Question

The input and response of a LTI system are

x(t) = δ(t) - 2e⁻³ᵗu(t)
y(t) = e⁻²ᵗ u(t).
Find
A. the transfer function
B. the impulse responsea
C. the LCCDE description of the system.

Answer 1

To find the transfer function, impulse response, and the LCCDE description of a LTI system, use the input-output relationship. Take the Laplace transform of the output and divide it by the Laplace transform of the input to find the transfer function. The impulse response is the inverse Laplace transform of the transfer function. The LCCDE description can be obtained by equating the derivative of the output with the Laplace transform of the input multiplied by the transfer function.

Step-by-step explanation:

To find the transfer function, impulse response, and the LCCDE description of the LTI system, we need to use the input-output relationship. The input is x(t) = δ(t) - 2e-3tu(t), and the output is y(t) = e-2tu(t). The transfer function, H(s), can be found by taking the Laplace transform of the output and dividing it by the Laplace transform of the input. The impulse response, h(t), is the inverse Laplace transform of the transfer function. The LCCDE (Linear Constant Coefficient Differential Equation) description of the system can be obtained by equating the derivative of y(t) with the Laplace transform of the input multiplied by the transfer function.