Digital Signal Processing Multiple Choice Questions 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
Answer: A
Clarification: 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
Answer: D
Clarification: 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
Answer: C
Clarification: 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
Answer: B
Clarification: 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
Answer: D
Clarification: 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
Answer: A
Clarification: 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
Answer: C
Clarification: 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
Answer: B
Clarification: 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
Answer: A
Clarification: 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