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

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

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

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. z^{M+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

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 multiplying 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

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

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

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

Answer: A

When h(n)=-h(M-1-n), the unit sample response is anti-symmetric. For M odd, the center point of the anti-symmetric is n=M-1/2. Consequently, h(M-1/2)=0.

9. What is the number of filter coefficients that specify the frequency response for h(n) symmetric?

A. (M-1)/2 when M is odd and M/2 when M is even
B. (M-1)/2 when M is even and M/2 when M is odd
C. (M+1)/2 when M is even and M/2 when M is odd
D. (M+1)/2 when M is odd and M/2 when M is even

Answer: D

We know that, for a symmetric h(n), the number of filter coefficients that specify the frequency response is (M+1)/2 when M is odd and M/2 when M is even.

10. What is the number of filter coefficients that specify the frequency response for h(n) anti-symmetric?

A. (M-1)/2 when M is even and M/2 when M is odd
B. (M-1)/2 when M is odd and M/2 when M is even
C. (M+1)/2 when M is even and M/2 when M is odd
D. (M+1)/2 when M is odd and M/2 when M is even

Answer: B

We know that for an anti-symmetric h(n) h(M-1/2)=0 and thus the number of filter coefficients that specify the frequency response is (M-1)/2 when M is odd and M/2 when M is even.