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物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代考|Amplitude modulator

Modulation is the process used to shift the frequency content of an informationbearing signal such that the resulting modulated signal occupies a higher frequency range. Modulation is the key component in modern-day communication systems for two main reasons. One reason is that the frequency components of the human voice are limited to a range of around $4 \mathrm{kHz}$. If a human voice signal is transmitted directly by propagating electromagnetic radio waves, the communication antennas required to transmit and receive these radio signals would be impractically long. A second reason for modulation is to allow for simultaneous transmission of several voice signals within the same geographic region. If two signals within the same frequency range are transmitted together, they will interfere with each other. Modulation provides us with the means of separating the voice signals in the frequency domain by shifting each voice signal to a different frequency band. There are different techniques used to modulate a signal. Here we introduce the simplest form of modulation referred to as amplitude modulation (AM).

Consider an information-bearing signal $m(t)$ applied as an input to an AM system, referred to as an amplitude modulator. In communications, the input $m(t)$ to a modulator is called the modulating signal, while its output $s(t)$ is called the modulated signal. The steps involved in an amplitude modulator are illustrated in Fig. 2.5, where the modulating signal $m(t)$ is first processed by attenuating it by a factor $k$ and adding a dc offset such that the resulting signal $(1+k m(t))$ is positive for all time $t$. The modulated signal is produced by multiplying the processed input signal $(1+k m(t))$ with a high-frequency carrier $c(t)=A \cos \left(2 \pi f_{\mathrm{c}} t\right)$. Multiplication by a sinusoidal wave of frequency $f_{\mathrm{c}}$ shifts the frequency content of the modulating signal $m(t)$ by an additive factor of $f_{\mathrm{c}}$. Mathematically, the amplitude modulated $\operatorname{signal} s(t)$ is expressed as follows:
$$s(t)=A[1+k m(t)] \cos \left(2 \pi f_{\mathrm{c}} t\right),$$
where $A$ and $f_{\mathrm{c}}$ are, respectively, the amplitude and the fundamental frequency of the sinusoidal carrier.

物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代考|Mechanical water pump

The mechanical pump shown in Fig. $2.6$ is another example of a linear CT system. Water flows into the pump through a valve $\mathrm{V} 1$ controlled by an electrical circuit. A second valve V2 works mechanically as the outlet. The rate of the outlet flow depends on the height of the water in the mechanical pump. A higher level of water exerts more pressure on the mechanical valve V2, creating a wider opening in the valve, thus releasing water at a faster rate. As the level of water drops, the opening of the valve narrows, and the outlet flow of water is reduced.

A mathematical model for the mechanical pump is derived by assuming that the rate of flow $F_{\text {in }}$ of water at the input of the pump is a function of the input voltage $x(t)$ :
$$F_{\text {in }}=k x(t),$$
where $k$ is the linearity constant. Valve $\mathrm{V} 2$ is designed such that the outlet flow rate $F_{\text {out }}$ is given by
$$F_{\text {out }}=c h(t),$$
where $c$ denotes the outlet flow constant and $h(t)$ is the height of the water level. Denoting the total volume of the water inside the tank by $V(t)$, the rate of change in the volume of the stored water is $\mathrm{d} V / \mathrm{d} t$, which must be equal to the difference between the input flow rate, Eq. (2.11), and the outlet flow rate, Eq. (2.12). The resulting equation is as follows:
$$\frac{\mathrm{d} V}{\mathrm{~d} t}=F_{\text {in }}-F_{\text {out }}=k x(t)-c h(t) .$$

物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代写|Amplitude modulator

$$s(t)=A[1+k m(t)] \cos \left(2 \pi f_c t\right),$$

物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代 考|Mechanical water pump

$$F_{\text {in }}=k x(t),$$

$$F_{\text {out }}=\operatorname{ch}(t),$$

$$\frac{\mathrm{d} V}{\mathrm{~d} t}=F_{\text {in }}-F_{\text {out }}=k x(t)-c h(t)$$

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。

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物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代考|Time inversion

The time inversion (also known as time reversal or reflection) operation reflects the input signal about the vertical axis $(t=0)$. When a CT signal $x(t)$ is timereversed, the inverted signal is denoted by $x(-t)$. Likewise, when a DT signal $x[k]$ is time-reversed, the inverted signal is denoted by $x[-k]$. In the following we provide examples of time inversion in both CT and DT domains.
Example $1.19$
Sketch the time-inverted version of the causal decaying exponential signal
$$x(t)=\mathrm{e}^{-t} u(t)= \begin{cases}\mathrm{e}^{-t} & t \geq 0 \ 0 & \text { elsewhere }\end{cases}$$
which is plotted in Fig. 1.28(a).
Solution
To derive the expression for the time-inverted signal $x(-t)$, substitute $t=-\alpha$ in Eq. (1.56). The resulting expression is given by
$$x(-\alpha)=\mathrm{e}^\alpha u(-\alpha)= \begin{cases}\mathrm{e}^\alpha & -\alpha \geq 0 \ 0 & \text { elsewhere. }\end{cases}$$
Simplifying the above expression and expressing it in terms of the independent variable $t$ yields
$$x(-t)= \begin{cases}\mathrm{e}^t & t \leq 0 \ 0 & \text { elsewhere }\end{cases}$$

物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代考|Energy and power signals

In Sections 1.3.1-1.3.3, we presented three basic time-domain transformations. In many signal processing applications, these operations are combined. An arbitrary linear operation that combines the three transformations is expressed as $x(\alpha t+\beta)$, where $\alpha$ is the time-scaling factor and $\beta$ is the time-shifting factor. If $\alpha$ is negative, the signal is inverted along with the time-scaling and time-shifting operations. By expressing the transformed signal as
$$x(\alpha t+\beta)=x\left(\alpha\left[t+\frac{\beta}{\alpha}\right]\right)$$

we can plot the waveform graphically for $x(\alpha t+\beta)$ by following steps (i)-(iii) outlined below.
(i) Scale the signal $x(t)$ by $|\alpha|$. The resulting waveform represents $x(|\alpha| t)$.
(ii) If $\alpha$ is negative, invert the scaled signal $x(|\alpha| t)$ with respect to the $t=0$ axis. This step produces the waveform for $x(\alpha t)$.
(iii) Shift the waveform for $x(\alpha t)$ obtained in step (ii) by $|\beta / \alpha|$ time units. Shift towards the right-hand side if $(\beta / \alpha)$ is negative. Otherwise, shift towards the left-hand side if $(\beta / \alpha)$ is positive. The waveform resulting from this step represents $x(\alpha t+\beta)$, which is the required transformation.

物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代 考|Time inversion

$$x(t)=\mathrm{e}^{-t} u(t)=\left{\begin{array}{ll} \mathrm{e}^{-t} & t \geq 00 \end{array}\right. \text { elsewhere }$$

$$x(-\alpha)=\mathrm{e}^\alpha u(-\alpha)=\left{\mathrm{e}^\alpha \quad-\alpha \geq 00 \quad\right. \text { elsewhere. }$$

$$x(-t)= \begin{cases}\mathrm{e}^t \quad t \leq 00 \quad \text { elsewhere }\end{cases}$$

物理代写连续时间信号和系统代写Continuous Time Signals and Systems代 考|Energy and power signals

$$x(\alpha t+\beta)=x\left(\alpha\left[t+\frac{\beta}{\alpha}\right]\right)$$

(i) 缩放信昊 $x(t)$ 经过 $|\alpha|$. 产生的波形表示 $x(|\alpha| t)$.
(ii) 如果 $\alpha$ 为负，反转缩放信昊 $x(|\alpha| t)$ 相对于该 $t=0$ 轴。这一步产生的波形 $x(\alpha t)$.
(iii) 移动波形 $x(\alpha t)$ 在步骤 (ii) 中获得 $|\beta / \alpha|$ 时间单位。向右移动，如果 $(\beta / \alpha)$ 是负的。否则，向左移动，如果 $(\beta / \alpha)$ 是积极 的。此步䁃产生的波形表示 $x(\alpha t+\beta)$ ，这是所需的转换。

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。

Posted on Categories:模拟电路, 物理代写

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物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代考|Periodic and aperiodic signals

A CT signal $x(t)$ is said to be periodic if it satisfies the following property:
$$x(t)=x\left(t+T_0\right),$$
at all time $t$ and for some positive constant $T_0$. The smallest positive value of $T_0$ that satisfies the periodicity condition, Eq. (1.3), is referred to as the fundamental period of $x(t)$.
Likewise, a DT signal $x[k]$ is said to be periodic if it satisfies
$$x[k]=x\left[k+K_0\right]$$
at all time $k$ and for some positive constant $K_0$. The smallest positive value of $K_0$ that satisfies the periodicity condition, Eq. (1.4), is referred to as the fundamental period of $x[k]$. A signal that is not periodic is called an aperiodic or non-periodic signal. Figure $1.6$ shows examples of both periodic and aperiodic signals. The reciprocal of the fundamental period of a signal is called the fundamental frequency. Mathematically, the fundamental frequency is expressed as follows
$f_0=\frac{1}{T_0}$, for CT signals, $\quad$ or $\quad f_0=\frac{1}{K_0}$, for DT signals,
where $T_0$ and $K_0$ are, respectively, the fundamental periods of the CT and DT signals. The frequency of a signal provides useful information regarding how fast the signal changes its amplitude. The unit of frequency is cycles per second (c/s) or hertz $(\mathrm{Hz})$. Sometimes, we also use radians per second as a unit of frequency. Since there are $2 \pi$ radians (or $360^{\circ}$ ) in one cycle, a frequency of $f_0$ hertz is equivalent to $2 \pi f_0$ radians per second. If radians per second is used as a unit of frequency, the frequency is referred to as the angular frequency and is given by
$\omega_0=\frac{2 \pi}{T_0}$, for CT signals, or $\quad \Omega_0=\frac{2 \pi}{K_0}$, for DT signals.

物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代考|Energy and power signals

Before presenting the conditions for classifying a signal as an energy or a power signal, we present the formulas for calculating the energy and power in a signal. The instantaneous power at time $t=t_0$ of a real-valued CT signal $x(t)$ is given by $x^2\left(t_0\right)$. Similarly, the instantaneous power of a real-valued DT signal $x[k]$ at time instant $k=k_0$ is given by $x^2[k]$. If the signal is complex-valued, the expressions for the instantaneous power are modified to $\left|x\left(t_0\right)\right|^2$ or $\left|x\left[k_0\right]\right|^2$, where the symbol $|\cdot|$ represents the absolute value of a complex number.
The energy present in a CT or DT signal within a given time interval is given by the following:

CT signals $\quad E_{\left(T_1, T_2\right)}=\int_{T_1}^{T_2}|x(t)|^2 \mathrm{~d} t$ in interval $t=\left(T_1, T_2\right)$ with $T_2>T_1 ;$
DT sequences $E_{\left[N_1, N_2\right]}=\sum_{k=N_1}^{N_2}|x[k]|^2$ in interval $k=\left[N_1, N_2\right]$ with $N_2>N_1$.
(1.10b)

物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代 考|Periodic and aperiodic signals

CT信昊 $x(t)$ 如果满足以下性质，则称其为周期性的:
$$x(t)=x\left(t+T_0\right)$$

$$x[k]=x\left[k+K_0\right]$$

物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代 考|Energy and power signals

CT信昊 $\quad E_{\left(T_1, T 2\right)}=\int_{T_1}^{T_2}|x(t)|^2 \mathrm{~d} t$ 在间隔 $t=\left(T_1, T_2\right)$ 和 $T_2>T_1$
$\mathrm{DT}$ 序列 $E_{\left[N_1, N_2\right]}=\sum_{k=N_1}^{N_2}|x[k]|^2$ 在间隔 $k=\left[N_1, N_2\right]$ 和 $N_2>N_1$.
(1.10b)

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。

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物理代写|模拟电路代写Analog Circuit代考|Support User-Assisted Placement Generation

Beyond the efforts made towards the generation of a fully automatic layout capable of competing with expert-made layouts, it is possible to notice that EDA tools are moving in a different direction from two decades ago. There is now a strong attempt to recycle existing layouts, migrating them to new technologies or optimize the old design. Many of the circuits manufactured today are the ones developed and implemented years ago, so it is extremely important to take advantage of the knowledge embedded in their layouts and follow the advances in the integration technologies, instead of going through all the design process again. For this, the idea of parameterized model/template is present in the most recent successful approaches in the past few years. Increasing the designer’s active part in the generation of the floorplan is not necessarily a drawback, since the inclusion of his knowledge increases the floorplan quality and allows the tool to easily generate a solution that fully meets analog designers’ expectations/needs.

物理代写|模拟电路代写Analog Circuit代考|Support Fully-Automatic Placement Generation

The main drawback of using topological representations for the exploration of analog placement is the difficulty found on satisfying and maintaining the layout design constraints [18]. While symmetric-feasible conditions were derived for most of the representations, proximity constraints are barely supported and often considered as objective in the weighted single-objective cost functions. Absolute representation is the most intuitive manner of implementing analog constraints, as symmetry and proximity can be forced in any combination, without relying in special symmetric-feasible conditions of the topological structure that encodes it. No packing, structural scan or post-processing time is required to fulfill/implement the analog constraints, the algorithm moves the cells explicitly and implements the requirements inherently.

Topological floorplan representations became a trend on analog design automation in the last two decades. Research community became extremely focused on developing innovative and problem-oriented topological representations and left for background the optimization algorithms used. A summary of the optimization algorithms used in the overviewed tools for analog placement automation is sketched on Table 2.5. The SA-based kernel used nowadays to perturb most of the topological representations is nearly identical from the one used in the disruptive approaches in the early $1980 \mathrm{~s}$. However, the truth is that published absolute representations’ floorplans are rather sparse, which further degrade with the scalability of the problem and the results are no match for the most recent topological representations. Nevertheless, recent efforts on stochastic/evolutionary algorithms suggest that searching the extensive search space derived from the use of absolute coordinates can compete with a search within the reduced solution space produced by a topological structure.

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。

Posted on Categories:模拟电路, 物理代写

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物理代写|模拟电路代写Analog Circuit代考|From Netlist to Pathfinding

While the netlist is available for any circuit, it does not provide any information on how the devices’ terminals should be connect, that is, the terminal-to-terminal connectivity is also needed. Starting from the netlist, if the terminal-to-terminal connectivity of each net is unknown, the routing of single-port multiterminal signal nets is usually addressed as the classical Steiner minimal tree (SMT) problem. Where, a set of Steiner points must be found in order to minimize the SMT total interconnect length that contains all the terminals of the net. This problem is often generalized as the rectilinear Steiner minimal tree (RSMT), when only edges defined by vertical or horizontal segments are considered, however, the problem is still NP-complete [50].

A RSMT global router is used in [51] for wiring length estimation, and, when the terminal-to-terminal connectivity is found, the traditional pathfinding algorithm is used. These deterministic pathfinding algorithms developed in the late $1980 \mathrm{~s}$, are variations of the classic maze algorithm [51-53], which is the most common approach, but line-expansion techniques [4] can also be found. Each instance of those approaches is used to generate a wire that connects two different terminals in the presence of obstacles (e.g., devices placed on the floorplan or other wires), usually by means of a grid-based or tile-based representation to ensure no overlaps or design rule violations, where the routing of all nets is done by iterating the different wires. The design rule validations are usually forced in the path-finding algorithms, e.g., by expanding the grids or routing channels with the minimum space requirements. Since an analog cell has a considerable number of conflicting nets, each one containing multiple wires, heuristics for net (re)ordering, backtracking and re-routing must be used to obtain valid solutions [1].

A different approach to automatic routing is the template adjustment techniques $[54,55]$, these have the highest setup times, but outperform the remaining by its fast and user-defined generation. A detailed state-of-the-art on the pathfinding algorithms and analog layout routing difficulties can be found in the Chapter Routing Analog Circuits by Dündar and Unutulmaz of [1].

物理代写|模拟电路代写Analog Circuit代考|Electromigration and IR-Drop

AMS ICs suffer from diverse non-idealities that became increasingly more relevant with the reduction of the circuit sizes in the last years, and may cause catastrophic circuit failures. These non-idealities must be taken into account during the circuit design in order to mitigate their effect on the product reliability [56]. Two of these non-idealities are: electromigration, which refers to the material migration in the power networks and signal wires that are stressed with high current-densities, deteriorating the interconnect lifetime; and IR-Drop, that consists of a fluctuation of the net voltage due to the interconnect resistances, affecting circuit behavior and performance $[57,58]$.

Analyzing the electromigration physical phenomenon, J. R. Black, in 1969 , was the first to developed a theoretical model to estimate the median time to failure (MTF) in hours of an interconnect in an IC [59], as presented in Eq. (2.5). The model was developed for the aluminum conductors used during the early years of integration industry.
$$M T F=\frac{A}{J^n} \exp \left(-\frac{\Phi}{k \cdot T}\right)$$
where, $A$ is a constant that contains a factor involving the cross-sectional area of the interconnect, $J$ the current-density in amperes per square centimeter, $\Phi$ the activation energy in electron volts for the interconnect material, $k$ the Boltzmann constant, $T$ the working temperature of the interconnect, and $n$ is used as a scaling factor. By observing Eq. (2.5) it is notorious that only two parameters can be changed by the designer: the current-density, which can be controlled by designing the proper interconnects for the current imposed on them, i.e., the wider the interconnect is assigned, the smaller is the current-density and subsequently electromigration resistance [58]; and the temperature, which may be indirectly controlled by assigning power and thermally-sensitive devices and interconnects in different areas of the chip, and by considering worst-case conditions in the determination of the current-densities.
While the accuracy of the model of Eq. (2.5) is outdated for today’s integration technologies, the principles that the combined effects of current-density and temperature are responsible for the gradual degradation of the interconnect is still valid.

物理代写|模拟电路代写Analog Circuit代考|From Netlist to Pathfinding

[51]中使用 RSMT 全局路由楍进行布线长度估计，并且当发现終端到終煓的连接时，使用传统的寻路算法。这些确定性寻路算法 是在后期开发的 $1980 \mathrm{~s}$ ，是经典迷宆算法 [51-53] 的变体，这是最常见的方法，但也可以找到线扩展技术 [4]。这些方法的每个实 例都用于在存在障碍物（例如，放置在平面图上的设备或其他电线）的情况下生成连接两个不同終端的电线，通常通过基于网格或 基于图块的表示来确保不会重叕或违反设计规则，其中所有网絡的布线都是通过迭代不同的电线来完成的。设计规则验证通常在寻 路算法中强制执行，例如，通过扩展网格或以最小空间要求布线通道。由于一个模拟单元有相当数量的冲突网絡，每个网络包含多 条线，网络 (重新) 排序的试探去，

物理代写|模拟电路代写Analog Circuit代考|Electromigration and IR-Drop

AMS IC 存在各种非理想性，这些非理想性在过去几年随着电路尺寸的缩小而变得越来越重要，并且可能导致灾难性的电路故障。 在电路设计过程中必须考虑这些非理想因羏，以减轻它们对产品可靠性的影响 [56]。其中两个非理想情况是：电迁移，指在高电 流密度下受力的电源网络和信号线中的材料迁移，会摍短互连寿命；和 IR-Drop，它由互连电阻引起的净电压波动组成，影响电 路行为和性能 $[57,58]$.
1969 年，JR Black 分析了电迁移物理现象，率先开发了一个理论模型来估计 IC [59] 互连的中位故障时间 (MTF)，如等式 1 所 示。(2.5)。该模型是为集成行业早期使用的铝导体开发的。
$$M T F=\frac{A}{J^n} \exp \left(-\frac{\Phi}{k \cdot T}\right)$$

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。

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物理代写|模拟电路代写Analog Circuit代考|Analog Layout Automation

The methodologies proposed and presented in this book focus on the layout generation task of analog ICs. Analog design automation has been intensively studied in academia for more than two decades (as overviewed later in Chap. 2 of this book) and is still an intensive research topic [20-22, 26]. Although much has been accomplished, the fact is that there is still no mature tool in the industrial environment and the analog layout is mostly done manually using time-consuming layout editors. Applications that provide some kind of user-assisted functionalities found their way into commercial EDA tools, however, the automatic functionalities are limited, far from perfect and lots of problems remain unsolved [16]. Allied to the fact that these functionalities are constantly discredited by analog designers, EDA vendors do not put enough effort on their continuous development and improvement. The onset of more efficient and user-oriented tools is mandatory in order to boost analog designers’ productivity and ease this time-consuming task.

In the traditional design flow, layout generation is only triggered when circuit sizing is complete. However, to achieve post-layout successful designs that meet all specifications, time-consuming and non-systematic iterations between these electrical and physical design phases are required. Without them, performance overdesign results in wasted power and area, and if underestimated, the circuits’ post-layout performance can be compromised [18, 23]. Automatic circuit sizing methodologies in both research environment as in the industry are way more developed than layout generators, and the urge for the so called layout-aware or layout-driven methodologies (i.e., that include layout-related data, e.g., geometrical information from the layout or parasitic components, into the sizing task) to close the gap between electrical and physical design steps, really enforces the need of automatic procedures to generate the layout. Fast, flexible and as robust as possible layout generators are mandatory to include precise layout-related data into the sizing process, and, eventually, obtain a final layout simultaneously with sizing.

The main innovative contributions of the methodologies proposed in this book were integrated in the AIDA framework $[24,25]$ and can be summarized as follows:

Hierarchical combination of Pareto fronts of placements: analog IC floorplan automation is complex as multiple requirements, which appear mainly in the form of topological constraints, must be dealt simultaneously along with several objectives and/or considerations for a robust floorplan. Absolute coordinates is the most practical and intuitive manner of implementing those layout constraints/ requirements. However, a complete study of previous absolute floorplanners suggests that illegal overlaps and other constraints have been improperly weighted in a single-objective cost function for optimization along with other objectives. The problem of analog floorplan automation in absolute coordinates is here reformulated, and, since it is impossible to determine a single best floorplan for all of the design objectives, a constrained multi-objective optimization algorithm is proposed to solve it. In order to reduce the problem’s complexity, the concept of proximity group is introduced, and the Pareto fronts of placements representing the tradeoffs between the optimization objectives for each group are combined bottom-up through the design hierarchy. Current-flow and current-density considerations are taken during the optimization to improve routing quality, reliability and, attempting to reduce routing-induced parasitics for better post-layout circuit performance.

Electromigration-aware routing with multilayer multiport terminal structures: in real analog cells, a complicated terminal geometry can easily have tens of ports (electrically equivalent locations) on multiple fabrication layers that are consider in manual design and, should also be considered in automatic approaches to increase routing efficiency. However, in the state-of-the-art of analog layout automation only simplistic single-ports/’dot-models’ to represent these complex multiport structures are considered. Instead of assuming that an unknown shape of a group of ports is an element of a ‘dot-model’, in the proposed methodology the solution is found considering every port available, over different fabrication layers, of the terminal geometry, which further increases the complexity of the original NP-complete Steiner problem. In addition, electromigration and voltage drop in the wires are taken in consideration, without adding setup complexity. The Router module inputs only the netlist and the electric-currents for each terminal and introduces a new degree of freedom by automatically exploring all the available electrically-equivalent ports of a terminal to connect a wire, in both global and detailed routing phases. No additional setup/tuning is required for a symmetric electromigration- and IR-drop-reliable solution.

Evolutionary multi-objective multi-constraint detailed Router: unlike the available deterministic approaches, wires are not restricted to limitative representations and are represented by their absolute coordinates, which expand the effectiveness of the exploration of the solution space. An in-loop internal procedure is used to evaluate each generation of layout solutions, instead of forcing the design rules in the path-finding algorithms by expanding the grids or routing channels with the minimum space requirements. This robust but lightweight builtin layout evaluation procedure is compliant with industrial grade validation tools. All nets are optimized simultaneously to achieve a solution in the conditions imposed by the problem definition, which eliminates the need for deterministic and error-prone backtracking, rip-up, re-routing and net-ordering considerations.

Parasitic extraction performed over a semi-complete layout: to address postlayout performance degradation and geometric requirements earlier in the design flow, the layout-aware design approaches include layout effects during the automatic sizing loop. However, both complete automatic layout generation and exhaustive parasitic extraction are still time consuming and hard to setup operations. By using a lightweight built-in extractor it is possible to accurately compute the impact of layout parasitics for both floorplan and early-stages of routing in-loop, without requiring a detailed and final layout (i.e., a layout with all DRC and LVS errors solved), unlike previous approaches. Traditionally, the last step required in the automatic layout generation flow is the detailed routing, which is by far the most computational intensive and time consuming step. By avoiding the need of a detailed layout and consequently an external extractor, the overall optimization time is greatly reduced.

物理代写|模拟电路代写Analog Circuit代考|Analog Layout Automation

Evolutionary multi-objective multi-constraint detailed Router：与可用的确定性方法不同，wire 不局限于限制性表示，而是用它们的绝对坐标表示，这扩大了探索解决方案空间的有效性。环内内部程序用于评估每一代布局解决方案，而不是通过以最小空间要求扩展网格或路由通道来强制路径查找算法中的设计规则。这种强大但轻量级的内置布局评估程序符合工业级验证工具。同时优化所有网络以在问题定义强加的条件下实现解决方案，从而消除了确定性和容易出错的回溯、撕裂、

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。