# Preliminaries

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## 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 real-time tracking problems of the Kalman filter.

### Keywords

• Probability Density Function
• Schwarz Inequality
• Joint Probability Density Function
• Fair Coin
• Conditional Probability Density Function

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

Authors

### Corresponding author

Correspondence to Charles K. Chui .

## Exercises

### Exercises

1. 1.1

Prove Lemma 1.4.

2. 1.2

Prove Lemma 1.6.

3. 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.

4. 1.4

Prove Lemma 1.8.

5. 1.5

Show that $$\int \nolimits _{-\infty }^{\infty }e^{-y^{2}}dy=\sqrt{\pi }$$.

6. 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 $$x\rightarrow 1$$.)

7. 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

1. (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

2. (b)
\begin{aligned} Var(X)=E(X-\underline{\mu })(X-\underline{\mu })^{\top }=R. \end{aligned}
8. 1.8

Verify the properties (1.32a1.32e) of the expectation, variance, and covariance.

9. 1.9

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

10. 1.10

Verify (1.35).

11. 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 n-vectors $$\mathbf {y}_{k}$$, 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 scalar-valued function $$F(\mathbf {y})$$ with respect to the n-vector $$\mathbf {y}=[y_{1}\cdots y_{n}]^{\top }$$ 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}
12. 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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