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
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Exercises
Exercises

11.1.
The following Haar and triangle functions are typical window functions in the time domain:
$$\begin{aligned}\begin{gathered} \phi _{H}(t)=\left\{ \begin{array}{l@{\quad }l} 1 &{} 0\le t<1\\ 0 &{} otherwise; \end{array}\right. \\ \phi _{T}(t)=\left\{ \begin{array}{r@{\quad }l} t &{} 0\le t<1,\\ 2t &{} 1\le t<2,\\ 0 &{} otherwise. \end{array}\right. \end{gathered}\end{aligned}$$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
$$\begin{aligned} \phi _{n}(t)=\int _{0}^{1}\phi _{n1}(t\tau )d\tau ,\quad n=2, 3, \cdots . \end{aligned}$$Calculate and sketch \(\phi _{2}(t)\) and \(\phi _{3}(t)\). (Note that \(\phi _{0}=\phi _{H}\) and \(\phi _{1}=\phi _{T}.)\)

11.2.
Find the Fourier transforms of \(\phi _{H}(t)\) and \(\phi _{T}(t)\) defined above.

11.3.
Based on the graphs of \(\phi _{H}(t)\) and \(\phi _{T}(t)\), sketch
$$\begin{aligned} \phi _{1, 0}(t)=\phi _{H}(2t),\quad \phi _{1, 1}(t)=\phi _{H}(2t1),\quad and\quad \mathrm {\Phi }(t)=\phi _{1, 0}(t)+\phi _{1, 1}(t). \end{aligned}$$Moreover, sketch the wavelets
$$\begin{aligned} \psi _{H}(t)=\phi _{H}(2t)\phi _{H}(2t1) \end{aligned}$$and
$$\begin{aligned} \psi _{T}(t)=\frac{1}{2}\phi _{T}(2t)\frac{1}{2}\phi _{T}(2t2)+\phi _{T}(2t1). \end{aligned}$$ 
11.4.
Verify (11.16).
 11.5.
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Chui, C.K., Chen, G. (2017). Wavelet Kalman Filtering. In: Kalman Filtering. Springer, Cham. https://doi.org/10.1007/9783319476124_11
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DOI: https://doi.org/10.1007/9783319476124_11
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