# 电子工程代写|电子系统工程代写Digital Systems Engineering代考|EE273

## 电子工程代写|电子系统工程代写Digital Systems Engineering代考|Signal Transmission Through Linear Systems

Linear time-invariant (LTI) systems are those systems which satisfy the following three criteria mathematically:

1. If the system has as an input $x(t)$ and its output $y(t)$
$$x(t) \Rightarrow y(t)$$
then when we use an input $a x(t)$, the output is $a y(t)$
$$a x(t) \Rightarrow a y(t)$$
Here, the arrow $\Rightarrow$ is used to designate the output of the system. This property is called the scaling property.
2. If the system has two separate inputs $x_{1}(t)$ and $x_{2}(t)$ so that the outputs to these two separate signals individually are $y_{1}(t)$ and $y_{2}(t)$, then if the input is $x_{1}(t)+x_{2}(t)$ the output is $y_{1}(t)+y_{2}(t)$,
\begin{aligned} x_{1}(t) & \Rightarrow y_{1}(t) \quad \text { and } \ x_{2}(t) & \Rightarrow y_{2}(t) \quad \text { then } \ x_{1}(t)+x_{2}(t) & \Rightarrow y_{1}(t)+y_{2}(t) \end{aligned}
This property is called the additive property. Both these properties together are called the superposition properties.
3. If the system has an input $x(t)$ which gives an output $y(t)$ then if the input is given at a later time $t_{0}$, then the output is also delayed by time $t_{0}$. Or more formally,

## 电子工程代写|电子系统工程代写Digital Systems Engineering代考|Paley–Wiener Criterion

The necessary and sufficient condition that $H(\omega)$ is the frequency response of a filter is
$$\int_{-\infty}^{\infty} \frac{\ln |H(\omega)|}{1+\omega^{2}} d \omega<\infty$$ Thus, for example, the gain $|H(\omega)|$ may be zero only for a finite number of frequencies, but it cannot be zero for a band of frequencies. By this criterion, the gain may approach zero over a band of frequencies but not be zero over that band. By this criterion, it is impossible to realise a filter such as $$H(\omega)= \begin{cases}1 & -2 \pi B \leq \omega \leq 2 \pi B \ 0 & \text { elsewhere }\end{cases}$$ since, then $\ln |H(\omega)|$ will be $-\infty$ over those portions of the frequency where $|H(\omega)|$ will be zero. However, the filter $$H(\omega)= \begin{cases}1 & -2 \pi B \leq \omega \leq 2 \pi B \ >\varepsilon & \text { elsewhere }\end{cases}$$
where $\varepsilon$ is a small positive value, is a realisable filter, $\ln (\varepsilon) \neq-\infty$.

## 电子工程代写|电子系统工程代写Digital Systems Engineering代考|Signal Transmission Through Linear Systems

1. 如果系统有作为输入 $x(t)$ 及其输出 $y(t)$
$$x(t) \Rightarrow y(t)$$
那么当我们使用输入时 $a x(t)$ ，输出为 $a y(t)$
$$a x(t) \Rightarrow a y(t)$$
在这里，箭头 $\Rightarrow$ 用于指定系统的输出。此属性称为缩放属性。
2. 如果系统有两个独立的输入 $x_{1}(t)$ 和 $x_{2}(t)$ 因此，这两个独立信号的输出分别是 $y_{1}(t)$ 和 $y_{2}(t)$, 那么如果输 入是 $x_{1}(t)+x_{2}(t)$ 输出是 $y_{1}(t)+y_{2}(t)$,
$$x_{1}(t) \Rightarrow y_{1}(t) \quad \text { and } x_{2}(t) \quad \Rightarrow y_{2}(t) \quad \text { then } x_{1}(t)+x_{2}(t) \Rightarrow y_{1}(t)+y_{2}(t)$$
这种性质称为加性性质。这两个属性一起称为菷加属性。
3. 如果系统有输入 $x(t)$ 这给出了一个输出 $y(t)$ 那么如果输入是在以后给出的 $t_{0}$ ，那么输出也会被时间延迟 $t_{0}$. 或者更正式地说，

## 电子工程代写|电子系统工程代写Digital Systems Engineering代考|Paley–Wiener Criterion

$$\int_{-\infty}^{\infty} \frac{\ln |H(\omega)|}{1+\omega^{2}} d \omega<\infty$$ 因此，例如，增益 $|H(\omega)|$ 仅对于有限数量的频率可能为零，但对于一个频带，它不能为零。通过这个标 准，增益可能在一个频帯上接近零，但在该频帯上不是零。按照这个标准，不可能实现像这样的过滤器 $$H(\omega)= \begin{cases}1 & -2 \pi B \leq \omega \leq 2 \pi B 0 \quad \text { elsewhere }\end{cases}$$ 自那时候起 $\ln |H(\omega)|$ 将会 $-\infty$ 在频率的那些部分 $|H(\omega)|$ 将为霎。然而，过滤器 $$H(\omega)= \begin{cases}1 & -2 \pi B \leq \omega \leq 2 \pi B>\varepsilon \quad \text { elsewhere }\end{cases}$$

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