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is minimized when h(n) is designed using the rectangular window design method. If ER is the squared error using a rectangular window, find the excess squared emor that results when a Hanning window is used instead of a rectangular window; that is, find an expression for
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where EH is the squared error using a Hanning window.
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Using Parseval's theorem. it is more convenient to express the least-squares error in the time domain as follows:
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FlLTER DESIGN Because e(n) = hd(n) - h(n) = hd(n) for n c 0 and n > N.
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where wH(n) and wx(n) are the Hanning and rectangular windows, respectively. However, the second sum is equal to zero. Therefore, the excess squared error is simply
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which is the desired relationship.
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Consider the following specifications for a low-pass filter:
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Design a linear phase FIR filter to meet these specifications using the window design method. Designing a low-pass filter with the window design method generally produces a filter with ripples of the same amplitude in the passband and stopband. Therefore, because the passband and stopband ripples in the filter specifications are the same, we only need to be concerned about the slopband ripple requirement. A stopband ripple of 6 , = 0.01 corresponds to a stopband attenuation of -40 dB. Therefore. froin Table 9-2 it follows that we may use a Hanning window, which provides an attenuation of approximately 44 dB. The specification on the transition band is that Aw = 0.05rr, or A f = 0.025. Therefore, the required filter order is
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and we have
With an ideal low-pass filter that has a cutoff frequency of w,. = 0.325 (the midpoint of the transition band), and a delay of N/2 = 62 so that hd(n) is placed symmetrically within the interval [O, 1241, we have
Therefore, the filter is 0.5 - 0.5 cos -
sin[0.325~(n 62)] ~ ( - 62) n
O S n s 124
Note that if we were to use a Hamming or a Blackman window instead of a Hanning window, the stopband and passband ripple requirements would have been exceeded, and the required filter order would have been larger. With a Blackman window, for example, the filter order required to meet the transition band requirement is
W e would like t o filter an analog signal x,(t) with a n analog low-pass filter that has a cutoff frequency f,. = 2 kHz, a transition width Af = 500 Hz, a n d a stopband attenuation o f 50 dB. This filter is t o b e implemented digitally, a s illustrated in the following figure:
CHAP. 91
FILTER DESIGN
4 1 )
y(n)
H(ejw)
yo ( t )
Design a digital filter to meet the analog filter specifications with a sampling frequency f, = 10 kHz.
With a sampling frequency of 10 kHz, the digital filter should have a cutoff frequency w, = 2 n f,./f, = 0.41~ and a transition bandwidth Aw = 2nAf/f, = O.lrr. For a stopband attenuation of 50 dB, we may use a Kaiser window with B = 0.l l02(50 - 8.7) = 4.55 For the length of the window, we have
or N = 59. Finally, the unit sample response of the ideal filter that is to be windowed is a low-pass filter with a ~ cutoff frequency w,. = 0 . 4 and a delay N / 2 = 29.5. Therefore,
where w ( n ) is a Kaiser window with N = 59 and
B = 4.55. and
Find the Kaiser window parameters, B and N , to designa low-pass filter with acutofffrequency w, = n / 2 , a stopband ripple 6 = 0.002, and a transition bandwidth no 1a.rger than 0.117. ,
The parameter
B for the Kaiser window depends only on the stopband ripple requirements.
With 6, = 0.002,
and we have = 0.1 102(0r, - 8.7) = 4.99 The window length, N , on the other hand, is determined by the stopband ripple, 6,. and the transition width as follows:
Therefore, the required filter order is N = 65
Consider the following specifications for a bandpass filter:
(a) Design a linear phase FIR filter to meet these specifications using a Blackman window
(b) Repeat part (a) using a Kaiser window.
(a) For this filter, the width of each transition band is Aw = 0 . 1 ~ The ripples in the lower stopband, passband, . and upper stopband are 4 = 0.01. a2 = 0.05, and a3 = 0.02, respectively, and are all different. Because the ripples produced with the window design method will be approximately the same in all three bands, the filter must be designed so that it has a maximum ripple of 8, = 0.01 in all three bands. With
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