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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|ECE2237 Periodic and aperiodic signals

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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|Periodic and aperiodic signals

A CT signal $x(t)$ is said to be periodic if it satisfies the following property:
$$x(t)=x\left(t+T_0\right),$$
at all time $t$ and for some positive constant $T_0$. The smallest positive value of $T_0$ that satisfies the periodicity condition, Eq. (1.3), is referred to as the fundamental period of $x(t)$.
Likewise, a DT signal $x[k]$ is said to be periodic if it satisfies
$$x[k]=x\left[k+K_0\right]$$
at all time $k$ and for some positive constant $K_0$. The smallest positive value of $K_0$ that satisfies the periodicity condition, Eq. (1.4), is referred to as the fundamental period of $x[k]$. A signal that is not periodic is called an aperiodic or non-periodic signal. Figure $1.6$ shows examples of both periodic and aperiodic signals. The reciprocal of the fundamental period of a signal is called the fundamental frequency. Mathematically, the fundamental frequency is expressed as follows
$f_0=\frac{1}{T_0}$, for CT signals, $\quad$ or $\quad f_0=\frac{1}{K_0}$, for DT signals,
where $T_0$ and $K_0$ are, respectively, the fundamental periods of the CT and DT signals. The frequency of a signal provides useful information regarding how fast the signal changes its amplitude. The unit of frequency is cycles per second (c/s) or hertz $(\mathrm{Hz})$. Sometimes, we also use radians per second as a unit of frequency. Since there are $2 \pi$ radians (or $360^{\circ}$ ) in one cycle, a frequency of $f_0$ hertz is equivalent to $2 \pi f_0$ radians per second. If radians per second is used as a unit of frequency, the frequency is referred to as the angular frequency and is given by
$\omega_0=\frac{2 \pi}{T_0}$, for CT signals, or $\quad \Omega_0=\frac{2 \pi}{K_0}$, for DT signals.

## 物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代考|Energy and power signals

Before presenting the conditions for classifying a signal as an energy or a power signal, we present the formulas for calculating the energy and power in a signal. The instantaneous power at time $t=t_0$ of a real-valued CT signal $x(t)$ is given by $x^2\left(t_0\right)$. Similarly, the instantaneous power of a real-valued DT signal $x[k]$ at time instant $k=k_0$ is given by $x^2[k]$. If the signal is complex-valued, the expressions for the instantaneous power are modified to $\left|x\left(t_0\right)\right|^2$ or $\left|x\left[k_0\right]\right|^2$, where the symbol $|\cdot|$ represents the absolute value of a complex number.
The energy present in a CT or DT signal within a given time interval is given by the following:

CT signals $\quad E_{\left(T_1, T_2\right)}=\int_{T_1}^{T_2}|x(t)|^2 \mathrm{~d} t$ in interval $t=\left(T_1, T_2\right)$ with $T_2>T_1 ;$
DT sequences $E_{\left[N_1, N_2\right]}=\sum_{k=N_1}^{N_2}|x[k]|^2$ in interval $k=\left[N_1, N_2\right]$ with $N_2>N_1$.
(1.10b)

## 物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代 考|Periodic and aperiodic signals

CT信昊 $x(t)$ 如果满足以下性质，则称其为周期性的:
$$x(t)=x\left(t+T_0\right)$$

$$x[k]=x\left[k+K_0\right]$$

## 物理代写|连续时间信号和系统代写Continuous Time Signals and Systems代 考|Energy and power signals

CT信昊 $\quad E_{\left(T_1, T 2\right)}=\int_{T_1}^{T_2}|x(t)|^2 \mathrm{~d} t$ 在间隔 $t=\left(T_1, T_2\right)$ 和 $T_2>T_1$
$\mathrm{DT}$ 序列 $E_{\left[N_1, N_2\right]}=\sum_{k=N_1}^{N_2}|x[k]|^2$ 在间隔 $k=\left[N_1, N_2\right]$ 和 $N_2>N_1$.
(1.10b)

## MATLAB代写

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