# Interpolation Factor MCQ [Free PDF] – Objective Question Answer for Interpolation Factor Quiz

1. Which of the following operation has to be performed to increase the sampling rate by an integer factor I?

A. Interpolating I+1 new samples
B. Interpolating I-1 new samples
C. Extrapolating I+1 new samples
D. Extrapolating I-1 new samples

An increase in the sampling rate by an integer factor of I can be accomplished by interpolating I-1 new samples between successive values of the signal.

2. In one of the interpolation processes, we can preserve the spectral shape of the signal sequence x(n).

A. True
B. False

The interpolation process can be accomplished in a variety of ways. Among them, there is a process that preserves the spectral shape of the signal sequence x(n).

3. If v(m) denotes a sequence with a rate Fy=I.Fx which is obtained from x(n), then which of the following is the correct definition for v(m)?

A.x(mI), m=0,±I,±2I…. 0, otherwise
B. x(mI), m=0,±I,±2I…. x(m/I), otherwise
C. x(m/I), m=0,±I,±2I….
0, otherwise
D. None of the mentioned

If v(m) denote a sequence with a rate Fy=I.Fx which is obtained from x(n) by adding I-1 zeros between successive values of x(n). Thus
v(m)= x(m/I), m=0,±I,±2I….
0, otherwise.

4. If X(z) is the z-transform of x(n), then what is the z-transform of interpolated signal v(m)?

A. X(zI)
B. X(z+I)
C. X(z/I)
D. X(zI)

By taking the z-transform of the signal v(m), we get
V(z)=(sum_{m=-∞}^∞ v(m)z^{-m})
=(sum_{m=-∞}^∞ x(m)z^{-mI})
= X(z-I)

5. If x(m) and v(m) are the original and interpolated signals and ωy denotes the frequency variable relative to the new sampling rate, then V(ωy)= X(ωyI).

A. True
B. False

The spectrum of v(m) is obtained by evaluating V(z) = X(zI) on the unit circle. Thus V(ωy)= X(ωyI), where ωy denotes the frequency variable relative to the new sampling rate.

6. What is the relationship between ωx and ωy?

A. ωy= ωx.I
B. ωy= ωx/I
C. ωy= ωx+I
D. None of the mentioned

We know that the relationship between sampling rates is Fy=IFx and hence the frequency variables ωx and ωy are related according to the formula
ωy= ωx/I.

7. The following sampling rate conversion technique is an interpolation by a factor I.

A. True
B. False

From the diagram, the values are interpolated between two successive values of x(n), thus it is called as sampling rate conversion using interpolation by a factor I.

8. The following sampling rate conversion technique is an interpolation by a factor I.

A. True
B. False

The sampling rate conversion technique given in the diagram is decimation by a factor D.

9. Which of the following is true about the interpolated signal whose spectrum is V(ωy)?

A. (I-1)-fold non-periodic
B. (I-1)-fold periodic repetition
C. I-fold non periodic
D. I-fold periodic repetition

We observe that the sampling rate increase, obtained by the addition of I-1 zero samples between successive values of x(n), results in a signal whose spectrum is an I-fold periodic repetition of the input signal spectrum.

10. C=I is the desired normalization factor.

A. True
B. False

The amplitude of the sampling rate converted signal should be multiplied by a factor C, whose value when equal to I is called the desired normalization factor W.

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

A. Parallel
C. Convolution
D. None of the mentioned

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

12. 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

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

13. Which of the following operation is performed by the blocks given in 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

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

14. The Nth root of unity WN is given as ______

A. ej2πN
B. e-j2πN
C. e-j2π/N
D. ej2π/N

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

X(k)=$$\sum_{n=0}^{N-1} x(n)e^{-j2πkn/N}=\sum_{n=0}^{N-1} x(n) W_N^{kn}$$

Thus we get Nth rot of unity WN= e-j2π/N

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

A. N2 complex multiplications and N(N-1) complex additions
B. N2 complex additions and N(N-1) complex multiplications
C. N2 complex multiplications and N(N+1) complex additions
D. N2 complex additions and N(N+1) complex multiplications

The formula for calculating N point DFT is given as

X(k)=$$\sum_{n=0}^{N-1} x(n)e^{-j2πkn/N}$$

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 N2 complex multiplications and N(N-1) complex additions.

16. Which of the following is true?

A. $$W_N^*=\frac{1}{N} W_{N^{-1}}$$

B. $$W_N-1=\frac{1}{N} W_{N^*}$$

C. $$W_N-1=W_{N^*}$$

D. None of the mentioned

If XN represents the N point DFT of the sequence xN in the matrix form, then we know that XN = WN.xN
By pre-multiplying both sides by WN-1, we get
xN=WN-1.XN

But we know that the inverse DFT of XN is defined as
xN=1/N*XN

Thus by comparing the above two equations we get
WN-1=1/N WN*

17. 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}

The first step is to determine the matrix W4. By exploiting the periodicity property of W4 and the symmetry property
$$W_{N}^{k+N/2}=-W_{N^k}$$

The matrix W4 may be expressed as

W4=$$\begin{bmatrix}W_4^0&W_4^0&W_4^0&W_4^1\\W_4^0&W_4^0&W_4^2&W_4^3\\W_4^0&W_4^2&W_4^0&W_4^3\\W_4^4&W_4^6&W_4^6&W_4^9\end{bmatrix}=\begin{bmatrix}W_4^0&W_4^0&W_4^0&W_4^1\\W_4^0&W_4^0&W_4^2&W_4^3\\W_4^0&W_4^2&W_4^0&W_4^3\\W_4^0&W_4^2&W_4^2&W_4^1\end{bmatrix}$$
=$$\begin{bmatrix}1&1&1&1\\1&-j&-1&j\\1&-1&1&-1\\1&j&-1&-j\end{bmatrix}$$

Then X4=W4.x4=$$\begin{bmatrix}6\\-2+2j\\-2\\-2-2j\end{bmatrix}$$

18. If X(k) is the N point DFT of a sequence whose Fourier series coefficients is given by ck, then which of the following is true?

A. X(k)=Nck
B. X(k)=ck/N
C. X(k)=N/ck
D. None of the mentioned

The Fourier series coefficients are given by the expression

ck=$$\frac{1}{N} \sum_{n=0}^{N-1} x(n)e^{-j2πkn/N} = \frac{1}{N}X(k)=> X(k)=Nc_k$$

19. 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}

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

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

X(k)=$$\sum_{n=0}^{N-1} x(n)e^{-j2πkn/N}$$

In this case N=4

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

20. If W4100=Wx200, then what is the value of x?

A. 2
B. 4
C. 8
D. 16

We know that according to the periodicity and symmetry property,
100/4=200/x=>x=8.

21. The frequency transformation in the digital domain involves replacing the variable z-1 by a rational function g(z-1).

A. True
B. False

As in the analog domain, frequency transformations can be performed on a digital low pass filter to convert it to either a bandpass, band stop, or high pass filter. The transformation involves the replacing of the variable z-1 with a rational function g(z-1).

22. The mapping z-1 → g(z-1) must map inside the unit circle in the z-plane into __________

A. Outside the unit circle
B. On the unit circle
C. Inside the unit circle
D. None of the mentioned

The map z-1 → g(z-1) must map inside the unit circle in the z-plane into itself to apply digital frequency transformation.

23. The unit circle must be mapped outside the unit circle.
A. True
B. False

For the map z-1 → g(z-1) to be a valid digital frequency transformation, then the unit circle also must be mapped inside the unit circle.

24. The mapping z-1 → g(z-1) must be __________

A. Low pass
B. High pass
C. Bandpass
D. All-pass

We know that the unit circle must be mapped inside the unit circle.
Thus it implies that for r=1, e-jω = g(e-jω)=|g(ω)|.ej arg [ g(ω) ]
It is clear that we must have |g(ω)|=1 for all ω. That is, the mapping is all-pass.

25. What should be the value of |ak| to ensure that a stable filter is transformed into another stable filter?

A. < 1
B. =1
C. > 1
D. 0

The value of |ak| < 1 to ensure that a stable filter is transformed into another stable filter to satisfy condition 1.

26. Which of the following methods are inappropriate to design high pass and many bandpass filters?

A. Impulse invariance
B. Mapping of derivatives
C. Impulse invariance & Mapping of derivatives
D. None of the mentioned

We know that the impulse invariance method and mapping of derivatives are inappropriate to use in the designing of high pass and bandpass filters.

27. The impulse invariance method and mapping of derivatives are inappropriate to use in the designing of high pass and bandpass filters due to the aliasing problems.

A. True
B. False

We know that the impulse invariance method and mapping of derivatives are inappropriate to use in the designing of high pass and bandpass filters due to the aliasing problems.

28. We can employ the analog frequency transformation followed by conversion of the result into the digital domain by use of impulse invariance and mapping the derivatives.

A. True
B. False

Since there is a problem with aliasing in designing high pass and many bandpass filters using impulse invariance and mapping of derivatives, we cannot employ the analog frequency transformation followed by conversion of the result into the digital domain by use of these two mappings.

29. It is better to perform the mapping from an analog low pass filter into a digital low pass filter by either of these mappings and then perform the frequency transformation in the digital domain.

A. True
B. False

It is better to perform the mapping from an analog low pass filter into a digital low pass filter by either of these mappings and then perform the frequency transformation in the digital domain because, by this kind of frequency transformation, the problem of aliasing is avoided.

30. In which of the following transformations, it doesn’t matter whether the frequency transformation is performed in the analog domain or in the frequency domain?

A. Impulse invariance
B. Mapping of derivatives
C. Bilinear transformation
D. None of the mentioned