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
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
Digital speech processing
Adaptive filter
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.
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.
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
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?
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.