Test: FIR Filters Windows Design - 1

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.

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).

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

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.

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.

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

Explanation: The Bartlett window which is also called as triangular window has a time domain sequence as
, 0≤n≤M-1.

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.

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.

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

For a rectangular window, w(n)=1 for n=0,1,2….M-1
Thus we get