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• 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|数字系统设计作业代写Digital System Design代考|NORMALIZED FREQUENCY REPRESENTATION

Generally, the discrete time samples resulting from sampling a continuous time signal can be represented by replacing the variable $t$ by $n T$ for the case where $t$ is the independent variable for the continuous time system and $n$ is the independent variable for the discrete time system. The parameter $T$ is the sampling interval. A normalized frequency representation for the discrete time signal can then be obtained by replacing the continuous time radial frequency $\Omega$ by the discrete time frequency $\omega=\Omega T$.
Consider the signal
$$x(t)=A e^{j \Omega t} u(t) .$$

The corresponding discrete time signal, after sampling, can be represented as
$$x(n)=A e^{j \Omega n T} u(n)=A e^{\omega n t} u(n) .$$
Thus, the normalized discrete time frequency can be obtained by multiplying the continuous time frequency $\Omega$ by the sampling interval $T$.

The Sampling Theorem states that the largest frequency that can be represented in the sampled sequence is given by
$$\Omega_{N}<2 \pi f_{N}=\left(\frac{1.0}{2.0}\right)\left(\frac{2.0 \pi}{T_{\max }}\right)=\frac{\pi}{T_{\max }}$$
It follows that the largest possible sampling interval to avoid aliasing is
$$T_{\max }=\frac{\pi}{\Omega_{N}}$$
Thus, the normalized discrete time Nyquist frequency is always
$$\omega_{N}=\Omega_{N} T_{\max }=\Omega_{N}\left(\frac{\pi}{\Omega_{N}}\right)=\pi$$
The appropriate fundamental radial frequency interval of concern for discrete time normalized frequencies is the range
$$-\pi \leq \omega \leq \pi$$

## 电子工程代写|数字系统设计作业代写Digital System Design代考|PERIODICITY FOR DISCRETE TIME SEQUENCES

The sample sequence obtained from sampling a periodic, continuous time signal is not necessarily periodic. The ratio between the sampling frequency and the fundamental frequency of the original, periodic, continuous time signal must be a rational number in order for the corresponding sampled sequence to be periodic [2]. This can be shown by considering the continuous time sinusoidal signal
$$x(t)=A \cos (\Omega t+\phi) .$$
The corresponding discrete time signal, after sampling $x(t)$, can be represented by
$$x(n T)=A \cos (\Omega n T+\phi) .$$
The requirement for periodicity can be stated as
$$x(n T)=A \cos (\Omega n T+\Omega N T+\phi)=A \cos (\Omega n T+\phi)$$

where $N$ is either a positive or negative integer. The following relationship must be true in order for $x(n T)$ to be periodic:
$$\Omega N T=2 \pi k,$$
or
$$\Omega T=\frac{2 \pi k}{N}$$
where $k$ is also an integer. If $f$ is the frequency of the original, continuous time, signal and the sampling frequency $F_{s}=\frac{1}{T}$, then the requirement for periodicity is that
$$\Omega T=\frac{2 \pi f}{\Gamma_{s}}=\frac{2 \pi k}{N},$$
or
$$\frac{f}{F_{s}}=\frac{k}{N} .$$
This means that the ratio of the sampling frequency and the fundamental frequency of the original, periodic signal must be a rational number. Thus, if a sampled, continuous time signal is periodic, the period, $N$, can be determined by finding the smallest integer values of $k$ and $N$ for which
$$\frac{f}{F_{s}}=\frac{k}{N}$$
Note that if normalized frequencies are used, where $F_{s}=\frac{1}{T}=1$, then the requirement for periodicity becomes
$$\omega_{0}=\frac{2 \pi k}{N},$$
or
$$N=\frac{2 \pi k}{\omega_{0}}$$
where $\omega_{0}$ is the normalized, radial frequency of the original, periodic, continuous time signal. Example $2.13$ illustrates this point.

# 数字系统设计代考

## 电子工程代写|数字系统设计作业代写Digital System Design代考|NORMALIZED FREQUENCY REPRESENTATION

$$x(t)=A e^{j \Omega t} u(t)$$

$$x(n)=A e^{j \Omega n T} u(n)=A e^{\omega n t} u(n)$$

$$\Omega_{N}<2 \pi f_{N}=\left(\frac{1.0}{2.0}\right)\left(\frac{2.0 \pi}{T_{\max }}\right)=\frac{\pi}{T_{\max }}$$

$$T_{\max }=\frac{\pi}{\Omega_{N}}$$

$$\omega_{N}=\Omega_{N} T_{\max }=\Omega_{N}\left(\frac{\pi}{\Omega_{N}}\right)=\pi$$

$$-\pi \leq \omega \leq \pi$$

## 电子工程代写|数字系统设计作业代写Digital System Design代考|PERIODICITY FOR DISCRETE TIME SEQUENCES

$$x(t)=A \cos (\Omega t+\phi) .$$

$$x(n T)=A \cos (\Omega n T+\phi) .$$

$$x(n T)=A \cos (\Omega n T+\Omega N T+\phi)=A \cos (\Omega n T+\phi)$$

$$\Omega N T=2 \pi k,$$

$$\Omega T=\frac{2 \pi k}{N}$$

$$\Omega T=\frac{2 \pi f}{\Gamma_{s}}=\frac{2 \pi k}{N}$$
$$\frac{f}{F_{s}}=\frac{k}{N} .$$

$$\frac{f}{F_{s}}=\frac{k}{N}$$

$$\omega_{0}=\frac{2 \pi k}{N},$$

$$N=\frac{2 \pi k}{\omega_{0}}$$

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