This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Structures for FIR Systems-1”.
1. In general, an FIR system is described by the difference equation y(n)=\(\sum_{k=0}^{M-1}b_k x(n-k)\).
a) True
b) False
View Answer
Explanation: The difference equation y(n)=\(\sum_{k=0}^{M-1}b_k x(n-k)\) describes the FIR system.
2. What is the general system function of an FIR system?
a) \(\sum_{k=0}^{M-1}b_k x(n-k)\)
b) \(\sum_{k=0}^M b_k z^{-k}\)
c) \(\sum_{k=0}^{M-1}b_k z^{-k}\)
d) None of the mentioned
View Answer
Explanation: We know that the difference equation of an FIR system is given by y(n)=\(\sum_{k=0}^{M-1}b_k x(n-k)\).
=>h(n)=bk=>\(\sum_{k=0}^{M-1}b_k z^{-k}\).
3. Which of the following is an method for implementing an FIR system?
a) Direct form
b) Cascade form
c) Lattice structure
d) All of the mentioned
View Answer
Explanation: There are several structures for implementing an FIR system, beginning with the simplest structure, called the direct form. There are several other methods like cascade form realization, frequency sampling realization and lattice realization which are used for implementing and FIR system.
4. How many memory locations are used for storage of the output point of a sequence of length M in direct form realization?
a) M+1
b) M
c) M-1
d) None of the mentioned
View Answer
Explanation: The direct form realization follows immediately from the non-recursive difference equation given by y(n)=\(\sum_{k=0}^{M-1}b_k x(n-k)\).
We observe that this structure requires M-1 memory locations for storing the M-1 previous inputs.
5. The direct form realization is often called a transversal or tapped-delay-line filter.
a) True
b) False
View Answer
Explanation: The structure of the direct form realization, resembles a tapped delay line or a transversal system.
6. What is the condition of M, if the structure according to the direct form is as follows?
a) M even
b) M odd
c) All values of M
d) Doesn’t depend on value of M
View Answer
Explanation: When the FIR system has linear phase, the unit sample response of the system satisfies either the symmetry or asymmetry condition, h(n)=±h(M-1-n)
For such a system the number of multiplications is reduced from M to M/2 for M even and to (M-1)/2 for M odd. Thus for the structure given in the question, M is odd.
7. By combining two pairs of poles to form a fourth order filter section, by what factor we have reduced the number of multiplications?
a) 25%
b) 30%
c) 40%
d) 50%
View Answer
Explanation: We have to do 3 multiplications for every second order equation. So, we have to do 6 multiplications if we combine two second order equations and we have to perform 3 multiplications by directly calculating the fourth order equation. Thus the number of multiplications are reduced by a factor of 50%.
8. The desired frequency response is specified at a set of equally spaced frequencies defined by the equation?
a) \(\frac{\pi}{2M}\)(k+α)
b) \(\frac{\pi}{M}\)(k+α)
c) \(\frac{2\pi}{M}\)(k+α)
d) None of the mentioned
View Answer
Explanation: To derive the frequency sampling structure, we specify the desired frequency response at a set of equally spaced frequencies, namely ωk=\(\frac{2\pi}{M}\)(k+α), k=0,1…(M-1)/2 for M odd
k=0,1….(M/2)-1 for M even
α=0 or 0.5.
9. The realization of FIR filter by frequency sampling realization can be viewed as cascade of how many filters?
a) Two
b) Three
c) Four
d) None of the mentioned
View Answer
Explanation: In frequency sampling realization, the system function H(z) is characterized by the set of frequency samples {H(k+ α)} instead of {h(n)}. We view this FIR filter realization as a cascade of two filters. One is an all-zero or a comb filter and the other consists of parallel bank of single pole filters with resonant frequencies.
10. What is the system function of all-zero filter or comb filter?
a) \(\frac{1}{M}(1+z^{-M} e^{j2πα})\)
b) \(\frac{1}{M}(1+z^M e^{j2πα})\)
c) \(\frac{1}{M}(1-z^M e^{j2πα})\)
d) \(\frac{1}{M}(1-z^{-M} e^{j2πα})\)
View Answer
Explanation: The system function H(z) which is characterized by the set of frequency samples is obtained as
H(z)=\(\frac{1}{M}(1-z^{-M} e^{j2πα})\sum_{k=0}^{M-1}\frac{H(k+α)}{1-e^{j2π(k+α)/M} z^{-1}}\)
We view this FIR realization as a cascade of two filters, H(z)=H1(z).H2(z)
Here H1(z) represents the all-zero filter or comb filter whose system function is given by the equation
H1(z)=\(\frac{1}{M}(1-z^{-M} e^{j2πα})\).
11. The zeros of the system function of comb filter are located at ______________
a) Inside unit circle
b) On unit circle
c) Outside unit circle
d) None of the mentioned
View Answer
Explanation: The system function of the comb filter is given by the equation
H1(z)=\(\frac{1}{M}(1-z^{-M} e^{j2πα})\)
Its zeros are located at equally spaced points on the unit circle at
zk=ej2π(k+α)/M k=0,1,2….M-1
12. What is the system function of the second filter other than comb filter in the realization of FIR filter?
a) \(\sum_{k=0}^M \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}\)
b) \(\sum_{k=0}^{M-1} \frac{H(k+α)}{1+e^{\frac{j2π(k+α)}{M}} z^{-1}}\)
c) \(\sum_{k=0}^{M-1} \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}\)
d) None of the mentioned
View Answer
Explanation: The system function H(z) which is characterized by the set of frequency samples is obtained as
H(z)=\(\frac{1}{M}(1-z^{-M} e^{j2πα})\sum_{k=0}^{M-1}\frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}}z^{-1}}\)
We view this FIR realization as a cascade of two filters, H(z)=H1(z).H2(z)
Here H1(z) represents the all-zero filter or comb filter, and the system function of the other filter is given by the equation
H2(z)=\(\sum_{k=0}^{M-1} \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}\)
13. Where does the poles of the system function of the second filter locate?
a) ej2π(k+α)M
b) ej2π(k+α)/M
c) ej2π(k-α)/M
d) ejπ(k+α)/M
View Answer
Explanation: The system function of the second filter in the cascade of an FIR realization by frequency sampling method is given by
H2(z)=\(\sum_{k=0}^{M-1} \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}\)
We obtain the poles of the above system function by equating the denominator of the above equation to zero.
=>\(1-e^{\frac{j2π(k+α)}{M}} z^{-1}\)=0
=>z=pk=\(e^{\frac{j2π(k+α)}{M}}\), k=0,1….M-1
14. When the desired frequency response characteristic of the FIR filter is narrowband, most of the gain parameters {H(k+α)} are zero.
a) True
b) False
View Answer
Explanation: When the desired frequency response characteristic of the FIR filter is narrowband, most of the gain parameters {H(k+α)} are zero. Consequently, the corresponding resonant filters can be eliminated and only the filters with nonzero gains need be retained.
15. Which of the following filters have a cascade realization as shown below?
a) IIR filter
b) Comb filter
c) High pass filter
d) FIR filter
View Answer
Explanation: The system function of the FIR filter according to the frequency sampling realization is given by the equation
H(z)=\(\frac{1}{M}(1-z^{-M} e^{j2πα})\sum_{k=0}^{M-1}\frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}\)
The above system function can be represented in the cascade form as shown in the above block diagram.
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