U. Glaeser

is the FIR filter order. This method of filter design turns out to give the optimal least-squares filter. However,
the least-squares filter is not an acceptable filter, especially when the application calls for a frequency
selective filter. The reason is that the least-squares approximation exhibits an overshoot called the Gibbs’
phenomenon, which means that the magnitude of the error is large at the cutoff frequency regardless of
the filter order. To reduce the magnitude error near the cutoff frequency, the strict truncation (done by
applying a rectangular window) can be replaced by other windowing. Windowing for filter design involves
the multiplication of a finite-length window shape times the ideal impulse response. For example, the
1
ideal lowpass filter with delay µ = -- M has an impulse response that is infinitely long:
so the windowed filter coefficients are b[n] = w[n]h[n] for n = 0, 1, 2,…, M.
Different windows generate filter responses that allow a trade-off between the sharpness of transition
region and the error magnitude. Popular windows are: Bartlett, Hamming, vonHann (or Hanning), and
Kaiser, but for filter design the only important one is the Kaiser window, which is based on the modified
Bessel function. The Kaiser window is defined as
where I 0(x) is the modified Bessel function, and the parameter β is chosen to control the ripple height
in the stopband with the relationship:
where δ dB = −20 log 10 (δ stopband) is the ripple height in dB. The design of the Kaiser window is illustrated
in Fig. 26.6. Examples of digital filters designed via windowing are shown in Fig. 26.7.
Stop band Attenuation in dB
80
70
60
50
40
30
FIGURE 26.6 The Kaiser window: (a) shows the relationship between β and the ripple height in the stop band;
(b) shows examples of length-51 Kaiser windows (i.e., filter order = 50) with different parameters β. Note that, with
β = 0, the window is the rectangular window and, with β = 5, the window is very similar to the Hamming window.
© 2001 by CRC Press LLC
β =
w[ n]
2
h[ n]
sin (ω c( n – µ ))
= -----------------------------------, – ∞ < n < ∞
π ( n – µ )
I0( β 1 – n – µ )
= ---------------------------------------------------- , n = 0, 1, 2,…, M
I0( β)
( ) 2 /µ 2
0, δdB < 21
0.5842( δdB – 21)
0.4
⎧
⎪
⎨
+ 0.07886( δdB – 21),
21≤δdB ≤50
⎪
⎩ 0.1102( δdB – 8.7),
δdB > 50
Kaiser Window FIR Design: δ dB VS β
20
0 1 2 3 4 5 6 7 8
Design Parameter, β
Amplitude
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Kaiser(0.0)
Kaiser Windows: β = 0, 3, 5, 8
Kaiser(8.0)
Kaiser(3.0)
Kaiser(5.0)
0
0 10 20 30 40 50
Index (n)
(a) (b)