lINEAR TIME INVARIEANT SYSTEM , IMPULSE RESPONSE

lINEAR TIME INVARIEANT SYSTEM , IMPULSE RESPONSE

1. Unit –III Linear Time- Invariant (LTI) system

2. Linear Time Invariant System(LTI) A system with linear and time invariant called LTI system. Obeys principle of superposition and homogeneity. Input and output characteristics do not change with time. y (t, T) = y(t- T) If input is delayed by T units then output also delayed by T units.

3. LTI System properties Commutative property Distributive property Associative property System with and without Memory Invertibility Causality Stability Unit step response

4. Commutative property x(t) ∗ h(t) = h(t) ∗ x(t)

5. Distributive property x(t) ∗ [h1(t) + h2(t)] = x(t) ∗ h1(t) + x(t) ∗ h2(t)

6. Associative property x(t) ∗ [h1(t) * h2(t)] = [x(t) ∗ h1(t)] ∗ h2(t)

7. System with and without memory Without memory system: Also called as static system. It depends on present inputs. With memory system: Also called as dynamic system. It depends upon past and future inputs.

8. Invertibility A system is invertible if the input of the system can be recovered from the output of the system. h(t) * h1(t) = δ(t)

9. ∞ Stability: In a stable system, a bounded input results in a bounded output. ∫ −∞ |ℎ 𝞃 |𝑑𝞃 < ∞ Causality: A causal system depends only on the present and past values of the input to the system. Unit Step Response: It can be obtained by using convoluting unit step input u(t) with impulse response h(t) s(t)=h(t)*u(t)

10. Transfer Function of an LTI system

11. Transfer function defined by Fourier or Laplace transform. Ratio of Laplace transform of o/p signal to Laplace transform of i/p signal when initial conditions are zero. 𝐻 𝑠 = 𝑦 𝑠 𝑥 𝑠 Or 𝐻 𝑠 = 𝐿[ℎ(𝑡)] Impulse response is nothing but inverse Laplace transform of transfer function. h 𝑡 = 𝐿 −1 [𝐻(𝑠)]

12. Ratio of Fourier transform of o/p signal to Fourier transform of i/p signal when initial conditions are zero. 𝐻 ω = 𝑦 ω 𝑥 ω Or 𝐻 ω = 𝐹[ℎ(𝑡)] Impulse response is nothing but inverse Laplace transform of transfer function. h 𝑡 = 𝐹 −1 [𝐻(ω)]

13. Write the condition for LTI system to be stable and causal.

14. Given the differential equation representation of the system d 2 y(t)/ dt 2 +2dy(t )/ dt-3y(t )=2x(t ). Examine the frequency response.

15. Identify the differential equation relating the input and output a CT system represented by 𝐻(𝑗𝛺) =1 / [ ( 𝑗𝞨) 2 +8(𝑗𝞨)+ 1]

16. Given the input x(t) =u(t) and h(t) = δ(t-1). Find the response y(t).

17. List the properties for convolution integral.

18. The input - output relationship of the system is described as, d 2 y/dt 2 +3dy/ dt +2y=dx/ dt. Find the system function H(s) of the system.

19. Summarize impulse response of an LTI system.

20. Given H(s) =1/[ 𝑠 2 +2𝑠+ 1]. Express the differential equation representation of the system.

21. Examine the Convolution of following signals. x(t )= u(t) and h(t)= u(t) u(t-2)

22. A stable LTI system is characterized by the differential equation d 2 y(t)/dt 2 + 4dy(t)/ dt + 3y(t) = dx(t)/ dt + 2x(t). Derive its frequency response & impulse response using Fourier transform.

23. Identify the impulse response h(t) of the system given by the differential equation d 2 y(t)/dt 2 + 3dy(t)/ dt + 2y(t) = x(t) with all initial conditions to be zero.

24. Derive the output expression of the system described by the differential equation d 2 y(t)/dt 2 + 6dy(t)/ dt +8y(t) = dx(t)/ dt + x(t), when the input signal is x(t) =u(t) and the initial conditions are y(0 + )=1 , dy (0 + )/ dt =1

25. A system is described by the differential equation d 2 y(t)/dt 2 +6dy(t)/ dt + 8y(t) = dx(t)/ dt + x(t). Evaluate the transfer function and the output signal y(t) for x(t) = δ(t).