Discrete Fourier transform - Continuous and Discrete Time Signals and Systems - page 532

p

5

3

0.5 p

4

2

3

0

1

2

0

− p

1

−1

k

r

r

−0.5 p

0

0 1 2 3

0 1 2 3

0 1 2 3

(a) (b) (c)

Example 12.1
Calculate the four-point DFT of the aperiodic sequence x [ k ] of length N

Fig. 12.2. (a) DT sequence x [ k ];
(b) magnitude spectrum and
(c) phase spectrum of its DTFT
X [ r ] computed in Example 12.1.

= 4,

which is defined as follows:

2 k = 0

= 1

3 k

x [ k ] =

− 1 = 2

k

1 k = 3 .

Solution
Using Eq. (12.15), the four-point DFT of x [ k ]isgivenby

3

− j(2 π kr / 4)

X [ r ] =

x [ k ]e

k = 0

− j(2 π r / 4) − 1 e − j(2 π (2) r / 4) + 1 e − j(2 π (3) r / 4) ,

= 2 + 3 e

for 0 ≤ r ≤ 3. Substituting different values of r , we obtain

r = 0 X [0] = 2 + 3 − 1 + 1 = 5;

X [1] = 2 + 3e − j(2 π/ 4) − e − j(2 π (2) / 4) + e − j(2 π (3) / 4)

r = 1

= 2 + 3( − j) − 1( − 1) + 1(j) = 3 − 2j;

X [2] = 2 + 3e − j(2 π (2) / 4) − e − j(2 π (2)(2) / 4) + e − j(2 π (3)(2) / 4) r = 2

= 2 + 3( − 1) − 1(1) + 1( − 1) =− 3;

X [3] = 2 + 3e − j(2 π (3) / 4) − e − j(2 π (2)(3) / 4) + e − j(2 π (3)(3) / 4) r = 3

= 2 + 3( j) − 1( − 1) + 1( − j) = 3 + j2 .

The magnitude and phase spectra of the DFT are plotted in Figs. 12.2(b) and
(c), respectively.