# Digital Signal Processing Questions and Answers – Characteristics of Commonly Used Analog Filters

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This set of Digital Signal Processing aptitude tests focuses on “Characteristics of Commonly Used Analog Filters”.

1. Low pass Butterworth filters are also called as _____________
a) All-zero filter
b) All-pole filter
c) Pole-zero filter
d) None of the mentioned

Explanation: Low pass Butterworth filters are also called as all-pole filters because it has only non-zero poles.

2. What is the equation for magnitude square response of a low pass Butterworth filter?
a) $$\frac{1}{\sqrt{1+(\frac{Ω}{Ω_C})^{2N}}}$$
b) $$1+(\frac{Ω}{Ω_C})^{2N}$$
c) $$\sqrt{1+(\frac{Ω}{Ω_C})^{2N}}$$
d) None of the mentioned

Explanation: A Butterworth is characterized by the magnitude frequency response
|H(jΩ)| = $$\frac{1}{\sqrt{1+(\frac{Ω}{Ω_C})^{2N}}}$$
where N is the order of the filter and ΩC is defined as the cutoff frequency.

3. What is the transfer function of magnitude squared frequency response of the normalized low pass Butterworth filter?
a) $$\frac{1}{1+(s/j)^{-2N}}$$
b) $$1+(\frac{s}{j})^{-2N}$$
c) $$1+(\frac{s}{j})^{2N}$$
d) $$\frac{1}{1+(\frac{s}{j})^{2N}}$$

Explanation: We know that the magnitude squared frequency response of a normalized low pass Butterworth filter is given as
|H(jΩ)|2=$$\frac{1}{1+Ω^{2N}}$$ => HN(jΩ).HN(-jΩ)=$$\frac{1}{1+Ω^{2N}}$$
Replacing jΩ by ‘s’ and hence Ω by s/j in the above equation, we get
HN(s).HN(-s)=$$\frac{1}{1+(\frac{s}{j})^{2N}}$$ which is called the transfer function.

4. Which of the following is the band edge value of |H(Ω)|2?
a) (1+ε2)
b) (1-ε2)
c) 1/(1+ε2)
d) 1/(1-ε2)

Explanation: 1/(1+ε2) gives the band edge value of the magnitude square response |H(Ω)|2.

5. The magnitude square response shown in the below figure is for which of the following given filters?

a) Butterworth
b) Chebyshev
c) Elliptical
d) None of the mentioned

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

6. What is the order of a low pass Butterworth filter that has a -3dB bandwidth of 500Hz and an attenuation of 40dB at 1000Hz?
a) 4
b) 5
c) 6
d) 7

Explanation: 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.

7. Which of the following is true about type-1 chebyshev filter?
a) Equi-ripple behavior in pass band
b) Monotonic characteristic in stop band
c) Equi-ripple behavior in pass band & Monotonic characteristic in stop band
d) None of the mentioned

Explanation: Type-1 chebyshev filters are all-pole filters that exhibit equi-ripple behavior in pass band and a monotonic characteristic in the stop band.

8. Type-2 chebyshev filters consists of ______________
a) Only poles
b) Both poles and zeros
c) Only zeros
d) Cannot be determined

Explanation: Type-1 chebyshev filters are all-pole filters where as the family of type-2 chebyshev filters contains both poles and zeros.

9. Which of the following is false about the type-2 chebyshev filters?
a) Monotonic behavior in the pass band
b) Equi-ripple behavior in the stop band
c) Zero behavior
d) Monotonic behavior in the stop band

Explanation: Type-2 chebyshev filters exhibit equi-ripple behavior in stop band and a monotonic characteristic in the pass band.

10. The zeros of type-2 class of chebyshev filters lies on ___________
a) Imaginary axis
b) Real axis
c) Zero
d) Cannot be determined

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

11. Which of the following defines a chebyshev polynomial of order N, TN(x)?
a) cos(Ncos-1x) for all x
b) cosh(Ncosh-1x) for all x
c)

cos(Ncos-1x), |x|≤1
cosh(Ncosh-1x), |x|>1

d) None of the mentioned

Explanation: In order to understand the frequency-domain behavior of chebyshev filters, it is utmost important 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

12. The frequency response shown in the figure below belongs to which of the following filters?

a) Type-1 chebyshev
b) Type-2 chebyshev
c) Butterworth
d) Elliptical

Explanation: Since the pass band is monotonic in behavior and the stop band exhibit equi-ripple behavior, it is the magnitude square response of a type-2 chebyshev filter.

13. What is the order of the type-2 chebyshev filter whose magnitude square response is as shown in the following figure?

a) 2
b) 4
c) 6
d) 3

Explanation: Since from the magnitude square response of the type-2 chebyshev filter, it has odd number of maxima and minima in the stop band, the order of the filter is odd i.e., 3.

14. Which of the following is true about the magnitude square response of an elliptical filter?
a) Equi-ripple in pass band
b) Equi-ripple in stop band
c) Equi-ripple in pass band and stop band
d) None of the mentioned

Explanation: An elliptical filter is a filter which exhibit equi-ripple behavior in both pass band and stop band of the magnitude square response.

15. Bessel filters exhibit a linear phase response over the pass band of the filter.
a) True
b) False