One of the advantages of defining the (t) functions is that we are now able to find

One of the advantages of defining the δ(t) functions is that we are now able to find the derivative of discontinuous signals. Consider a periodic sinusoid defined for all times

x(t) = cos(Ω₀ t)      −∞

and a causal sinusoid defined as x₁ (t) = cos(Ω₀ t) u(t), where the unit-step function indicates that the function has a discontinuity at zero, since for t = 0+ the function is close to 1 and for t = 0− the function is zero.

(a) Find the derivative y(t) = dx(t)/dt and plot it.

(b) Find the derivative z(t) = dx₁ (t)/dt(treat x₁ (t) as the product of two functions cos(Ω₀ t)and u(t)) and plot it. Express z(t) in terms of y(t).

(c) Verify that the integral ʃ^t_−∞ z(τ )dτ gives back x₁(t).