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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 Continuous-time and discrete-time 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|Continuous-time and discrete-time signals

If a signal is defined for all values of the independent variable $t$, it is called a continuous-time (CT) signal. Consider the signals shown in Figs. 1.1(b) and (d). Since these signals vary continuously with time $t$ and have known magnitudes for all time instants, they are classified as CT signals. On the other hand, if a signal is defined only at discrete values of time, it is called a discretetime (DT) signal. Figure 1.1(h) shows the output temperature of a room measured at the same hour every day for one week. No information is available for the temperature in between the daily readings. Figure $1.1(\mathrm{~h})$ is therefore an example of a DT signal. In our notation, a CT signal is denoted by $x(t)$ with regular parenthesis, and a DT signal is denoted with square parenthesis as follows:
$$x[k T], \quad k=0, \pm 1, \pm 2, \pm 3, \ldots,$$
where $T$ denotes the time interval between two consecutive samples. In the example of Fig. 1.1(h), the value of $T$ is one day. To keep the notation simple, we denote a one-dimensional (1D) DT signal $x$ by $x[k]$. Though the sampling interval is not explicitly included in $x[k]$, it will be incorporated if and when required.

Note that all DT signals are not functions of time. Figure 1.1(f), for example, shows the output of a CCD camera, where the discrete output varies spatially in two dimensions. Here, the independent variables are denoted by $(m, n)$, where $m$ and $n$ are the discretized horizontal and vertical coordinates of the picture element. In this case, the two-dimensional (2D) DT signal representing the spatial charge is denoted by $x[m, n]$.

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

A second classification of signals is based on their amplitudes. The amplitudes of many real-world signals, such as voltage, current, temperature, and pressure, change continuously, and these signals are called analog signals. For example, the ambient temperature of a house is an analog number that requires an infinite number of digits (e.g., $24.763578 \ldots$ ) to record the readings precisely. Digital signals, on the other hand, can only have a finite number of amplitude values. For example, if a digital thermometer, with a resolution of $1{ }^{\circ} \mathrm{C}$ and a range of $\left[10^{\circ} \mathrm{C}, 30^{\circ} \mathrm{C}\right]$, is used to measure the room temperature at discrete time instants, $t=k T$, then the recordings constitute a digital signal. An example of a digital signal was shown in Fig. 1.1(h), which plots the temperature readings taken once a day for one week. This digital signal has an amplitude resolution of $0.1^{\circ} \mathrm{C}$, and a sampling interval of one day.

Figure $1.5$ shows an analog signal with its digital approximation. The analog signal has a limited dynamic range between $[-1,1]$ but can assume any real value (rational or irrational) within this dynamic range. If the analog signal is sampled at time instants $t=k T$ and the magnitude of the resulting samples are quantized to a set of finite number of known values within the range $[-1,1]$, the resulting signal becomes a digital signal. Using the following set of eight uniformly distributed values,
$$[-0.875,-0.625,-0.375,-0.125,0.125,0.375,0.625,0.875] \text {, }$$
within the range $[-1,1]$, the best approximation of the analog signal is the digital signal shown with the stem plot in Fig. 1.5.

Another example of a digital signal is the music recorded on an audio compact disc (CD). On a CD, the music signal is first sampled at a rate of 44100 samples per second. The sampling interval $T$ is given by $1 / 44100$, or $22.68$ microseconds ( $\mu \mathrm{s})$. Each sample is then quantized with a 16-bit uniform quantizer. In other words, a sample of the recorded music signal is approximated from a set of uniformly distributed values that can be represented by a 16-bit binary number. The total number of values in the discretized set is therefore limited to $2^{16}$ entries.

Digital signals may also occur naturally. For example, the price of a commodity is a multiple of the lowest denomination of a currency. The grades of students on a course are also discrete, e.g. 8 out of 10 , or $3.6$ out of 4 on a 4-point grade point average (GPA). The number of employees in an organization is a non-negative integer and is also digital by nature.

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

$$x[k T], \quad k=0, \pm 1, \pm 2, \pm 3, \ldots,$$

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

[-0.875,-0.625,-0.375,-0.125,0.125,0.375,0.625,0.875] \text {, }


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