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Which of the following is the application of lattice filter?
Explanation: Lattice filters are used extensively in digital signal processing and in the implementation of adaptive filters.
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)?
Explanation: Consider a sequence of FIR filer with system function Hm(z)=Am(z), m=0,1,2…M1
where, by definition, Am(z) is the polynomial
What is the unit sample response of the m^{th} filter?
Explanation: We know that Hm(z)=Am(z) and Am(z) is a polynomial whose equation is given as
The FIR filter whose direct form structure is as shown below is a prediction error filter.
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.
What is the output of the single stage lattice filter if x(n) is the input?
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(n1).
What is the output from the second stage lattice filter when two single stage lattice filers are cascaded with an input of x(n)?
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(n1)
g_{1}(n)=K_{1}x(n)+x(n1)
The output from the second filter is obtained as
f_{2}(n)=f_{1}(n)+K_{2}g_{1}(n1)
=x(n)+K_{2}[K_{1}x(n1)+x(n2)]+ K_{1}x(n1)
= x(n)+K_{1}(1+K_{2})x(n1)+K_{2}x(n2).
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}?
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(n1)+ α_{2}(2)x(n2)
The output from the double stage lattice structure is given by the equation,
f_{2}(n)= x(n)+K_{2}(1+K2)x(n1)+K_{2}x(n2)
By comparing the coefficients of both the equations, we get
α_{2}(1)= K_{1}(1+K_{2}).
The constants K_{1} and K_{2} of the lattice structure are called as reflection coefficients.
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(n1)+K_{2}x(n2)
In the above equation, the constants K_{1} and K_{2} are called as reflection coefficients.
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?
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).
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}?
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).
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