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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) = 2𝑟^{2}(𝑡)

𝑐(𝑛) = 𝑟(𝑛)

𝑐(𝑛) = 5𝑟^{2}(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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