In the generation of dc from ac voltage, the ”half-wave” rectified signal is an important part. Suppose the ac voltage is x(t) = sin(2π t) u(t) and y(t) is the half-wave rectified signal.

(a) Let y₁(t) be the period of y(t) between 0 ≤ t ≤ 1. Show that y₁(t) can be written equivalently as either

y₁(t) = sin(2π t) u(t) + sin(2π (t − 0.5)) u(t − 0.5) or

y₁(t) = sin(2π t) [u(t) − u(t − 0.5)]

Use MATLAB to verify this. Find the Laplace transform X₁(s)of x₁(t).

(b) Express y(t) in terms of x₁(t) and find the Laplace transform Y(s) of y(t).