Please provide a detailed answer Find the convolution yt xt

Please provide a detailed answer

Find the convolution y(t) = x(t) * h(t) of the following two signals

Solution

If y(t) = x(t) * h(t), then y(2t) = 2x(2t) * h(2t)

y(t) = Z x( )h(t )d = Z x(t 1)h(t (t 1))d(1) = Z x(t 1)h(1)d1

If y(t) = x(t) * h(t), then Ev{y(t)} = x(t) * Ev{h(t)} + Ev{x(t)} * h(t).

y(t) = Z x( )h(t ) d = (x h)(t) or y = x h. This is also often written as y(t) = x(t) h(t) which is potentially confusing, since the t’s have different interpretations on the left and right sides of the equation

If we make the substitution 1 = t , then = t 1, and d = d1. (f g)(t) = Z f (t 1)g(1) (d1) = Z g( )f (t 1) d1 = (g f )(t) This means that convolution is commutative

While convolution is conceptually simple, it can be practically difficult. It can be tedious to convolve your way through a complex system.