Final answer:
To evaluate the convolution using the graphical method, we need to graphically represent the functions and determine the resulting convolution by sliding one function over the other and calculating the areas of overlap. The convolution will be a piecewise function defined over different intervals.
Step-by-step explanation:
Convolution is a mathematical operation that combines two functions to produce a third function. To evaluate the convolution y(t) = h(t) * x(t) using the graphical method, we need to graphically represent the functions h(t) and x(t) and then determine the resulting convolution y(t) by sliding one of the functions over the other and calculating the areas of overlap at each point.
(a) For h(t) = u(t) and x(t) = r(at), the convolution y(t) can be found by graphically sliding x(t) over h(t) and calculating the areas of overlap. The resulting convolution will be a piecewise function defined over different intervals of t.
(b) For h(t) = r(at) and x(t) = r(t), the convolution y(t) can be found by graphically sliding x(t) over h(t) and calculating the areas of overlap. The resulting convolution will be a piecewise function defined over different intervals of t.
(c) For h(t) = rect(at) and x(t) = rect(t), the convolution y(t) can be found by graphically sliding x(t) over h(t) and calculating the areas of overlap. The resulting convolution will be a piecewise function defined over different intervals of t.
(d) For h(t) = u(-t) and x(t) = Λ(at), the convolution y(t) can be found by graphically sliding x(t) over h(t) and calculating the areas of overlap. The resulting convolution will be a piecewise function defined over different intervals of t.
(e) For h(t) = rect(t) and x(t) = e^(-2|t|), the convolution y(t) can be found by graphically sliding x(t) over h(t) and calculating the areas of overlap. The resulting convolution will be a piecewise function defined over different intervals of t.