Signals in Natural Domain
Chapter 3 :  Discrete-Time Systems
 
Linearity :
This is an important property of the system. We will see later that if we have system which is linear and time invariant then it has a very compact representation. A linear system possesses the important property of superposition: if an input consists of weighted sum of several signals, the nthe output is also weighted sum of the responses of the system to each of those input signals. Mathematically let $ \{y_1[n]\}$ be the response of the system to the input $ \{x_1[n]\}$ and let $ \{y_2[n]\}$ be the response of the system to the input. Then the system is linear if:
Additivity: The response to $ \{x_[n]\}+\{x_2[n]\}$ is $ \{y_1[n]\}+\{y_2[n]\}$
Homogeneity: The response to $ a\{x_1[n]\}$ is $ a\{y_1[n]\}$, where $ a$ is any real number if we are considering only real signals and $ a$ is any complex number if we are considering complex valued signals.
Continuity: Let us consider $ \{x_1[n]\},\{x_2[n]\},...\{x_k[n]\}...$ be countably infinite number of signals such that

$\displaystyle \lim\limits_{k\rightarrow\infty}\{x_k[n]\}=\{x[n]\}$

Let the corresponding output signals be denoted by $ \{y_n[n]\}$ and $ \lim\limits_{k \rightarrow \infty}\{y_n[n]\}=\{y[n]\}$
We say that system posseses the continuity property if the response of the system to the limiting input $ \{x[n]\}$ is limit of the responses.
$\displaystyle T(\lim\limits_{k\rightarrow \infty}\{x_k[n]\})=\lim\limits_{k\rightarrow \infty}T(\{X_k[n]\})$
The additivity and continuity properties can be replaced by requiring that system is additive for countably infinite number of signals i.e. response to
$ \{x_1[n]\}+\{x_2[n]\}+...+\{x_n[n]\}+...\qquad$        is      $ \{y_1[n]\}+\{y_2[n]\}+...+\{y_k[n]\}+....$
Most of the books do not mention the continuity property. They state only finite additivity and homogeneity. But from finite additivity we can not deduce countable additivity. This distinction becomes very important in continuous time systems.
A system can be linear without being time invariant and it can be time invariant without being linear. If a system is linear, an all zero input sequence will produce a all zero output sequence. Let $ \{0\}$ denote the all zero sequence, then. If $ T(\{x[n]\}=\{y[n]\})$ then by homogeneity property
$ T(0.\{x[n]\})=0.\{y[n]\}$
or,   $\displaystyle T(\{0\})=\{0\}$
Consider the system defined by     $\displaystyle y[n]=2x[n]+3$

This system is not linear. This can be verified in several ways. If the input is all zero sequence $ \{0\}$, the output is not an all zero sequence. Although the defining equation is a linear equation is x and y the system is nonlinear. The output of this system can be represented as sum of a linear system and another signal equal to the zero input response. In this case the linear system is
$\displaystyle y[n]=2x[n]$
and the zero-input response is                $\displaystyle y_0[n]=3$    for all n
Such systems correspond to the class of incrementally linear system. System is linear in term of differnce signal i.e if we define $ \{x_d[n]\}=\{x_1[n]\}-\{X_2[n]\}$ and. Then in terms of $ \{x_d[n]\}$ and $ \{y_d[n]\}$ the system is linear.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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