This set of tough Digital Signal Processing Questions & Answers focuses on “Sampling Rate Conversion by a Rational Factor I/D”.

1. Sampling rate conversion by the rational factor I/D is accomplished by what connection of interpolator and decimator?

a) Parallel

b) Cascade

c) Convolution

d) None of the mentioned

View Answer

Explanation: A sampling rate conversion by the rational factor I/D is accomplished by cascading an interpolator with a decimator.

2. Which of the following has to be performed in sampling rate conversion by rational factor?

a) Interpolation

b) Decimation

c) Either interpolation or decimation

d) None of the mentioned

View Answer

Explanation: We emphasize that the importance of performing the interpolation first and decimation second, is to preserve the desired spectral characteristics of x(n).

3. Which of the following operation is performed by the blocks given the figure below?

a) Sampling rate conversion by a factor I

b) Sampling rate conversion by a factor D

c) Sampling rate conversion by a factor D/I

d) Sampling rate conversion by a factor I/D

View Answer

Explanation: In the diagram given, a interpolator is in cascade with a decimator which together performs the action of sampling rate conversion by a factor I/D.

^{th}root of unity W

_{N}is given as:

a) e

^{j2πN}

b) e-j2πN

c) e

^{-j2π/N}

d) e

^{j2π/N}

View Answer

Explanation: We know that the Discrete Fourier transform of a signal x(n) is given as

Thus we get Nth rot of unity W

_{N}= e

^{-j2π/N}

5. Which of the following is true regarding the number of computations requires to compute an N-point DFT?

a) N^{2} complex multiplications and N(N-1) complex additions

b) N^{2} complex additions and N(N-1) complex multiplications

c) N^{2} complex multiplications and N(N+1) complex additions

d) N^{2} complex additions and N(N+1) complex multiplications

View Answer

Explanation: The formula for calculating N point DFT is given as

From the formula given at every step of computing we are performing N complex multiplications and N-1 complex additions. So, in a total to perform N-point DFT we perform N

^{2}complex multiplications and N(N-1) complex additions.

6. Which of the following is true?

View Answer

Explanation: If XN represents the N point DFT of the sequence xN in the matrix form, then we know that

7. What is the DFT of the four point sequence x(n)={0,1,2,3}?

a) {6,-2+2j-2,-2-2j}

b) {6,-2-2j,2,-2+2j}

c) {6,-2+2j,-2,-2-2j}

d) {6,-2-2j,-2,-2+2j}

View Answer

Explanation: The first step is to determine the matrix W4. By exploiting the periodicity property of W4 and the symmetry property

_{k}, then which of the following is true?

a) X(k)=Nc

_{k}

b) X(k)=c

_{k}/N

c) X(k)=N/c

_{k}

d) None of the mentioned

View Answer

9. What is the DFT of the four point sequence x(n)={0,1,2,3}?

a) {6,-2+2j-2,-2-2j}

b) {6,-2-2j,2,-2+2j}

c) {6,-2-2j,-2,-2+2j}

d) {6,-2+2j,-2,-2-2j}

View Answer

Answer: Given x(n)={0,1,2,3}

We know that the 4-point DFT of the above given sequence is given by the expression

In this case N=4

=>X(0)=6,X(1)=-2+2j,X(2)=-2,X(3)=-2-2j.

10. If W_{4}^{100}=W_{x}^{200}, then what is the value of x?

a) 2

b) 4

c) 8

d) 16

View Answer

Explanation: We know that according to the periodicity and symmetry property,

100/4=200/x=>x=8.

**Sanfoundry Global Education & Learning Series – Digital Signal Processing.**

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