## 电子工程代写|数字信号处理代写Digital Signal Processing代考|ECE310

statistics-lab™ 为您的留学生涯保驾护航 在代写数字信号处理Digital Signal Processing方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数字信号处理Digital Signal Processing方面经验极为丰富，各种代写数字信号处理Digital Signal Processing相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|数字信号处理代写Digital Signal Processing代考|Primary Transmitter Detection

Transmitter detection techniques emphasize detecting low power signals from any PU. Low power signals mix with noise from the environment and make it hard for the CR user to detect primary signals. A low signal-to-noise ratio, multipath fading effects, and time depression make primary transmissions detection very difficult for the $\mathrm{CR}$ user. We discuss some primary transmitter detection techniques including energy detection, coherent detection, and matched filter detection.

This technique does not require CR users to have knowledge of PU signal characteristics, and it is easy to implement. Because of this, it is widely used to detect primary transmissions. Let us assume $S(n)$ is the signal received by the CR user, $W(n)$ is white Gaussian noise, and $P(n)$ is the original signal from the PU.
$$\begin{gathered} H_0: S(n)=W(n) \ H_1: S(n)=W(n)+h P(n) \end{gathered}$$
Hypothesis $H_0$ indicates the absence of a PU and hypothesis $H_1$ indicates the presence of PU transmissions. $h$ denotes the channel gain between the primary and secondary transmissions. Then, the average energy $S$ of $N$ samples is
$$S=1 / N \sum_{n=1}^N S(n)^2$$
The CR user collects $N$ samples, calculates the average energy, and compares it with a threshold $\lambda$. If the average energy is greater than the threshold, $\lambda$, then the CR user concludes that primary transmissions are present. To measure the performance, we denote the probability of the false positive (CR detects the presence of PU transmissions when there is no PU transmission) as $P_f$ and probability of the detection as $P_d$
\begin{aligned} &P_f=P\left(S>\lambda \mid H_0\right) \ &P_d=P\left(S>\lambda \mid H_1\right) \end{aligned}
To improve the performance, we need to keep the PU’s transmission secured. Therefore, the false positive probability should be less than $0.1$ and the detection probability should be greater than $0.9$.

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|Primary Receiver Detection

The most effective way to detect PU transmissions is to detect the primary receivers who are receiving from the primary channel. The circuit in Fig. 5 shows a simple RF receiver. It has a local oscillator that emits a very low power signal for its leakage current in the circuit. A CR user can detect the leakage signals from the RF receiver circuit and identify the presence of primary transmissions. This detection technique solves both the hidden terminal and shadowing effect problems. Since the signal power is very low, it is very challenging and costly to implement the circuit for primary receiver detection.

When primary signal features like modulation type, pulse shape, operating frequency, packet format, noise statistics, etc., are known, matched filter detection can be an optimal detection technique. If these parameters are known, the CR user only needs to calculate a small number of samples. As the signal-to-noise ratio decreases, the $\mathrm{CR}$ user needs to calculate a greater number of samples. The disadvantages of this technique are the complexities in low signal-to-noise ratio, the high cost of implementation, and the very poor performance if the features are incorrect.

In a broader sense, a signal can be called a cyclostationary process if its statistical properties vary cyclically with time. In [6], the authors presented a signal classification procedure that extracts cyclic frequency domain profiles and classifies them by comparing their log-likelihood with the signal type in the database. This technique can work very well in a low SNR. The drawback of this technique is that it needs a huge amount of computation and thus, a high-speed sensing is hard to achieve [7].

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|Primary Transmitter Detection

$$H_0: S(n)=W(n) H_1: S(n)=W(n)+h P(n)$$

$$S=1 / N \sum_{n=1}^N S(n)^2$$
CR用户收藏 $N$ 采样，计算平均能量，并将其与阈值进行比较 $\lambda$. 如果平均能量大于阈值， $\lambda$ ，则 CR 用户断定存在 主要传输。为了衡量性能，我们将误报的概率表示为 (CR 在没有 $\mathrm{PU}$ 传输时检测到 $\mathrm{PU}$ 传输的存在) 为 $P_f$ 和检 测概率为 $P_d$
$$P_f=P\left(S>\lambda \mid H_0\right) \quad P_d=P\left(S>\lambda \mid H_1\right)$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|EE615

statistics-lab™ 为您的留学生涯保驾护航 在代写数字信号处理Digital Signal Processing方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数字信号处理Digital Signal Processing方面经验极为丰富，各种代写数字信号处理Digital Signal Processing相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|数字信号处理代写Digital Signal Processing代考|Network Architecture of Cognitive Radio Networks

This subsection describes the network architecture and components of a CRN. Figure 1 depicts the whole network system. User devices, primary base stations, and CR base stations are the components of a basic CRN. In Fig. 1, there are two channels: channel 1 and channel 2. One primary base station operates in channel 1 and another in channel 2. Transmissions with the primary base station are done through licensed channels by mobile users, and the transmissions are called primary transmissions (denoted by solid lines). Transmissions with the CR base station can be done through either licensed or unlicensed channels and these transmissions are called secondary transmissions (marked by dotted lines). There is also another kind of trañsmission in which any usēr devicee can transmit directly to anothèr userr device.. Therefore, transmissions in a CRN can be grouped into three classes:

• Primary transmissions: Primary transmissions are most prioritized transmissions and cannot be compromised by other transmissions. These transmissions are done in a licensed channel between primary base stations and PUs. Primary transmissions are denoted by solid lines in Fig. 1.
• Secondary transmissions: Secondary transmissions are done in the absence of primary transmissions. Transmissions between the CR base station and the $\mathrm{CR}$ user are usually secondary transmissions.
• Secondary ad hoc transmissions: User-to-user communications are called ad hoc transmissions. These transmissions can continue without base stations or other components of the network architecture. Users create their own network topology and adapt any routing protocols of ad hoc networks. Users in the gray area form an ad hoc network in Fig. 1. There are a lot of routing protocols for mobile ad hoc networks. For example, the proposed routing algorithm in [1], which ensures a fair amount of communications among nodes and improves the load concentration problem, can be used in secondary ad hoc networks. The on-demand cluster-based hybrid routing protocol proposed in [2] is also applicable here.

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|Spectrum Sensing

Secondary transmissions depend on spectrum sensing information, so this step should be done very accurately. Inaccurate sensing detection can lead to interferences with the PU that are highly unexpected. Though false alarms (in which channel is not occupied, but is detected as occupied) do not create interferences with the primary transmissions, it makes the CR user choose a channel from a narrower range of channels. As a result, a channel must be shared with many CR users and there would be increased competition among $\mathrm{CR}$ users to access the channel. The authors of [3] present a classification of spectrum sensing techniques. First, they classify sensing techniques into three groups: noncooperative sensing, cooperative sensing, and interference-based sensing. Noncooperative sensing is again classified into three groups: energy detection, matched filter detection, and cyclostationary feature detection. The classification is depicted in Fig. 2.

In noncooperative sensing, CR users do not share sensing information with one another. A CR user makes a decision about the PU’s presence using its own sensing information. We discuss primary transmitter detection and primary receiver detection, which are presented in $[4,5]$, in the following subsection.

Transmitter detection techniques emphasize detecting low power signals from any PU. Low power signals mix with noise from the environment and make it hard for the $\mathrm{CR}$ user to detect primary signals. A low signal-to-noise ratio, multipath fading effects, and time depression make primary transmissions detection very difficult for the CR user. We discuss some primary transmitter detection techniques including energy detection, coherent detection, and matched filter detection.

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|Network Architecture of Cognitive Radio Networks

• 主传输：主传输是最优先的传输，不能被其他传输影响。这些传输是在主基站和 PU 之间的许可信道中完成的。初级传输在图 1 中用实线表示。
• 二次传播：二次传播是在没有一次传播的情况下进行的。CR基站与CR用户通常是二次传输。

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|ELEC3104

statistics-lab™ 为您的留学生涯保驾护航 在代写数字信号处理Digital Signal Processing方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数字信号处理Digital Signal Processing方面经验极为丰富，各种代写数字信号处理Digital Signal Processing相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|数字信号处理代写Digital Signal Processing代考|Need for Advancement in Wireless Technologies

The performance metrics such as packet loss, throughput and delay of WiMAX are measured on the basis of optimal boundary per WiMAX cell under different WiMAX network models. The performance metrics considered are spectral efficiency, throughput, transmit power, percentage of successful links, PAPR, BER.

SNR and CINR. This chapter mainly focusses on spectrum sensing techniques to achieve better spectral efficiency [5].

Recently, there is a lot of demand for tremendous technologies such as 3G, 4G and $5 \mathrm{G}$, where voice-only communications are transitioned into multimedia type applications $[6,7]$. These applications may be mobile TV, mobile P2P, streaming multimedia, video games, video monitors, interactive video, 3D services and video sharing. These high data rate applications consume more and more energy to guarantee quality of service [8]. However, the current frequency allocation schemes are unable to handle the requirements of recent higher data rate systems due to the limitations of the frequency spectrum.

Therefore, more efforts are kept on efficient frequency spectrum usage, and then a solution is found by Joseph Mittola [9], in the name of cognitive radio. The basic definition given by him is that cognitive radio (CR) is a type of a transceiver which can intelligently sense or detect unusable communication channel, and instantly allocate those channels to the unlicensed users without disturbing occupied channels [10]. Though there is no formal meaning of cognitive radio, various definitions can be seen in several contexts. A cognitive radio is, as defined by the researchers at Virginia Tech, ‘A software defined radio with a cognitive engine brain’ [11, 12]. The evolution of SDR in current technologies is provided in Fig. 2. The physical, data link and network layers of OSI model can be implemented by using SDR as shown in Fig. 3. The SDR Forum proposed a multi-tiered definition of SDR by providing the use of open architectures for advanced wireless systems and supports deployment and development [13-15]. An abstraction of the five-tier definition is illustrated in Fig. 4, where the length of the arrow represents the distribution of the software content within the radio [16].

Software-defined radio architecture comprises three sections such as radio frequency (RF), intermediate frequency (IF) and baseband section [17, 18]. It is observed from Fig. 5 that an RF signal received by smart antenna is sent to the hardware (here USRP) in which various components are inbuilt such as daughterboard, ADC/DAC, FPGAs, DSPs and ASICs. This hardware converts RF signal to IF signal and then to low-frequency baseband signal (digitized) and that will be sent to a personal computer (PC) for baseband signal processing in the transmitter (Tx) path. In this experimentation, an open-source software, GNU Radio, is employed as a software to perform baseband processing in which most of the signal processing blocks are inbuilt. All the reverse operations are performed in receiver (Rx) path such that baseband signal is converted to analogue by DAC and then sent into the air by RF hardware.

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|Results and Discussion

Generally, energy detection performance is measured in terms of probability of false alarm $P_{f a}$ (detection algorithm falsely decides that $\mathrm{PU}$ is present when it actually is absent) and probability of detection $P_d$ (correctly detecting the PU signal). Mathematically, $P_{f a}$ and $P_d$ can be expressed as [16]:
\begin{aligned} &P_{f a}=P_r\left(\text { signal is detectedl } H_0=P_r\left(u>\lambda \mid H_0\right)=\int_\lambda^{\infty} f\left(u \mid H_0\right) d u\right. \ &P_d=P_r \text { (signal is detected } H_1=P_r\left(u>\lambda \mid H_1\right)=\int_\lambda^{\infty} f\left(u \mid H_1\right) d u \end{aligned}
where $f\left(u \mid H_i\right.$ ) denotes the probability density function (pdf) of test statistic under hypothesis $H_i$ with $i=0,1$.

Thus, we target at maximizing $P_d$ while minimizing $P_{f a} . P_d$ versus $P_{f a}$ plot depicts receiver operating characteristics (ROC) and is considered as an important performance indicator. The receiver operating characteristics $(\mathrm{ROC})$ for various number of sensing samples, such as 10,50, 100 and 200, are presented in Fig. 7a, b, c and d, respectively [16]. It can be observed from Fig. 7 that the probability of detection $\left(p_d\right)$ is increased with the number of sensing samples. In our simulations, some assumptions are made such as the primary signal is deterministic, and noise is real Gaussian with mean 0 and variance 1 [17]. The probability of detection for Rayleigh channel is calculated by the averaging the probability of detection for AWGN channel [18].

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|Need for Advancement in Wireless Technologies

WiMAX 的丢包率、吞吐量和延迟等性能指标是在不同WiMAX 网络模型下每个WiMAX 小区的最佳边界的基础上测量的。考虑的性能指标是频谱效率、吞吐量、发射功率、成功链接的百分比、PAPR、BER。

SNR 和 CINR。本章主要关注频谱感知技术，以实现更好的频谱效率[5]。

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|Results and Discussion

$P_{f a}=P_r\left(\right.$ signal is detectedl $H_0=P_r\left(u>\lambda \mid H_0\right)=\int_\lambda^{\infty} f\left(u \mid H_0\right) d u \quad P_d=P_r$ (signal is

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|ECE310

statistics-lab™ 为您的留学生涯保驾护航 在代写数字信号处理Digital Signal Processing方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数字信号处理Digital Signal Processing方面经验极为丰富，各种代写数字信号处理Digital Signal Processing相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|数字信号处理代写Digital Signal Processing代考|The Basic Network

Consider the simplest configuration for impedance matching, namely, the L-network shown in Fig. 3.1. $Z_{\mathrm{s}}$ is an $L C$ impedance and $Y_{p}$ is an $L C$ admittance such that when the network is terminated in $R_{L}$, the input impedance is $R_{S}$ at the frequencies of interest. Let, at $s=j \omega, Z_{s}=j X_{s}$ and $Y_{P}=j B_{P}$, where $X$ denotes reactance and $B$ denotes susceptance. Then, at the frequencies of interest, we should have
$$R_{S}=j X_{s}+1 /\left(j B_{p}+G_{L}\right)$$
where $G_{L}=1 / R_{L}$. Cross multiplying, simplifying, and equating the real and imaginary parts on both sides give the two equations
$$X_{S} B_{p}=1-R_{S} G_{L} \text { and } R_{S} B_{p}=X_{S} G_{L}$$
The second equation shows that $X_{s}$ and $B_{p}$ must be of the same sign, both positive or both negative. Combining this fact with the first Equation in (3.2), we note that $R_{S} G_{L}$ must be less than unity, i.e. $R_{S}$ must be less than $R_{L}$. However, this is no restriction because the other situation, i.e. $R_{S}>R_{L}$, can be taken care of by simply interchanging the positions of $R_{S}$ and $R_{L}$ in Fig. 3.1. Eliminating $B_{p}$ from the two Equations in (3.2), we get
$$X_{s}^{2}=R_{S}\left(R_{L}-R_{S}\right)=R_{1}^{2}, \text { say }$$

Or,
$$X_{\mathrm{s}}=\pm R_{1}$$
Combining this with the second Equation in (3.2) gives
$$B_{p}=\pm R_{1} /\left(R_{L} R_{S}\right)=\pm G_{2} \text {, say }$$
As already stated, the signs in Eqs. (3.4) and (3.5) should be either both positive or both negative.

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|Impedance Matching at a Single Frequency

For matching $R_{L}$ to $R_{S}$ at a single frequency $\omega_{0}$, we can choose either an inductance $L_{s}$ for $X_{s}$ and a capacitance $C_{p}$ for $B_{p}$, or a capacitance $C_{s}$ for $X_{s}$ and an inductance $L_{p}$ for $B_{p}$. In the first case, to be referred to as Design 1 (D1)
$$L_{s}=R_{1} / \omega_{0} \text { and } C_{p}=G_{2} / \omega_{0}$$
while for the alternative design, to be called Design 2 (D2),
$$C_{s}=1 /\left(R_{1} \omega_{0}\right) \text { and } L_{p}=1 /\left(G_{2} \omega_{0}\right)$$
We shall mostly use D1 in our further discussions, it is being implied that D2 is equally applicable, giving another set of solutions.

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|The Basic Network

$$R_{S}=j X_{s}+1 /\left(j B_{p}+G_{L}\right)$$

$$X_{S} B_{p}=1-R_{S} G_{L} \text { and } R_{S} B_{p}=X_{S} G_{L}$$

$$X_{s}^{2}=R_{S}\left(R_{L}-R_{S}\right)=R_{1}^{2} \text {, say }$$

$$X_{\mathrm{s}}=\pm R_{1}$$

$$B_{p}=\pm R_{1} /\left(R_{L} R_{S}\right)=\pm G_{2}, \text { say }$$

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|Impedance Matching at a Single Frequency

$$L_{s}=R_{1} / \omega_{0} \text { and } C_{p}=G_{2} / \omega_{0}$$

$$C_{s}=1 /\left(R_{1} \omega_{0}\right) \text { and } L_{p}=1 /\left(G_{2} \omega_{0}\right)$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|EE615

statistics-lab™ 为您的留学生涯保驾护航 在代写数字信号处理Digital Signal Processing方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数字信号处理Digital Signal Processing方面经验极为丰富，各种代写数字信号处理Digital Signal Processing相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|数字信号处理代写Digital Signal Processing代考|BWER of the Asymmetrical BTC

For a general transfer function of the form of Eq. (2.14), combining Eqs. (2.17) and (2.18) with Eq. (2.15), we get the following expression for the BWER of the asymmetrical BTC:
$$\eta=(1+b+2 m) \sqrt{\frac{b+m}{(1+m)\left(b-m^{2}\right)}} \times \sqrt{\left(1-2 \varsigma^{2}\right)+\sqrt{\left(1-2 \varsigma^{2}\right)^{2}+1}}$$
For MFM, $5=1 / \sqrt{2}$ and Eq. (2.24) becomes
$$\eta=(1+b+2 m)\left(\frac{b+m}{(1+m)\left(b-m^{2}\right)}\right)^{\frac{1}{2}}$$
Further, putting $\zeta=1 / \sqrt{2}$, we get
$$2=\frac{(b+m)^{3}}{(1+m)\left(b+m^{2}\right)}$$
Thus for MFM response, $b$ and $m$ have to satisfy Eq. (2.26) and for any such set, $\eta$ is given by Eq. (2.25). Combining Eqs. (2.25) with (2.26), the expression for $\eta$ gets further simplified to the following:
$$\eta=\sqrt{2}\left(1+\frac{1+m}{b+m}\right)$$
It is difficult to find, analytically, the variation of $b$ with $m$ from Eq. (2.26) or that of $\eta$ with $m$ from Eq. (2.27). One way is to simplify Eq. (2.26) to get a cubic equation in $b$ and to solve it; however, as is well known $[35,36]$, an explicit expression for $b$ in terms of $m$ cannot be written and one has to take recourse to numerical computations. An alternative, and a simpler way, is to introduce the variable $\beta=b / m$.

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|Technology Simulation Results

Using Agilent’s Advanced Design System tools with models of United Microelectronics $130 \mathrm{~nm}$ (1P8M) technology, simulation studies were carried out for a few specific designs of Table $2.1$ with $R_{T}=50 \Omega$ and $C_{L}=5 \mathrm{pF}$. The results of simulation are rather disappointing and indicate that parasitics cause appreciable deviations from the ideal characteristics and that the distortions increase with increasing coefficient of coupling. Simulations with only the device parasitics indicate that the BTC parasitics have dominant effect on the deviations from maximal flatness and reduction of the BWER. Realizing the full potential of the theoretical results derived here will therefore have to wait for improved technologies for inductor fabrication in future.

A comprehensive theoretical analysis has been carried out in this chapter of the general asymmetrical BTC network used as the load of a wide-band amplifier, and it has been shown that the BWER achievable is unlimited, the limit being set only by practical considerations of tight coupling, large spreads in inductance and capacitance values, and of course, parasitics. Unlimited bandwidth has never been reported earlier, by either the BTC or any other network, and it is believed that the results of this chapter will set a new trend in the design of wide-band and ultra wide-band amplifiers.

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|BWER of the Asymmetrical BTC

$$\eta=(1+b+2 m) \sqrt{\frac{b+m}{(1+m)\left(b-m^{2}\right)}} \times \sqrt{\left(1-2 \varsigma^{2}\right)+\sqrt{\left(1-2 \varsigma^{2}\right)^{2}+1}}$$

$$\eta=(1+b+2 m)\left(\frac{b+m}{(1+m)\left(b-m^{2}\right)}\right)^{\frac{1}{2}}$$

$$2=\frac{(b+m)^{3}}{(1+m)\left(b+m^{2}\right)}$$

$$\eta=\sqrt{2}\left(1+\frac{1+m}{b+m}\right)$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|ELEC3104

statistics-lab™ 为您的留学生涯保驾护航 在代写数字信号处理Digital Signal Processing方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数字信号处理Digital Signal Processing方面经验极为丰富，各种代写数字信号处理Digital Signal Processing相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|数字信号处理代写Digital Signal Processing代考|Analysis of the General BTC

The small-signal equivalent circuit of Fig. $2.1$ is shown in Fig. 2.2, where $g_{m}$ is the transconductance of the NMOS transistor. In [16], this circuit was analysed by using the extra-element theorem [33] whereas in [17]. a much simpler analysis was carried out by using classical techniques.
In either case, the result obtained is the following:
\begin{aligned} Z_{T}=& V_{0} / I_{i}=R_{T} N(s) / D(s) \ N(s)=& s^{2} C_{c}\left(L_{a}+L_{b}+2 M\right)+s\left(\left(L_{b}+M\right) / R_{T}\right)+1 \ D(s)=& s^{4} C_{c} C_{L}\left(L_{a} L_{b}-M^{2}\right)+s^{3} C_{c} C_{L} R_{T}\left(L_{a}+L_{b}+2 M\right) \ &+s^{2}\left(C_{c}\left(L_{a}+L_{b}+2 M\right)+C_{L} L_{b}\right)+s C_{L} R_{T}+1 \end{aligned}
The transfer impedance $Z_{T}$ is of the fourth order, and it is difficult to proceed further analytically. Following [16], we, therefore, convert it to a second-order one by forcing pole-zero cancellation, i.e. we write $$D(s)=\left(p s^{2}+q s+1\right) N(s)$$
and determine the constraints on the element values by equating the coefficients of powers of $s$ on both sides. Carrying out these steps, we get [16]
$$\begin{gathered} p=C_{L}\left(L_{a} L_{b}-M^{2}\right) /\left(L_{a}+L_{b}+2 M\right) \ q=C_{L} R_{T}-\frac{C_{L}\left(L_{a} L_{b}-M^{2}\right)\left(L_{b}+M\right)}{C_{c} R_{T}\left(L_{a}+L_{b}+2 M\right)^{2}} \ \frac{q\left(L_{b}+M\right)}{R_{T}}+p-C_{L} L_{b} \end{gathered}$$
and
$$q=C_{L} R_{T}-\frac{L_{b}+M}{R_{T}}$$
Comparing Eqs. (2.4) and (2.6) gives
$$\frac{C_{c}}{C_{L}}=\frac{L_{a} L_{b}-M^{2}}{\left(L_{a}+L_{b}+2 M\right)^{2}}$$

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|Symmetrical BTC

For the symmetrical BTC, $b=1$, and Eqs. (2.15) and (2.16) simplify to the following:
$$\omega_{3}=\frac{2}{C_{L} R_{T}} \sqrt{\frac{1+m}{1-m}}$$
and
$$2 \varsigma=\sqrt{\frac{1+m}{1-m}}$$
With $b=1$, Eq. (2.9) gives $k=m$, and from Eq. (2.20), we get
$$k=\frac{4 \varsigma^{2}-1}{4 \zeta^{2}+1}$$
Putting $b=1$ in Eqs. (2.10) and (2.11), and using Eq. (2.21), we get
$$L=\frac{C_{L} R_{T}^{2}}{4}\left(1+\frac{1}{4 \varsigma^{2}}\right) \text { and } C_{c}=C_{L} /\left(16 \varsigma^{2}\right) .$$
These results agree with those derived by many authors earlier.
Now combine Eqs. (2.18) with (2.17) and then use Eqs. (2.19)-(2.21), along with the fact that $k=m$. The result is
$$\eta=4 \varsigma \sqrt{\left(1-2 \varsigma^{2}\right)+\sqrt{\left(1-2 \varsigma^{2}\right)^{2}+1}}$$
Equation (2.17) shows that as $\zeta$ increases from 0 to $1, \omega_{3}$ decreases monotonically from $(\sqrt{2}+1)^{1 / 2} \omega_{0}$ to $(\sqrt{2}-1)^{1 / 2} \omega_{0}$, passing through the value $\omega_{0}$ at $\zeta=1 / \sqrt{2}$.

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|Analysis of the General BTC

$$Z_{T}=V_{0} / I_{i}=R_{T} N(s) / D(s) N(s)=s^{2} C_{c}\left(L_{a}+L_{b}+2 M\right)+s\left(\left(L_{b}+M\right) / R_{T}\right)+1 D(s)=s^{4} C_{\text {}}$$

$$D(s)=\left(p s^{2}+q s+1\right) N(s)$$

$$p=C_{L}\left(L_{a} L_{b}-M^{2}\right) /\left(L_{a}+L_{b}+2 M\right) q=C_{L} R_{T}-\frac{C_{L}\left(L_{a} L_{b}-M^{2}\right)\left(L_{b}+M\right)}{C_{c} R_{T}\left(L_{a}+L_{b}+2 M\right)^{2}} \frac{q\left(L_{b}+M\right)}{R_{T}}+p$$

$$q=C_{L} R_{T}-\frac{L_{b}+M}{R_{T}}$$

$$\frac{C_{c}}{C_{L}}=\frac{L_{a} L_{b}-M^{2}}{\left(L_{a}+L_{b}+2 M\right)^{2}}$$

## 电子工程代写|数字信号处理代写Digital Signal Processing代考|Symmetrical BTC

$$\omega_{3}=\frac{2}{C_{L} R_{T}} \sqrt{\frac{1+m}{1-m}}$$

$$2 \varsigma=\sqrt{\frac{1+m}{1-m}}$$

$$k=\frac{4 \varsigma^{2}-1}{4 \zeta^{2}+1}$$

$$L=\frac{C_{L} R_{T}^{2}}{4}\left(1+\frac{1}{4 \varsigma^{2}}\right) \text { and } C_{c}=C_{L} /\left(16 \varsigma^{2}\right) .$$

$$\eta=4 \varsigma \sqrt{\left(1-2 \varsigma^{2}\right)+\sqrt{\left(1-2 \varsigma^{2}\right)^{2}+1}}$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 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

statistics-lab™ 为您的留学生涯保驾护航 在代写信号处理与线性系统Signal Processing and Linear Systems方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信号处理与线性系统Signal Processing and Linear Systems方面经验极为丰富，各种代写信号处理与线性系统Signal Processing and Linear Systems相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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 $A^{\prime} T=T A, \quad \widetilde{A}^{\prime} \tilde{T}=\widetilde{T} \tilde{A}, \quad B^{\prime} \tilde{T}=T B, \quad C^{\prime} T=\tilde{T} C, \quad \Pi^{\prime} T=\Pi, \quad \widetilde{\Pi}^{\prime} \tilde{T}=\widetilde{\Pi} .$

## 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 $\Theta_{P}(z):=P(z) \Theta(z) P(z)^{-1}=\tilde{\Upsilon}(z) \bar{N}$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 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|EE483

statistics-lab™ 为您的留学生涯保驾护航 在代写信号处理与线性系统Signal Processing and Linear Systems方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信号处理与线性系统Signal Processing and Linear Systems方面经验极为丰富，各种代写信号处理与线性系统Signal Processing and Linear Systems相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|The focus here

However our focus here is not on interpolation aspects but rather on the intrinsic structure of the associated reproducing kernel Hilbert spaces. The main objective of the present paper is to find Stieltjes-class counterparts of Theorems $1.3$ and 1.4. Specifically, in Section 3 we shall consider the following:

Problem 1.10. Given two reproducing kernel Hilbert spaces $\mathcal{H}$ and $\tilde{\mathcal{H}}$ of $\widehat{\mathcal{G}}$-valued functions meromorphic in $\Omega$, find necessary and sufficient conditions for the existence of a function $\Theta \in \mathcal{M S}(\mathcal{G}, \Omega)$ such that $\mathcal{H}=\mathcal{H}(\Theta)$ and $\widetilde{\mathcal{H}}=\mathcal{H}\left(P \Theta P^{-1}\right)$. In case $\mathcal{H}$ and $\widetilde{\mathcal{H}}$ are presented as ranges of observability operators
$$\mathcal{H}=\operatorname{Ran} \mathcal{O}{\Pi, A, \mu} \quad \text { and } \quad \tilde{\mathcal{H}}=\operatorname{Ran} \mathcal{O}{\tilde{\Pi}, \tilde{A}, \mu},$$ find necessary and sufficient conditions directly in terms of the operators $\Pi, A, \widetilde{\Pi}, \widetilde{A}$ for it to happen that $\mathcal{H}=\mathcal{H}(\Theta)$ and $\tilde{\mathcal{H}}=\mathcal{H}\left(\Theta_{P}\right)$ for some $\Theta$.

Solutions to these problems are presented in Theorem $3.1$ (the Stieltjes analogue of Theorem 1.3) and Theorem $4.1$ (the Stieltjes analogue of Theorem 1.4).
Finally we note that the reproducing kernel space $\mathcal{H}(\Theta)$ determines the function $\Theta \in \mathcal{M} \mathcal{P}(\mathcal{G}, \Omega)$ only up to a unitary constant right factor $\Upsilon$. While $\Theta \Upsilon$ is in the Pick class $\mathcal{M} \mathcal{P}(\mathcal{G}, \Omega)$ whenever $\Theta \in \mathcal{M} \mathcal{P}(\mathcal{G}, \Omega)$ for any constant $J$-unitary operator $\Upsilon$, the corresponding property for the multiplicative Stieltjes class fails in general. Thus it is a subtle but nontrivial point to show that, if $\Theta$ is such that $\mathcal{H}=\mathcal{H}(\Theta)$ and $\widetilde{\mathcal{H}}=\mathcal{H}\left(\Theta_{P}\right)$, then there is a choice of constant J-unitary operators $\Upsilon$ and $\tilde{\Upsilon}$ so that $(\Theta \cdot \Upsilon){P}=\Theta{P} \cdot \widetilde{\Upsilon}$, in which case we then have $\Theta^{\prime}:=\Theta \cdot \Upsilon \in \mathcal{M} \mathcal{S}(\mathcal{G}, \Omega)$ as well as $\mathcal{H}=\mathcal{H}\left(\Theta^{\prime}\right)$ and $\tilde{\mathcal{H}}=\mathcal{H}\left(\left(\Theta^{\prime}\right)_{P}\right)$. This issue is addressed in Section $4.2$ below.

The paper is organized as follows. Section 2 presents some material on the simultaneous $J$-unitary equivalence of a pair of Krein-space operators as well as some identities involving the operators $R_{\alpha}$ and $\left[\begin{array}{cc}R_{\alpha} & 0 \ 0 & I+\alpha R_{\alpha}\end{array}\right]$ needed in the proof of the characterization of a pair of reproducing kernel Hilbert spaces of the form $\mathcal{H}(\Theta)$ and $\mathcal{H}\left(\Theta_{P}\right)$. Section 3 gives an intrinsic structural characterization of pairs of reproducing kernel Hilbert spaces of the form $\left(\mathcal{H}(\Theta), \mathcal{H}\left(\Theta_{P}\right)\right)$ in intrinsic geometric, structural form, while in Section 4, these results are reformulated in explicit state-space coordinates.

## 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|Characterization of Stieltjes reproducing-kernel

In this section we characterize pairs $\left{\mathcal{H}(\Theta), \mathcal{H}\left(P \Theta P^{-1}\right)\right}$ in terms of invariance properties and structure identities.

Theorem 3.1. Let $\mathcal{H}$ and $\widetilde{\mathcal{H}}$ be two reproducing kernel Hilbert spaces whose elements are $\widehat{\mathcal{G}}$ valued functions which are meromorphic in $\Omega$. In order that $\mathcal{H}$ and $\widetilde{\mathcal{H}}$ be spaces $\mathcal{H}(\Theta)$ and $\mathcal{H}\left(P \Theta P^{-1}\right)$ it is necessary and sufficient that

1. For each $\alpha \in \Omega$, the invariance conditions
$$R_{\alpha} \mathcal{H} \subset \mathcal{H}, \quad R_{\alpha} \tilde{\mathcal{H}} \subset \widetilde{\mathcal{H}}$$
hold as well as the coupled invariance conditions
$$\left[\begin{array}{cc} I+\alpha R_{\alpha} & 0 \ 0 & R_{\alpha} \end{array}\right] \mathcal{H} \subset \tilde{\mathcal{H}} \text { and }\left[\begin{array}{cc} R_{\alpha} & 0 \ 0 & I+\alpha R_{\alpha} \end{array}\right] \tilde{\mathcal{H}} \subset \mathcal{H} .$$
2. The following four identities hold for all functions
3. $F=\left[\begin{array}{l}F_{1} \ F_{2}\end{array}\right] \in \mathcal{H}, \quad G=\left[\begin{array}{l}G_{1} \ G_{2}\end{array}\right] \in \mathcal{H}, \quad \widetilde{F}=\left[\begin{array}{c}\widetilde{F}{1} \ \widetilde{F}{2}\end{array}\right] \in \widetilde{\mathcal{H}}, \quad \widetilde{G}=\left[\begin{array}{c}\widetilde{G}{1} \ \widetilde{G}{2}\end{array}\right] \in \widetilde{\mathcal{H}}$
4. and for all $\alpha, \beta \in \Omega$ :
5. $\left\langle R_{\alpha} F,\left(I+\beta R_{\beta}\right) G\right\rangle_{\mathcal{H}}-\left\langle\left[\begin{array}{cc}I+\alpha R_{\alpha} & 0 \ 0 & R_{\alpha}\end{array}\right] F,\left[\begin{array}{cc}I+\beta R_{\beta} & 0 \ 0 & R_{\beta}\end{array}\right] G\right\rangle_{\tilde{\mathcal{H}}}$
6. $=G_{2}(\beta)^{} F_{1}(\alpha)$, $\left\langle\left[\begin{array}{cc}R_{\alpha} & 0 \ 0 & I+\alpha R_{\alpha}\end{array}\right] \widetilde{F},\left[\begin{array}{cc}R_{\beta} & 0 \ 0 & I+\beta R_{\beta}\end{array}\right] \widetilde{G}\right\rangle_{\mathcal{H}}-\left\langle\left(I+\alpha R_{\alpha}\right) \widetilde{F}, R_{\beta} \widetilde{G}\right\rangle_{\tilde{\mathcal{H}}}$ $=\widetilde{G}{2}(\beta)^{} \widetilde{F}{1}(\alpha)$,
7. $\left\langle\left[\begin{array}{cc}R_{\alpha} & 0 \ 0 & I+\alpha R_{\alpha}\end{array}\right] \widetilde{F}, R_{\beta} G\right\rangle_{\mathcal{H}}-\left\langle R_{\alpha} \widetilde{F},\left[\begin{array}{cc}I+\beta R_{\beta} & 0 \ 0 & R_{\beta}\end{array}\right] G\right\rangle_{\tilde{\mathcal{H}}}$
8. $=G_{1}(\beta)^{} \widetilde{F}{2}(\alpha)$, $\left\langle\left[\begin{array}{cc}R{\alpha} & 0 \ 0 & I+\alpha R_{\alpha}\end{array}\right] \widetilde{F},\left(I+\beta R_{\beta}\right) G\right\rangle_{\varkappa}-\left\langle\left(I+\alpha R_{\alpha}\right) \widetilde{F},\left[\begin{array}{cc}I+\beta R_{\beta} & 0 \ 0 & R_{\beta}\end{array}\right] G\right\rangle_{\tilde{\varkappa}}$
9. $=G_{2}(\beta)^{} \widetilde{F}_{1}(\alpha)$

## 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|Characterization of Stieltjes reproducing-kernel

1. 对于每个 $\alpha \in \Omega$, 不变条件
$$R_{\alpha} \mathcal{H} \subset \mathcal{H}, \quad R_{\alpha} \tilde{\mathcal{H}} \subset \widetilde{\mathcal{H}}$$
保持以及耦合不变条件
$$\left[\begin{array}{llll} I+\alpha R_{\alpha} & 0 & 0 & R_{\alpha} \end{array}\right] \mathcal{H} \subset \tilde{\mathcal{H}} \text { and }\left[\begin{array}{llll} R_{\alpha} & 0 & 0 & I+\alpha R_{\alpha} \end{array}\right] \tilde{\mathcal{H}} \subset \mathcal{H} .$$
2. 以下四个恒等式适用于所有功能
3. $F=\left[\begin{array}{ll}F_{1} & F_{2}\end{array}\right] \in \mathcal{H}, \quad G=\left[\begin{array}{ll}G_{1} & G_{2}\end{array}\right] \in \mathcal{H}, \quad \widetilde{F}=[\widetilde{F} 1 \widetilde{F} 2] \in \widetilde{\mathcal{H}}, \quad \widetilde{G}=[\widetilde{G} 1 \widetilde{G} 2] \in \widetilde{\mathcal{H}}$
4. 并为所有人 $\alpha, \beta \in \Omega$ :
5. $\left\langle R_{\alpha} F,\left(I+\beta R_{\beta}\right) G\right\rangle_{\mathcal{H}}-\left\langle\left[I+\alpha R_{\alpha} \quad 0 \quad 0 \quad R_{\alpha}\right] F,\left[I+\beta R_{\beta} \quad 0 \quad 0 \quad R_{\beta}\right] G\right\rangle_{\tilde{\mathcal{H}}}$
6. $=G_{2}(\beta) F_{1}(\alpha)$,
$\left\langle\left[\begin{array}{llll}R_{\alpha} & 0 & 0 & I+\alpha R_{\alpha}\end{array}\right] \widetilde{F},\left[\begin{array}{llll}R_{\beta} & 00 & I+\beta R_{\beta}\end{array}\right] \widetilde{G}\right\rangle_{\mathcal{H}}-\left\langle\left(I+\alpha R_{\alpha}\right) \widetilde{F}, R_{\beta} \widetilde{G}\right\rangle_{\tilde{\mathcal{H}}}$ $=\widetilde{G} 2(\beta) \widetilde{F} 1(\alpha)$,
7. $\left\langle\left[\begin{array}{llll}R_{\alpha} & 0 & 0 & I+\alpha R_{\alpha}\end{array}\right] \widetilde{F}, R_{\beta} G\right\rangle_{\mathcal{H}}-\left\langle R_{\alpha} \widetilde{F},\left[I+\beta R_{\beta} \quad 00 \quad R_{\beta}\right] G\right\rangle_{\tilde{\mathcal{H}}}$
8. $=G_{1}(\beta) \widetilde{F} 2(\alpha)$, 9. $=G_{2}(\beta) \widetilde{F}_{1}(\alpha)$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 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|ELEN30012

statistics-lab™ 为您的留学生涯保驾护航 在代写信号处理与线性系统Signal Processing and Linear Systems方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信号处理与线性系统Signal Processing and Linear Systems方面经验极为丰富，各种代写信号处理与线性系统Signal Processing and Linear Systems相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|The Stieltjes and multiplicative Stieltjes classes

An important subclass of the Pick class is the Stieltjes class denoted here by $\mathcal{S}(\widehat{\mathcal{G}})$, consisting of functions $S$ in the Pick class $\mathcal{P}(\widehat{\mathcal{G}})$ with analytic continuation across the negative half-axis $\mathbb{R}^{-}$and taking positive semidefinite values on $\mathbb{R}^{-}$:
$$\frac{S(z)-S(z)^{}}{z-\bar{z}} \succeq 0(z \notin \mathbb{R}), \quad S(x) \succeq 0 \quad(x<0) .$$ Stieltjes functions made their first explicit appearance in [44] as continued fractions of certain type and as Cauchy transforms of positive measures on $\mathbb{R}^{+}=[0, \infty)$. Being special instances of absolutely monotone functions, operator monotone functions and Pick functions, they have been extensively studied in various contexts $[12,29,30,34,33,37,43,45]$. Such functions have the alternative characterization as being those functions $S \in \mathcal{P}(\widehat{\mathcal{G}})$ such that the function $z \mapsto z S(z)$ is also in $\mathcal{P}(\widehat{\mathcal{G}})$ (see [33] for the scalar case – the operator-valued case is similar). This leads to the kernel characterization of the Stieltjes class: an $\mathcal{L}(\mathcal{G})$-valued function $S$ is in $\mathcal{S}(\mathcal{G})$ if and only if both kernels $$\mathfrak{K}(z, \omega)=\frac{S(z)-S(\omega)^{}}{z-\bar{\omega}} \text { and } \widetilde{\mathfrak{K}}(z, \omega)=\frac{z S(z)-\bar{\omega} S(\omega)^{*}}{z-\bar{\omega}}$$
are positive on the upper half-plane.

## 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|Connections with interpolation theory

The importance of multiplicative Pick functions for interpolation theory arises from the fact that the linear fractional map based on a function $\Theta \in \mathcal{M} \mathcal{P}(\widehat{\mathcal{G}})$ maps the class $\mathcal{P}(\mathcal{G})$ into itself. Choosing $\Theta$ with a suitable pole/zero structure then implies that the linear-fractional map based on $\Theta$ gives rise to a parametrization (with free parameter from the Pick class $\mathcal{P}(\mathcal{G})$ ) of the solution set of a given interpolation problem in the class $\mathcal{P}(\mathcal{G})$; we refer to $[11,42]$ for specific examples. It turns out the multiplicative Stieltjes class $\mathcal{M S}(\mathcal{G}, \mathbb{C})$ has similar applications in interpolation theory for the additive Stieltjes class $\mathcal{S}(\mathcal{G})$ as the linear fractional map based on a function $\Theta \in \mathcal{M} \mathcal{S}(\mathcal{G})$ not only maps the class $\mathcal{P}(\mathcal{G})$ into itself, but also the class $\mathcal{S}(\mathcal{G})$ into itself. In the context of the Nevanlinna-Pick interpolation problem, multiplicative Stieltjes functions appeared explicitly in the series of papers $[23,25,26]$; see also $[2,13,14,15,24,25,26]$ for other examples and far-reaching generalizations. From the integral representation (1.27) for the Stieltjes class, we see that the Stieltjes moment problem going back to the nineteenth century [44] can be seen as a boundary version of a Stieltjes interpolation problem. The Stieltjes class also arises in the recent work of Agler-Tully-Doyle-Young [1] on characterizing boundary directional derivatives of Schur-class functions on the bidisk.

## 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|The Stieltjes and multiplicative Stieltjes classes

Pick 类的一个重要子类是 Stieltjes 类，在此表示为 $S(\widehat{\mathcal{G}})$ ，由函数组成 $S$ 在 Pick 类中 $\mathcal{P}(\widehat{\mathcal{G}})$ 在负半轴上具有解析延拓 $\mathbb{R}^{-}$并取半正定值 $\mathbb{R}^{-}$:
$$\frac{S(z)-S(z)}{z-\bar{z}} \succeq 0(z \notin \mathbb{R}), \quad S(x) \succeq 0 \quad(x<0) .$$
Stieltjes 函数在 [44] 中作为某种类型的连分数和正测量的柯西变换在 [44] 中首次明确出现 $\mathbb{R}^{+}=[0, \infty)$. 作为绝 对单调函数、算子单调函数和 Pick 函数的特例，它们在各种情况下得到了广泛的研究
$[12,29,30,34,33,37,43,45]$. 此类功能具有作为这些功能的替代特征 $S \in \mathcal{P}(\widehat{\mathcal{G}})$ 使得函数 $z \mapsto z S(z)$ 也在 $\mathcal{P}(\widehat{\mathcal{G}})$ (有关标量情况，请参见 [33] – 运算符值情况类似) 。这导致了 Stieltjes 类的内核特征: $\mathcal{L}(\mathcal{G})$ 值函数 $S$ 在 $\mathcal{S}(\mathcal{G})$ 当且仅当两个内核
$$\mathfrak{K}(z, \omega)=\frac{S(z)-S(\omega)}{z-\bar{\omega}} \text { and } \widetilde{\Re}(z, \omega)=\frac{z S(z)-\bar{\omega} S(\omega)^{*}}{z-\bar{\omega}}$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 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|ELECENG4112

statistics-lab™ 为您的留学生涯保驾护航 在代写信号处理与线性系统Signal Processing and Linear Systems方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信号处理与线性系统Signal Processing and Linear Systems方面经验极为丰富，各种代写信号处理与线性系统Signal Processing and Linear Systems相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|Reproducing kernel Hilbert spaces with additional structure

In this paper we shall be interested in how additional properties of the positive kernel $K$ translate to additional structural properties of the reproducing kernel Hilbert space $\mathcal{H}{K}$. A specific form for the positive kernel $K$ of interest for us can be explained as follows. Given a Hilbert space $\mathcal{G}$, we define the unitary selfadjoint operator $$J=\left[\begin{array}{cc} 0 & i I{\mathcal{G}} \ -i I_{\mathcal{G}} & 0 \end{array}\right] \in \mathcal{L}(\mathcal{G} \oplus \mathcal{G})$$
To distinguish the summands in the direct sum $\widehat{\mathcal{G}}=\mathcal{G} \oplus \mathcal{G}$, we identify the first summand with the subspace $\mathcal{G}=\left{\left[\begin{array}{c}x \ 0\end{array}\right], x \in \mathcal{G}\right}$ of $\widehat{\mathcal{G}}$ and represent $\widehat{\mathcal{G}}$ as
$$\widehat{\mathcal{G}}=\mathcal{G} \oplus J \mathcal{G} .$$
We choose and fix a non-empty open subset $\Omega \subset \mathbb{C}$ which is symmetric about the real axis $\mathbb{R}$ and consider a Hilbert space $\mathcal{H}$ whose elements are $\widehat{\mathcal{G}}$-valued functions meromorphic in $\Omega$. Any reference to the value of a meromorphic function at $\alpha \in \Omega$ assumes that the function is analytic at $\alpha$.

Definition 1.2. We say that $\mathcal{H}$ is a space $\mathcal{H}(\Theta)$ if it admits a reproducing kernel $K_{\Theta}$ of the form
$$K_{\Theta}(z, \omega):=\frac{J-\Theta(z) J \Theta(\omega)^{}}{i(\bar{\omega}-z)}$$ for some function $\Theta$ meromorphic on $\Omega$, subject to $$\Theta(z) J \Theta(\bar{z})^{}=\Theta(\bar{z})^{*} J \Theta(z)=J \quad \text { for all } \quad z \in \Omega,$$ i.e., if H = HKΘ =: H(Θ) where KΘ is as in (1.4)–(1.5).

## 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|The Pick class and connections

Let us recall the Pick class $\mathcal{P}(\mathcal{G})$ (in the literature also known as NevanlinnaHerglotz class and sometimes also simply as $R$-class) consisting of $\mathcal{L}(\mathcal{G})$-valued functions holomorphic on the upper half-plane $\mathbb{C}{+}$with values there having positive semidefinite imaginary part, i.e., the functions $S: \mathbb{C}{+} \rightarrow \mathcal{L}(\mathcal{G})$ such that the kernel
$$\mathfrak{K}{S}(z, \omega)=\frac{S(z)-S(\omega)^{}}{z-\bar{\omega}}$$ is positive on $\mathbb{C}{+} .$In fact, if the kernel (1.18) is positive on a domain $\Omega \subset \mathbb{C}{+}$, it can be (uniquely) extended as a positive kernel to all of $\mathbb{C}{+}$due to the Pick interpolation theorem. It is convenient (and is consistent with Nevanlinna-Herglotz integral formula) furthermore to extend Pick functions to the lower half-plane by reflection: define $S(z)=S(\bar{z})^{}$ for $z \in \mathbb{C}^{-}$.

Let us note that the kernel $\mathfrak{K}{S}$ can be rewritten in a more aggregate form as \begin{aligned} \mathfrak{K}{S}(z, \bar{\omega}) &=\frac{\left[\begin{array}{ll} I & S(z) \end{array}\right]\left[\begin{array}{cc} 0 & i I_{\mathcal{G}} \ -i I_{\mathcal{G}} & 0 \end{array}\right]\left[\begin{array}{c} I \ S(\omega)^{} \end{array}\right]}{i(\bar{\omega}-z)} \ &=\frac{\left[\begin{array}{ll} I & S(z) \end{array}\right] \mathcal{J}{\mathcal{P}}\left[\begin{array}{c} I \ S(\omega)^{} \end{array}\right]}{i(\bar{\omega}-z)}, \quad \text { where } \quad \mathcal{J}{\mathcal{P}}=\left[\begin{array}{cc} 0 & i I_{\widehat{\mathcal{G}}} \ i I_{\widehat{\mathcal{G}}} & 0 \end{array}\right] . \end{aligned}
In case we replace $\mathcal{G}$ with $\widehat{\mathcal{G}}=\mathcal{G} \oplus J \mathcal{G}$, comparison of (1.19) with (1.6) suggests the close connection between the multiplicative Pick class $\mathcal{M} \mathcal{P}(\mathcal{G})$ and the Pick class over $\widehat{\mathcal{G}}$, i.e., $\mathcal{P}(\widehat{\mathcal{G}})$; the kernel $K_{\Theta}$ built from $\Theta$ appearing in (1.6) has exactly the same form as the kernel $\mathfrak{K}{S}$ built from $S$ appearing in (1.19), but with the aggregate signature matrix $\mathcal{J}{\mathcal{M} \mathcal{P}}$ for the class $\mathcal{M} \mathcal{P}(\mathcal{G})$ replaced by the aggregate signature matrix $\mathcal{J}{\mathcal{P}}$ for the class $\mathcal{P}(\widehat{\mathcal{G}})$. In fact there is a simple linear-fractional transformation $T{\mathcal{P G}}$ (called the Potapov-Ginzburg transformation (see [27]) which maps $\mathcal{P}(\widehat{\mathcal{G}})$ bijectively to $\mathcal{M P}(\mathcal{G})$ and which can be derived as follows.

## 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|Reproducing kernel Hilbert spaces with additional structure

$$J=\left[\begin{array}{lll} 0 & i I \mathcal{G}-i I_{\mathcal{G}} & 0 \end{array}\right] \in \mathcal{L}(\mathcal{G} \oplus \mathcal{G})$$

$$\widehat{\mathcal{G}}=\mathcal{G} \oplus J \mathcal{G} .$$

$$K_{\Theta}(z, \omega):=\frac{J-\Theta(z) J \Theta(\omega)}{i(\bar{\omega}-z)}$$

$$\Theta(z) J \Theta(\bar{z})=\Theta(\bar{z})^{*} J \Theta(z)=J \quad \text { for all } \quad z \in \Omega,$$

## 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|The Pick class and connections

$$\mathfrak{K} S(z, \omega)=\frac{S(z)-S(\omega)}{z-\bar{\omega}}$$

$\mathbb{C}+$ 由于 Pick 揷值定理。此外，通过反射将 Pick 函数扩展到下半平面很方便 (并且与 Nevanlinna-Herglotz 积分 公式一致)：定义 $S(z)=S(\bar{z})$ 为了 $z \in \mathbb{C}^{-}$.

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。