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200 Chapter 5 THE DISCRETE FOURIER TRANSFORM
5.7 PROBLEMS
P5.1 Compute the DFS coefficients of the following periodic sequences using the DFS definition,
and then verify your answers using MATLAB.
1. ˜x 1(n) ={4, 1, −1, 1}, N =4
2. ˜x 2(n) ={2, 0, 0, 0, −1, 0, 0, 0}, N =8
3. ˜x 3(n) ={1, 0, −1, −1, 0}, N =5
4. ˜x 4(n) ={0, 0, 2j, 0, 2j, 0}, N =6
5. ˜x 5(n) ={3, 2, 1}, N =3
P5.2 Determine the periodic sequences given the following periodic DFS coefficients. First use
the IDFS definition and then verify your answers using MATLAB.
1. ˜X1(k) ={4, 3j, −3j}, N =3
2. ˜X2(k) ={j, 2j, 3j, 4j}, N =4
3. ˜X3(k) ={1, 2+3j, 4, 2 − 3j}, N =4
4. ˜X4(k) ={0, 0, 2, 0, 0}, N =5
5. ˜X5(k) ={3, 0, 0, 0, −3, 0, 0, 0}, N =8
P5.3 Let ˜x 1(n) beperiodic with fundamental period N =40where one period is given by
{
5 sin(0.1πn), 0 ≤ n ≤ 19
˜x 1(n) =
0, 20 ≤ n ≤ 39
and let ˜x 2(n) beperiodic with fundamental period N = 80, where one period is given by
{
5 sin(0.1πn), 0 ≤ n ≤ 19
˜x 2(n) =
0, 20 ≤ n ≤ 79
These two periodic sequences differ in their periodicity but otherwise have the same
nonzero samples.
1. Compute the DFS ˜X 1(k) of˜x 1(n), and plot samples (using the stem function) of its
magnitude and angle versus k.
2. Compute the DFS ˜X 2(k) of˜x 2(n), and plot samples of its magnitude and angle versus k.
3. What is the difference between the two preceding DFS plots?
P5.4 Consider the periodic sequence ˜x 1(n) given in Problem P5.3. Let ˜x 2(n) beperiodic with
fundamental period N = 40, where one period is given by
{
˜x 1(n), 0 ≤ n ≤ 19
˜x 2(n) =
−˜x 1(n − 20), 20 ≤ n ≤ 39
1. Determine analytically the DFS ˜X 2(k) interms of ˜X1(k).
2. Compute the DFS ˜X 2(k) of˜x 2(n) and plot samples of its magnitude and angle versus k.
3. Verify your answer in part 1 using the plots of ˜X1(k) and ˜X 2(k)?
P5.5 Consider the periodic sequence ˜x 1(n) given in Problem P5.3. Let ˜x 3(n) beperiodic with
period 80, obtained by concatenating two periods of ˜x 1(n), i.e.,
˜x 3(n) =[˜x 1(n), ˜x 1(n)] PERIODIC
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