# Digital Signal Processing Questions and Answers – Design of Optimum Equi Ripple Linear Phase FIR Filters – 2

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This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Design of Optimum Equi Ripple Linear Phase FIR Filters-2”.

1. If δ1 represents the ripple in the pass band for a chebyshev filter, then which of the following conditions is true?
a) 1-δ1 ≤ Hr(ω) ≤ 1+δ1; |ω|≤ωP
b) 1+δ1 ≤ Hr(ω) ≤ 1-δ1; |ω|≥ωP
c) 1+δ1 ≤ Hr(ω) ≤ 1-δ1; |ω|≤ωP
d) 1-δ1 ≤ Hr(ω) ≤ 1+δ1; |ω|≥ωP

Explanation: Let us consider the design of a low pass filter with the pass band edge frequency ωP and the ripple in the pass band is δ1, then from the general specifications of the chebyshev filter, in the pass band the filter frequency response should satisfy the condition
1- δ1 ≤ Hr(ω) ≤ 1+δ1; |ω|≤ωP

2. If the filter has symmetric unit sample response with M odd, then what is the value of Q(ω)?
a) cos(ω/2)
b) sin(ω/2)
c) 1
d) sinω

Explanation: If the filter has a symmetric unit sample response, then we know that
h(n)=h(M-1-n)
and for M odd in this case, Q(ω)=1.

3. If the filter has anti-symmetric unit sample response with M odd, then what is the value of Q(ω)?
a) cos(ω/2)
b) sin(ω/2)
c) 1
d) sinω

Explanation: If the filter has a anti-symmetric unit sample response, then we know that
h(n)= -h(M-1-n)
and for M odd in this case, Q(ω)=sin(ω).
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4. In which of the following way the real valued desired frequency response is defined?
a) Unity in stop band and zero in pass band
b) Unity in both pass and stop bands
c) Unity in pass band and zero in stop band
d) Zero in both stop and pass band

Explanation: The real valued desired frequency response Hdr(ω) is simply defined to be unity in the pass band and zero in the stop band.

5. The error function E(ω) should exhibit at least how many extremal frequencies in S?
a) L
b) L-1
c) L+1
d) L+2

Explanation: According to Alternation theorem, a necessary and sufficient condition for P(ω) to be unique, best weighted chebyshev approximation, is that the error function E(ω) must exhibit at least L+2 extremal frequencies in S.
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6. The filter designs that contain maximum number of alternations are called as ______________
a) Extra ripple filters
b) Maximal ripple filters
c) Equi ripple filters
d) None of the mentioned

Explanation: In general, the filter designs that contain maximum number of alternations or ripples are called as maximal ripple filters.

7. Remez exchange algorithm is an iterative algorithm used in error approximation.
a) True
b) False

Explanation: Initially, we neither know the set of external frequencies nor the parameters. To solve for the parameters, we use an iterative algorithm called the Remez exchange algorithm, in which we begin by guessing at the set of extremal frequencies.

8. When |E(ω)|≤δ for all frequencies on the dense set, the optimal solution has been found in terms of the polynomial H(ω).
a) True
b) False

Explanation: |E(ω)|≥δ for some frequencies on the dense set, then a new set of frequencies corresponding to the L+2 largest peaks of |E(ω)| are selected and computation is repeated. Since the new set of L+2 extremal frequencies are selected to increase in each iteration until it converges to the upper bound, this implies that when |E(ω)|≤δ for all frequencies on the dense set, the optimal solution has been found in terms of the polynomial H(ω).

9. In Parks-McClellan program, an array of maximum size 10 that specifies the weight function in each band is denoted by?
a) WTX
b) FX
c) EDGE
d) None of the mentioned

Explanation: FX denotes an array of maximum size 10 that specifies the weight function in each band.

10. The filter designs which are formulated using chebyshev approximating problem have ripples in?
a) Pass band
b) Stop band
c) Pass & Stop band
d) Restart band

Explanation: The chebyshev approximation problem is viewed as an optimum design criterion on the sense that the weighted approximation error between the desired frequency response and the actual frequency response is spread evenly across the pass band and evenly across the stop band of the filter minimizing the maximum error. The resulting filter designs have ripples in both pass band and stop band.

11. If the filter has symmetric unit sample response with M even, then what is the value of Q(ω)?
a) cos(ω/2)
b) sin(ω/2)
c) 1
d) sinω

Explanation: If the filter has a symmetric unit sample response, then we know that
h(n)=h(M-1-n)
and for M even in this case, Q(ω)=cos(ω/2).

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