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# 电气工程代写|模拟和数字通信代写Analogue and Digital Communications代考|ECE461 Analysis of Signals

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## 电气工程代写|模拟和数字通信代写Analogue and Digital Communications代考|Signal Transformations

When we study engineering communication, a great deal of manipulation of signals have to be performed. With this in mind, we need to understand the transformation of signals, both of the dependent variable and independent variable. We take a look at these two types of transformations.

If we consider a real signal $y(t)$, which is a function of time, then $y$ is the dependant variable and $t$ is the independent variable. Generally, the independent variable is on the $\mathrm{x}$-axis while the dependent variable is on the $\mathrm{y}$-axis. The independent variable values may be chosen in an independent manner, while the ones for the dependent variable are obtained from the independent variable according to a rule. (i.e. the function $y(t))$.
We will be considering two types of transformations:

1. Transformations of the dependent variable.
2. Transformations of the independent variable.

## 电气工程代写|模拟和数字通信代写Analogue and Digital Communications代考|Transformation of the Dependent Variable

The simple transformations of the dependent variables are

1. Multiplication of the signal $f(t)$ by a constant: $a f(t)$, where $a$ may be a complex number.
2. Addition or subtraction of a constant $f(t) \pm a$ where $a$ in general may be complex.
These types of transformations are shown in Fig. 2.3. In the (a) part of the figure, the original signal, $f(t)$, is shown, in the (b) part, a real positive constant $a$ is added to $f(t)$ and in the (c) part, the signal $a f(t)$ is shown where $a$ is a real number greater than one.

Similarly, two complex signals, $f_1(t)$ and $f_2(t)$, may added, subtracted or multiplied:

1. $f(t)=f_1(t) \pm f_2(t)$ and
2. $f(t)=f_1(t) f_2(t)$

A simple case where two signals are added is shown in Fig. 1.5(b), where noise is added to a signal when it passes through a channel. Similarly, the multiplication of two signals is shown in Fig. 1.3, where amplitude modulation is shown, and the two signals $f_1(t)=\left(1+0.1 t^2\right)$ and $f_2(t)=\cos (3 t)$ are multiplied.

We may also consider more general cases where complicated operations may be done on the dependant variable, $y=f(t)$, such as squaring of a given real signal
$$g(t)=[f(t)]^2$$
or taking the logarithm of a given complex signal
$$g(t)=\ln [|f(t)|]$$

## 电气工程代写模拟和数字通信代写Analogue and Digital Communications代 考|Transformation of the Dependent Variable

$f(t)=f_1(t) \pm f_2(t)$ 和

$f(t)=f_1(t) f_2(t)$

$$g(t)=[f(t)]^2$$

$$g(t)=\ln [|f(t)|]$$

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