### 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|ECE310

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Realization formulas

To present the Stieltjes-class analog of Theorem 1.4, we start with the collection
$$\Lambda={\mu, \mathcal{\mathcal { X }}, \tilde{\mathcal{X}}, \widehat{\mathcal{G}}, A, \widetilde{A}, B, C, \Pi, \widetilde{\Pi}}$$
consisting of a point $\mu \in \mathbb{C}$, three Hilbert spaces $\mathcal{X}, \tilde{\mathcal{X}}, \widehat{\mathcal{G}}=\mathcal{G} \oplus \mathcal{G}$, and bounded operators
\begin{aligned} &A \in \mathcal{L}(\mathcal{X}), \quad \tilde{A} \in \mathcal{L}(\tilde{\mathcal{X}}), \quad B \in \mathcal{L}(\tilde{\mathcal{X}}, \mathcal{X}), \quad C \in \mathcal{L}(\mathcal{X}, \tilde{\mathcal{X}}), \ &\Pi=\left[\begin{array}{l} \Pi_{1} \ \Pi_{2} \end{array}\right] \in \mathcal{L}(\mathcal{X}, \widehat{\mathcal{G}}), \quad \tilde{\Pi}=\left[\begin{array}{c} \widetilde{\Pi}{1} \ \widetilde{\Pi}{2} \end{array}\right] \in \mathcal{L}(\tilde{\mathcal{X}}, \widehat{\mathcal{G}}) \end{aligned}
and we call this collection admissible if the pairs $(\Pi, A)$ and $(\tilde{\Pi}, \tilde{A})$ are observable and the following equalities hold:
\begin{aligned} &A(I+\mu A)=B C, \quad \tilde{A}(I+\mu \tilde{A})=C B, \quad C A=\tilde{A} C, \quad A B=B \tilde{A} \ &\Pi_{1}\left[\begin{array}{ll} I+\mu A & B \end{array}\right]=\widetilde{\Pi}{1}\left[\begin{array}{ll} C & \tilde{A} \end{array}\right], \quad \Pi{2}\left[\begin{array}{ll} A & B \end{array}\right]=\widetilde{\Pi}_{2}\left[\begin{array}{ll} C & I+\mu \tilde{A} \end{array}\right] . \end{aligned}
As a model for an admissible collection, consider the choice based on a $\mathcal{L}(\widehat{\mathcal{G}})$-valued function $\Theta$ meromorphic on the domain $\Omega$ and a fixed point $\mu$ in $\Omega$ where $\Theta$ is analytic:
$\mathcal{X}=\mathcal{H}(\Theta), \quad \tilde{\mathcal{X}}=\mathcal{H}\left(\Theta_{P}\right), \quad A=\left.R_{\mu}\right|{\mathcal{H}(\Theta)}, \quad \widetilde{A}=\left.R{\mu}\right|{\mathcal{H}\left(\Theta{P}\right)}$,
$B=\left.\left[\begin{array}{cc}R_{\mu} & 0 \ 0 & I+\mu R_{\mu}\end{array}\right]\right|{\mathcal{H}\left(\Theta{P}\right)}, \quad C=\left.\left[\begin{array}{cc}I+\mu R_{\mu} & 0 \ 0 & R_{\mu}\end{array}\right]\right|{\mathcal{H}(\Theta)}$, $\Pi=E{\mu}\left|\mathcal{H}(\Theta), \quad \tilde{\Pi}=E_{\mu}\right|{\mathcal{H}\left(\Theta{P}\right)} .$
It is a consequence of Theorem $3.1$ that the mapping properties (4.2) work out with this specification. The remaining identities (4.3)-(4.4) follow from the definitions or straightforward algebra.
We will say that the collection (4.1) is similar to the collection
$$\Lambda=\left{\mu, \mathcal{X}^{\prime}, \widetilde{\mathcal{X}}^{\prime}, \widehat{\mathcal{G}}^{\prime}, A^{\prime}, \widetilde{A}^{\prime}, B^{\prime}, C^{\prime}, \Pi^{\prime}, \widetilde{\Pi}^{\prime}\right}$$
if there exist invertible operators $T \in \mathcal{L}\left(\mathcal{X}, \mathcal{X}^{\prime}\right)$ and $\widetilde{T} \in \mathcal{L}\left(\tilde{\mathcal{X}}, \tilde{\mathcal{X}}^{\prime}\right)$ such that $A^{\prime} T=T A, \quad \widetilde{A}^{\prime} \tilde{T}=\widetilde{T} \tilde{A}, \quad B^{\prime} \tilde{T}=T B, \quad C^{\prime} T=\tilde{T} C, \quad \Pi^{\prime} T=\Pi, \quad \widetilde{\Pi}^{\prime} \tilde{T}=\widetilde{\Pi} .$
It is readily seen that a collection similar to an admissible one is also admissible.

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Explicit formulas for Θ

Let us assume now that the gramians $\mathcal{G}{\Pi, A, \mu}$ and $\mathcal{G}{\tilde{\Pi}, \tilde{A}, \mu}$ are invertible. By the geneneral princíples of reproducíng kernèl Hilbert spaces, it follows from the reepresentations (4.5) that reproducing kernels $K_{\Theta}$ and $K_{P \Theta P^{-1}}$ for $\mathcal{H}$ and $\mathcal{H}$ are equal to
$$K_{P \Theta P^{-1}}(z, \omega)=\frac{J-\Theta_{P}(z) J \Theta_{P}(\omega)^{}}{i(\bar{\omega}-z)}=\left[\begin{array}{l} \widetilde{\Pi}{1} \ \widetilde{\Pi}{2} \end{array}\right] \widetilde{\Gamma}(z) \mathcal{G}{\widetilde{\Pi}, \widetilde{A}, \mu}^{-1} \widetilde{\Gamma}(\omega)^{}\left[\begin{array}{ll} \widetilde{\Pi}{1}^{} & \widetilde{\Pi}_{2}^{} \end{array}\right]$$
The next question is to find a fairly satisfactory formula for $\Theta$ satisfying the kernel identities (4.35), (4.36).

Theorem 4.5. Given an admissible collection (4.1) with $\mu \in \mathbb{R}$ and subject to the identity (4.30). Then:

1. The functions
\begin{aligned} &\Upsilon(z)=I_{\widehat{\mathcal{G}}}+i(z-\mu) \Pi \Gamma(z) \mathcal{G}{\Pi, A, \mu}^{-1} \Pi^{} J, \ &\widetilde{\Upsilon}(z)=I{\widehat{\mathcal{G}}}+i(z-\mu) \widetilde{\Pi} \widetilde{\Gamma}(z) \mathcal{G}{\widetilde{\Pi}, \widetilde{A}, \mu}^{-1} \widetilde{\Pi}^{} J \end{aligned}
belong to the class $\mathcal{M \mathcal { P }}(\mathcal{G})$ and the kernels $K{\Upsilon}(z, \omega)$ and $K_{\tilde{\Upsilon}}(z, \omega)$ are equal to the right-hand side expressions in (4.35), (4.36):
\begin{aligned} &K_{\curlyvee}(z, \omega)=\left[\begin{array}{l} \Pi_{1} \ \Pi_{2} \end{array}\right] \Gamma(z) \mathcal{G}{\Pi, A, \mu}^{-1} \Gamma(\omega)^{}\left[\begin{array}{ll} \Pi{1}^{} & \Pi_{2}^{} \end{array}\right], \ &K_{\widetilde{\Upsilon}}(z, \omega)=\left[\begin{array}{l} \widetilde{\Pi}{1} \ \widetilde{\Pi}{2} \end{array}\right] \widetilde{\Gamma}(z) \mathcal{G}{\widetilde{\Pi}, \widetilde{A}, \mu}^{-1} \widetilde{\Gamma}(\omega)^{}\left[\begin{array}{ll} \widetilde{\Pi}{1}^{} & \widetilde{\Pi}_{2}^{} \end{array}\right] . \end{aligned}
2. Furthermore, there exist J-unitary operators $N, \widetilde{N} \in \mathcal{L}(\widehat{\mathcal{G}})$ such that the function $\Theta(z)=\Upsilon(z) N$ belongs to the class $\mathcal{M S}(\mathcal{G})$ and the associated function $\Theta_{P}$ is equal to $\Theta_{P}(z):=P(z) \Theta(z) P(z)^{-1}=\tilde{\Upsilon}(z) \bar{N}$.

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Realization formulas

$$\Lambda=\mu, \mathcal{X}, \tilde{\mathcal{X}}, \widehat{\mathcal{G}}, A, \widetilde{A}, B, C, \Pi, \widetilde{\Pi}$$

$A \in \mathcal{L}(\mathcal{X}), \quad \tilde{A} \in \mathcal{L}(\tilde{\mathcal{X}}), \quad B \in \mathcal{L}(\tilde{\mathcal{X}}, \mathcal{X}), \quad C \in \mathcal{L}(\mathcal{X}, \tilde{\mathcal{X}}), \quad \Pi=\left[\Pi_{1} \Pi_{2}\right] \in \mathcal{L}(\mathcal{X}, \widehat{\mathcal{G}}), \quad \tilde{\Pi}=[\tilde{\Pi}$

$$A(I+\mu A)=B C, \quad \tilde{A}(I+\mu \tilde{A})=C B, \quad C A=\tilde{A} C, \quad A B=B \tilde{A} \quad \Pi_{1}[I+\mu A \quad B]=\widetilde{\Pi} 1[C$$

$$\mathcal{X}=\mathcal{H}(\Theta), \quad \tilde{\mathcal{X}}=\mathcal{H}\left(\Theta_{P}\right), \quad A=R_{\mu}|\mathcal{H}(\Theta), \quad \widetilde{A}=R \mu| \mathcal{H}(\Theta P) \text {, }$$
$B=\left[\begin{array}{llll}R_{\mu} & 0 & 0 & I+\mu R_{\mu}\end{array}\right]\left|\mathcal{H}(\Theta P), \quad C=\left[I+\mu R_{\mu} \quad 0 \quad 0 \quad R_{\mu}\right]\right| \mathcal{H}(\Theta)$,
$\Pi=E \mu\left|\mathcal{H}(\Theta), \quad \tilde{\Pi}=E_{\mu}\right| \mathcal{H}(\Theta P)$.

$A^{\prime} T=T A, \quad \tilde{A}^{\prime} \tilde{T}=\widetilde{T} \tilde{A}, \quad B^{\prime} \tilde{T}=T B, \quad C^{\prime} T=\tilde{T} C, \quad \Pi^{\prime} T=\Pi, \quad \tilde{\Pi}^{\prime} \tilde{T}=\widetilde{\Pi} .$

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Explicit formulas for Θ

$$K_{P \Theta P^{-1}}(z, \omega)=\frac{J-\Theta_{P}(z) J \Theta_{P}(\omega)}{i(\bar{\omega}-z)}=[\widetilde{\Pi} 1 \widetilde{\Pi} 2] \widetilde{\Gamma}(z) \mathcal{G} \widetilde{\Pi}, \widetilde{A}, \mu^{-1} \widetilde{\Gamma}(\omega)\left[\widetilde{\Pi} 1 \quad \widetilde{\Pi}_{2}\right]$$

1. 功能
$$\Upsilon(z)=I_{\widehat{\mathcal{G}}}+i(z-\mu) \Pi \Gamma(z) \mathcal{G} \Pi, A, \mu^{-1} \Pi J, \quad \widetilde{\Upsilon}(z)=I \widehat{\mathcal{G}}+i(z-\mu) \widetilde{\Pi} \widetilde{\Gamma}(z) \mathcal{G} \widetilde{\Pi}, \widetilde{A}, \mu^{-1} \widetilde{\Pi} J$$
属于类 $\mathcal{M} \mathcal{P}(\mathcal{G})$ 和内核 $K \Upsilon(z, \omega)$ 和 $K_{\tilde{\Upsilon}}(z, \omega)$ 等于 (4.35), (4.36) 中的右侧表达式:
$$K_{\curlyvee}(z, \omega)=\left[\Pi_{1} \Pi_{2}\right] \Gamma(z) \mathcal{G} \Pi, A, \mu^{-1} \Gamma(\omega)\left[\begin{array}{ll} \Pi 1 & \Pi_{2} \end{array}\right], \quad K_{\tilde{\Upsilon}}(z, \omega)=[\widetilde{\Pi} 1 \widetilde{\Pi} 2] \widetilde{\Gamma}(z) \mathcal{G} \widetilde{\Pi}, \widetilde{A}, \mu^{-1}$$
2. 此外，存在J-酉算子 $N, \widetilde{N} \in \mathcal{L}(\widehat{\mathcal{G}})$ 使得函数 $\Theta(z)=\Upsilon(z) N$ 属于类 $\mathcal{M} \mathcal{S}(\mathcal{G})$ 和相关的功能 $\Theta_{P}$ 等于 $\Theta_{P}(z):=P(z) \Theta(z) P(z)^{-1}=\tilde{\Upsilon}(z) \bar{N}$

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