Digital Signal Processing MCQ [Free PDF] – Objective Question Answer for Digital Signal Processing Quiz

The magnitude square response shown in the given figure is for the Butterworth filter.

Given Ωc=1000π and Ωs=2000π

For an attenuation of 40dB, δ2=0.01.

We know that

N=\(\frac{log⁡[(\frac{1}{δ_2^2})-1]}{2log⁡[\frac{Ω_s}{Ω_s}]}\)

Thus by substituting the corresponding values in the above equation, we get N=6.64

To meet the desired specifications, we select N=7.

Type-1 Chebyshev filters are all-pole filters that exhibit equi-ripple behavior in the passband and a monotonic characteristic in the stopband.

Type-1 Chebyshev filters are all-pole filters whereas the family of type-2 Chebyshev filters contains both poles and zeros.

Type-2 Chebyshev filters exhibit equi-ripple behavior in the stopband and a monotonic characteristic in the passband.

The zeros of this class of filters lie on the imaginary axis in the s-plane.

In order to understand the frequency-domain behavior of Chebyshev filters, it is of utmost importance to define a Chebyshev polynomial and then its properties. A Chebyshev polynomial of degree N is defined as

TN(x) = cos(Ncos-1x), |x|≤1

cosh(Ncosh-1x), |x|>1

Since the passband is monotonic in behavior and the stopband exhibit equi-ripple behavior, it is the magnitude square response of a type-2 Chebyshev filter.

Since the magnitude square response of the type-2 Chebyshev filter, it has an odd number of maxima and minima in the stopband, the order of the filter is odd i.e., 3.

An elliptical filter is a filter that exhibits equi-ripple behavior in both passband and stopband of the magnitude square response.

An important characteristic of the Bessel filter is the linear phase response over the passband of the filter. As a consequence, Bessel filters have a larger transition bandwidth, but their phase is linear within the passband.

The frequency characteristic given in the figure is the magnitude response of a 13-order type-2 Chebyshev filter.

If Ωu is the desired passband edge frequency of the new high pass filter, then the transfer function of this new high pass filter is obtained by using the transformation s→Ωu/s. If ΩS and Ω’S are the stopband frequencies of prototype and transformed filters respectively, then the backward design equation is given by

ΩS=\(\frac{Ω_u}{Ω’_S}\)

The phase response given in the figure belongs to the frequency characteristic of a 7-order elliptic filter.

Let H(s) denote the transfer function of a low pass analog filter with a passband edge frequency ΩP equal to 1 rad/sec. This filter is known as the analog low pass normalized prototype.

The low pass-to-high pass transformation is simply achieved by replacing s by 1/s. If the desired high pass filter has the passband edge frequency Ωu, then the transformation is

s → Ωu / s.

If Ωu is the desired passband edge frequency of the new low pass filter, then the transfer function of this new low pass filter is obtained by using the transformation s → s / Ωu. If ΩS and Ω’S are the stopband frequencies of prototype and transformed filters respectively, then the backward design equation is given by

ΩS=\(\frac{Ω’_S}{Ω_u}\)

If Ωu is the desired passband edge frequency of the new low pass filter, then the transfer function of this new low pass filter is obtained by using the transformation s → s / Ωu.

If Ωu and Ωl are the upper and lower cutoff pass band frequencies of the desired bandpass filter, then the transformation to be performed on the normalized low pass filter is

s→\(\frac{s^2+Ω_u Ω_l}{s(Ω_u-Ω_l)}\)

If Ωu and Ωl are the upper and lower cutoff pass band frequencies of the desired bandpass filter and Ω1 and Ω2 are the lower and upper cutoff stopband frequencies of the desired bandpass filter, then the backward design equation is
ΩS=Min{|A|,|B|}

where, A=\(\frac{-Ω_1^2+Ω_u Ω_l}{Ω_1 (Ω_u-Ω_l)}\) and B=\(\frac{Ω_2^2-Ω_u Ω_l}{Ω_2 (Ω_u-Ω_l)}\).