Finite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 230

7.3 WINDOW METHOD
In this section, the window method (Fourier transform design with window functions) is developed to
remedy the undesirable Gibbs oscillations in the passband and stopband of the designed FIR filter.
Recall that the Gibbs oscillations originate from the abrupt truncation of the infinite-length coeffi-
cient sequence. Then it is natural to seek a window function, which is symmetrical and can gradually
weight the designed FIR coefficients down to zeros at both ends for the range M n M .
Applying the window sequence to the filter coefficients gives

h w ðnÞ¼hðnÞ$wðnÞ
where wðnÞ designates the window function. Common window functions used in the FIR filter design
are as follows:

1. Rectangular window:

w rec ðnÞ¼ 1 ; M n M (7.15)

2. Triangular (Bartlett) window:

w tri ðnÞ¼ 1 jnj

M ; M n M

(7.16)

3. Hanning window:

w han ðnÞ¼ 0 : 5 þ 0 : 5 cos np
M

; M n M (7.17)

4. Hamming window:

w ham ðnÞ¼ 0 : 54 þ 0 : 46 cos np
M

; M n M (7.18)

5. Blackman window:

þ 0 : 08 cos 2 np
M

w black ðnÞ¼ 0 : 42 þ 0 : 5 cos np
M

; M n M (7.19)

In addition, there is another popular window function, called the Kaiser window (detailed
information can be found in Oppenheim, Shaffer, and Buck [1999]). As we expected, the rect-
angular window function has a constant value of 1 within the window, and hence only does
truncation. For comparison, shapes of the other window functions from Equations (7.16) to (7.19)
are plotted in Figure 7.9 for the case of 2 M þ 1 ¼ 81.
We apply the Hamming window function in Example 7.4.