101. 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
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)
102. 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).
103. 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.
104. 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.
105. 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.
106. 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.
107. 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.
108. Which of the following is not suitable either as a low pass or a high pass filter?
A. h(n) symmetric and M odd
B. h(n) symmetric and M even
C. h(n) anti-symmetric and M odd
D. h(n) anti-symmetric and M even
Answer: C
If h(n)=-h(M-1-n) and M is odd, we get H(0)=0 and H(π)=0. Consequently, this is not suitable as either a low pass filter or a high pass filter.
109. The anti-symmetric condition with M even is not used in the design of which of the following linear-phase FIR filter?
A. Low pass
B. High pass
C. Bandpass
D. Bans stop
Answer: A
When h(n)=-h(M-1-n) and M is even, we know that H(0)=0. Thus it is not used in the design of a low pass linear phase FIR filter.
110. The anti-symmetric condition is not used in the design of a low pass linear phase FIR filter.
A. True
B. False
Answer: A
We know that if h(n)=-h(M-1-n) and M is odd, we get H(0)=0 and H(π)=0. Consequently, this is not suitable as either a low pass filter or a high pass filter and when h(n)=-h(M-1-n) and M is even, we know that H(0)=0.
Thus it is not used in the design of a low pass linear phase FIR filter. Thus the anti-symmetric condition is not used in the design of low pass linear phase FIR filter.
111. Which of the following defines the rectangular window function of length M-1?
A. w(n)=1, n=0,1,2…M-1 =0, else where
B. w(n)=1, n=0,1,2…M-1 =-1, else where
C. w(n)=0, n=0,1,2…M-1 =1, else where
D. None of the mentioned
Answer: A
We know that the rectangular window of length M-1 is defined as
w(n)=1, n=0,1,2…M-1
=0, elsewhere.
112. The multiplication of the window function w(n) with h(n) is equivalent to the multiplication of H(w) and W(w).
A. True
B. False
Answer: B
According to the basic formula of convolution, the multiplication of two signals w(n) and h(n) in the time domain is equivalent to the convolution of their respective Fourier transforms W(w) and H(w).
113. What is the Fourier transform of the rectangular window of length M-1?
A. \(e^{jω(M-1)/2} \frac{sin(\frac{ωM}{2})}{sin(\frac{ω}{2})}\)
B. \(e^{jω(M+1)/2} \frac{sin(\frac{ωM}{2})}{sin(\frac{ω}{2})}\)
C. \(e^{-jω(M+1)/2} \frac{sin(\frac{ωM}{2})}{sin(\frac{ω}{2})}\)
D. \(e^{-jω(M-1)/2} \frac{sin(\frac{ωM}{2})}{sin(\frac{ω}{2})}\)
Answer: D
We know that the Fourier transform of a function w(n) is defined as
115. What is the width of the main lobe of the frequency response of a rectangular window of length M-1?
A. π/M
B. 2π/M
C. 4π/M
D. 8π/M
Answer: C
The width of the main lobe width is measured to the first zero of W(ω)) is 4π/M.
116. The width of each side lobes decreases with an increase in M.
A. True
B. False
Answer: A
Since the width of the main lobe is inversely proportional to the value of M if the value of M increases then the main lobe becomes narrower. In fact, the width of each side lobes decreases with an increase in M.
117. With an increase in the value of M, the height of each side lobe ____________
A. Do not vary
B. Does not depend on value of M
C. Decreases
D. Increases
Answer: D
The height of each side lobes increases with an increase in M such a manner that the area under each side lobe remains invariant to changes in M.
118. As M is increased, W(ω) becomes wider and the smoothening produced by the W(ω) is increased.
A. True
B. False
Answer: B
As M is increased, W(ω) becomes narrower and the smoothening produced by the W(ω) is reduced.
119. Which of the following windows has a time-domain sequence h(n)=(1-frac{2|n-frac{M-1}{2}|}{M-1})?
A. Bartlett window
B. Blackman window
C. Hanning window
D. Hamming window
Answer: A
The Bartlett window which is also called as triangular window has a time domain sequence as
h(n)=(1-frac{2|n-frac{M-1}{2}|}{M-1}), 0≤n≤M-1.
120. The width of each side lobes decreases with a decrease in M.
A. True
B. False
Answer: B
Since the width of the main lobe is inversely proportional to the value of M if the value of M increases then the main lobe becomes narrower. In fact, the width of each side lobes decreases with an increase in M.
121. What is the approximate transition width of the main lobe of a Hamming window?
A. 4π/M
B. 8π/M
C. 12π/M
D. 2π/M
Answer: B
The transition width of the main lobe in the case of Hamming window is equal to 8π/M where M is the length of the window.