This set of Digital Signal Processing Questions & Answers for experienced focuses on “Structures for FIR Systems”.

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

a) Digital speech processing

b) Adaptive filter

c) Electroencephalogram

d) All of the mentioned

View Answer

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

2. If we consider a sequence of FIR filer with system function H_{m}(z)=A_{m}(z), then what is the definition of the polynomial A_{m}(z)?

View Answer

Explanation: 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

3. What is the unit sample response of the m^{th} filter?

View Answer

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

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

a) True

b) False

View Answer

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

5. 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)

View Answer

Explanation: The single stage lattice filter is as shown below.

Here both the inputs are excited and 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).

6. 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) K_{1}K_{2}x(n-1)+K_{2}x(n-2)

b) x(n)+K_{1}x(n-1)

c) x(n)+K_{1}K_{2}x(n-1)+K_{2}x(n-2)

d) x(n)+K_{1}(1+K_{2})x(n-1)+K_{2}x(n-2)

View Answer

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

f

_{1}(n)= x(n)+K

_{1}x(n-1)

g

_{1}(n)=K

_{1}x(n)+x(n-1)

The output from the second filter is obtained as

f

_{2}(n)=f

_{1}(n)+K

_{2}g

_{1}(n-1)

=x(n)+K

_{2}[K

_{1}x(n-1)+x(n-2)]+ K

_{1}x(n-1)

= x(n)+K

_{1}(1+K

_{2})x(n-1)+K

_{2}x(n-2).

7. 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 K_{1} and K_{2}?

a) K_{1}(K_{2})

b) K_{1}(1-K_{2})

c) K_{1}(1+K_{2})

d) None of the mentioned

View Answer

Explanation: The equation for the output of an FIR filter represented in the direct form structure is given as

y(n)=x(n)+ α

_{2}(1)x(n-1)+ α

_{2}(2)x(n-2)

The output from the double stage lattice structure is given by the equation,

f

_{2}(n)= x(n)+K

_{2}(1+K2)x(n-1)+K

_{2}x(n-2)

By comparing the coefficients of both the equations, we get

α

_{2}(1)= K

_{1}(1+K

_{2}).

8. The constants K_{1} and K_{2} of the lattice structure are called as reflection coefficients.

a) True

b) False

View Answer

Explanation: The equation of the output from the second stage lattice filter is given by

f

_{2}(n)= x(n)+K

_{1}(1+K

_{2})x(n-1)+K

_{2}x(n-2)

In the above equation, the constants K

_{1}and K

_{2}are called as reflection coefficients.

9. If a three stage lattice filter with coefficients K_{1}=1/4, K_{2}=1/2 K_{3}=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)

View Answer

Explanation: 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).

10. 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

View Answer

Explanation: 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 is (1/4,1/2,1/3).

**Sanfoundry Global Education & Learning Series – Digital Signal Processing.**

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