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.
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Exercises
Exercises
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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 \(\phi _{1}=\phi _{T}.)\)
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11.2.
Find the Fourier transforms of \(\phi _{H}(t)\) and \(\phi _{T}(t)\) defined above.
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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
$$\begin{aligned} \psi _{T}(t)=-\frac{1}{2}\phi _{T}(2t)-\frac{1}{2}\phi _{T}(2t-2)+\phi _{T}(2t-1). \end{aligned}$$ -
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/10.1007/978-3-319-47612-4_11
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DOI: https://doi.org/10.1007/978-3-319-47612-4_11
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Online ISBN: 978-3-319-47612-4
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