11. 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
Answer: A
The difference equation y(n)=\(\sum_{k=0}^{M-1}b_k x(n-k)\) describes the FIR system.
12. 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
Answer: C
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=>\(\sum_{k=0}^{M-1}b_k z^{-k}\).
13. Which of the following is a method for implementing an FIR system?
A. Direct form
B. Cascade form
C. Lattice structure
D. All of the mentioned
Answer: D
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 an FIR system.
14. 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
Answer: C
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.
15. The direct form realization is often called a transversal or tapped-delay-line filter.
A. True
B. False
Answer: A
The structure of the direct form realization resembles a tapped delay line or a transversal system.
16. 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 the value of M
Answer: B
When the FIR system has a 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.
17. 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%
Answer: D
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 is reduced by a factor of 50%.
18. 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
Answer: C
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.
19. The realization of the FIR filter by frequency sampling realization can be viewed as a cascade of how many filters?
A. Two
B. Three
C. Four
D. None of the mentioned
Answer: A
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 a parallel bank of single-pole filters with resonant frequencies.
20. 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. \(\frac{1}{M}(1-z^{-M} e^{j2πα})\)
Answer: D
The system function H(z) which is characterized by the set of frequency samples is obtained as
