2. 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.

3. 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).

4. 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.

5. 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.

6. 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

7. 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.

8. 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.

9. 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.

10. 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