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电子代写|数字信号处理代写Digital Signal Processing代考|The Frequency Sampling Method of FIR Filter Design

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电子代写|数字信号处理代写Digital Signal Processing代考|The Frequency Sampling Method of FIR Filter Design

The frequency sampling method $[10,11]$ of FIR filter design consists of the following steps.
$1 \rightarrow$ Specify the desired frequency response, $H[m]$.
$2 \rightarrow$ Take the inverse discrete Fourier transform that yields the impulse response, $h[n]$.
$3 \rightarrow$ Multiply this impulse response by one of many possible windowing functions.
$4 \rightarrow$ Normalize the magnitude of the impulse response to the overall desired DC gain.
$5 \rightarrow$ Compare the results with the specified frequency response using the discrete time FT and if desired pass a test signal through a filter with the final coefficients using convolution.
Defining the Frequency Response of the FIR Filter
In order to correctly define the frequency response, there are three separate vectors that we must set. We need a vector indicating the independent variable, which is the normalized frequency. Further, we require a vector specifying the magnitude response at the given frequencies and another specifying the phase response. Clearly, the vector specifying the frequencies at which we will define our response must be formatted such that the IDFT can correctly compute the impulse response. The vector defining normalized frequency obeys the format $m / N$ where $m=0,1 \ldots N-1$, and $N$ represents the tap length of the FIR filter.

Example 2.27: (Part 1) Defining the Frequency Response, $H[m]$, of a Low Pass FIR Filter
Assume a low pass FIR filter of tap length $N=13$ running at a sample rate of $20 \mathrm{MHz}$. It is our goal to force a response that passes frequency content below $3.5 \mathrm{MHz}$ (normalized frequency $=$ $3.5 / 20 \mathrm{MHz}=0.175 \mathrm{~Hz}$ ) and blocks it everywhere else. According to the rule regarding frequency assignments above, we set the frequency vector as follows.

电子代写|数字信号处理代写Digital Signal Processing代考|Understanding the Phase and Group Delay

The phase and group delays are two common metrics that illustrate phase distortion in linear systems such as filters. These metrics calculate the transit time that a sinusoid at a particular frequency or groups of sinusoids at different but close frequencies experience as they traverse a filter. In communication systems, a great number of filters are tasked to attenuate interference and noise outside the bandwidth of the signal of interest. The time that it takes each frequency to traverse these filters should be the same. If this is not the case, then the different frequency components of the signal in the pass band will reassemble out of phase at the output, causing linear distortion. The phase and group delay function are tools that allow us to easily visualize the presence of phase distortion. The phase response of a linear system that does not introduce phase distortion and thus delays all frequency components equally is a straight line as is suggested by the time shifting property of the Fourier transform (see Section 3.1 of this chapter.)
$$\begin{array}{ll} x(t) \quad \stackrel{F T}{\rightarrow} X(f) \ x\left(t-t_0\right) & \stackrel{F T}{\rightarrow} X(f) \cdot e^{-j 2 \pi f t_o} \end{array}$$
Phase and group delay are functions of frequency and can be calculated directly from the phase response of the linear time invariant system.
Phase Delay
The phase delay is a measure of transit time, $t_0$, experienced by a complex sinusoid, $\exp (j 2 \pi f t)$, as it travels through a linear time-invariant system such as a filter. The transit time is calculated by comparing the input and output phases of the complex sinusoid.
$$\begin{gathered} \theta(f)=\angle \text { Output }(f)-\angle \operatorname{Input}(f)=2 \pi f\left(t-t_o\right)-2 \pi f t \ \theta(f)=-2 \pi f t_o \ \text { PhaseDelay }(f)=t_o=-\frac{\theta(f)}{2 \pi f} \text { seconds } \end{gathered}$$

电子代写|数字信号处理代写Digital Signal Processing代考|The Frequency Sampling Method of FIR Filter Design

FIR滤波器设计的频率采样方法$[10,11]$包括以下步骤。
$1 \rightarrow$指定所需的频率响应，$H[m]$。
$2 \rightarrow$求离散傅里叶反变换得到脉冲响应$h[n]$。
$3 \rightarrow$将这个脉冲响应乘以许多可能的窗口函数之一。
$4 \rightarrow$将脉冲响应的幅度归一化到所需的总体直流增益。
$5 \rightarrow$使用离散时间傅里叶变换将结果与指定的频率响应进行比较，如果需要的话，可以使用卷积将测试信号通过带有最终系数的滤波器。

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。