Question

For the following s-domain representation of a causal signal X(s) = (s² - 3)/((s + 1)(s + 2)) find the corresponding time-domain signal x(t)

Answer 1

The time-domain signal x(t) corresponding to the given s-domain representation X(s) = (s² - 3)/((s + 1)(s + 2)) is x(t) =
`(1 - e^(-t) - 2e^(-2t))u(t)`, where u(t) is the unit step function.

Step-by-step explanation:

To find the time-domain signal x(t) from the given s-domain representation X(s), we can use inverse Laplace transform. The given X(s) can be expressed as X(s) = (s² - 3)/((s + 1)(s + 2)). We factorize the denominator to obtain partial fraction decomposition:

`\[X(s) = (A)/(s + 1) + (B)/(s + 2)\]`

Multiplying both sides by the common denominator (s + 1)(s + 2) and equating coefficients, we can solve for A and B. The resulting expression for X(s) will be in a form suitable for inverse Laplace transform.

`\[X(s) = (1)/(s + 1) - (2)/(s + 2)\]`

Now, applying the inverse Laplace transform to each term, we get:

`\[x(t) = e^(-t) - 2e^(-2t)\]`

However, this solution assumes a causality condition, so we include the unit step function u(t) to account for the signal being zero for t < 0. Therefore, the final time-domain signal is:

`\[x(t) = (1 - e^(-t) - 2e^(-2t))u(t)\]`

This expression represents the time-domain signal corresponding to the given s-domain representation.