Digital Signal Processing

Bilinear Transformations

Question 1
Marks : +2 | -2
Pass Ratio : 100%
Which of the following substitution is done in Bilinear transformations?
s = \\(\\frac{2}{T}[\\frac{1+z^{-1}}{1-z^1}]\\)
s = \\(\\frac{2}{T}[\\frac{1+z^{-1}}{1+}]\\)
s = \\(\\frac{2}{T}[\\frac{1-z^{-1}}{1+z^{-1}}]\\)
None of the mentioned
Explanation:
In bilinear transformation of an analog filter to digital filter, using the trapezoidal rule, the substitution for ‘s’ is given as
Question 2
Marks : +2 | -2
Pass Ratio : 100%
What is the value of y(n)-y(n-1) in terms of input x(n)?
\\([\\frac{x(n)+x(n-1)}{2}]T\\)
\\([\\frac{x(n)-x(n-1)}{2}]T\\)
\\([\\frac{x(n)-x(n+1)}{2}]T\\)
\\([\\frac{x(n)+x(n+1)}{2}]T\\)
Explanation:
We know that the derivative equation is
Question 3
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If s=σ+jΩ and z=rejω and r=1, then which of the following inference is correct?
LHS of the s-plane is mapped inside the circle, |z|=1
RHS of the s-plane is mapped outside the circle, |z|=1
Imaginary axis in the s-plane is mapped to the circle, |z|=1
None of the mentioned
Explanation:
We know that if =σ+jΩ and z=rejω, then by substituting the values in the below expression
Question 4
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What is the expression for system function in z-domain?
\\(\\frac{2}{T}[\\frac{1+z^{-1}}{1-z^1}]\\)
\\(\\frac{2}{T}[\\frac{1+z^{-1}}{1-z^1}]\\)
\\(\\frac{T}{2}[\\frac{1+z^{-1}}{1-z^1}]\\)
\\(\\frac{T}{2}[\\frac{1-z^{-1}}{1+z^{-1}}]\\)
Explanation:
We know that
Question 5
Marks : +2 | -2
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Which of the following rule is used in the bilinear transformation?
Simpson’s rule
Backward difference
Forward difference
Trapezoidal rule
Explanation:
Bilinear transformation uses trapezoidal rule for integrating a continuous time function.
Question 6
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The equation s = \\(\\frac{2}{T}[\\frac{1-z^{-1}}{1+z^{-1}}]\\) is a true frequency-to-frequency transformation.
True
False
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 s = \\(\\frac{2}{T}[\\frac{1-z^{-1}}{1+z^{-1}}]\\) is a true frequency-to-frequency transformation.
Question 7
Marks : +2 | -2
Pass Ratio : 100%
In bilinear transformation, the left-half s-plane is mapped to which of the following in the z-domain?
Entirely outside the unit circle |z|=1
Partially outside the unit circle |z|=1
Partially inside the unit circle |z|=1
Entirely inside the unit circle |z|=1
Explanation:
In bilinear transformation, the z to s transformation is given by the expression
Question 8
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If s=σ+jΩ and z=rejω, then what is the condition on σ if r<1?
σ > 0
σ < 0
σ > 1
σ < 1
Explanation:
We know that if = σ+jΩ and z=rejω, then by substituting the values in the below expression
Question 9
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What is the value of \\(\\int_{(n-1)T}^{nT} x(t)dt\\) according to trapezoidal rule?
\\([\\frac{x(nT)-x[(n-1)T]}{2}]T\\)
\\([\\frac{x(nT)+x[(n-1)T]}{2}]T\\)
\\([\\frac{x(nT)-x[(n+1)T]}{2}]T\\)
\\([\\frac{x(nT)+x[(n+1)T]}{2}]T\\)
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
Question 10
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Pass Ratio : 100%
Bilinear Transformation is used for transforming an analog filter to a digital filter.
True
False
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.