Test: FIR System Structures - 2 - Electrical Engineering (EE) MCQ

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

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

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

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.

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).

Explanation: 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).

Explanation: 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).

Explanation: 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 as reflection coefficients.

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).

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).