# Static System & Dynamic System

A static system is a system in which output at any instant of time depends on the input sample at the same time. In other words, the system in which output depends only on the present input at any instant of time then this system is known as the static system. A static system is a memoryless system.

A dynamic system is a system in which output at any instant of time depends on the input sample at the same time as well as at other times. In other words, the system in which output depends on the past and/or future input at any instant of time then this system is known as the dynamic system. A dynamic system possesses memory.

Consider the system. x(t) is the input and y(t) is the output of the system. Examples

1. y(t) = 2 x(t)

put t = 0, y(0) = 2 x(0)

put t = 1, y(1) = 2 x(1)

put t = −2, y(−2) = 2 x(−2)

In the above example, the output y(t) at ‘t’ instant depends on the input x(t) at the same time instant ‘t’ (present input). So this system is a static system.

2. y(t) = x2(t)

put t = 0, y(0) = x2(0)

put t = 1, y(1) = x2(1)

put t = −1, y(−1) = x2(−1)

In the above example, the output y(t) at ‘t’ instant depends on the input x(t) at the same time instant ‘t’ (present input). So this system is a static system.

3. y(t) = t x(t)

put t = 0, y(0) = 0 x(0) = 0

put t = 1, y(1) = 1 .x(0) = x(1)

put t = −1, y(−1) = −x(−1)

In the above example, the output y(t) at ‘t’ instant depends on the input x(t) at the same time instant ‘t’ (present input). So this system is a static system.

4. y(t) = log [x(t)]

put t = 0, y(0) = log [x(0)]

put t = 1, y(1) = log [x(1)]

put t = −1, y(−1) = log [x(−1)]

In the above example, the output y(t) at ‘t’ instant depends on the input x(t) at the same time instant ‘t’(present input). So this system is a static system.

5. y(t) = Cos[x(t)]

put t = 0, y(0) = Cos[x(0)]

put t = 1, y(1) = Cos[x(1)]

put t = −1, y(−1) = Cos[x(−1)]

In the above example, the output y(t) at ‘t’ instant depends on the input x(t) at the same time instant ‘t’ (present input). So this system is a static system.

6. y(t) = ax(t)

put t = 0, y(0) = ax(0)

put t = 1, y(1) = ax(1)

put t = −1, y(−1) = ax(−1)

In the above example, the output y(t) at ‘t’ instant depends on the input x(t) at the same time instant ‘t’ (present input). So this system is a static system.

7. y(t) = 2 x(t) − 3 x(t−1)

put t = 0, y(0) = 2 x(0) − 3 x(−1)

put t = −2, y(−2) = 2 x(−2) − 3 x(−3)

In the above example, the output y(t) depends on present input as well as past input. So this system is a dynamic system.

8. y(t) = − 3 x(−t)

put t = 0, y(1) = − 3 x(0)

put t = 1, y(1) = − 3 x(−1)

put t = −2, y(−2) = − 3 x(2)

In the above example, the output y(t) depends on present input, past input and future input. So this system is a dynamic system.

9. y(t) = x(2t)

put t = 0, y(0) = x(0)

put t = 1, y(1) = x(2)

put t = −1, y(−1) = x(−2)

In the above example, the output y(t) depends on present input, past input and future input. So this system is a dynamic system.

10. y(t) = x(t2)

put t = 0, y(t) = x(0)

put t = 1, y(t) = x(1)

put t = −2, y(t) = x((−2)2) = x(4)

In the above example, the output y(t) depends on present input, past input and future input. So this system is a dynamic system.

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