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Signals and Systems: Static and Dynamic System
Static System
A system whose response or output is due to present input alone is known as static system. The static system is also called the memoryless system. For a static or memoryless system, the output of the system at any instant of time (t for continuous-time system or n for discrete-time system) depends only on the input applied at that instant of time (t or n), but not on the past or future values of the input.
A purely resistive electrical circuit is an example of static system. Some examples of continuous-time and discrete-time static systems are given below −
𝑐(𝑡) = 𝑟(𝑡)
𝑐(𝑡) = 2𝑟2(𝑡)
𝑐(𝑛) = 𝑟(𝑛)
𝑐(𝑛) = 5𝑟2(𝑛)
Where, c(t) or c(n) and r(t) or r(n) are the output and input of the system respectively.
Dynamic System
A system whose response or output depends upon the past or future inputs in addition to the present input is called the dynamic system. The dynamic systems are also known as memory systems. Any continuous-time dynamic system can be described by a differential equation or any discrete-time dynamic system by a difference equation.
An electric circuit containing inductors and (or) capacitors is an example of dynamic system. Also, a summer or accumulator, a delay circuit, etc. are some examples of discrete-time dynamic systems.
Some more examples of dynamic systems are given below −
𝑐(𝑡) = 𝑟(𝑡 − 3)
$$\mathrm{c(t)=\frac{\mathrm{d} ^{2}r(t)}{\mathrm{d} t^{2}}+r\left ( t \right )}$$
𝑐(𝑡) = 𝑟(𝑡) + 𝑟(𝑡 + 1)
𝑐(𝑛) = 𝑟(5𝑛)
𝑐(𝑛) = 𝑟(𝑛) + 𝑟(𝑛 + 5)
Numerical Example
Find whether the following systems are static or dynamic −
𝑦(𝑡) = 𝑥(𝑡 − 4)
𝑦(𝑛) = 𝑥(6𝑛)
$\mathrm{y(t)=\frac{\mathrm{d} ^{2}x(t)}{\mathrm{d} t^{2}}+3x\left ( t \right )}$
𝑦(𝑛) = 𝑥(𝑛 − 1) + 𝑥(𝑛)
Solution
Given system is,
𝑦(𝑡) = 𝑥(𝑡 − 4)
From the equation of the system, it is clear that the output depends upon the past values of the input. Hence, the given system is a dynamic system.
Given system is,
𝑦(𝑛) = 𝑥(6𝑛)
For this system, the output depends upon the present inputs only. Thus, it is a static system.
Given system is,
$$\mathrm{y(t)=\frac{\mathrm{d} ^{2}x(t)}{\mathrm{d} t^{2}}+3x\left ( t \right )}$$Here the system is described by a differential equation. Therefore, it is a dynamic system.
Given,
𝑦(𝑛) = 𝑥(𝑛 − 1) + 𝑥(𝑛)
The given discrete-time system is described by a difference equation. Hence, the system is a dynamic system.
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