Page 299 - Schaum's Outline of Theory and Problems of Signals and Systems

286 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS [CHAP. 5

5.73. Let x(t) be a real signal with the Fourier transform X(w). The analytical signal x+(t)
associated with x(t) is a complex signal defined by
x+(t) =x(t) +ji(t)
where is the Hilbert transform of x(t 1.
(a) Find the Fourier transform X+(w) of x+(t 1.
(6) Find the analytical signal x+(t) associated with cos w,t and its Fourier transform X+(w).

5.74. Consider a continuous-time LTI system with frequency response H(w). Find the Fourier
transform S(w) of the unit step response s(t) of the system.

Hint: Use Eq. (2.12) and the integration property (5.57).
Am. S(w) = .rrH(O)G(o) + (l/jw)H(w)

5.75. Consider the RC filter shown in Fig. 5-39. Find the frequency response H(w) of this filter and
discuss the type of filter.
iw
Ans. H(o)= , high-pass filter
(l/RC) + jw

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Fig. 5-39

5.76. Determine the 99 percent energy containment bandwidth for the signal

Ans. W,, = 2.3/a radians/second or f, = 0.366/a hertz

5.77. The sampling theorem in the frequency domain states that if a real signal x(t) is a duration-
limited signal. that is,