# Digital Signal Processing - Dynamic Systems

If a system depends upon the past and future value of the signal at any instant of the time then it is known as dynamic system. Unlike static systems, these are not memory less systems. They store past and future values. Therefore, they require some memory. Let us understand this theory better through some examples.

## Examples

Find out whether the following systems are dynamic.

a) $y(t) = x(t+1)$

In this case if we put t = 1 in the equation, it will be converted to x(2), which is a future dependent value. Because here we are giving input as 1 but it is showing value for x(2). As it is a future dependent signal, so clearly it is a dynamic system.

b) $y(t) = Real[x(t)]$

$$= \frac{[x(t)+x(t)^*]}{2}$$

In this case, whatever the value we will put it will show that time real value signal. It has no dependency on future or past values. Therefore, it is not a dynamic system rather it is a static system.

c) $y(t) = Even[x(t)]$

$$= \frac{[x(t)+x(-t)]}{2}$$

Here, if we will substitute t = 1, one signal shows x(1) and another will show x(-1) which is a past value. Similarly, if we will put t = -1 then one signal will show x(-1) and another will show x(1) which is a future value. Therefore, clearly it is a case of Dynamic system.

d) $y(t) = \cos [x(t)]$

In this case, as the system is cosine function it has a certain domain of values which lies between -1 to +1. Therefore, whatever values we will put we will get the result within specified limit. Therefore, it is a static system

From the above examples, we can draw the following conclusions −

• All time shifting cases signals are dynamic signals.
• In case of time scaling too, all signals are dynamic signals.
• Integration cases signals are dynamic signals.