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[CHAP. 9
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Mainlobe 'IT Width Fig. 9-2. The DTFT of a typical window, which is characterized by the width of its main lobe. A. and the peak amplitude of its side lobes, A, relative to the amplitude of W ( d ' " )at o = 0. Ideally, the main-lobe width should be narrow, and the side-lobe amplitude should be small. However, for a fixed-length window, these cannot be minimized independently. Some general properties of windows are as follows:
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1. As the length N of the window increases, the width of the main lobe decreases, which results in a decrease
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in the transition width between passbands and stopbands. This relationship is given approximately by
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where A f is the transition width, and c is a parameter that depends on the window.
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2. The peak side-lobe amplitude of the window is determined by the shape of the window, and it is essentially independent of the window length. 3. If the window shape is changed to decrease the side-lobe amplitude, the width of the main lobe will generally increase.
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Listed in Table 9.2 are the side-lobe amplitudes of several windows along with the approximate transition width and stopband attenuation that results when the given window is used to design an N th-order low-pass filter.
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Table 9-1 Some Common Windows
Rectangular
w(n) =
O s n s N
else
Hanning'
else
Hamming
else
Blackman
'In the literature, this window is also called a Hann window or a von Hann window.
else
CHAP. 91
FILTER DESIGN
Table 9-2 The Peak Side-Lobe Amplitude of Some Common Windows and the Approximate Transition Width and Stopband Attenuation of an Nth-Order Low-Pass Filter Designed Using the Given Window.
Window Side-Lobe Amplitude (dB)
- 13 -31 -41 -57
Transition Width (.Af ) 0.9/N 3.1IN 3.3/N 5S/N
Stopband Attenuation (dB) -21 -44 -53 - 74
Rectangular Hanning Hamming Blackman
EXAMPLE 9.3.1 specifications:
Suppose that we would like to design an FIR linear phase low-pass filter according to the following
For a stopband attenuation of 20 log(0.O I ) = -40 dB. we may use a Hanning window. Although we could also use a Hamming or a Blackman window, these windows would overdesign the filter and produce a larger stopband attenuation at the expense of an increase in the transition width. Because the specification calls for a transition width of Aw = w, - w, = 0.02n, or Af = 0.01, with NAf = 3.1 for a Hanning window (see Table 9.2), an estimate of the required filter order is
The last step is to find the unit sample response of the ideal low-pass filter that is to be windowed. With a cutoff frequency of w,. = (w, w,)/2 := 0.2n, and a delay of cr = N/2 = 155, the unit sample response is
In addition to the windows listed in Table 9-1, Kaiser developed a family of windows that are defined by
is where a = N / 2 , and l o ( . ) a zeroth-order modified Bessel function of the first kind, which may be easily generated using the power series expansion
The parameter determines the shape of the window and thus controls the trade-off between main-lobe width and side-lobe amplitude. A Kaiser window is nearly optimum in the sense of having the most energy in its main lobe for a given side-lobe amplitude. Table 9-3 illustrates the effect of changing the parameter /3. There are two empirically derived relationships for the Kaiser window that facilitate the use of these windows to design FIR filters. The first relates the stopband ripple of a low-pass filter, a, = -20 log(6,), to the parameter B,
FILTER DESIGN
[CHAP. 9
Table 9-3 Characteristics of the Kaiser Window as a Function of 0
Parameter Side Lobe (dB) Transition Width (NAf Stopband Attenuation (dB)
The second relates N to the transition width Af and the stopband attenuation a,,
Note that if a, < 21 dB, a rectangular window may be used
(B = O ) , and N
= O.9/Af.
EXAMPLE 9 3 2 Suppose that we would like to design a low-pass filter with a cutoff frequency w, = n / 4 , a transition .. width Aw = 0.02n, and a stopband ripple 6, = 0.01. Because a, = -20 log(O.01)= -40. the Kaiser window parameter is
With A f = Aw/2n = 0.01, we have
where is the unit sample response of the ideal low-pass filter.
Although it is simple to design a filter using the window design method, there are some limitations with this method. First, it is necessary to find a closed-form expression for hd(n)(or it must be approximated using a very long DFT). Second, for a frequency selective filter, the transition widths between frequency bands, and the ripples within these bands, will be approximately the same. As a result, the window design method requires that the filter be designed to the tightest tolerances in all of the bands by selecting the smallest transition width and the smallest ripple. Finally, window design filters are not, in general, optimum in the sense that they do not have the smallest possible ripple for a given filter order and a given set of cutoff frequencies.
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