B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

82 ANALYSIS AND TRANSMISSION OF SIGNALS
Figure 3.18
Physical
explanation of
the time-shifting
property.
higher phase shifts to achieve the same time delay. This effect is shown in Fig. 3. 18 with two
sinusoids, the frequency of the lower sinusoid being twice that of the upper. The same time
delay to amounts to a phase shift of n /2 in the upper sinusoid and a phase shift of n in the
lower sinusoid. This verifies that to achieve the same time delay, higher frequency sinusoids
must undergo proportionately higher phase shifts.
Example 3. 10 Find the Fourier transform of e-alt-tol _
This function, shown in Fig. 3.19a, is a time-shifted version of e-a\tl (shown in Fig. 3. 17a).
From Eqs. (3.31) and (3.32) we have
e - alt-tol {=::}
2a
e - j 2 1rfto
(3.33)
a 2 + (2nf) 2
The spectrum of e-alt-to\ (Fig. 3. 19b) is the same as that of e-altl (Fig. 3.17b), except for
an added phase shift of -2nfto.
Figure 3.19
Effect of time
shifting on the
Fourier spectrum
of a signal.
0 t---,..
(a)
·- J-,,..
/··---------.
0 g(f) = -2rc_ft0 ·------. __
(b)
Observe that the time delay to causes a linear phase spectrum -2nfto. This example
clearly demonstrates the effect of time shift.