11. If s=σ+jΩ and z=rejω 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
Answer: C
We know that if =σ+jΩ and z=rejω, then by substituting the values in the below expression
s = \(\frac{2}{T}[\frac{1-z^{-1}}{1+z^{-1}}]\)
=>σ = \(\frac{2}{T}[\frac{r^2-1}{r^2+1+2rcosω}]\)
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.
12. 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: A
We know that if = σ+jΩ and z=rejω, then by substituting the values in the below expression
s = \(\frac{2}{T}[\frac{1-z^{-1}}{1+z^{-1}}]\)
=>σ = \(\frac{2}{T}[\frac{r^2-1}{r^2+1+2rcosω}]\)
When r>1 => σ > 0.
13. What is the expression for the digital frequency when r=1?
A. \(\frac{1}{T} tan(\frac{ΩT}{2})\)
B. \(\frac{2}{T} tan(\frac{ΩT}{2})\)
C. \(\frac{1}{T} tan^{-1}(\frac{ΩT}{2})\)
D. \(\frac{2}{T} tan^{-1}(\frac{ΩT}{2})\)
Answer: D
When r=1, we get σ=0 and
Ω = \(\frac{2}{T} [\frac{2 sinω}{1+1+2 cosω}]\)
=>ω=\(\frac{2}{T} tan^{-1}(\frac{ΩT}{2})\).
14. 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
Answer: C
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.
15. The system function of a general IIR filter is given as
H(z)=\(\frac{\sum_{k=0}^M b_k z^{-k}}{1+\sum_{k=1}^N a_k z^{-k}}\).
A. True
B. False
Answer: A
If ak and bk are the filter coefficients, then the transfer function of a general IIR filter is given by the expression
16. If ak is the filter coefficient and āk represents the quantized coefficient with Δak as the quantization error, then which of the following equation is true?
A. āk = ak.Δak
B. āk = ak/Δak
C. āk = ak + Δak
D. None of the mentioned
Answer: C
The quantized coefficient āk can be related to the un-quantized coefficient ak by the relation
āk = ak + Δak
where Δak represents the quantization error.
17. Which of the following is the equivalent representation of the denominator of the system function of a general IIR filter?
A. \(\prod_{k=1}^N (1+p_k z^{-1})\)
B. \(\prod_{k=1}^N (1+p_k z^{-k})\)
C. \(\prod_{k=1}^N (1-p_k z^{-k})\)
D. \(\prod_{k=1}^N (1-p_k z^{-1})\)
Answer: D
We know that the system function of a general IIR filter is given by the equation
The denominator of H(z) may be expressed in the form
D(z)=\(1+\sum_{k=1}^N a_k z^{-k}=\prod_{k=1}^N (1-p_k z^{-1})\)
where pk are the poles of H(z).
18. If pk is the set of poles of H(z), then what is Δpk that is the error resulting from the quantization of filter coefficients?
A. Pre-turbation
B. Perturbation
C. Turbation
D. None of the mentioned
Answer: B
We know that &pmacr;k = pk + Δpk, k=1,2…N and Δpk that is the error resulting from the quantization of filter coefficients, which is called perturbation error.
19. What is the expression for the perturbation error Δpi?
A. \(\sum_{k=1}^N \frac{∂p_i}{∂a_k} \Delta a_k\)
B. \(\sum_{k=1}^N p_i \Delta a_k\)
C. \(\sum_{k=1}^N \Delta a_k\)
D. None of the mentioned
Answer: A
The perturbation error Δpi can be expressed as
Δpi=\(\sum_{k=1}^N \frac{∂p_i}{∂a_k} \Delta a_k\)
Where \(\frac{∂p_i}{∂a_k}\), the partial derivative of pi with respect to ak, represents the incremental change in the pole pi due to a change in the coefficient ak. Thus the total error Δpi is expressed as a sum of the incremental errors due to changes in each of the coefficients ak.
20. Which of the following is the expression for \(\frac{∂p_i}{∂a_k}\)?
A. \(\frac{-p_i^{N+k}}{\prod_{l=1}^n p_i-p_l}\)
B. \(\frac{p_i^{N-k}}{\prod_{l=1}^n p_i-p_l}\)
C. \(\frac{-p_i^{N-k}}{\prod_{l=1}^n p_i-p_l}\)
D. None of the mentioned
Answer: C
The expression for \(\frac{∂p_i}{∂a_k}\) is given as follows
21. If the poles are tightly clustered as they are in a narrow band filter, the lengths of |pi-pl| are large for the poles in the vicinity of pi.
A. True
B. False
Answer: B
If the poles are tightly clustered as they are in a narrow band filter, the lengths of |pi-pl| are small for the poles in the vicinity of pi. These small lengths will contribute to large errors and hence a large perturbation error results.
22. Which of the following operation has to be done on the lengths of |pi-pl| in order to reduce the perturbation errors?
A. Maximize
B. Equalize
C. Minimize
D. None of the mentioned
Answer: A
The perturbation error can be minimized by maximizing the lengths of |pi-pl|. This can be accomplished by realizing the high order filter with either single pole or double pole filter sections.
23. The sensitivity analysis made on the poles of a system results in which of the following of the IIR filters?
A. Poles
B. Zeros
C. Poles & Zeros
D. None of the mentioned
Answer: B
The sensitivity analysis made on the poles of a system results in the zeros of the IIR filters.
24. Which of the following is the equivalent representation of the denominator of the system function of a general IIR filter?
A. \(\prod_{k=1}^N (1+p_k z^{-1})\)
B. \(\prod_{k=1}^N (1+p_k z^{-k})\)
C. \(\prod_{k=1}^N (1-p_k z^{-k})\)
D. \(\prod_{k=1}^N (1-p_k z^{-1})\)
Answer: D
We know that the system function of a general IIR filter is given by the equation
25. The quantization inherent in the finite precision arithmetic operations renders the system linear.
A. True
B. False
Answer: B
In the realization of a digital filter, either in digital hardware or in software on a digital computer, the quantization inherent in the finite precision arithmetic operations renders the system linear.
