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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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  • DOI: 10.1007/978-3-319-47612-4_1
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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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Chui, C.K., Chen, G. (2017). Preliminaries. In: Kalman Filtering. Springer, Cham. https://doi.org/10.1007/978-3-319-47612-4_1

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