Exponential Myriad Smoothing Algorithm for Robust Signal Processing in $\alpha$-Stable Noise Environments

Abstract

The sequential sample myriad has been proposed recently to estimate an unknown location parameter in real time by updating the current estimate when a new input sample is available. However, the algorithm is only capable of estimating an unknown constant (i.e., a time-invariant location parameter). In this paper, we propose a sequential myriad smoothing approach for tracking a time-varying location parameter corrupted by impulsive symmetric $\alpha$-stable noise. By incorporating exponential weighting factor to the sequential algorithm, the new algorithm weighs the recent samples more heavily to provide effective tracking capability. Simulation results show that the proposed method outperforms the classical exponential smoothing and is as good as the running myriad smoother.

Appendix

1.1 Derivation of Exponential Myriad Smoothing

The myriad smoother ${\hat{\theta }}\left[ n \right]$ based on n observations is the solution of the following equation

$$\begin{aligned} \mathop \sum \limits _{i=1}^n \frac{\eta _i \left( {y_i -\theta } \right) }{k^{2}+\left( {y_i -\theta } \right) ^{2}}=0 \end{aligned}$$

(11)

Substituting $\eta _i =\lambda ^{n-i}$, (11) can be written as

$$\begin{aligned} \mathop \sum \limits _{i=1}^n \frac{\lambda ^{n-i}\left( {y_i -\theta } \right) }{k^{2}+\left( {y_i -\theta } \right) ^{2}}=0 \end{aligned}$$

(12)

Let

$$\begin{aligned} \psi _\theta \left[ n \right] =\mathop \sum \limits _{i=1}^n \frac{\lambda ^{n-i}\left( {y_i -\theta } \right) }{k^{2}+\left( {y_i -\theta } \right) ^{2}} \end{aligned}$$

(13)

The first-order Taylor series of $\psi _\theta \left[ {n+1} \right]$ about the point $\theta ={\hat{\theta }}\left[ n \right]$ is given by

$$\begin{aligned} \psi _\theta \left[ {n+1} \right] \approx \psi _{{\hat{\theta }}\left[ n \right] } \left[ {n+1} \right] +{\psi }^{\prime }_{{\hat{\theta }}\left[ n \right] } \left( {n+1} \right) \left( \theta -{\hat{\theta }}[n]\right) \end{aligned}$$

(14)

where

$$\begin{aligned} {\psi }^{\prime }_{\hat{\theta }[n]} [n+1] =\frac{\partial \psi _{\theta }[n+1]}{\partial \theta }\Bigg |_{\theta ={\hat{\theta }}[n]} \end{aligned}$$

(15)

Replacing $\theta ={\hat{\theta }}\left[ {n+1} \right]$ in (14) and since $\psi _{{\hat{\theta }}\left[ {n+1} \right] } \left[ {n+1} \right] \approx 0$, (14) can be rewritten as follows:

$$\begin{aligned} {\hat{\theta }}\left[ {n+1} \right] \approx {\hat{\theta }}\left[ n \right] -\frac{\psi _{{\hat{\theta }}\left[ n \right] } \left[ {n+1} \right] }{{\psi }^{\prime }_{\hat{\theta } \left[ n \right] }\left[ {n+1} \right] } \end{aligned}$$

(16)

By using the definition of $\psi _\theta \left[ n \right]$in (13), we have

$$\begin{aligned} \psi _\theta \left[ {n+1} \right]= & {} \mathop \sum \limits _{i=1}^{n+1} \frac{\lambda ^{n+1-i}\left( {y_i -\theta } \right) }{k^{2}+\left( {y_i -\theta } \right) ^{2}}\nonumber \\= & {} \lambda \psi _\theta \left[ n \right] +\frac{y_{n+1} -\theta }{k^{2}+\left( {y_{n+1} -\theta } \right) ^{2}} \end{aligned}$$

(17)

Substituting $\theta$ with ${\hat{\theta }}\left[ n \right]$ and since $\psi _{{\hat{\theta }}\left[ n \right] } \left[ n \right] \approx 0$, (17) can be rewritten as

$$\begin{aligned} \psi _{{\hat{\theta }}\left[ n \right] } \left[ {n+1} \right] \approx \frac{y_{n+1} -{\hat{\theta }}\left[ n \right] }{k^{2}+\left( {y_{n+1} -{\hat{\theta }}\left[ n \right] } \right) ^{2}} \end{aligned}$$

(18)

Now, the problem is to find the updates that define $\psi ^{{\prime }}_{{\hat{\theta }}\left[ n \right] } \left[ {n+1} \right]$ in terms of $\psi ^{{\prime }}_{{\hat{\theta }}\left[ n-1 \right] } \left[ {n} \right]$. From (13) and (15), we have

$$\begin{aligned} {\psi }^{\prime }_{\theta } [n+1]= & {} {\mathop \sum \limits _{i=1}^{n+1}} \frac{\lambda ^{n+1-i}[-k^{2}+(y_{i}-\theta )^{2}]}{[k^{2}+(y_{i} -\theta )^{2}]^{2}}\nonumber \\= & {} \lambda \psi ^{\prime }_{\theta }[n]+\frac{-k^{2}+(y_{n+1}-\theta )^{2}}{[k^{2}+(y_{n+1} -\theta )^{2}]^{2}} \end{aligned}$$

(19)

Substituting $\theta ={\hat{\theta }}\left[ n \right]$ in (19) yields

$$\begin{aligned} \psi ^{\prime }_{\hat{\theta }[n]}[n+1]= & {} \lambda \psi ^{{\prime }}_{{\hat{\theta }}\left[ {n}\right] } \left[ n \right] +\frac{-k^{2}+\left( {y_{n+1} -{\hat{\theta }}\left[ n \right] } \right) ^{2}}{\left[ {k^{2}+\left( {y_{n+1} -{\hat{\theta }}\left[ n \right] } \right) ^{2}} \right] ^{2}} \end{aligned}$$

(20)

$$\begin{aligned}\approx & {} \lambda \psi ^{{\prime }}_{{\hat{\theta }}\left[ {n-1} \right] } \left[ n \right] +\frac{-k^{2}+\left( {y_{n+1} -{\hat{\theta }}\left[ n \right] } \right) ^{2}}{\left[ {k^{2}+\left( {y_{n+1} -{\hat{\theta }}\left[ n \right] } \right) ^{2}} \right] ^{2}} \end{aligned}$$

(21)

1.2 Convergence of Exponential Myriad Smoothing to Classical Exponential Smoothing

From (7), let $\hbox {k}\rightarrow \infty$, and we have

$$\begin{aligned} {\hat{J}} \left[ n \right]= & {} \mathop \sum \limits _{i=1}^n \frac{-k^{2}+\left( {y_i -{\hat{\theta }}\left[ {n-1} \right] } \right) ^{2}}{\left[ {k^{2}+\left( {y_i -{\hat{\theta }}\left[ {n-1} \right] } \right) ^{2}} \right] ^{2}} \nonumber \\\approx & {} \mathop \sum \limits _{i=1}^n \frac{-1}{k^{2}} \nonumber \\= & {} -\frac{n}{k^{2}} \end{aligned}$$

(22)

Substituting (22) into (8), the following expression for ${\hat{J}} \left[ {n+1} \right]$ can be simplified as

$$\begin{aligned} {\hat{J}} \left[ {n+1} \right]= & {} \lambda {\hat{J}} \left[ n \right] +\frac{-k^{2}+\left( {y_{n+1} -{\hat{\theta }}\left[ n \right] } \right) ^{2}}{\left[ {k^{2}+\left( {y_{n+1} -{\hat{\theta }}\left[ n \right] } \right) ^{2}} \right] ^{2}}\nonumber \\\approx & {} -\frac{n\lambda }{k^{2}}-\frac{1}{k^{2}}\nonumber \\= & {} -\frac{n\lambda +1}{k^{2}} \end{aligned}$$

(23)

Similarly, for the next iteration, ${\hat{J}} \left[ {n+2} \right]$ will be updated as

$$\begin{aligned} {\hat{J}} \left[ {n+2} \right]\approx & {} \lambda {\hat{J}} \left[ {n+1} \right] -\frac{1}{k^{2}}\nonumber \\= & {} -\frac{n\lambda ^{2}+\lambda +1}{k^{2}} \end{aligned}$$

(24)

Hence, in general, for $i\ge 1,$

$$\begin{aligned} {\hat{J}} \left[ {n+i} \right]\approx & {} \lambda {\hat{J}} \left[ {n+i-1} \right] -\frac{1}{k^{2}}\nonumber \\= & {} -\frac{n\lambda ^{i}+\mathop \sum \nolimits _{m=0}^{i-1} \lambda ^{m}}{k^{2}} \end{aligned}$$

(25)

and

$$\begin{aligned} {\hat{\theta }}\left[ {n+i} \right]= & {} {\hat{\theta }}\left[ {n+i-1} \right] -\left( {{\hat{J}} \left[ {n+i} \right] } \right) ^{-1}\frac{y_{n+i} -{\hat{\theta }}\left[ {n+i-1} \right] }{k^{2}+\left( {y_{n+i} -{\hat{\theta }}\left[ {n+i-1} \right] } \right) ^{2}}\nonumber \\\approx & {} {\hat{\theta }}\left[ {n+i-1} \right] +\left( {\frac{k^{2}}{n\lambda ^{i}+\mathop \sum \nolimits _{m=0}^{i-1} \lambda ^{m}}} \right) \frac{y_{n+i} -{\hat{\theta }}\left[ {n+i-1} \right] }{k^{2}}\nonumber \\= & {} {\hat{\theta }}\left[ {n+i-1} \right] +\left( {\frac{1}{n\lambda ^{i}+\mathop \sum \nolimits _{m=0}^{i-1} \lambda ^{m}}} \right) \left( {y_{n+i} -{\hat{\theta }}\left[ {n+i-1} \right] } \right) \end{aligned}$$

(26)

Consider $i\rightarrow \infty$ and since $\lambda <1$, then $n\lambda ^{i}\rightarrow 0$ and $\mathop \sum \nolimits _{m=0}^{i-1} \lambda ^{m}\rightarrow \frac{1}{1-\lambda }$

$$\begin{aligned} {\hat{\theta }}\left[ {n+i} \right] \approx {\hat{\theta }}\left[ {n+i-1} \right] +\left( {1-\lambda } \right) \left( {y_{n+i} -{\hat{\theta }}\left[ {n+i-1} \right] } \right) \end{aligned}$$

(27)

1.3 Asymptotic Variance of the Exponential Myriad Smoothing

From (13), we have

$$\begin{aligned} \mathop \sum \limits _{i=1}^n \psi _\theta \left( {y_i -\theta } \right) =0 \end{aligned}$$

(28)

where $\psi _\theta \left( x \right) =\frac{\eta x}{k^{2}+x^{2}}$.

The asymptotic variance of exponential myriad smoother at a distribution F can be defined as

$$\begin{aligned} \sigma _\theta ^2 \left( {\psi _\theta ;F} \right) =\frac{\smallint \psi _\theta ^2 \left( x \right) \hbox {d}F}{\left( {\smallint \psi _\theta ^{\prime } \left( x \right) \hbox {d}F} \right) ^{2}} \end{aligned}$$

(29)

and $\psi _\theta ^{\prime } \left( x \right) =\hbox {d}\psi _\theta \left( x \right) /\hbox {d}x$ [12].

Substituting $\psi _\theta \left( x \right) =\frac{\eta x}{k^{2}+x^{2}}$ in (29) yields

$$\begin{aligned} \sigma _\theta ^2 \left( {\psi _\theta ;F} \right)= & {} \frac{\displaystyle \int \frac{\eta ^{2}x^{2}}{\left( {k^{2}+x^{2}} \right) ^{2}}f_\alpha \left( x \right) \hbox {d}x}{\left( {\displaystyle \int \frac{\eta \left( {k^{2}-x^{2}} \right) }{\left( {k^{2}+x^{2}} \right) ^{2}}f_\alpha \left( x \right) \hbox {d}x} \right) ^{2}} \end{aligned}$$

(30)

$$\begin{aligned}= & {} \frac{\displaystyle \int \frac{x^{2}}{\left( {k^{2}+x^{2}} \right) ^{2}}f_\alpha \left( x \right) \hbox {d}x}{\left( {\displaystyle \int \frac{\left( {k^{2}-x^{2}} \right) }{\left( {k^{2}+x^{2}} \right) ^{2}}f_\alpha \left( x \right) \hbox {d}x} \right) ^{2}} \end{aligned}$$

(31)

where $f_\alpha \left( x \right)$ is the distribution function. Eq. (31) shows that the exponential myriad smoother has the same asymptotic variance as the sample myriad given in [12].