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# 物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代考|TSTE93 Linear and non-linear systems

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## 物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代考|Linear and non-linear systems

A CT system with the following set of inputs and outputs:
$$x_1(t) \rightarrow y_1(t) \text { and } x_2(t) \rightarrow y_2(t)$$
is linear iff it satisfies the additive and the homogeneity properties described below:
additive property $\quad x_1(t)+x_2(t) \rightarrow y_1(t)+y_2(t)$
homogeneity property $\quad \alpha x_1(t) \rightarrow \alpha y_1(t)$;
for any arbitrary value of $\alpha$ and all possible combinations of inputs and outputs. The additive and homogeneity properties are collectively referred to as the principle of superposition. Therefore, linear systems satisfy the principle of superposition. Based on the principle of superposition, the properties in Eqs. (2.28) and (2.29) can be combined into a single statement as follows. A CT system with the following sets of inputs and outputs:
$$x_1(t) \rightarrow y_1(t) \quad \text { and } \quad x_2(t) \rightarrow y_2(t)$$
is linear iff
$$\alpha x_1(t)+\beta x_2(t) \rightarrow \alpha y_1(t)+\beta y_2(t)$$
for any arbitrary set of values for $\alpha$ and $\beta$, and for all possible combinations of inputs and outputs.

## 物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代考|Time-varying and time-invariant systems

A system is said to be time-invariant (TI) if a time delay or time advance of the input signal leads to an identical time-shift in the output signal. In other words, except for a time-shift in the output, a TI system responds exactly the same way no matter when the input signal is applied. We now define a TI system formally.

A CT system with $x(t) \rightarrow y(t)$ is time-invariant iff
$$x\left(t-t_0\right) \rightarrow y\left(t-t_0\right)$$
for any arbitrary time-shift $t_0$. Likewise, a DT system with $x[k] \rightarrow y[k]$ is time-invariant iff
$$x\left[k-k_0\right] \rightarrow y\left[k-k_0\right]$$
for any arbitrary discrete shift $k_0$.
Example 2.4
Consider two CT systems represented mathematically by the following inputoutput relationship:
(i) system I
$$y(t)=\sin (x(t))$$
(ii) system II $y(t)=t \sin (x(t))$.
Determine if systems (i) and (ii) are time-invariant.
Solution
(i) From Eq. (2.42), it follows that:
$$x(t) \rightarrow \sin (x(t))=y(t)$$
and
$$x\left(t-t_0\right) \rightarrow \sin \left(x\left(t-t_0\right)\right)=y\left(t-t_0\right)$$

## 物理代写|连续时间信号和系统代写CONTINUOUS TIME SIGNALS AND SYSTEMS代考|LINEAR AND NON-LINEAR SYSTEMS

$$x_1(t) \rightarrow y_1(t) \text { and } x_2(t) \rightarrow y_2(t)$$

$$x_1(t) \rightarrow y_1(t) \quad \text { and } \quad x_2(t) \rightarrow y_2(t)$$

$$\alpha x_1(t)+\beta x_2(t) \rightarrow \alpha y_1(t)+\beta y_2(t)$$

## 物理代写|连续时间信号和系统代写CONTINUOUS TIME SIGNALS AND SYSTEMS代考|TIME-VARYING AND TIMEINVARIANT SYSTEMS

$\mathrm{CT}$ 系统与 $x(t) \rightarrow y(t)$ 是时不变的当且仅当
$$x\left(t-t_0\right) \rightarrow y\left(t-t_0\right)$$

$$x\left[k-k_0\right] \rightarrow y\left[k-k_0\right]$$

$i$ 系统一
$$y(t)=\sin (x(t))$$
$i i$ 系统二 $y(t)=t \sin (x(t))$.

$i$ 从等式。 $2.42$ ，它䢗循:
$$x(t) \rightarrow \sin (x(t))=y(t)$$

$$x\left(t-t_0\right) \rightarrow \sin \left(x\left(t-t_0\right)\right)=y\left(t-t_0\right)$$

## Matlab代写

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