Discrete Time Systems Implementation MCQ Quiz – Objective Question with Answer for Discrete Time Systems Implementation

21. The zeros of the system function of the comb filter are located at _______

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

Answer: B

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

 

22. 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

Answer: C

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)=\(\sum_{k=0}^{M-1} \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}\)

 

23. Where do 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

Answer: B

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

 

24. 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

Answer: A

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 to be retained.

 

25. Which of the following is the application of lattice filter?

A. Digital speech processing
B. Adaptive filter
C. Electroencephalogram
D. All of the mentioned

Answer: D

Lattice filters are used extensively in digital signal processing and in the implementation of adaptive filters.

The application of lattice filters are

  1. Digital speech processing
  2. Adaptive filter
  3. Electroencephalogram

 

2. If we consider a sequence of FIR filers with system function Hm(z)=Am(z), then what is the definition of the polynomial Am(z)?

A. \(1+\sum_{k=0}^m α_m (k)z^{-k}\)

B. \(1+\sum_{k=1}^m α_m (k)z^{-k}\)

C. \(1+\sum_{k=1}^m α_m (k)z^k \)

D. \(\sum_{k=0}^m α_m (k)z^{-k}\)

Answer: B

Consider a sequence of FIR filer with system function Hm(z)=Am(z), m=0,1,2…M-1
where, by definition, Am(z) is the polynomial

Am(z)=\(1+\sum_{k=1}^m α_m (k)z^{-k}\), m≥1 and A0(z)=1.

 

26. What is the unit sample response of the mth filter?

A. hm(0)=0 and hm(k)=αm(k), k=1,2…m
B. hm(k)=αm(k), k=0,1,2…m(αm(0)≠1)
C. hm(0)=1 and hm(k)=αm(k), k=1,2…m
D. none of the mentioned

Answer: C

We know that Hm(z)=Am(z) and Am(z) is a polynomial whose equation is given as

Am(z)=\(1+\sum_{k=1}^m α_m (k)z^{-k}\), m≤1 and A0(z)=1

A0(z)=1 => hm(0)=1 and

Am(z)=\(\sum_{k=1}^m α_m (k)z^{-k}\)(m≤1)=> hm(k)=αm(k) for k=1,2…m.

 

27. The FIR filter whose direct form structure is as shown below is a prediction error filter.

The FIR filter whose direct form structure is as shown below is a prediction error filter.


A. True
B. False

Answer: A

The FIR structure shown in the above figure is intimately related to the topic of linear prediction. Thus the top filter structure shown in the above figure is called a prediction error filter.

 

28. What is the output of the single stage lattice filter if x(n) is the input?

A. x(n)+Kx(n+1)
B. x(n)+Kx(n-1)
C. x(n)+Kx(n-1)+Kx(n+1)
D. Kx(n-1)

Answer: B

The single-stage lattice filter is shown below.

the output of the single stage lattice filter if x(n) is the input?


Here both the inputs are excited and the output is selected from the top branch.

Thus the output of the single-stage lattice filter is given by y(n)= x(n)+Kx(n-1).

 

29. What is the output from the second-stage lattice filter when two single-stage lattice filers are cascaded with an input of x(n)?

A. K1K2x(n-1)+K2x(n-2)
B. x(n)+K1x(n-1)
C. x(n)+K1K2x(n-1)+K2x(n-2)
D. x(n)+K1(1+K2)x(n-1)+K2x(n-2)

Answer: D

When two single stage lattice filters are cascaded, then the output from the first filter is given by the equation

f1(n)= x(n)+K1x(n-1)
g1(n)=K1x(n)+x(n-1)

The output from the second filter is obtained as

f2(n)=f1(n)+K2g1(n-1)
=x(n)+K2[K1x(n-1)+x(n-2)]+ K1x(n-1)
= x(n)+K1(1+K2)x(n-1)+K2x(n-2).

 

30. What is the value of the coefficient α2(1) in the case of FIR filter represented in direct form structure with m=2 in terms of K1 and K2?

A. K1(K2)
B. K1(1-K2)
C. K1(1+K2)
D. None of the mentioned

Answer: C

The equation for the output of an FIR filter represented in the direct form structure is given as
y(n)=x(n)+ α2(1)x(n-1)+ α2(2)x(n-2)

The output from the double stage lattice structure is given by the equation,
f2(n)= x(n)+K2(1+K2)x(n-1)+K2x(n-2)
By comparing the coefficients of both the equations, we get

α2(1)= K1(1+K2).

 

31. The constants K1 and K2 of the lattice structure are called reflection coefficients.

A. True
B. False

Answer: A

The equation of the output from the second stage lattice filter is given by
f2(n)= x(n)+K1(1+K2)x(n-1)+K2x(n-2)
In the above equation, the constants K1 and K2 are called reflection coefficients.

 

32. If a three-stage lattice filter with coefficients K1=1/4, K2=1/2 K3=1/3, then what are the FIR filter coefficients for the direct form structure?

A. (1,8/24,5/8,1/3)
B. (1,5/8,13/24,1/3)
C. (1/4,13/24,5/8,1/3)
D. (1,13/24,5/8,1/3)

Answer: D

We get the output from the third stage lattice filter as
A3(z)=1+(13/24)z-1+(5/8)z-2+(1/3)z-3.

Thus the FIR filter coefficients for the direct form structure are (1,13/24,5/8,1/3).

 

33. What are the lattice coefficients corresponding to the FIR filter with system function H(z)= 1+(13/24)z-1+(5/8)z-2+(1/3)z-3?

A. (1/2,1/4,1/3)
B. (1,1/2,1/3)
C. (1/4,1/2,1/3)
D. None of the mentioned

Answer: C

Given the system function of the FIR filter is

H(z)= 1+(13/24)z-1+(5/8)z-2+(1/3)z-3

Thus the lattice coefficients corresponding to the given filter are (1/4,1/2,1/3).

 

34. If M and N are the orders of numerator and denominator of rational system function respectively, then how many multiplications are required in direct form-I realization of that IIR filter?

A. M+N-1
B. M+N
C. M+N+1
D. M+N+2

Answer: C

From the direct form-I realization of the IIR filter, if M and N are the orders of numerator and denominator of rational system function respectively, then M+N+1 multiplications are required.

 

35. If M and N are the orders of numerator and denominator of rational system function respectively, then how many additions are required in direct form-I realization of that IIR filter?

A. M+N-1
B. M+N
C. M+N+1
D. M+N+2

Answer: B

From the direct form-I realization of the IIR filter, if M and N are the orders of numerator and denominator of rational system function respectively, then M+N additions are required.

 

36. If M and N are the orders of numerator and denominator of rational system function respectively, then how many memory locations are required in direct form-I realization of that IIR filter?

A. M+N+1
B. M+N
C. M+N-1
D. M+N-2

Answer: A

From the direct form-I realization of the IIR filter, if M and N are the orders of numerator and denominator of rational system function respectively, then M+N+1 memory locations are required.

 

37. In direct form-I realization, the all-pole system is placed before the all-zero system.

A. True
B. False

Answer: B

In direct form-I realization, the all-zero system is placed before the all-pole system.

 

38. If M and N are the orders of numerator and denominator of rational system function respectively, then how many memory locations are required in direct form-II realization of that IIR filter?

A. M+N+1
B. M+N
C. Min [M, N]
D. Max [M, N]

Answer: D

From the direct form-II realization of the IIR filter, if M and N are the orders of numerator and denominator of rational system function respectively, then Max[M, N] memory locations are required.

 

39. The basic elements of a flow graph are branches and nodes.

A. True
B. False

Answer: A

A signal flow graph provides an alternative, but an equivalent graphical representation to a block diagram structure that we have been using to illustrate various system realizations. The basic elements of a flow graph are branches and nodes.

 

40. Which of the following is true for the given signal flow graph?

Which of the following is true for the given signal flow graph?

A. Two-pole system
B. Two zero system
C. Two pole and two zero system
D. None of the mentioned

Answer: C

The equivalent filter structure of the given signal flow graph in the direct form-II is given as

Thus from the above structure, the system has two zeros and two poles.

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