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# 信号处理代写signal processing代考|Stochastic processes

##### 代写数字信号处理signal processing

In comparison, the output viewpoint examines how a single point in the output signal is determined by the various values from the input signal. Just as with discrete signals, each instantaneous value in the output signal is affected by a section of the input signal, weighted by the impulse response flipped left-for-right. In the discrete case, the signals are multiplied and summed. In the continuous case, the signals are multiplied and integrated. In equation form:

$$y(t)=\int_{-\infty}^{+\infty} x(\tau) h(t-\tau) d \tau$$

The convolution integral. This equation defines the meaning of: $y(t)=x(t) * h(t)$.

This equation is called the convolution integral, and is the twin of the convolution sum ) used with discrete signals.shows how this equation can be understood. The goal is to find an expression for calculating the value of the output signal at an arbitrary time, $t$. The first step is to change the independent variable used to move through the input signal and the impulse response. That is, we replace $t$ with $\tau$ (a lower case Greek tau). This makes $x(t)$ and $h(t)$ become $x(\tau)$ and $h(\tau)$, respectively. This change of variable names is needed because $t$ is already being used to represent the point in the output signal being calculated. The next step is to flip the impulse response left-for-right, turning it into $h(-\tau)$. Shifting the flipped impulse response to the location $t$, results in the expression becoming $h(t-\tau)$. The input signal is then weighted by the flipped and shifted impulse response by multiplying the two, i.e., $x(\tau) h(t-\tau)$. The value of the output signal is then found by integrating this weighted input signal from negative to positive infinity.

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• 微分方程 Differential equations
• 递归关系 Recurrence relations
• 变换理论 Time-frequency analysis – for dealing with non-stationary signals [14]
• 时频分析 Transformation theory Time-frequency analysis – for dealing with non-stationary signals
• 频谱估计 Spectral estimation – for determining the spectral content
• 统计信号处理 Statistical signal processing – for analyzing and extracting information based on the stochastic properties of signals and noise
• 线性时不变系统理论和变换理论 Linear time-invariant systems theory and transformation theory
• 多项式信号处理 Polynomial signal processing – analysis of systems related to inputs and outputs using polynomials

## 信号处理代写signal processing代考|deterministic

In real-world applications, time-varying quantities are usually modelled by a stochastic process, that is, an ordered collection of random variables. For ergodic processes, the theoretical mean can be approximated by time averages, however, for non-ergodic processes time averages do not necessarily match averages in the probability space and therefore the theoretical statistics are not always well approximated using observed samples.

To illustrate this concept, consider the deterministic sinusoidal signal $x[n]=\sin \left(2 \pi 5 \times 10^{-3} n\right)$ corrupted by independent and identically distributed (i.i.d.) Gaussian noise $\eta[n] \sim \mathcal{N}(0,1)$, to give $y[n]=x[n]+\eta[n]$. By averaging multiple realisations of the process $\mathbf{y}=[y[1], y[2], \ldots, y[N]]^{T}$, we aim to obtain reduced-noise estimates of the process $\mathbf{x}$, in terms of the Signal-to-Noise (SNR) ratio. If $M$ independent realisations of the process $\mathbf{y}$, denoted by $\mathbf{y}{1: M}$, are considered to compute an ensemble estimate, the variance of such an estimate is given by \begin{aligned} \sigma{M}^{2} &=\mathbb{E}\left{\left(\mathbb{E}{\mathbf{y}}-\frac{1}{M} \sum_{i=1}^{M} \mathbf{y}{i}\right)^{2}\right}=\mathbb{E}\left{\left(\frac{1}{M} \sum{i=1}^{M} \boldsymbol{\eta}{i}\right)^{2}\right} \ &=\frac{1}{M^{2}} \mathbb{E}\left{\left(\sum{i=1}^{M} \sum_{j=1}^{M} \boldsymbol{\eta}{i}^{T} \boldsymbol{\eta}{j}\right)\right}=\frac{1}{M^{2}}\left(\sum_{i=1}^{M} \sum_{j=1}^{M} \mathbb{E}\left{\boldsymbol{\eta}{i}^{T} \boldsymbol{\eta}{j}\right}\right) \end{aligned}
where every noise sequence $\boldsymbol{\eta}{j}$ comprises realisations of zero-mean and uncorrelated random variables $\eta[n]$. We know that $\mathbb{E}\left{\boldsymbol{\eta}{i}^{T} \boldsymbol{\eta}{j}\right}=\sigma{\boldsymbol{\eta}}^{2}$ iff $i=j$, and zero otherwise, hence
$$\sigma_{M}^{2}=\frac{1}{M^{2}}\left(M \sigma_{\eta}^{2}\right)=\frac{\sigma_{\eta}^{2}}{M} .$$
Therefore, the SNR of an $M$-member ensemble estimate increases linearly with the number of members of the ensemble
$$S N R=\frac{\sigma_{\mathbf{y}}^{2}}{\sigma_{M}^{2}}=\frac{\sigma_{\mathbf{y}}^{2}}{\sigma_{\boldsymbol{\eta}}^{2}} M, \text { and in } \mathrm{dB}: S N R_{d B}=\log {10}\left(\frac{\sigma{\mathbf{y}}^{2}}{\sigma_{\boldsymbol{\eta}}^{2}} M\right)[d B] .$$
Figure 1 shows a realisation of $\mathbf{y}$, together with ensemble averages for $M=10,50,200,1000$ and the original deterministic signal x. Additionally, the bottom plot shows the SNR computed from the ensemble averages and its theoretical value in Eq. (5). Observe that for nonstationary signals, time-average will not provide meaningful approximations of the process statistics (e.g. sample mean).

We now study three stochastic processes generated by the following MATLAB codes, which give an ensemble of $M$ realisations of $N$ samples for each stochastic process.

## 信号处理代写SIGNAL PROCESSING代考|MATLA

Run the above MATLAB codes and explain the differences between the time averages and ensemble averages, together with the stationarity and ergodicity of the process generated by the following steps:

1. Compute the ensemble mean and standard deviation for each process and plot them as a function of time. For all
[10] the above random processes, use $M=100$ members of the ensemble, each of length $N=100$. Comment on the stationarity of each process.
2. Generate $M=4$ realisations of length $N=1000$ for each process, and calculate the mean and standard deviation
[10] for each realisation. Comment on the ergodicity of each process.
3. Write a mathematical description of each of the three stochastic processes. Calculate the theoretical mean and
[10] variance for each case and compare the theoretical results with those obtained by sample averaging.

## 信号处理代写SIGNAL PROCESSING代考|MATLA

1. 计算每个过程的整体平均值和标准差，并将它们绘制为时间的函数。对所有人
10上述随机过程，使用米=100乐团的成员，每个人的长度ñ=100. 评论每个过程的平稳性。
2. 产生米=4长度的实现ñ=1000对于每个过程，并计算均值和标准差
10对于每个实现。评论每个过程的遍历性。
3. 写出三个随机过程中每一个的数学描述。计算理论平均值和
10每个案例的方差，并将理论结果与通过样本平均获得的结果进行比较。

## 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。