This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Bilinear Transformations”.

1. Bilinear Transformation is used for transforming an analog filter to a digital filter.

a) True

b) False

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Explanation: The bilinear transformation can be regarded as a correction of the backward difference method. The bilinear transformation is used for transforming an analog filter to a digital filter.

2. Which of the following rule is used in the bilinear transformation?

a) Simpson’s rule

b) Backward difference

c) Forward difference

d) Trapezoidal rule

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Explanation: Bilinear transformation uses trapezoidal rule for integrating a continuous time function.

3. Which of the following substitution is done in Bilinear transformations?

d) None of the mentioned

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Explanation: In bilinear transformation of an analog filter to digital filter, using the trapezoidal rule, the substitution for ‘s’ is given as

.

4. What is the value ofaccording to trapezoidal rule?

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Explanation: The given integral is approximated by the trapezoidal rule. This rule states that if T is small, the area (integral) can be approximated by the mean height of x(t) between the two limits and then multiplying by the width. That is

.

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6. What is the expression for system function in z-domain?

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Explanation: We know that

y(n)-y(n-1)= [(x(n)+x(n-1))/2]T

Taking z-transform of the above equation gives

=>Y(z)[1-z

^{-1}]=([1+z

^{-1}]/2).TX(z)

=>H(z)=Y(z)/X(z)= T/2[(1+z

^{-1})/(1-z^1 )].

7. In bilinear transformation, the left-half s-plane is mapped to which of the following in the z-domain?

a) Entirely outside the unit circle |z|=1

b) Partially outside the unit circle |z|=1

c) Partially inside the unit circle |z|=1

d) Entirely inside the unit circle |z|=1

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Explanation: In bilinear transformation, the z to s transformation is given by the expression

z=[1+(T/2)s]/[1-(T/2)s].

Thus unlike the backward difference method, the left-half s-plane is now mapped entirely inside the unit circle, |z|=1, rather than to a part of it.

8. The equation is a true frequency-to-frequency transformation.

a) True

b) False

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Explanation: Unlike the backward difference method, the left-half s-plane is now mapped entirely inside the unit circle, |z|=1, rather than to a part of it. Also, the imaginary axis is mapped to the unit circle. Therefore, equation is a true frequency-to-frequency transformation.

9. If s=σ+jΩ and z=re^{jω}, then what is the condition on σ if r<1?

a) σ > 0

b) σ < 0

c) σ > 1

d) σ < 1

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Explanation: We know that if =σ+jΩ and z=re

^{jω}, then by substituting the values in the below expression

When r<1 => σ < 0.

^{jω}and r=1, then which of the following inference is correct?

a) LHS of the s-plane is mapped inside the circle, |z|=1

b) RHS of the s-plane is mapped outside the circle, |z|=1

c) Imaginary axis in the s-plane is mapped to the circle, |z|=1

d) None of the mentioned

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Explanation: We know that if =σ+jΩ and z=re

^{jω}, then by substituting the values in the below expression

=>σ = 2/T[(r

^{2}-1)/(r

^{2}+1+2rcosω)] When r=1 => σ = 0.

This shows that the imaginary axis in the s-domain is mapped to the circle of unit radius centered at z=0 in the z-domain.

11. If s=σ+jΩ and z=re^{jω}, then what is the condition on σ if r>1?

a) σ > 0

b) σ < 0

c) σ > 1

d) σ < 1

View Answer

Explanation: We know that if =σ+jΩ and z=rejω, then by substituting the values in the below expression

s = 2/T[(1-z

^{-1})/(1+z

^{-1})] =>σ = 2/T[(r

^{2}-1)/(r

^{2}+1+2rcosω)] When r>1 => σ > 0.

12. What is the expression for the digital frequency when r=1?

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13. What is the kind of relationship between Ω and ω?

a) Many-to-one

b) One-to-many

c) One-to-one

d) Many-to-many

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Explanation: The analog frequencies Ω=±∞ are mapped to digital frequencies ω=±π. The frequency mapping is not aliased; that is, the relationship between Ω and ω is one-to-one. As a consequence of this, there are no major restrictions on the use of bilinear transformation.

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

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