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# 信号处理代写signal processing代考|MLE for the Frequency of a Signal (Optional – Extra Credit)

##### 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|MLE for the Frequency of a Signal (Optional – Extra Credit)

The pdf of a real signal $\mathbf{x}=[x[0], x[1], \ldots, x[N-1]]^{T}$ parametrized by $\boldsymbol{\theta}=\left[A, f_{0}, \phi\right]^{T}$ can be written as
$$p(\mathbf{x} ; \boldsymbol{\theta})=\frac{1}{\left(2 \pi \sigma^{2}\right)^{N / 2}} \exp \left{-\frac{1}{2 \sigma^{2}} \sum_{n=0}^{N-1}\left(x[n]-A \cos \left(2 \pi f_{0} n+\phi\right)\right)^{2}\right}$$
where $A>0$ and $0<f_{0}<1 / 2$. From the pdf, the maximum-likelihood estimate (MLE), $\hat{f}{0}$, of the frequency $f{0}$ is found by minimizing
$$J(\boldsymbol{\theta})=\sum_{n=0}^{N-1}\left(x[n]-A \cos \left(2 \pi f_{0} n+\phi\right)\right)^{2}$$

1. By letting $\alpha_{1}=A \cos \phi$ and $\alpha_{2}=-A \sin \phi$ and by defining $\mathbf{c}=\left[1, \cos \left(2 \pi f_{0}\right), \ldots, \cos \left(2 \pi f_{0}(N-1)\right)\right]^{T}$, s $=$ [5] $\left[0, \sin \left(2 \pi f_{0}\right), \ldots, \sin \left(2 \pi f_{0}(N-1)\right)\right]^{T}$ show that $J(\theta)$ can be mapped to
$$J^{\prime}\left(\alpha_{1}, \alpha_{2}, f_{0}\right)=\left(\mathbf{x}-\alpha_{1} \mathbf{c}-\alpha_{2} \mathbf{s}\right)^{T}\left(\mathbf{x}-\alpha_{1} \mathbf{c}-\alpha_{2} \mathbf{s}\right)=(\mathbf{x}-\mathbf{H} \boldsymbol{\alpha})^{T}(\mathbf{x}-\mathbf{H} \boldsymbol{\alpha})=J^{\prime}\left(\boldsymbol{\alpha}, f_{0}\right)$$
where $\boldsymbol{\alpha}=\left[\alpha_{1}, \alpha_{2}\right]^{T}$ and $\mathbf{H}=[\mathbf{c}, \mathbf{s}]$.
2. Find the minimizing solution, $\hat{\boldsymbol{\alpha}}$, to (46), and hence show that minimizing $J^{\prime}\left(\hat{\boldsymbol{\alpha}}, f_{0}\right)$ is equivalent to maximizing [5] $\mathbf{x}^{T} \mathbf{H}\left(\mathbf{H}^{T} \mathbf{H}\right)^{-1} \mathbf{H}^{T} \mathbf{x}$
3. Using the definition of $\mathbf{H}$ and the result from Exercise 2, show that the MLE of the frequency $f_{0}$ is the value that [5] maximizes
$$\left[\begin{array}{c} \mathbf{c}^{T} \mathbf{x} \ \mathbf{s}^{T} \mathbf{x} \end{array}\right]^{T}\left[\begin{array}{cc} \mathbf{c}^{T} \mathbf{c} & \mathbf{c}^{T} \mathbf{s} \ \mathbf{s}^{T} \mathbf{c} & \mathbf{s}^{T} \mathbf{s} \end{array}\right]^{-1}\left[\begin{array}{c} \mathbf{c}^{T} \mathbf{x} \ \mathbf{s}^{T} \mathbf{x} \end{array}\right]$$
Under the condition that $f_{0}$ is not close to 0 or $1 / 2$ (normalized frequency), expression (47) can be shown to become approximately
$$\left[\begin{array}{c} \mathbf{c}^{T} \mathbf{x} \ \mathbf{s}^{T} \mathbf{x} \end{array}\right]^{T}\left[\begin{array}{cc} \frac{N}{2} & 0 \ 0 & \frac{N}{2} \end{array}\right]^{-1}\left[\begin{array}{l} \mathbf{c}^{T} \mathbf{x} \ \mathbf{s}^{T} \mathbf{x} \end{array}\right]$$
4. Justify why it is required for $f_{0}$ not to be close to 0 or $1 / 2$, and, using the approximation given in (48), show that [10] the MLE $\hat{f}{0}$ is obtained by maximizing the periodogram $$\hat{P}{X}(f)=\frac{1}{N}\left|\sum_{n=0}^{N-1} x[n] e^{-\jmath 2 \pi f \frac{n}{N}}\right|^{2}$$
5. Using the noiseless data
$$x[n]=\cos \left(2 \pi f_{0} n\right), \quad n=0,1, \ldots, N-1$$
plot the periodogram and the MLE estimate for varying $0<f_{0}<1 / 2$, for $N=10$. Explain the behaviour when $f_{0}$ approaches 0 or $1 / 2$ ? Comment on your results.

## 信号处理代写SIGNAL PROCESSING代考|MLE FOR THE FREQUENCY OF A SIGNAL (OPTIONAL – EXTRA CREDIT)

p(\mathbf{x} ; \boldsymbol{\theta})=\frac{1}{\left(2 \pi \sigma^{2}\right)^{N / 2}} \exp \left{- \frac{1}{2 \sigma^{2}} \sum_{n=0}^{N-1}\left(x[n]-A \cos \left(2 \pi f_{0} n+\ phi\right)\right)^{2}\right}p(\mathbf{x} ; \boldsymbol{\theta})=\frac{1}{\left(2 \pi \sigma^{2}\right)^{N / 2}} \exp \left{- \frac{1}{2 \sigma^{2}} \sum_{n=0}^{N-1}\left(x[n]-A \cos \left(2 \pi f_{0} n+\ phi\right)\right)^{2}\right}


5. 使用无噪声数据
X[n]=某物⁡(2圆周率F0n),n=0,1,…,ñ−1
绘制不同的周期图和 MLE 估计值0<F0<1/2， 为了ñ=10. 解释行为时F0接近 0 或1/2? 评论你的结果。

## 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。