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

A. True
B. False

Answer: A

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

Answer: D
The bilinear transformation uses the trapezoidal rule for integrating a continuous time function.

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

A. s = \(\frac{2}{T}[\frac{1+z^{-1}}{1-z^1}]\)

B. s = \(\frac{2}{T}[\frac{1+z^{-1}}{1+}]\)

C. s = \(\frac{2}{T}[\frac{1-z^{-1}}{1+z^{-1}}]\)

D. None of the mentioned

Answer: C

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

s = \(\frac{2}{T}[\frac{1-z^{-1}}{1+z^{-1}}]\).

4. What is the value of \(\int_{(n-1)T}^{nT} x(t)dt\) according to trapezoidal rule?

A. \([\frac{x(nT)-x[(n-1)T]}{2}]T\)

B. \([\frac{x(nT)+x[(n-1)T]}{2}]T\)

C. \([\frac{x(nT)-x[(n+1)T]}{2}]T\)

D. \([\frac{x(nT)+x[(n+1)T]}{2}]T\)

Answer: B

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

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

Answer: D

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 s = \(\frac{2}{T}[\frac{1-z^{-1}}{1+z^{-1}}]\) is a true frequency-to-frequency transformation.
A. True
B. False

Answer: A

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 s = \(\frac{2}{T}[\frac{1-z^{-1}}{1+z^{-1}}]\) is a true frequency-to-frequency transformation.

9. If s=σ+jΩ and z=rejω, then what is the condition on σ if r<1?

A. σ > 0
B. σ < 0
C. σ > 1
D. σ < 1

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