# Sampling Rate Conversion by a Rational Factor I/D MCQ [Free PDF] – Objective Question Answer for Sampling Rate Conversion by a Rational Factor I/D Quiz

1. 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.

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

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 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.

4. 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

5. 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.

6. 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*

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}

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=$$\begin{bmatrix}6\\-2+2j\\-2\\-2-2j\end{bmatrix}$$

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

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}

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.

10. 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.

11. 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).

12. 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.

13. 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.

14. 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.

15. 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.

16. 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.

17. 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.

18. 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.

19. 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.

20. 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

In the case of bilinear transformation, where aliasing is not a problem, it does not matter whether the frequency transformation is performed in the analog domain or in the frequency domain.

21. Which of the following is the first method proposed for the design of FIR filters?

A. Chebyshev approximation
B. Frequency sampling method
C. Windowing technique
D. None of the mentioned

The design method based on the use of windows to truncate the impulse response h(n) and obtain the desired spectral shaping, was the first method proposed for designing linear phase FIR filters.

22. The lack of precise control of cutoff frequencies is a disadvantage of which of the following designs?

A. Window design
B. Chebyshev approximation
C. Frequency sampling
D. None of the mentioned

The major disadvantage of the window design method is the lack of precise control of the critical frequencies.

23. The values of cutoff frequencies, in general, depend on which of the following?

A. Type of the window
B. Length of the window
C. Type & Length of the window
D. None of the mentioned

The values of the cutoff frequencies of a filter in general by the windowing technique depend on the type of the filter and the length of the filter.

24. In the frequency sampling method, the transition band is a multiple of which the following?

A. π/M
B. 2π/M
C. π/2M
D. 2πM

In the frequency sampling technique, the transition band is a multiple of 2π/M.

25. The frequency sampling design method is attractive when the FIR filter is realized in the frequency domain by means of the DFT.

A. True
B. False

The frequency sampling design method is particularly attractive when the FIR is realized either in the frequency domain by means of the DFT or in any of the frequency sampling realizations.

26. Which of the following values can a frequency response take in the frequency sampling technique?

A. Zero
B. One
C. Zero or One
D. None of the mentioned

The attractive feature of the frequency sampling design is that the frequency response can take either zero or one at all frequencies, except in the transition band.

27. Which of the following technique is more preferable for the design of a linear phase FIR filter?

A. Window design
B. Chebyshev approximation
C. Frequency sampling
D. None of the mentioned

The Chebyshev approximation method provides total control of the filter specifications, and as a consequence, it is usually preferable to the other two methods.

28. By optimal filter design, the maximum sidelobe level is minimized.

A. True
B. False

By spreading the approximation error over the passband and stopband of the filter, this method results in an optimal filter design and using this the maximum sidelobe level is minimized.

29. Which of the following is the correct expression for transition band Δf?

A. (ωp– ωs)/2π
B. (ωp+ωs)/2π
C. (ωp.ωs)/2π
D. (ωs– ωp)/2π

The expression for Δf i.e., for the transition band is given as
Δf=(ωs-ωp)/2π.

30. If the resulting δ exceeds the specified δ2, then the length can be increased until we obtain a side lobe level that meets the specification.

A. True
B. False