• Problems
1. A continuous-time signal x(t) is shown in Figure 1. Sketch and label carefully each of the follow-ing signals:
Figure 1: x(t)
(a) x(t − 1)
(b) x(4 − t)
(c) [x(t − 2) − 2x(−t + 1)] u(t − 12 )
2. (a) Using complex exponentials, prove that:
a(t) = cos(θt) sin(ψt) = 12(sin((θ + ψ)t) − sin((θ − ψ)t))
(b) Can a(t) be periodic? If so, use θ = 2π to find the value of ψ where a(t) has a period of 3.
(c) Determine the fundamental period of the signal x(t) = 2 cos(10t + 1) − sin(4t − 1)
3. Consider the periodic signal x(t) = cos (3Ωot) + 5 cos (Ωot) , −∞ < t < ∞, and Ωo = π. The frequencies of the two sinusoids are harmonically related (that is, one is a multiple of the other).
(a) Determine the period To of x(t).
(b) Compute the power Px of x(t).
(c) Verify that the power Px is the sum of the powers P1 of x1(t) = cos (3Ωot) and P2 of x2(t) = 5 cos (Ωot), for Ωo = π.
1
(d) In the above case, we see that there is superposition of the powers because the frequencies are harmonically related. Suppose that γ(t) = cos(t) + cos(π2 t) where the frequencies are not harmonically related. Find out whether γ(t) is periodic or not. Indicate how you would find the power Pγ of γ(t). Would Pγ = P1 + P2 where P1 is the power of cos(t) and P2 is the power of cos(π2 t)? Explain what is the difference with respect to the case of harmonic frequencies.
4. (a) Determine and sketch the even and odd parts of the signal depicted in the figure below. Label your sketches carefully.
(b) Show that the energy of a general continuous time signal x(t) can be expressed as the sum of the energies of its even and odd components. That is,
∞
∞
|xe(t)|2dt +
∞
−∞
|x(t)|2dt =
|xo(t)|2dt
−∞
−∞
2
