Problems Based on Static (or) Dynamic System

Problems Based on Static (or) Dynamic System

(I) Determine whether the following CT systems are static (or) dynamic.

1. y(t) = cos[x(t)]

Solution:

Output y(t) depends upon present input x(t). Hence the system is memory less (or) Static system.

2. y(t) = 20 x(t) + 15

Solution:

Output y(t) depends upon present input x(t). Hence the given system is memory less (or) static system.

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

Solution:-

Output y(t) depends upon present input x(t). Hence the given system is memory less (or) static system.

4. y(t) = x(t+5)+ x²(t)

Solution:

Output y(t) depends upon the future input x(t + 5). Hence the given system is dynamic system.

5. y(t) = x(t⁴)

Solution:

If t = 2, y(2) = x(16)

Output y(t) depends upon future input x(t4). Hence the given system is dynamic system.

6. 

Solution:

This system requires memory to store the integration of all earlier terms. Hence the given system is dynamic system.

7. 

Solution:

Differentiation makes system dynamic. Hence the given system is dynamic system.

8. y(t) = dx(t)/dt

Solution:

Differentiation makes system dynamic. Hence the given system is dynamic system.

9. y₁(t) = x(t - 5) + x(5 - t)

Solution:

Output y(t) depends upon past and future input. Hence the given system is dynamic system.

10. y(t) = e^t x(t)

Solution:

Output y(t) depends upon present input x(t). Hence the given system is memoryless (or) static system

11. y(t) = x(t) cos 100πt

Solution:

Output y(t) depends upon present input x(t). Hence the given system is memoryless (or) static system.

12. y(t) = x(t/2)

Solution:

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

Output y(t) depends upon past input. Hence the given system is dynamic

13. y(t) = x(t+10) + x²(t)

Solution:

Output y(t) depends upon the future input x(t+10). Hence the given system requires memory to store the future input. This is dynamic system.

14. y(t) = x(-t)

Solution:

when t = 2, y(2) = x(−2)

Output y(t) depends upon past input. Hence the given system is dynamic system.

15. 

Solution:

Given system requires memory to store all earlier terms. Hence this is dynamic system.

(II) Determine whether the following systems are static (or) dynamic.

1. y(n) = 5x(n) + 8x²(n)

Solution:

Output depends upon present input only. Hence the given system is static system.

2. y(n) = cos[4πx (n+1)]

Solution:

Output depends upon future input x(n+1). Hence the given system is dynamic system.

3. y(n) = log₁₀(|x(n)|)

Solution:

Output depends upon present input x(n). Hence the given system is static system.

4. 

Solution:

Output y(n) depends upon the input ranges from x(-∞) to x(n+1). Hence the given system is dynamic system.

5. y(n) = x(4n)

Solution:

Output y(n) depends upon future input. Hence the given system is dynamic system.

6. y(n) = n x(n)

Solution:

Output y(n) depends upon present input x(n). Hence the given system is static system.

7. y(n) = x(n) cos (ω₀n)

Solution:

Output y(n) depends upon present input x(n). Hence the given system is static system.

8. y(n) = x(n³)

Solution:

Output y(n) depends upon future input. Hence the given system is dynamic system.

9. y(n) = x(n) + 5u (n+5)

Solution:

Output y(n) depends upon present input x(n). Hence the given system is static system.

10. y(n) = sgn [x(n)]

Solution:

Output y(n) depends upon present input x(n). Hence the given system is static system.

11. y(n) = 2x(2ⁿ)

Solution:

For n = 2, y(2) = 2x(4)

System requires memory to store the future input. Hence the given system is dynamic system.

12. y(n) = sin x(n)

Solution:

Output y(n) depends upon present input x(n). Hence the given system is static system.

13. y(n) = ax(n)

Solution:

Output y(n) of the given system depends upon present input x(n). Hence the given system is static system.

14. y(n) = x(2n)

Solution:

Example: when n = 3, y(3) = x(6).

Output y(n) depends upon future input. Hence the given system is dynamic system.

15. y(n) = x(n) + nx(n+1)

Solution:

Output y(n) depends upon future input x(n+1). Hence the given system is dynamic system.

16. y(n) = n x(n) + bx²(n)

Solution:

Output y(n) depends upon present input x(n) alone. Hence the given system is static system.

17. y(n)=|x(n)|

Solution:

Output y(n) depends upon present input x(n). Hence the given system is static system.

18. y(n) = x(n) + x²(n)

Solution:

Output y(n) depends upon present input. Hence the given system is static system.

19. y(n) = x(n) u(n)

Solution:

Output y(n) depends upon present input x(n). Hence the given system is static system.