# FIR Filters Windows Design MCQ [Free PDF] – Objective Question Answer for FIR Filters Windows Design Quiz

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
D. None of the mentioned

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

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

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).

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

We know that the Fourier transform of a function w(n) is defined as

$$\sum_{n=0}^{M-1} w(n) e^{-jωn}$$

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

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

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

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

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.

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

A. True
B. False

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

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 a decrease in M.

A. True
B. False