IIR Filter Design

Digital Communications Systems

This set of tricky Digital Signal Processing Questions & Answers focuses on “IIR Filter Design by the Bilinear Transformation”.

1. In IIR Filter design by the Bilinear Transformation, the Bilinear Transformation is a mapping from
a) Z-plane to S-plane
b) S-plane to Z-plane
c) S-plane to J-plane
d) J-plane to Z-plane

2. In Bilinear Transformation, aliasing of frequency components is been avoided.
a) True
b) False

3. Is IIR Filter design by Bilinear Transformation is the advanced technique when compared to other design techniques?
a) True
b) False

4. The approximation of the integral in y(t) = ∫tt0y′(τ)dt+y(t0) by the Trapezoidal formula at t = nT and t0=nT-T yields equation?
a) y(nT) = T2[y‘(nT)+y‘(T−nT)]+y(nT−T)
b) y(nT) = T2[y‘(nT)+y‘(nT−T)]+y(nT−T)
c) y(nT) = T2[y‘(nT)+y‘(T−nT)]+y(T−nT)
d) y(nT) = T2[y‘(nT)+y‘(nT−T)]+y(T−nT)

5. We use y{‘}(nT)=-ay(nT)+bx(nT) to substitute for the derivative in y(nT) = T2[y‘(nT)+y‘(nT−T)]+y(nT−T) and thus obtain a difference equation for the equivalent discrete-time system. With y(n) = y(nT) and x(n) = x(nT), we obtain the result as of the following?
a) (1+aT2)Y(z)−(1−aT2)y(n−1)=bT2[x(n)+x(n−1)]
b) (1+aTn)Y(z)−(1−aTn)y(n−1)=bTn[x(n)+x(n−1)]
c) (1+aT2)Y(z)+(1−aT2)y(n−1)=bT2(x(n)−x(n−1))
d) (1+aT2)Y(z)+(1−aT2)y(n−1)=bT2(x(n)+x(n+1))

6. The z-transform of below difference equation is?
(1+aT2)Y(z)−(1−aT2)y(n−1)=bT2[x(n)+x(n−1)]
a) (1+aT2)Y(z)−(1−aT2)z−1Y(z)=bT2(1+z−1)X(z)
b) (1+aTn)Y(z)−(1−aT2)z−1Y(z)=bTn(1+z−1)X(z)
c) (1+aT2)Y(z)+(1−aTn)z−1Y(z)=bT2(1+z−1)X(z)
d) (1+aT2)Y(z)−(1+aT2)z−1Y(z)=bT2(1+z−1)X(z)

7. What is the system function of the equivalent digital filter? H(z) = Y(z)/X(z) = ?
a) (bT2)(1+z−1)1+aT2−(1−aT2)z−1
b) (bT2)(1−z−1)1+aT2−(1+aT2)z−1
c) b2T(1−z−11+z−1+a)
d) (bT2)(1−z−1)1+aT2−(1+aT2)z−1 & b2T(1−z−11+z−1+a)

8. In the Bilinear Transformation mapping, which of the following are correct?
a) All points in the LHP of s are mapped inside the unit circle in the z-plane
b) All points in the RHP of s are mapped outside the unit circle in the z-plane
c) All points in the LHP & RHP of s are mapped inside & outside the unit circle in the z-plane
d) None of the mentioned

9. In Nth order differential equation, the characteristics of bilinear transformation, let z=rejw,s=o+jΩ Then for s = 2T(1−z−11+z−1), the values of Ω, ℴ are
a) ℴ = 2T(r2−11+r2+2rcosω), Ω = 2T(2rsinω1+r2+2rcosω)
b) Ω = 2T(r2−11+r2+2rcosω), ℴ = 2T(2rsinω1+r2+2rcosω)
c) Ω=0, ℴ=0
d) None

10. In equation ℴ = 2T(r2−11+r2+2rcosω) if r < 1 then ℴ < 0 and then mapping from s-plane to z-plane occurs in which of the following order?
a) LHP in s-plane maps into the inside of the unit circle in the z-plane
b) RHP in s-plane maps into the outside of the unit circle in the z-plane
c) All of the mentioned
d) None of the mentioned

11. In equation ℴ = 2T(r2−11+r2+2rcosω), if r > 1 then ℴ > 0 and then mapping from s-plane to z-plane occurs in which of the following order?
a) LHP in s-plane maps into the inside of the unit circle in the z-plane
b) RHP in s-plane maps into the outside of the unit circle in the z-plane
c) All of the mentioned
d) None of the mentioned

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