Question

Given

x(t) = 4(t + 2)u(t + 2) - 4tu(t) - 4ult - 2) - 4(t - 4)u(t - 4) + 4t - 5)ult - 5),
find and sketch x(2t - 4).

Answer 1

To compute and sketch x(2t - 4), the time transformation τ = 2t - 4 is applied to the original signal x(t), followed by adjustments to the Heaviside step functions, ultimately resulting in the transformed signal which can be graphed.

Step-by-step explanation:

The question involves finding and sketching the transformed signal x(2t - 4) based on the given signal x(t). The original signal is described as a piecewise function involving Heaviside step functions (u(t)). To find x(2t - 4), we must substitute 2t - 4 into the t variable of the original x(t) equation and simplify accordingly.

First, we identify the transformation of the time variable t into the new time variable τ = 2t - 4, and then we apply this substitution to each term in the x(t) equation. Following the substitution, we must also adjust the Heaviside step functions accordingly, as they are defined by shifts in the argument, i.e., u(τ + a) = u(2t - 4 + a).

Once we have the new function x(2t - 4), we can then proceed to sketch the signal by noting the key points where the function changes due to the Heaviside step functions.