Question

Find the Laplace transforms of the following time-domain signals. The unit-step function is given by u(t).

1. u(t)−2u(t−1)+u(t−2)
2. e⁻ᵗtu(t−1)
3. cos(ωt+b) where b∈ℝ

Answer 1

To find the Laplace transforms, we apply properties and standard pairs of the Laplace transformation to the given time-domain signals that involve the unit-step function.

Step-by-step explanation:

The student has asked to find the Laplace transforms of three given time-domain signals using the unit-step function u(t). The three signals are: 1. u(t) - 2u(t - 1) + u(t - 2), 2. e−tu(t - 1), and 3. cos(ωt + b), where b ∈ ℝ.

To approach this, we use the properties of the Laplace transform for the unit step function and use them to determine the transforms of the given functions. For example, the Laplace transform of u(t - a) is e−as/s. Using standard Laplace transform pairs and properties, such as the delay theorem and the transform of exponential functions, the transforms for these specific functions can be determined.