# Digital Signal Processing Questions and Answers – Design of Linear Phase FIR Filters Using Windows – 1

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This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Design of Linear Phase FIR Filters Using Windows-1”.

1. 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
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d) None of the mentioned

Explanation: We know that the rectangular window of length M-1 is defined as
w(n)=1, n=0,1,2…M-1
=0, else where.

2. 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

Explanation: According to the basic formula of convolution, the multiplication of two signals w(n) and h(n) in time domain is equivalent to the convolution of their respective Fourier transforms W(w) and H(w).

3. 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) $$e^{-jω(M-1)/2} \frac{sin⁡(\frac{ωM}{2})}{sin⁡(\frac{ω}{2})}$$

Explanation: We know that the Fourier transform of a function w(n) is defined as
W(ω)=$$\sum_{n=0}^{M-1} w(n) e^{-jωn}$$
For a rectangular window, w(n)=1 for n=0,1,2….M-1
Thus we get
W(ω)=$$\sum_{n=0}^{M-1} w(n) e^{-jωn}=e^{-jω(M-1)/2} \frac{sin⁡(\frac{ωM}{2})}{sin⁡(\frac{ω}{2})}$$

4. What is the magnitude response |W(ω)| of a rectangular window function?
a) $$\frac{|sin(ωM/2)|}{|sin(ω/2)|}$$
b) $$\frac{|sin(ω/2)|}{|sin(ωM/2)|}$$
c) $$\frac{|cos(ωM/2)|}{|sin(ω/2)|}$$
d) None of the mentioned

Explanation: We know that for a rectangular window
W(ω)=$$\sum_{n=0}^{M-1} w(n) e^{-jωn}=e^{-jω(M-1)/2} \frac{sin⁡(\frac{ωM}{2})}{sin⁡(\frac{ω}{2})}$$
Thus the window function has a magnitude response
|W(ω)|=$$\frac{|sin(ωM/2)|}{|sin(ω/2)|}$$

5. 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

Explanation: The width of the main lobe width is measured to the first zero of W(ω)) is 4π/M.

6. The width of each side lobes decreases with an increase in M.
a) True
b) False

Explanation: 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.

7. 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

Explanation: The height of each side lobes increase with an increase in M such a manner that the area under each side lobe remains invariant to changes in M.

8. As M is increased, W(ω) becomes wider and the smoothening produced by the W(ω) is increased.
a) True
b) False

Explanation: As M is increased, W(ω) becomes narrower and the smoothening produced by the W(ω) is reduced.

9. 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

Explanation: 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.

10. The width of each side lobes decreases with an decrease in M.
a) True
b) False

Explanation: 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.

11. What is the approximate transition width of main lobe of a Hamming window?
a) 4π/M
b) 8π/M
c) 12π/M
d) 2π/M

Explanation: 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.

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