Digital Filters Design MCQ [Free PDF] – Objective Question Answer for Digital Filters Design Quiz

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.

When r=1, we get σ=0 and

Ω = \(\frac{2}{T} [\frac{2 sin⁡ω}{1+1+2 cos⁡ω}]\)

=>ω=\(\frac{2}{T} tan^{-1}⁡(\frac{ΩT}{2})\).

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.

If ak and bk are the filter coefficients, then the transfer function of a general IIR filter is given by the expression

H(z)=\(\frac{\sum_{k=0}^M b_k z^{-k}}{1+\sum_{k=1}^N a_k z^{-k}}\)

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.

We know that the system function of a general IIR filter is given by the equation

H(z)=\(\frac{\sum_{k=0}^M b_k z^{-k}}{1+\sum_{k=1}^N a_k z^{-k}}\)

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).

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.

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.

The expression for \(\frac{∂p_i}{∂a_k}\) is given as follows

\(\frac{∂p_i}{∂a_k}=\frac{-p_i^{N-k}}{\prod_{l=1}^n p_i-p_l}\)

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.

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.

The sensitivity analysis made on the poles of a system results in the zeros of the IIR filters.

We know that the system function of a general IIR filter is given by the equation

H(z)=\(\frac{\sum_{k=0}^M b_k z^{-k}}{1+\sum_{k=1}^N a_k z^{-k}}\)

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).

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.

In the recursive systems, the nonlinearities due to the finite-precision arithmetic operations often cause periodic oscillations to occur in the output even when the input sequence is zero or some non-zero constant value.

In the recursive systems, the nonlinearities due to the finite-precision arithmetic operations often cause periodic oscillations to occur in the output even when the input sequence is zero or some non-zero constant value. The oscillations thus produced in the output are known as ‘limit cycles’.

The oscillations in the output of the recursive system are called limit cycles and are directly attributable to round-off errors in multiplication and overflow errors in addition.

The amplitudes of the output during a limit circle are confined to a range of values that is called the ‘dead band’ of the filter.