Digital Signal Processing Multiple Choice Questions on “Design of FIR Filters”.
1. Which of the following is the difference equation of the FIR filter of length M, input x(n) and output y(n)?
A. y(n)=(sum_{k=0}^{M+1} b_k x(n+k))
B. y(n)=(sum_{k=0}^{M+1} b_k x(n-k))
C. y(n)=(sum_{k=0}^{M-1} b_k x(n-k))
D. None of the mentioned
Answer: C
Clarification: An FIR filter of length M with input x(n) and output y(n) is described by the difference equation
y(n)=(sum_{k=0}^{M-1} b_k x(n-k))
where {bk} is the set of filter coefficients.
2. The lower and upper limits on the convolution sum reflect the causality and finite duration characteristics of the filter.
A. True
B. False
Answer: A
Clarification: We can express the output sequence as the convolution of the unit sample response h(n) of the system with the input signal. The lower and upper limits on the convolution sum reflect the causality and finite duration characteristics of the filter.
3. Which of the following condition should the unit sample response of a FIR filter satisfy to have a linear phase?
A. h(M-1-n) n=0,1,2…M-1
B. ±h(M-1-n) n=0,1,2…M-1
C. -h(M-1-n) n=0,1,2…M-1
D. None of the mentioned
Answer: B
Clarification: An FIR filter has an linear phase if its unit sample response satisfies the condition
h(n)= ±h(M-1-n) n=0,1,2…M-1.
4. If H(z) is the z-transform of the impulse response of an FIR filter, then which of the following relation is true?
A. zM+1.H(z-1)=±H(z)
B. z-(M+1).H(z-1)=±H(z)
C. z(M-1).H(z-1)=±H(z)
D. z-(M-1).H(z-1)=±H(z)
Answer: D
Clarification: We know that H(z)=(sum_{k=0}^{M-1} h(k)z^{-k}) and h(n)=±h(M-1-n) n=0,1,2…M-1
When we incorporate the symmetric and anti-symmetric conditions of the second equation into the first equation and by substituting z-1 for z, and multiply both sides of the resulting equation by z-(M-1) we get z-(M-1).H(z-1)=±H(z)
5. The roots of the polynomial H(z) are identical to the roots of the polynomial H(z-1).
A. True
B. False
Answer: A
Clarification: We know that z-(M-1).H(z-1)=±H(z). This result implies that the roots of the polynomial H(z) are identical to the roots of the polynomial H(z-1).
6. The roots of the equation H(z) must occur in ________________
A. Identical
B. Zero
C. Reciprocal pairs
D. Conjugate pairs
Answer: C
Clarification: We know that the roots of the polynomial H(z) are identical to the roots of the polynomial H(z-1). Consequently, the roots of H(z) must occur in reciprocal pairs.
7. If the unit sample response h(n) of the filter is real, complex valued roots need not occur in complex conjugate pairs.
A. True
B. False
Answer: B
Clarification: We know that the roots of the polynomial H(z) are identical to the roots of the polynomial H(z-1). This implies that if the unit sample response h(n) of the filter is real, complex valued roots must occur in complex conjugate pairs.
8. What is the value of h(M-1/2) if the unit sample response is anti-symmetric?
A. 0
B. 1
C. -1
D. None of the mentioned