Digital Signal Processing Questions and Answers – IIR Filter Design by the Bilinear Transformation

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
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2. In Bilinear Transformation, aliasing of frequency components is been avoided.
a) True
b) False
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3. Is IIR Filter design by Bilinear Transformation is the advanced technique when compared to other design techniques?
a) True
b) False
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4. The approximation of the integral in y(t) = \(\int_{t_0}^t y'(τ)dt+y(t_0)\) by the Trapezoidal formula at t = nT and t₀=nT-T yields equation?
a) y(nT) = \(\frac{T}{2} [y^{‘} (nT)+y^{‘} (T-nT)]+y(nT-T)\)
b) y(nT) = \(\frac{T}{2} [y^{‘} (nT)+y^{‘} (nT-T)]+y(nT-T)\)
c) y(nT) = \(\frac{T}{2} [y^{‘} (nT)+y^{‘} (T-nT)]+y(T-nT)\)
d) y(nT) = \(\frac{T}{2} [y^{‘} (nT)+y^{‘} (nT-T)]+y(T-nT)\)
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5. We use y{‘}(nT)=-ay(nT)+bx(nT) to substitute for the derivative in y(nT) = \(\frac{T}{2} [y^{‘} (nT)+y^{‘} (nT-T)]+y(nT-T)\) 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) \((1+\frac{aT}{2})Y(z)-(1-\frac{aT}{2})y(n-1)=\frac{bT}{2} [x(n)+x(n-1)]\)
b) \((1+\frac{aT}{n})Y(z)-(1-\frac{aT}{n})y(n-1)=\frac{bT}{n} [x(n)+x(n-1)]\)
c) \((1+\frac{aT}{2})Y(z)+(1-\frac{aT}{2})y(n-1)=\frac{bT}{2} (x(n)-x(n-1))\)
d) \((1+\frac{aT}{2})Y(z)+(1-\frac{aT}{2})y(n-1)=\frac{bT}{2} (x(n)+x(n+1))\)
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6. The z-transform of below difference equation is?
\((1+\frac{aT}{2})Y(z)-(1-\frac{aT}{2})y(n-1)=\frac{bT}{2} [x(n)+ x(n-1)]\)
a) \((1+\frac{aT}{2})Y(z)-(1-\frac{aT}{2}) z^{-1} Y(z)=\frac{bT}{2} (1+z^{-1})X(z)\)
b) \((1+\frac{aT}{n})Y(z)-(1-\frac{aT}{2}) z^{-1} Y(z)=\frac{bT}{n} (1+z^{-1})X(z)\)
c) \((1+\frac{aT}{2})Y(z)+(1-\frac{aT}{n}) z^{-1} Y(z)=\frac{bT}{2} (1+z^{-1})X(z)\)
d) \((1+\frac{aT}{2})Y(z)-(1+\frac{aT}{2}) z^{-1} Y(z)=\frac{bT}{2} (1+z^{-1})X(z)\)
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7. What is the system function of the equivalent digital filter? H(z) = Y(z)/X(z) = ?
a) \(\frac{(\frac{bT}{2})(1+z^{-1})}{1+\frac{aT}{2}-(1-\frac{aT}{2}) z^{-1}}\)
b) \(\frac{(\frac{bT}{2})(1-z^{-1})}{1+\frac{aT}{2}-(1+\frac{aT}{2}) z^{-1}}\)
c) \(\frac{b}{\frac{2}{T}(\frac{1-z^{-1}}{1+z^{-1}}+a)}\)
d) \(\frac{(\frac{bT}{2})(1-z^{-1})}{1+\frac{aT}{2}-(1+\frac{aT}{2}) z^{-1}}\) & \(\frac{b}{\frac{2}{T}(\frac{1-z^{-1}}{1+z^{-1}}+a)}\)
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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
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9. In Nth order differential equation, the characteristics of bilinear transformation, let z=re^jw,s=o+jΩ Then for s = \(\frac{2}{T}(\frac{1-z^{-1}}{1+z^{-1}})\), the values of Ω, ℴ are
a) ℴ = \(\frac{2}{T}(\frac{r^2-1}{1+r^2+2rcosω})\), Ω = \(\frac{2}{T}(\frac{2rsinω}{1+r^2+2rcosω})\)
b) Ω = \(\frac{2}{T}(\frac{r^2-1}{1+r^2+2rcosω})\), ℴ = \(\frac{2}{T}(\frac{2rsinω}{1+r^2+2rcosω})\)
c) Ω=0, ℴ=0
d) None
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10. In equation ℴ = \(\frac{2}{T}(\frac{r^2-1}{1+r^2+2rcosω})\) 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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11. In equation ℴ = \(\frac{2}{T}(\frac{r^2-1}{1+r^2+2rcosω})\), 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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