Page 329 - Schaum's Outline of Theory and Problems of Signals and Systems

316 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6

FOURIER TRANSFORM

6.11. Find the Fourier transform of
x[n] = -anu[-n - 11 a real

From Eq. (4.12) the z-transform of x[n] is given by
1
X(z) = Izl< lal
I -az-'
Thus, X(eJf') exists for la1 > 1 because the ROC of X( z) then contains the unit circle. Thus,

6.12. Find the Fourier transform of the rectangular pulse sequence (Fig. 6-10)

x[n] = u[n] - u[n - N]
Using Eq. (1.90), the z-transform of x[nl is given by
N- 1 1 -ZN
X(Z) = C zn = - 1-4 > 0 (6.131)
n=O 1-2
Thus, ~(e'") exists because the ROC of X(z) includes the unit circle. Hence,

10. . . . .

...

- - - F

0 1 2 3 N- 1 n
Fig. 6-10

6.13. Verify the time-shifting property (6.431, that is,

By definition (6.27)
m
F(x[n - n,]) = C x[n - no] e-j""
n= -m