Problems 113(b)Show that the energy of the signalx(t)can be expressed as the sum of the energies of its even and
odd components—that is, that∫∞−∞x^2 (t)dt=∫∞−∞x^2 e(t)dt+∫∞−∞x^2 o(t)dt(c)Verify that the energy ofx(t)is equal to the sum of the energies ofxe(t)andxo(t).1.15. Generation of periodic signals
A periodic signal can be generated by repeating a period.
(a) Find the functiong(t), defined in 0 ≤t≤ 2 only, in terms of basic signals and such, that when repeated
using a period of 2 , generates the periodic signalx(t), as shown in Figure 1.19.
(b)Obtain an expression forx(t)in terms ofg(t)and shifted versions of it.
(c)Suppose we shift and multiply by a constant the periodic signalx(t)to get new signalsy(t)= 2 x(t− 2 ),
z(t)=x(t+ 2 ), andv(t)= 3 x(t). Are these signals periodic?
(d)Let thenw(t)=dx(t)/dt, and plot it. Isw(t)periodic? If so, determine its period.
FIGURE 1.19
Problem 1.15.10
t
12x(t)− 1− 1· · · · · ·1.16. Contraction and expansion and periodicity—MATLAB
Consider the periodic signalx(t)=cos(πt)of periodT 0 = 2 sec.
(a) Is the expanded signalx(t/ 2 )periodic? If so, indicate its period.
(b)Is the compressed signalx( 2 t)periodic? If so, indicate its period.
(c)Use MATLAB to plot the above two signals and verify your analytic results.
1.17. Derivatives and integrals of periodic signals
Consider the triangular train of pulsesx(t)in Figure 1.20.
FIGURE 1.20
Problem 1.17.10
t
1 2· · · · · ·x(t)− 1