Digital Signal Processing Questions and Answers – Chebyshev Filters – 1

This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Chebyshev Filters-1”.

1. 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
View Answer

Answer: c
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.

2. What is the formula for chebyshev polynomial TN(x) in recursive form?
a) 2TN-1(x) – TN-2(x)
b) 2TN-1(x) + TN-2(x)
c) 2xTN-1(x) + TN-2(x)
d) 2xTN-1(x) – TN-2(x)
View Answer

Answer: d
Explanation: We know that a chebyshev polynomial of degree N is defined as
TN(x) = cos(Ncos-1x), |x|≤1
cosh(Ncosh-1x), |x|>1
From the above formula, it is possible to generate chebyshev polynomial using the following recursive formula
TN(x)= 2xTN-1(x)-TN-2(x), N ≥ 2.
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3. What is the value of chebyshev polynomial of degree 0?
a) 1
b) 0
c) -1
d) 2
View Answer

Answer: a
Explanation: We know that a chebyshev polynomial of degree N is defined as
TN(x) = cos(Ncos-1x), |x|≤1
cosh(Ncosh-1x), |x|>1
For a degree 0 chebyshev filter, the polynomial is obtained as
T0(x)=cos(0)=1.

4. What is the value of chebyshev polynomial of degree 1?
a) 1
b) x
c) -1
d) -x
View Answer

Answer: b
Explanation: We know that a chebyshev polynomial of degree N is defined as
TN(x) = cos(Ncos-1x), |x|≤1
cosh(Ncosh-1x), |x|>1
For a degree 1 chebyshev filter, the polynomial is obtained as
T0(x)=cos(cos-1x)=x.
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5. What is the value of chebyshev polynomial of degree 3?
a) 3x3+4x
b) 3x3-4x
c) 4x3+3x
d) 4x3-3x
View Answer

Answer: d
Explanation: We know that a chebyshev polynomial of degree N is defined as
TN(x) = cos(Ncos-1x), |x|≤1; TN(x) = cosh(Ncosh-1x), |x|>1
And the recursive formula for the chebyshev polynomial of order N is given as
TN(x)=2xTN-1(x)-TN-2(x)
Thus for a chebyshev filter of order 3, we obtain
T3(x)=2xT2(x)-T1(x)=2x(2x2-1)-x=4x3-3x.

6. What is the value of chebyshev polynomial of degree 5?
a) 16x5+20x3-5x
b) 16x5+20x3+5x
c) 16x5-20x3+5x
d) 16x5-20x3-5x
View Answer

Answer: c
Explanation: We know that a chebyshev polynomial of degree N is defined as
TN(x) = cos(Ncos-1x), |x|≤1
= cosh(Ncosh-1x), |x|>1
And the recursive formula for the chebyshev polynomial of order N is given as
TN(x)= 2xTN-1(x)-TN-2(x)
Thus for a chebyshev filter of order 5, we obtain
T5(x)=2xT4(x)-T3(x)=2x(8x4-8x2+1)-(4x3-3x)=16x5-20x3+5x.
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7. For |x|≤1, |TN(x)|≤1, and it oscillates between -1 and +1 a number of times proportional to N.
a) True
b) False
View Answer

Answer: a
Explanation: For |x|≤1, |TN(x)|≤1, and it oscillates between -1 and +1 a number of times proportional to N.
The above is evident from the equation,
TN(x) = cos(Ncos-1x), |x|≤1.

8. Chebyshev polynomials of odd orders are _____________
a) Even functions
b) Odd functions
c) Exponential functions
d) Logarithmic functions
View Answer

Answer: b
Explanation: Chebyshev polynomials of odd orders are odd functions because they contain only odd powers of x.
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9. What is the value of TN(0) for even degree N?
a) -1
b) +1
c) 0
d) ±1
View Answer

Answer: d
Explanation: We know that a chebyshev polynomial of degree N is defined as
TN(x) = cos(Ncos-1x), |x|≤1
cosh(Ncosh-1x), |x|>1
For x=0, we have TN(0)=cos(Ncos-10)=cos(N.π/2)=±1 for N even.

10. TN(-x)=(-1)NTN(x).
a) True
b) False
View Answer

Answer: a
Explanation: We know that a chebyshev polynomial of degree N is defined as
TN(x) = cos(Ncos-1x), |x|≤1
cosh(Ncosh-1x), |x|>1
=> TN(-x)= cos(Ncos-1(-x))=cos(N(π-cos-1x))=cos(Nπ-Ncos-1x)=(-1)N cos(Ncos-1x)=(-1)NTN(x)
Thus we get, TN(-x)=(-1)NTN(x).

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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