# Wavelet Kalman Filtering

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## Abstract

In addition to the Kalman filtering algorithms discussed in the previous chapters, there are other computational schemes available for digital filtering performed in the time domain.

### Keywords

• Kalman Filter
• Filter Bank
• Basic Wavelet
• Random Signal
• Unconditional Basis

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Author information

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### Corresponding author

Correspondence to Charles K. Chui .

## Exercises

### Exercises

1. 11.1.

The following Haar and triangle functions are typical window functions in the time domain:

\begin{aligned}\begin{gathered} \phi _{H}(t)=\left\{ \begin{array}{l@{\quad }l} 1 &{} 0\le t<1\\ 0 &{} otherwise; \end{array}\right. \\ \phi _{T}(t)=\left\{ \begin{array}{r@{\quad }l} t &{} 0\le t<1,\\ 2-t &{} 1\le t<2,\\ 0 &{} otherwise. \end{array}\right. \end{gathered}\end{aligned}

Sketch these two functions, and verify that

\begin{aligned} \phi _{T}(t)=\int _{-\infty }^{\infty }\phi _{H}(\tau )\phi _{H}(t-\tau )d{\tau }. \end{aligned}

Here, $$\phi _{H}(t)$$ and $$\phi _{T}(t)$$ are also called $$\mathrm {B}$$-splines of degree zero and one, respectively. $$\mathrm {B}$$-spline of degree n can be generated via

\begin{aligned} \phi _{n}(t)=\int _{0}^{1}\phi _{n-1}(t-\tau )d\tau ,\quad n=2, 3, \cdots . \end{aligned}

Calculate and sketch $$\phi _{2}(t)$$ and $$\phi _{3}(t)$$. (Note that $$\phi _{0}=\phi _{H}$$ and $$\phi _{1}=\phi _{T}.)$$

2. 11.2.

Find the Fourier transforms of $$\phi _{H}(t)$$ and $$\phi _{T}(t)$$ defined above.

3. 11.3.

Based on the graphs of $$\phi _{H}(t)$$ and $$\phi _{T}(t)$$, sketch

\begin{aligned} \phi _{1, 0}(t)=\phi _{H}(2t),\quad \phi _{1, 1}(t)=\phi _{H}(2t-1),\quad and\quad \mathrm {\Phi }(t)=\phi _{1, 0}(t)+\phi _{1, 1}(t). \end{aligned}

Moreover, sketch the wavelets

\begin{aligned} \psi _{H}(t)=\phi _{H}(2t)-\phi _{H}(2t-1) \end{aligned}

and

\begin{aligned} \psi _{T}(t)=-\frac{1}{2}\phi _{T}(2t)-\frac{1}{2}\phi _{T}(2t-2)+\phi _{T}(2t-1). \end{aligned}
4. 11.4.

Verify (11.16).

5. 11.5.

Verify (11.24) and (11.26).

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