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[REFERENCES](#page-8-0)

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REFERENCES

    1. Papoulis, A., The Fourier Integral and Its Applications. McGraw-Hill, New York, 1962.
    1. Mason, S. J., Electronic Circuits, Signals, and Systems. Wiley, New York, 1960.
    1. Kailath, T., Linear Systems. Prentice-Hall, Englewood Cliffs, NJ, 1980.
    1. Lathi, B. P., Signals and Systems. Berkeley-Cambridge Press, Carmichael, CA, 1987.

PROBLEMS

  • 1.1-1 Find the energies of the signals illustrated in Fig. P1.1-1. Comment on the effect on energy of sign change, time shifting, or doubling of the signal. What is the effect on the energy if the signal is multiplied by k?

  • 1.1-2 Repeat Prob. 1.1-1 for the signals in Fig. P1.1-2.

  • 1.1-3 (a) Find the energies of the pair of signals x(t) and y(t) depicted in Figs. P1.1-3a and P1.1-3b. Sketch and find the energies of signals x(t) + y(t) and x(t) − y(t). Can you make any observation from these results?

    • (b) Repeat part (a) for the signal pair illustrated in Fig. P1.1-3c. Is your observation in part (a) still valid?

  • 1.1-4 Find the power of the periodic signal x(t) shown in Fig. P1.1-4. Find also the powers and the rms values of:

    • (a) −x(t)
    • (b) 2x(t)
    • (c) cx(t)
    • Comment.

  • 1.1-5 By original design, a system outputs a 10-volt pulse that is 3 seconds in duration. It is desired to upgrade the square-pulse output with a “soft-start” pulse that steps up to 10 volts in 1-volt increments spaced every 20 milliseconds. Determine the signal duration T so that the “soft-start” pulse has the same signal energy as the original square pulse.

Figure P1.1-2

-8

t 3

  • 1.1-6 Determine the power and the rms value for each of the following signals:
    • (a) 5+10 cos(100t +π/3)
    • (b) 10 cos(100t +π/3)+16 sin(150t +π/5)
    • (c) (10+2 sin 3t) cos 10t
    • (d) 10 cos 5t cos 10t
    • (e) 10 sin 5t cos 10t
    • (f) ejα*t* cosω0t
  • 1.1-7 Figure P1.1-7 shows a periodic 50% duty cycle dc-offset sawtooth wave x(t) with peak amplitude A. Determine the energy and power of x(t).
  • 1.1-8 Two periodic signals that differ only by a 90-degree phase shift are considered to be quadrature signals. For example, cos(2πt) and sin(2πt) are quadrature signals. Another pair of quadrature signals is x(t) = sgn[cos(2πt)] and

y(t) = sgn[sin(2πt)], where sgn is the sign (or signum) function.

  • (a) Plot x(t) and determine its power Px and energy Ex.
  • (b) Plot y(t) and determine its power Py and energy Ey.
  • (c) Consider the complex function f(t) = x(t)+ jy(t). Determine the power and energy of f(t).
  • (d) When real functions x(t) and y(t) are combined as f(t) = x(t) + jy(t), is it generally true that Ef = Ex + Ey and Pf = Px + Py? Prove your answer.
  • 1.1-9 There are many useful properties related to signal energy. Prove each of the following statements. In each case, let energy signal x1(t) have energy E[x1(t)], let energy signal x2(t) have

Figure P1.1-7

energy E[x2(t)], and let T be a nonzero, finite, real-valued constant.

  • (a) Prove E[Tx1(t)] = T2E[x1(t)]. That is, amplitude scaling a signal by constant T scales the signal energy by T2.