This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Design of FIR Differentiators”.

1. How is the frequency response of an ideal differentiator related to the frequency?

a) Inversely proportional

b) Linearly proportional

c) Quadratic

d) None of the mentioned

View Answer

Explanation: An ideal differentiator has a frequency response that is linearly proportional to the frequency.

2. Which of the following is the frequency response of an ideal differentiator, Hd(ω)?

a) -jω ; -π ≤ ω ≤ π

b) -jω ; 0 ≤ ω ≤ π

c) jω ; 0 ≤ ω ≤ π

d) jω ; -π ≤ ω ≤ π

View Answer

Explanation: An ideal differentiator is defined as one that has the frequency response

H

_{d}(ω)= jω ; -π ≤ ω ≤ π

3. What is the unit sample response corresponding to Hd(ω)?

View Answer

Explanation: We know that, for an ideal differentiator, the frequency response is given as

H

_{d}(ω)= jω ; -π ≤ ω ≤ π

Thus, we get the unit sample response corresponding to the ideal differentiator is given as

h(n)=cosπn/n

4. The ideal differentiator ahs which of the following unit sample response?

a) Symmetric

b) Anti-symmetric

c) Cannot be explained

d) None of the mentioned

View Answer

Explanation: We know that the unit sample response of an ideal differentiator is given as

h(n)=cosπn/n

So, we can state that the unit sample response of an ideal differentiator is anti-symmetric because cosπn is also an anti-symmetric function.

5. If h_{d}(n) is the unit sample response of an ideal differentiator, then what is the value of h_{d}(0)?

a) 1

b) -1

c) 0

d) 0.5

View Answer

Explanation: Since we know that the unit sample response of an ideal differentiator is anti-symmetric,

=>h

_{d}(0)=0.

a) True

b) False

View Answer

Explanation: In view of the fact that the ideal differentiator has an anti-symmetric unit sample response, we shall confine our attention to FIR designs in which h(n)=-h(M-1-n).

7. Which of the following is the condition that an differentiator should satisfy?

a) Infinite response at zero frequency

b) Finite response at zero frequency

c) Negative response at zero frequency

d) Zero response at zero frequency

View Answer

Explanation: For an FIR filter, when M is odd, the real valued frequency response of the FIR filter Hr(ω) has the characteristic that Hr(0)=0. A zero response at zero frequency is just the condition that the differentiator should satisfy.

8. Full band differentiators can be achieved with an FIR filters having odd number of coefficients.

a) True

b) False

View Answer

Explanation: Full band differentiators cannot be achieved with an FIR filters having odd number of coefficients, since Hr(π)=0 for M odd.

9. If f_{p} is the bandwidth of the differentiator, then the desired frequency characteristic should be linear in the range:

a) 0 ≤ ω ≤ 2π

b) 0 ≤ ω ≤ 2f_{p}

c) 0 ≤ ω ≤ 2πf_{p}

d) None of the mentioned

View Answer

Explanation: In most cases of practical interest, the desired frequency response characteristic need only be linear over the limited frequency range 0 ≤ ω ≤ 2πf

_{p}, where f

_{p}is the bandwidth of the differentiator.

10. What is the desired response of the differentiator in the frequency range 2πf_{p} ≤ ω ≤ π?

a) Left unconstrained

b) Constrained to be zero

c) Either of the mentioned

d) None of the mentioned

View Answer

Explanation: In the frequency range 2πf

_{p}≤ ω ≤ π, the desired response may be either left unconstrained or constrained to be zero.

11. What is the weighting function used in the design of FIR differentiators based on the chebyshev approximation criterion?

a) 1/ω

b) ω

c) 1+ω

d) 1-ω

View Answer

Explanation: In the design of FIR differentiators based on the chebyshev approximation criterion, the weighting function W(ω) is specified in the program as

W(ω)=1/ω

in order that the relative ripple in the pass band be a constant.

_{r}(ω) decreases as ω varies from 0 to 2πf

_{p}.

a) True

b) False

View Answer

Explanation: We know that the weighting function is

W(ω)=1/ω

in order that the relative ripple in the pass band be a constant. Thus, the absolute error between the desired response ω and the approximation H

_{r}(ω) increases as ω varies from 0 to 2πf

_{p}.

13. Which of the following is the important parameter in a differentiator?

a) Length

b) Bandwidth

c) Peak relative error

d) All of the mentioned

View Answer

Explanation: The important parameters in a differentiator are its length, its bandwidth and the peak relative error of the approximation. The inter relationship among these three parameters can be easily displayed parametrically.

14. In this section, we confine our attention to FIR designs in which h(n)=h(M-1-n).

a) True

b) False

View Answer

Explanation: In view of the fact that the ideal differentiator has an anti-symmetric unit sample response, we shall confine our attention to FIR designs in which h(n)=-h(M-1-n).

15. What is the maximum value of f_{p} with which good designs are obtained for M odd?

a) 0.25

b) 0.45

c) 0.5

d) 0.75

View Answer

Explanation: Designs based on M odd are particularly poor if the bandwidth exceeds 0.45. The problem is basically the zero in the frequency response at ω=π(f=1/2). When f

_{p}<0.45, good designs are obtained for M odd.

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