Page 335 - Schaum's Outline of Theory and Problems of Signals and Systems

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

From Eqs. (6.49) and (6.138) we have

The time-scaled sequence xo,[n] and its Fourier transform Xo,(R) are sketched in Fig.
6-16(b).
In a similar manner we get

The time-scaled sequence x(,,[n] and its Fourier transform X,,,(R) are sketched in Fig.
6-16(~).

6.23. Verify the differentiation in frequency property (6.55), that is,

From definition (6.27)

Differentiating both sides of the above expression with respect to R and interchanging the
order of differentiation and summation, we obtain

Multiplying both sides by j, we see that

Hence,

6.24. Verify the convolution theorem (6.581, that is,

x,[nl* x2bI -X,(~)x,(n)
By definitions (2.35) and (6.27), we have
(
F{x,[n] * x,[n]) = z z x,[k]x,[n - k]) e-j""
n- -m k= -m
Changing the order of summation, we get
m / m