ECE 310 Digital Signal Processing

Homework 8

Prof. Zhi-Pei Liang Due: April 9, 2021

1. Assume x[n] is a finite-duration sequence of length 40, and y[n] is obtained by zero-padding x[n] to

length 64. That is, y[n] = x[n], for n = 0, 1, . . . , 39, and y[n] = 0, n = 40, 41, . . . , 63. Let {X[m]}

39

m=0

and {Y [m]}

63

m=0 be the DFT of {x[n]}

39

n=0 and {y[n]}

63

n=0, respectively. Determine all the correct

relationships and justify your answers.

(a) X[0] = Y [0]

(b) X[5] = Y [8]

(c) X[10] = Y [16]

(d) X[12] = Y [18]

(e) X[39] = Y [63]

2. Given that the DFT of {2, 0, 6, 4} is {X0, X1, X2, X3}, determine the DFT of {2, 1, 0, 3} and express

the result in terms of X0, X1, X2, X3.

Hint: relate the two sequences using transformations we discussed in class (scaling, time reversal,

conjugation, circular shift, …) and use the corresponding properties of the DFT.

3. a) Find the inverse DFT of the sequence X[m] = n

1, e−j

3π

4 , 0, ej

3π

4

o

where the first entry of X[m]

corresponds to m = 0.

b) Without explicitly computing the inverse DFT sum, find the inverse DFT of the sequence

Y [m] = n

1, e−j

π

4 , 0, ej

π

4

o

where the first entry of Y [m] corresponds to m = 0, using your answer

to part (a).

4. You are given two sequences x[n] = [1, 2, 3, 4, 5, 6] and y[n] = [4, 5, 6, 1, 2, 3]. It is known that

Y [m] = X[m]e

−j

2π

6 mn0

. Find two values of n0 consistent with this information.

5. Let X[m] be the 8-point DFT of the sequence x[n] = [1, −1, 2, 3, −3, 0, 0, 0]. Let y[n] be a finite length

sequence whose DFT Y [m] is related to X[m] as Y [m] = X[m]e

−j

2π

6 mn0

, where n0 = 3. Determine

the sequence y[n].

6. Let X[m] be the 6-point DFT of x[n] = [1, 2, 3, 4, 5, 6]. Determine the sequence y[n] whose DFT

Y [m] = X[< −m >6].

7. Let X[m] denote the 80-point DFT of x[n], 0 ≤ n ≤ 79. The sequence y[n] is obtained by zero-padding

x[n] to length 128. Determine m0 such that Y [8] = X[m0].

8. Let X[m],(0 ≤ m ≤ 20) and Xd(ω) respectively be the 21-point DFT and DTFT of a real-valued

sequence {xn}

7

n=0 that is zero-padded to length 21. Determine all the correct relationships and

justify your answer.

(a) X[19] = Xd(−

4π

21 ).

(b) X[2] = X∗

d

(−

4π

21 )

(c) X[12] = Xd(−

4π

21 )

(d) X[4] = X∗

d

(−

4π

21 )

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