Abstract
The importance of Kalman filtering in engineering applications is well known, and its mathematical theory has been rigorously established. The main objective of this treatise is to present a thorough discussion of the mathematical theory, computational algorithms, and application to realtime tracking problems of the Kalman filter.
Keywords
 Probability Density Function
 Schwarz Inequality
 Joint Probability Density Function
 Fair Coin
 Conditional Probability Density Function
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Exercises
Exercises

1.1
Prove Lemma 1.4.

1.2
Prove Lemma 1.6.

1.3
Give an example of two matrices A and B such that \(A\ge B> 0\) but for which the inequality \(AA^{\top }\ge BB^{\top }\) is not satisfied.

1.4
Prove Lemma 1.8.

1.5
Show that \(\int \nolimits _{\infty }^{\infty }e^{y^{2}}dy=\sqrt{\pi }\).

1.6
Verify that \(\int \nolimits _{\infty }^{\infty }y^{2}e^{y^{2}}dy=\frac{1}{2}\sqrt{\pi }\). (Hint: Differentiate the integral \(\int \nolimits _{\infty }^{\infty }e^{xy^{2}}dy\) with respect to x and then let \(x\rightarrow 1\).)

1.7
Let
$$\begin{aligned} f(\mathbf {x})=\frac{1}{(2\pi )^{n/2}(detR)^{1/2}}exp\left\{ \frac{1}{2}(\mathbf {x}\underline{\mu })^{\top }R^{1}(\mathbf {x}\underline{\mu })\right\} . \end{aligned}$$Show that

(a)
$$\begin{aligned} E(X)=&\int \nolimits _{\infty }^{\infty }\mathbf {x}f(\mathbf {x})d\mathbf {x}\\ :=&\int \nolimits _{\infty }^{\infty }\cdots \int \nolimits _{\infty }^{\infty } \left[ \begin{array}{c} x_{1}\\ \vdots \\ x_{n} \end{array}\right] \ f(\mathbf {x})dx_{1}\cdots dx_{n}\\ =&\underline{\mu }, \end{aligned}$$
and

(b)
$$\begin{aligned} Var(X)=E(X\underline{\mu })(X\underline{\mu })^{\top }=R. \end{aligned}$$

(a)

1.8
Verify the properties (1.32a–1.32e) of the expectation, variance, and covariance.

1.9
Prove that two random vectors \(X_{1}\) and \(X_{2}\) with normal distributions are independent if and only if \(Cov(X_{1}, X_{2})=0\).

1.10
Verify (1.35).

1.11
Consider the minimization of the quantity
$$\begin{aligned} F(\mathbf {y}_{k})=E(\mathbf {z}_{k}C_{k}\mathbf {y}_{k})^{\top }W_{k}(\mathbf {z}_{k}C_{k}\mathbf {y}_{k}) \end{aligned}$$over all nvectors \(\mathbf {y}_{k}\), where \(\mathbf {z}_{k}\) is a \(q\,\times \, 1\) vector, \(C_{k}\), a \(q\,\times \, n\) matrix, and \(W_{k}\), a \(q\,\times \, q\) weight matrix, such that the matrix \((C_{k}^{\top }W_{k}C_{k})\) is nonsingular. By letting \(dF(\mathbf {y}_{k})/d\mathbf {y}_{k}=0\), show that the optimal solution \(\hat{\mathbf {y}}_{k}\) is given by
$$\begin{aligned} \hat{\mathbf {y}}_{k}=(C_{k}^{\top }W_{k}C_{k})^{1}C_{k}^{\top }W_{k}\mathbf {z}_{k}. \end{aligned}$$(Hint: The differentiation of a scalarvalued function \(F(\mathbf {y})\) with respect to the nvector \(\mathbf {y}=[y_{1}\cdots y_{n}]^{\top }\) is defined to be
$$\begin{aligned} \frac{dF(\mathbf {y})}{d\mathbf {y}}=\left[ \frac{\partial F}{\partial y_{1}}\cdots \frac{\partial F}{\partial y_{n}}\right] ^{\top }.) \end{aligned}$$ 
1.12
Verify that the estimate \(\hat{\mathbf {x}}_{k}\) given by (1.38) is an unbiased estimate of \(\mathbf {x}_{k}\) in the sense that \(E\hat{\mathbf {x}}_{k}=E\mathbf {x}_{k}\).
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Chui, C.K., Chen, G. (2017). Preliminaries. In: Kalman Filtering. Springer, Cham. https://doi.org/10.1007/9783319476124_1
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DOI: https://doi.org/10.1007/9783319476124_1
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Print ISBN: 9783319476100
Online ISBN: 9783319476124
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