Page 231 - Schaum's Outline of Theory and Problems of Signals and Systems

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

E. Time Reversal:

Thus, time reversal of x(t) produces a like reversal of the frequency axis for X(o).
Equation (5.53) is readily obtained by setting a = - 1 in Eq. (5.52).

F. Duality (or Symmetry):

The duality property of the Fourier transform has significant implications. This property
allows us to obtain both of these dual Fourier transform pairs from one evaluation of
Eq. (5.31) (Probs. 5.20 and 5.22).

G. Differentiation in the Time Domain:

Equation (5.55) shows that the effect of differentiation in the time domain is the
multiplication of X(w) by jw in the frequency domain (Prob. 5.28).

H. Differentiation in the Frequency Domain:

dX(4
(-P)x(t) - ,o

Equation (5.56) is the dual property of Eq. (5.55).

I. Integration in the Time Domain:

Since integration is the inverse of differentiation, Eq. (5.57) shows that the frequency-
domain operation corresponding to time-domain integration is multiplication by l/jw, but
an additional term is needed to account for a possible dc component in the integrator
output. Hence, unless X(0) = 0, a dc component is produced by the integrator (Prob. 5.33).

J. Convolution: