Page 230 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

The Fourier transform of u(r) is given by (Prob. 5.30)
1
F{u(t)} = ns(o) + -
10
Thus, the Fourier transform of u(t) cannot be obtained from its Laplace transform. Note that the unit
step function u(t) is not absolutely integrable.

5.4 PROPERTIES OF THE CONTINUOUS-TIME FOURIER TRANSFORM
Basic properties of the Fourier transform are presented in the following. Many of these
properties are similar to those of the Laplace transform (see Sec. 3.4).

B. Time Shifting:

Equation (5.50) shows that the effect of a shift in the time domain is simply to add a linear
term -ot, to the original phase spectrum 8(w). This is known as a linear phase shift of the
Fourier transform X( w).

C. Frequency Shifting:

The multiplication of x(t) by a complex exponential signal eJ"l)' is sometimes called
complex modulation. Thus, Eq. (5.51) shows that complex modulation in the time domain
corresponds to a shift of X(w) in the frequency domain. Note that the frequency-shifting
property Eq. (5.51) is the dual of the time-shifting property Eq. (5.50).

D. Time Scaling:

where a is a real constant. This property follows directly from the definition of the Fourier
transform. Equation (5.52) indicates that scaling the time variable t by the factor a causes
an inverse scaling of the frequency variable o by l/a, as well as an amplitude scaling of
X(o/a) by l/la(. Thus, the scaling property (5.52) implies that time compression of a
signal (a > 1) results in its spectral expansion and that time expansion of the signal (a < 1)
results in its spectral compression.