This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Backward Difference Method”.
1. The equation for Heq(s) is \(\frac{\sum_{K=0}^M b_K s^K}{\sum_{K=0}^N a_K s^K}\).
a) True
b) False
View Answer
Explanation: The analog filter in the time domain is governed by the following difference equation,
\(\sum_{K=0}^N a_K y^K (t)=\sum_{K=0}^M b_K x^K (t)\)
Taking Laplace transform on both the sides of the above differential equation with all initial conditions set to zero, we get
\(\sum_{K=0}^N a_K s^K Y(s)=\sum_{K=0}^M b_K s^K X(s)\)
=> Heq(s)=Y(s)/X(s)=\(\frac{\sum_{K=0}^M b_K s^K }{\sum_{K=0}^N a_K s^K}\).
2. What is the first backward difference of y(n)?
a) [y(n)+y(n-1)]/T
b) [y(n)+y(n+1)]/T
c) [y(n)-y(n+1)]/T
d) [y(n)-y(n-1)]/T
View Answer
Explanation: A simple approximation to the first order derivative is given by the first backward difference. The first backward difference is defined by
[y(n)-y(n-1)]/T.
3. Which of the following is the correct relation between ‘s’ and ‘z’?
a) z=1/(1+sT)
b) s=1/(1+zT)
c) z=1/(1-sT)
d) none of the mentioned
View Answer
Explanation: We know that s=(1-z-1)/T=> z=1/(1-sT).
4. What is the center of the circle represented by the image of jΩ axis of the s-domain?
a) z=0
b) z=0.5
c) z=1
d) none of the mentioned
View Answer
Explanation: Letting s=σ+jΩ in the equation z=1/(1-sT) and by letting σ=0, we get
|z-0.5|=0.5
Thus the image of the jΩ axis of the s-domain is a circle with centre at z=0.5 in z-domain.
5. What is the radius of the circle represented by the image of jΩ axis of the s-domain?
a) 0.75
b) 0.25
c) 1
d) 0.5
View Answer
Explanation: Letting s=σ+jΩ in the equation z=1/(1-sT) and by letting σ=0, we get
|z-0.5|=0.5
Thus the image of the jΩ axis of the s-domain is a circle of radius 0.5 centered at z=0.5 in z-domain.
6. The frequency response H(ω) will be considerably distorted with respect to H(jΩ).
a) True
b) False
View Answer
Explanation: Since jΩ axis is not mapped to the circle |z|=1, we can expect that the frequency response H(ω) will be considerably distorted with respect to H(jΩ).
7. The left half of the s-plane is mapped to which of the following in the z-domain?
a) Outside the circle |z-0.5|=0.5
b) Outside the circle |z+0.5|=0.5
c) Inside the circle |z-0.5|=0.5
d) Inside the circle |z+0.5|=0.5
View Answer
Explanation: The left half of the s-plane is mapped inside the circle of |z-0.5|=0.5 in the z-plane, which completely lies in the right half z-plane.
8. An analog high pass filter can be mapped to a digital high pass filter.
a) True
b) False
View Answer
Explanation: An analog high pass filter cannot be mapped to a digital high pass filter because the poles of the digital filter cannot lie in the correct region, which is the left-half of the z-plane(z < 0) in this case.
9. Which of the following is the correct relation between ‘s’ and ‘z’?
a) s=(1-z-1)/T
b) s=1/(1+zT)
c) s=(1+z-1)/T
d) none of the mentioned
View Answer
Explanation: We know that z=1/(1-sT)=> s=(1-z-1)/T.
10. What is the z-transform of the first backward difference equation of y(n)?
a) \(\frac{1+z^{-1}}{T}\) Y(z)
b) \(\frac{1-z^{-1}}{T}\) Y(z)
c) \(\frac{1+z^1}{T}\) Y(z)
d) None of the mentioned
View Answer
Explanation: The first backward difference of y(n) is given by the equation
[y(n)-y(n-1)]/T
Thus the z-transform of the first backward difference of y(n) is given as
\(\frac{1-z^{-1}}{T}\) Y(z).
Sanfoundry Global Education & Learning Series – Digital Signal Processing.
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