }_p\frac{E\left[X\left(n\right)X\left(n+m\right)\right]}{E\left[X^2\left(n\right)\right]}\end{array}$$

(13)

Then

$$g_{YZ}\left(m\right)=\frac{E\left[X\left(n\right)X\left(n+m\right)\right]}{E\left[X^2\left(n\right)\right]}=g_{XX}\left(m\right)$$

(14)

So $g_{YZ}\left(m\right)$ of observed data can be used as a substitute for $g_{XX}\left(m\right)$ of noiseless data. Similarly, $g_{ZY}\left(m\right)$ of observed data can also be used as a substitute for $g_{XX}\left(m\right)$ of noiseless data.

Properties of SNCC

Following are some of the significant properties of SNCC

If $Y$ , $Z_1$ , and $Z_2$ are jointly $S\alpha S$ , then SNCC of $g_{YZ}\left(m\right)$ is linear in $Z$

$$\begin{array}{l}{g}_{YZ}\left(m\right)=\frac{E\left[Y\left(n\right),A{Z}_1\left(n+m\right)+B{Z}_2\left(n+m\right)\right]}{E\left[Y^2\left(n\right)\right]}\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}=A\frac{E\left[Y\left(n\right),Z_1\left(n+m\right)\right]}{E\left[Y^2\left(n\right)\right]}+B\frac{E\left[Y\left(n\right),Z_2\left(n+m\right)\right]}{E\left[Y^2\left(n\right)\right]}\end{array}$$

for arbitrary real constants $A$ and $B$ .

Similarly, if $Z$ , $Y_1$ and $Y_2$ are jointly $S\alpha S$ , then SNCC of $g_{ZY}\left(m\right)$ is linear in $Y$

$$\begin{array}{l}{g}_{ZY}\left(m\right)=\frac{E\left[Z\left(n\right),C{Y}_1\left(n+m\right)+D{Y}_2\left(n+m\right)\right]}{E\left[Z^2\left(n\right)\right]}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}=C\frac{E\left[Z\left(n\right),Y_1\left(n+m\right)\right]}{E\left[Z^2\left(n\right)\right]}+D\frac{E\left[Z\left(n\right),Y_2\left(n+m\right)\right]}{E\left[Z^2\left(n\right)\right]}\end{array}$$

for arbitrary real constants $C$ and $D$ .

• 2. If $X$ and $U$ are independent and jointly $S\alpha S$ , according to Kanter and Steiger, ²⁴ then

  $$g_{XU}\left(m\right)=\frac{E\left[X\left(n\right)U\left(n+m\right)\right]}{E\left[X^2\left(n\right)\right]}=0$$

• 3. Let $X\left(n+m\right)={\displaystyle \sum _{k=1}^{j}a\left(k\right)}X\left(n+m-k\right)$ , then:

  $$\begin{array}{l}{g}_{XX}\left(m\right)=\frac{E\left[X\left(n\right)X\left(n+m\right)\right]}{E\left[X^2\left(n\right)\right]}=\frac{E\left\{X\left(n\right)\left[{\displaystyle {\sum }_{k=1}^{j}a\left(k\right)X\left(n+m-k\right)}\right]\right\}}{E\left[X^2\left(n\right)\right]}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}={\displaystyle \sum _{k=1}^{j}a\left(k\right)}\frac{E\left[X\left(n\right)X\left(n+m-k\right)\right]}{E\left[X^2\left(n\right)\right]}={\displaystyle \sum _{k=1}^{j}a\left(k\right)}{g}_{XX}\left(m-k\right)\end{array}$$

Consistency of parameters estimation for AR model

Theorem 1: The natural estimate ${\widehat{g}}_{XX}\left(m\right)$ of $g_{XX}\left(m\right)=\frac{E\left[X\left(n\right)X\left(n+m\right)\right]}{E\left[X^2\left(n\right)\right]}$ is consistent.

Prove: Since ${\widehat{g}}_{XX}\left(m\right)=\frac{{\displaystyle {\sum }_{n=1}^N\left[X\left(n\right)X\left(n+m\right)\right]}}{{\displaystyle {\sum }_{n=1}^N\left[X^2\left(n\right)\right]}}$ is absolutely summable, the order of $\lim$ and ${\displaystyle \sum }$ can be exchanged.

$$E\left[{\widehat{g}}_{XX}\left(m\right)\right]=\frac{1}{N}{\displaystyle \sum _{n=1}^N\left\{\frac{{\displaystyle {\sum }_{n=1}^N\left[X\left(n\right)X\left(n+m\right)\right]}}{{\displaystyle {\sum }_{n=1}^N\left[X^2\left(n\right)\right]}}\right\}}\underset{N\to \infty }{\to }\frac{1}{N}{\displaystyle \sum _{n=1}^N\left[g_{XX}\left(m\right)\right]}=g_{XX}\left(m\right)$$

(15)

So ${\widehat{g}}_{XX}\left(m\right)$ is an unbiased estimate of $g_{XX}\left(m\right)$ .

$$\begin{array}{l}Var\left[{\widehat{g}}_{XX}\left(m\right)\right]=E\left\{{\left[{\widehat{g}}_{XX}\left(m\right)-E\left[{\widehat{g}}_{XX}\left(m\right)\right]\right]}^2\right\}=E\left\{{\left[{\widehat{g}}_{XX}\left(m\right)\right]}^2\right\}-{\left\{E\left[{\widehat{g}}_{XX}\left(m\right)\right]\right\}}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{1em}=\underset{N\to \infty }{\lim }\frac{1}{N}{\displaystyle \sum _{n=1}^N\left\{{\left[{\widehat{g}}_{XX}\left(m\right)\right]}^2\right\}}-{\left\{E\left[{\widehat{g}}_{XX}\left(m\right)\right]\right\}}^2\end{array}$$

(16)

According to equation (12), then $\left|{\widehat{g}}_{XX}\left(m\right)\right|\le 1$ , so ${\left[{\widehat{g}}_{XX}\left(m\right)\right]}^2\le 1$ . Since ${\left[{\widehat{g}}_{XX}\left(m\right)\right]}^2$ is absolutely summable, the order of $\lim$ and ${\displaystyle \sum }$ can be exchanged, then

$$\begin{array}{l}Var\left[{\widehat{g}}_{XX}\left(m\right)\right]=\frac{1}{N}{\displaystyle \sum _{n=1}^N\underset{N\to \infty }{\lim }}\left\{{\left[{\widehat{g}}_{XX}\left(m\right)\right]}^2\right\}-{\left\{E\left[{\widehat{g}}_{XX}\left(m\right)\right]\right\}}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}=\frac{1}{N}{\displaystyle \sum _{n=1}^N\left\{{\left[g_{XX}\left(m\right)\right]}^2\right\}}-{\left[g_{XX}\left(m\right)\right]}^2={\left[g_{XX}\left(m\right)\right]}^2-{\left[g_{XX}\left(m\right)\right]}^2=0\end{array}$$

(17)

The above suffice to establish the consistency of parameters estimation for AR model.

Development of an SNCC-based parameter estimation algorithm and spectrum estimation

SNCC-based Yule–Walker equation of AR mode

The SNCC functions of processes $Y\left(n\right)$ and $Z\left(n\right)$ is

$$\begin{array}{l}{g}_{YZ}\left(m\right)=g_{ZY}\left(m\right)=g_{XX}\left(m\right)=\frac{E\left[X\left(n\right)X\left(n+m\right)\right]}{E\left[X^2\left(n\right)\right]}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}=\frac{E\left\{X\left(n\right)\left[{\displaystyle \sum _{k=1}^p-\left[a\left(k\right)X\left(n+m-k\right)\right]}+U\left(n+m\right)\right]\right\}}{E\left[X^2\left(n\right)\right]}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}=\frac{-a\left(1\right)E\left[X\left(n+m-1\right)X\left(n\right)\right]-\cdots -a\left(m\right)E\left[X^2\left(n\right)\right]}{E\left[X^2\left(n\right)\right]}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}+\frac{\left\{\begin{array}{l}-a\left(m+1\right)E\left[X\left(n-1\right)X\left(n\right)\right]-\cdots \\ -a\left(p\right)E\left[X\left(n+m-p\right)X\left(n\right)\right]+E\left[X\left(n\right)U\left(n+m\right)\right]\end{array}\right\}}{E\left[X^2\left(n\right)\right]}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}=-a\left(1\right)g_{XX}\left(m-1\right)-\cdots -a\left(m\right)-a\left(m+1\right)\left[g_{XX}\left(1\right)-\frac{X\left(N\right)X\left(N+1\right)}{E\left[X^2\left(n\right)\right]}\right]-\cdots \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}-a\left(p\right)\left[g_{XX}\left(p-m\right)-\frac{X\left(N\right)X\left(N+p-m\right)-\cdots -X\left(N-p+m+1\right)X\left(N+1\right)}{E\left[X^2\left(n\right)\right]}\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}+\frac{E\left[X\left(n\right)U\left(n+m\right)\right]}{E\left[X^2\left(n\right)\right]}\end{array}$$

(18)

According to Kanter and Steiger, ²⁴ $\frac{E\left[X\left(N\right)X\left(N+k\right)+\cdots +X\left(N-k+1\right)X\left(N+1\right)\right]}{E\left[X^2\left(n\right)\right]}{\to }_{p}0$ (convergence in probability), $\frac{E\left[X\left(n\right)U\left(n+m\right)\right]}{E\left[X^2\left(n\right)\right]}{\to }_{p}0$ (convergence in probability), then

$$\begin{array}{l}{g}_{XX}\left(m\right)=a\left(1\right)g_{XX}\left(m-1\right)+\cdots +a\left(m\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}+a\left(m+1\right)g_{XX}\left(1\right)+\cdots +a\left(p\right)g_{XX}\left(p-m\right),\hspace{1em}1\le m\le p\end{array}$$

(19)

Employing equation (19) to write in the form of Yule–Walker equation

$$\left[\begin{array}{ccccc}1& g_{XX}\left(1\right)& g_{XX}\left(2\right)& \cdots & g_{XX}\left(p-1\right)\\ g_{XX}\left(1\right)& 1& g_{XX}\left(1\right)& \cdots & g_{XX}\left(p-2\right)\\ g_{XX}\left(2\right)& g_{XX}\left(1\right)& 1& \cdots & g_{XX}\left(p-3\right)\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ g_{XX}\left(p-1\right)& g_{XX}\left(p-2\right)& g_{XX}\left(p-3\right)& \cdots & 1\end{array}\right]\left[\begin{array}{c}a\left(1\right)\\ a\left(2\right)\\ a\left(3\right)\\ ⋮\\ a\left(p\right)\end{array}\right]=-\left[\begin{array}{c}{g}_{XX}\left(1\right)\\ g_{XX}\left(2\right)\\ g_{XX}\left(3\right)\\ ⋮\\ g_{XX}\left(p\right)\end{array}\right]$$

(20)

Let $G=\left[\begin{array}{ccccc}1& g_{XX}\left(1\right)& g_{XX}\left(2\right)& \cdots & g_{XX}\left(p-1\right)\\ g_{XX}\left(1\right)& 1& g_{XX}\left(1\right)& \cdots & g_{XX}\left(p-2\right)\\ g_{XX}\left(2\right)& g_{XX}\left(1\right)& 1& \cdots & g_{XX}\left(p-3\right)\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ g_{XX}\left(p-1\right)& g_{XX}\left(p-2\right)& g_{XX}\left(p-3\right)& \cdots & 1\end{array}\right]$

Let $A={\left[\begin{array}{ccccc}a\left(1\right)& a\left(2\right)& a\left(3\right)& \cdots & a\left(p\right)\end{array}\right]}^T$ , $g=-{\left[\begin{array}{ccccc}{g}_{XX}\left(1\right)& g_{XX}\left(2\right)& g_{XX}\left(3\right)& \cdots & g_{XX}\left(p\right)\end{array}\right]}^T$ , then $GA=g$ .

The coefficients $a\left(1\right),\cdots ,a\left(p\right)$ are unique if and only if $g_{XX}\left(1\right),\cdots ,g_{XX}\left(p\right)$ are linearly independent elements in the space of integrable random variables.

Theorem 2: $G$ is a Full mark matrix.

Proof

$$\text{Let}{G}^M=\left[\begin{array}{ccccc}1& g_{XX}\left(1\right)& g_{XX}\left(2\right)& \cdots & g_{XX}\left(p-1\right)\\ g_{XX}\left(1\right)& 1& g_{XX}\left(1\right)& \cdots & g_{XX}\left(p-2\right)\\ g_{XX}\left(2\right)& g_{XX}\left(1\right)& 1& \cdots & g_{XX}\left(p-3\right)\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ g_{XX}\left(p-1\right)& g_{XX}\left(p-2\right)& g_{XX}\left(p-3\right)& \cdots & 1\\ g_{XX}\left(p\right)& g_{XX}\left(p-1\right)& g_{XX}\left(p-2\right)& \cdots & g_{XX}\left(1\right)\\ g_{XX}\left(p+1\right)& g_{XX}\left(p\right)& g_{XX}\left(p-1\right)& \cdots & g_{XX}\left(2\right)\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ g_{XX}\left(M\right)& g_{XX}\left(M-1\right)& g_{XX}\left(M-2\right)& \cdots & g_{XX}\left(M-p+1\right)\end{array}\right]$$

(21)

where $M$ is an arbitrary positive integer greater than $p$ , let $gmM$ be the $m$ th line row vector of the matrix $G^M$ , it is known by equation (19):

$g_{p+1}{\hspace{0.17em}}^M=\left[\begin{array}{cccc}{g}_p{\hspace{0.17em}}^M\cdot A& {\hspace{0.17em}}_{g}p-1{\hspace{0.17em}}^M\cdot A& \cdots & g_1{\hspace{0.17em}}^M\cdot A\end{array}\right]$ , where $g_{p+1}{\hspace{0.17em}}^M$ is the linear combination of vectors from $p$ th to the first row vector of the matrix $G^M$ .

$g_{p+2}{\hspace{0.17em}}^M=\left[\begin{array}{cccc}{g}_{p+1}{\hspace{0.17em}}^M\cdot A& g_p{\hspace{0.17em}}^M\cdot A& \cdots & g_2{\hspace{0.17em}}^M\cdot A\end{array}\right]$ , $g_{p+2}{\hspace{0.17em}}^M$ is the linear combination of vectors from ( $p+1$ )th to the second row vector of the matrix $G^M$ .

The rest of the form can be deduced as, $g_M{\hspace{0.17em}}^M=\left[\begin{array}{cccc}{g}_{M-1}{\hspace{0.17em}}^M\cdot A& g_{M-2}{\hspace{0.17em}}^M\cdot A& \cdots & g_{M-p+1}{\hspace{0.17em}}^M\cdot A\end{array}\right]$ _.

Then the nonsingularity of the matrix $G^M$ depends only on the former $p$ row vectors.

Through the reverse proving, suppose the former $p$ row vectors $g_1{\hspace{0.17em}}^M,g_2{\hspace{0.17em}}^M,\cdots ,g_p{\hspace{0.17em}}^M$ of matrix $G^M$ have a significant linear correlation, since $g_{p+1}{\hspace{0.17em}}^M$ is linearly dependent on $g_1{\hspace{0.17em}}^M,g_2{\hspace{0.17em}}^M,\cdots ,g_p{\hspace{0.17em}}^M$ , $g_{p+2}{\hspace{0.17em}}^M$ is linearly dependent on $g_1{\hspace{0.17em}}^M,g_2{\hspace{0.17em}}^M,\cdots ,g_p{\hspace{0.17em}}^M$ , in a similar fashion, matrix $G^M$ is only linearly independent of the front ( $p-1$ ) row vectors, so there exists such a $\left(p-1\right)\times 1$ dimension parameter vector

$$\begin{array}{l}{g}_{XX}\left(m\right)=-c\left(1\right)g_{XX}\left(m-1\right)-\cdots -c\left(m\right)-c\left(m+1\right)g_{XX}\left(1\right)\\ -\cdots -c\left(p-1\right)g_{XX}\left(p-m-1\right),1\le m\le p\end{array}$$

It obviously leads to the conclusion that $a\left(p\right)=0$ , which contradicts the AR model assumption [AS3]. So it is well explained that the former $p$ row vectors $g_1{\hspace{0.17em}}^M,g_2{\hspace{0.17em}}^M,\cdots ,g_p{\hspace{0.17em}}^M$ of the matrix $G^M$ must be linearly independent. As a result of matrix $G$ being symmetric, the former $p$ row vectors are linearly independent of each other. In summary, matrix $G$ is full rank with its rank $p$ .

SVD-TLS algorithm for parameter estimation of AR model

Selecting extended order $p_e$ to construct Yule–walker equation of matrix with extended order

$$\left[\begin{array}{ccccc}{g}_{XX}\left(1\right)& 1& g_{XX}\left(1\right)& \cdots & g_{XX}\left(p_e-1\right)\\ g_{XX}\left(2\right)& g_{XX}\left(1\right)& 1& \cdots & g_{XX}\left(p_e-2\right)\\ g_{XX}\left(3\right)& g_{XX}\left(2\right)& g_{XX}\left(1\right)& \cdots & g_{XX}\left(p_e-3\right)\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ g_{XX}\left(l\right)& g_{XX}\left(l-1\right)& g_{XX}\left(l-2\right)& \cdots & g_{XX}\left(l-p_e+1\right)\end{array}\right]\left[\begin{array}{c}1\\ a\left(1\right)\\ a\left(2\right)\\ ⋮\\ a\left(p_e\right)\end{array}\right]=\left[\begin{array}{l}0\\ 0\\ 0\\ ⋮\\ 0\end{array}\right]$$

(22)

where $p_e\ge p$ , $l\succ p+1$ .

Writing equation (22) in brief

$$G_e{A}_e=0$$

(23)

SVD decomposition of matrix $G_e$ can be obtained

$$G_e=U\Lambda V^{*}$$

(24)

$U$ and $V$ are respectively the left and right singular vectors of $G_{ge}$ , i.e.

$$G_e{G}_e{\hspace{0.17em}}^{*}{u}_i={\sigma }_i{u}_i,G_e{G}_e{\hspace{0.17em}}^{*}{v}_i={\sigma }_i{v}_i$$

$\Lambda =diag\left[{\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_{p_e+1}\right]$ ranges according to the descending order of ${\sigma }_i$ , i.e. ${\sigma }_1\succ {\sigma }_2\succ \cdots \succ {\sigma }_{p_{\text{e}}+1}$ .

Computing $G_e{\hspace{0.17em}}^{\left(p\right)}={\displaystyle {\sum }_{i=1}^p{\displaystyle {\sum }_{k=1}^{p_e-p+1}{\sigma }_i}{u}_i{\hspace{0.17em}}^k{v}_i{k}^{*}}$ , where $u_i^k$ represents an $\left(p+1\right)\times 1$ dimensional vector consisting of $\left(p+1\right)$ elements of the $i$ column vector in a matrix $U$ and $u_i^k=\left[u_i\left(k\right),u_i\left(k+1\right),\cdots ,u_i\left(k+p\right)\right]$ , where $1\le k\le p_e-p+1,1\le i\le p$ .

Similarly, $v_i^k=\left[v_i\left(k\right),v_i\left(k+1\right),\cdots ,v_i\left(k+p\right)\right],1\le k\le p_e-p+1,1\le i\le p$ .

Solving linear equations $G_e^{\left(p\right)}{A}_1=a{e}_1$ , where: $A_1={\left[\begin{array}{llll}1& a_1& \cdots & a_p\end{array}\right]}^T$ , $e_1={\left[\begin{array}{llll}1& 0& \cdots & 0\end{array}\right]}^T$ , $a$ is the value that makes the first element of the $A_1$ equal to 1. $A_1$ is the parameter vector of the AR model.

α-Spectrum

Since the problem we are confronted with is estimation of 1/f noise spectrum, the proposed SNCC method is used to suppress CGN of the dual-channel measured data. The spectrum estimation is specially towards the same diode 1/f noise produced by the dual-channel output, thus we can transform the cross-spectrum estimation problem of the dual-channel data into the problem of self-spectrum estimation.

Due to infinite variance of α-stable process $X\left(n\right)$ , its frequency characteristics cannot be represented by the power spectrum used in the Gaussian model. Literature ²⁶ proves the connection between α-spectrum and covariation spectrum, i.e. α-spectrum in unit circle is $q$ -order interceptive form of covariation vcspectrum

$${\phi }_{xx}\left(\omega \right)=\underset{q\to \infty }{\lim }{\left\{S_{\alpha }\left(z\right)\right\}}^{\ast }|z=e^{j\omega }$$

(25)

Dealing with the problem of spectrum estimation for an α-stable process, we adopt α-spectrum estimation method, ²⁷ due to its equivalent position to the power spectrum density estimation method based on the autocorrelation function in Gaussian distribution.

On the condition of α-stable distribution, a finite impulse response system with input a $S\alpha S$ of i.i.d. can be expressed as a general linear system

$$X\left(n\right)={\displaystyle \sum _{k=0}^{p}h\left(k\right)}\omega \left(n-k\right)$$

(26)

where $h\left(k\right)$ is the impulse response coefficient of FIR system, $\omega \left(n\right)$ is a stable distributed noise sequence of i.i.d. α-stable noise, α-spectrum is defined as follows

$$S_{\alpha }\left(z\right)={\gamma }_{\omega }H\left[{\left(z^{-1}\right)}^{〈\alpha -1〉}\right]{\left[H\left(z\right)\right]}^{〈\alpha -1〉}$$

(27)

So when the linear model of $X\left(n\right)$ is known previously, the spectrum of $X\left(n\right)$ can be obtained by using equation (29). Evaluating the α-spectrum on the unit circle $\left|z\right|=1$ based on the above-mentioned AR model yields

$$S_{\alpha }\left(e^{j\omega }\right)={\gamma }_{\omega }\frac{1}{{\left|1+{\displaystyle {\sum }_{k=1}^{p}a\left(k\right)e^{j\omega }}\right|}^{\alpha }}$$

(28)

When the value of $\alpha$ is known, by replacing AR coefficients $A=\left[\begin{array}{lllll}a\left(1\right)& a\left(2\right)& a\left(3\right)& \cdots & a\left(p\right)\end{array}\right]$ obtained from the previous section into equation (28), α-spectrum can be estimated.

Experiments

Simulation and experiments

The composition of the analog dual-channel output data includes the signal (1/f signal conforming to α-stable distribution) and the additive noise (CGN). The 1/f signal is generated according to literature. ²⁸ We substitute α-stable noise for Gaussian noise as AR system input. The generating method of α-stable noise that we adopted is Chambers–Mallows–Stuck method.^29,30 Thus, we generate a 1/f process, adding two different colored Gaussian processes, respectively.

According to an earlier section, the characteristic parameter of input $U\left(n\right)$ is set to $\alpha \left(1\le \alpha \prec 2\right)$ , the dispersion coefficient $\gamma$ is set to 1, symmetric parameter $\beta$ and location parameter $\delta$ are both set to 0. The characteristic parameter of the generated signal $X\left(n\right)$ is equal to that of $U\left(n\right)$ .

Mixed signal-to-noise

With nonexistence of finite variance, the variance of conventional signal-to-noise (SNR) ratio stands no point, based on that, we use mixed SNR^16,31 to define SNR under α-stable distribution.

$$R_{MSNR}=10\log 10\left(\gamma /{\sigma }_{WN}^2\right)=m$$

(29)

$\gamma$ is the dispersion coefficient of signal $X\left(n\right)$ , ${\sigma }_{WN}^2$ is the variance of CGN, $R_{MSNR}$ is set to $m$ dB.

Comparison between self-spectrum, cross-spectrum and the real spectrum

We refer to the AR spectrum estimation based on SNCC to cross-spectrum with dual-channel data, similarly, self-spectrum is AR spectrum estimation based on SNSC with single-channel data. The real spectrum, cross-spectrum, and self-spectrum are compared in Figure 2.

Figure 2.

Comparison between self-spectrum, cross-spectrum, and the real spectrum.

From Figure 2, we can see that the spectrum estimation of cross-spectrum method draws nearer to the real spectrum than that of the self-spectrum method.

Estimation of the characteristic parameter $\alpha$

Compared with the Gaussian noise, the non-Gaussian property of α-stable noise is stronger. Due to the non-Gaussian property, we use the method of a fractional lower order statistics-based independent component analysis (ICA) method to extract impulsive α-stable signal. Then, the value of characteristic parameter of extracted signal can be estimated according to Qiu et al. ³²

Figure 3(a) shows the original generated 1/f signal. Figure 3(b) shows the original 1/f signal with additive CGN. Figure 3(c) is the extracted signal from original 1/f signal and additive CGN by using the method of Qiu et al. ³²

Figure 3.

(a) The generated 1/f signal; (b) 1/f signal with additive CGN; (c) the extracted signal.

The true α-value is 1.5. Estimated α-value of generated 1/f signal is 1.4926, and that of extracted signal is 1.5888. $R_{MSNR}$ is set to 20 dB. Spectrum curves of true α-value (1.5) and estimated α-value (1.5888) are compared in Figure 4. We can conclude that despite the differences between true and estimated α-value, the spectrum curves are similar.

Figure 4.

Spectrum with true/estimated α-value.

The mean and standard deviation (std) of estimated value of $\alpha$ over 100 trails, each with 10,000 samples generated through Chambers–Mallows–Stuck method are presented in Table 1.

Table 1.

Mean (std) of the estimated value of $\alpha$ over 100 trails, with 10,000 samples.

	α = 1.2	α = 1.5	α = 1.7	α = 1.9
U	1.2099 (0.0166)	1.5043 (0.0226)	1.7038 (0.0261)	1.8967 (0.0280)
S	1.2553 (0.3135)	1.5427 (0.2417)	1.7058 (0.2559)	1.8859 (0.1344)
S1	1.2972 (0.3006)	1.5669 (0.2616)	1.7248 (0.2605)	1.8850 (0.1513)
S2	1.2973 (0.3001)	1.5676 (0.2623)	1.7251 (0.2606)	1.8846 (0.1510)

U:input $S\alpha S$ noise; S: noiseless generated 1/f noise; $S1$ and $S2$ : dual-channel 1/f signal extracted from that with additive CGN, with $R_{MSNR}$ =20.

We refer SNSC-ICA method as applying FLOS based ICA method to extract α-stable signal before SNSC method.

Error analysis of AR coefficients

The accuracy of AR parameters estimation is measured by root mean square error (RMSE). The RMSE of AR parameters estimated respectively by SNCC, SNSC, and SNSC-ICA method are illustrated by Figure 5.

Figure 5.

RMSE with α = 1.2; (b) RMSE with α = 1.8.

Figure 5 shows the RMSE of AR coefficients respectively with $\alpha =1.2$ and $\alpha =1.\text{8}$ . We run 100 times independently, samples number is equal to 10,000. Notice that the larger the $\alpha$ is, the more will be the influence on the accuracy of AR coefficients. From Figure 5, it can be implied that the estimated α-value of extracted signal is approximate to that of the original signal, while the spectrum estimation performance of SNSC-ICA is far less effective than that of the other two methods. This phenomenon is due to that SNSC-ICA method only retains most of the pulse characteristics; the representation of extracting signal is not accurate enough. So ICA-based method can only be used to extract signal other than the spectrum estimation.

The mean and standard deviation (std) of estimated AR coefficients of over 100 trails, each with 10,000 samples generated through Chambers–Mallows–Stuck method are presented in Table 2, where α = 1.8, $R_{MSNR}$ =20.

Table 2.

Mean (std) of AR coefficients over 100 trails, with 10,000 samples, α = 1.8, $R_{MSNR}$ =20.

	a(1)	a(2)	a(3)	a(4)	a(5)
True value	−0.5000	−0.1250	−0.0625	−0.0391	−0.0273
SNCC	−0.4967 (0.0233)	−0.1285 (0.0112)	−0.0594 (0.0086)	−0.0433 (0.0071)	−0.0264 (0.0172)
SNSC	−0.4871 (0.0246)	−0.1345 (0.0123)	−0.0953 (0.0100)	−0.0587 (0.0089)	−0.0125 (0.0118)
SNSC-ICA	−0.3070 (0.1023)	−0.1572 (0.1502)	−0.1 (0.0689)	−0.0640 (0.0538)	−0.0093 (0.0462)

SNCC: samples normalized cross-correlation; SNSC: samples normalized self-correlation; ICA: independent component analysis.

The RMSE of estimated AR coefficients of over 100 trails, each with 10,000 samples generated through Chambers–Mallows–Stuck method are presented in Table 2, where α = 1.5, $R_{MSNR}$ =15.

From Tables 2 and 3, we can draw the same conclusion that ICA-based method can only be used to extract signal other than spectrum estimation, where the performance of SNCC-based method is better than the SNSC-based method.

Table 3.

Root mean square error (RMSE) of AR coefficients over 100 trails, with 10,000 samples, α = 1.5, $R_{MSNR}$ =15.

	a(1)	a(2)	a(3)	a(4)
SNCC	0.0233	0.0111	0.0086	0.0070
SNSC	0.0246	0.0123	0.0100	0.0089
SNSC-ICA	0.2023	0.1501	0.0689	0.0537

SNCC: samples normalized cross-correlation; SNSC: samples normalized self-correlation; ICA: independent component analysis.

Spectrum estimation of real dual-channel data

The reason why the voltage regulator diode is chosen is that it does not have amplification function itself. It needs amplification by amplifier and noise of amplifier, which is challenging.

Noise measurement system

We choose STC5V1 zener diodes, with a rated power of 0.5 W and a regulated voltage of 5.1 V. Hundred zener diodes with the same type and batch are measured. The noise spectrum is measured in the frequency range of 0.4 Hz to 25 kHz and the specified current of 1 mA with a noise measurement system at room temperature, which consists of two amplifiers, DAQ, a personal computer.

The low noise amplifier selected by this system is two Model 5184 type amplifiers produced by Perkin Elmer. The amplifier is of medium impedance, AC-coupled voltage amplifier. With ultralow noise input amplification characteristics, frequency response is 0.5 Hz to 1 Mhz and magnification is 1000 times. In order to make the two LNA independent, using independent power supply, the experiment uses four batteries (9 V) supplied by itself.

The selected DAQ is ART USB2811, the input range is +5 V, the conversion accuracy is 16 bits, and the sampling frequency is 31 Hz to 250 kHz. The DAQ adopts pseudo-synchronous method for dual-channel acquisition, and the sampling frequency is 200 K. The number of sampling points is 10,000. A group of 100 zener diodes modeled ZW36 are used in our experiments.

Diodes noise measurement and spectrum estimation

We verified that time domain noise processes of 100 zener diodes conform to α-stable distribution, their characteristic exponent $\alpha$ range from 1.7682 to 1.9547. Taking one of the diodes for instance, we run 50 times independently to obtain estimated spectrum curves. The mean of $\alpha$ is 1.8798, with std 0.0411. Although the value of $\alpha$ is close to 2, it still can be distinguished from Gaussian distribution. ^33,34

Figure 6 is the time domain diagram of dual-channel noise measurement. Figure 7(a) to (e) demonstrates the influence of sample number $N$ on the spectrum estimation, $N$ =[1000 5000 10,000 50,000 100,000], and the order $p$ of AR model is set to 300. With increasing number of samples, performance of spectrum estimation is more stable. For the sake of improving efficiency, we choose 10,000 samples. Figure 8(a) to (g) shows the influence of order $p$ of AR model on the spectrum estimation, $p$ = [10 50 100 200 300 400 500], and $N$ is set to 10,000. In order to advance the operational speed, a lower $p$ is chosen from the range 100–300. Figure 9(a) shows the average spectrum curves of different $N$ from Figure 7(a) to (e). Figure 9(b) shows the average spectrum curves of different $p$ from Figure 8(a) to (g).

Figure 6.

(a) Channel 1; (b): Channel 2.

Figure 7.

(a) N = 1000; (b) N = 5000; (c) N = 10,000; (d) N = 50,000; (e) N = 100,000.

Figure 8.

The influence of order p of AR model on spectrum estimation. p = 10; (b) p = 50; (c) p = 100; (d) p = 200; (e) p = 300; (f) p = 400; (g) p = 500.

Figure 9

The average spectrum curves of different N and p.

Conclusions

CGN is inevitably introduced during the measurement of transistor 1/f noise, which causes the decline of spectrum estimation accuracy. Our research showed that the 1/f noises of transistors conform to α-stable distribution, so the problem of 1/f spectrum estimation is transformed into α-stable noise spectrum estimation problem. Previously, the estimation of α-stable noise spectrum is carried out under noise-free background. With the increasing of the background noise, the spectrum estimation accuracy of the existing methods is declined. Therefore, in view of the fact that there is no finite second-order characteristic of α-stable noise, we proposed a SNCC method to suppress the influence of CGN on the declining accuracy of spectrum estimation. The experimental results have shown that the estimated spectrum by our method has a high degree of accuracy.