CHAPTER 16
h[i]'sin(2B f
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Windowed-Sinc Filters
Windowed-sinc filters are used to separate one band of frequencies from another.  They are very stable, produce few surprises, and can be pushed to incredible performance levels.  These exceptional frequency domain characteristics are obtained at the expense of poor performance in the time domain, including excessive ripple and overshoot in the step response.  When carried out by standard convolution, windowed-sinc filters are easy to program, but slow to execute.  Chapter 18 shows how the FFT can be used to dramatically improve the computational speed of these filters.
Strategy of the Windowed-Sinc
Figure 16-1 illustrates the idea behind the windowed-sinc filter.  In (a), the
frequency response of the ideal low-pass filter is shown.  All frequencies below
the cutoff frequency, , are passed with unity amplitude, while all higher
f
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frequencies are blocked.  The passband is perfectly flat, the attenuation in the
stopband is infinite, and the transition between the two is infinitesimally small.
Taking the Inverse Fourier Transform of this ideal frequency response produces
the ideal filter kernel (impulse response) shown in (b).  As previously discussed
(see Chapter 11, Eq. 11-4), this curve is of the general form: , called
sin(x)/x
the sinc function, given by:
Convolving an input signal with this filter kernel provides a perfect low-pass
filter.  The problem is, the sinc function continues to both negative and positive
infinity without dropping to zero amplitude.  While this infinite length is not
a problem for mathematics, it is a show stopper for computers.
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286w [i ]'0.54&0.46cos(2B i /M )
EQUATION 16-1
The Hamming window. These windows run from  to M ,i '0for a total of  points.M %1w [i ]'0.42&0.5cos(2B i /M )%0.08cos(4B i /M )
EQUATION 16-2
The Blackman window.
FIGURE 16-1  (facing page)
Derivation of the windowed-sinc filter kernel.  The frequency response of the ideal low-pass filter is shown in (a), with the corresponding filter kernel in (b), a sinc function.  Since the sinc is infinitely long, it must be truncated to be used in a computer, as shown in (c).  However, this truncation results in undesirable changes in the frequency response, (d).  The solution is to multiply the truncated-sinc with a smooth window, (e),resulting in the windowed-sinc filter kernel, (f).  The frequency response of the windowed-sinc, (g), is smooth and well behaved. These figures are not to scale.
To get around this problem, we will make two modifications to the sinc function in (b), resulting in the waveform shown in (c).  First, it is truncated to  points, symmetrically chosen around the main lobe, where M  is an M %1even number.  All samples outside these  points are set to zero, or simply M %1ignored.  Second, the entire sequence is shifted to the right so that it runs from 0 to M .  This allows the filter kernel to be represented using only positive indexes.  While many programming languages allow negative  indexes, they are a nuisance to use.  The sole effect of this  shift in the filter kernel is to M /2shift the output signal by the same amount.
Since the modified filter kernel is only an approximation to the ideal filter kernel, it will  not have an ideal frequency response.  To find the frequency response that is obtained, the Fourier transform can be taken of the signal in (c), resulting in the curve in (d).  It's a mess!  There is excessive ripple in the passband and poor attenuation in the stopband (recall the Gibbs effect discussed in Chapter 11).  These problems result from the abrupt discontinuity at the ends of the truncated sinc function.  Increasing the length of the filter kernel does not reduce these problems; the discontinuity is significant no matter how long M  is made.
Fortunately, there is a simple method of improving this situation.  Figure (e)shows a smoothly tapered curve called a Blackman window .  Multiplying the truncated-sinc, (c), by the Blackman window, (e), results in the windowed-sinc  filter kernel shown in (f).  The idea is to reduce the abruptness of the truncated ends and thereby improve the frequency response.  Figure (g) shows this improvement.  The passband is now flat, and the stopband attenuation is so good it cannot be seen in this graph.
Several different windows are available, most of them named after their original developers in the 1950s.  Only two are worth using, the Hamming window  and the Blackman window  These are given by:
Figure 16-2a shows the shape of these two windows for  (i.e., 51 total M '50points in the curves).  Which of these two windows should you use?  It's a trade-off between parameters.  As shown in Fig. 16-2b, the Hamming window has about a 20% faster roll-off  than the Blackman.  However,
Chapter 16- Windowed-Sinc Filters 287
Time Domain
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FIGURE 16-2
Characteristics of the Blackman and Hamming windows.  The shapes of these two windows are shown in (a), and given by Eqs. 16-1 and 16-2.  As shown in (b), the Hamming window results in about 20% faster roll-off than the Blackman window.However, the Blackman window has better stop-band attenuation (Blackman: 0.02%, Hamming:0.2%), and a lower passband ripple (Blackman:0.02% Hamming: 0.2%).
A m l i t u d e  (d A m p l i t u d e
truncated 带whereA p l i t u d (c) shows that the Blackman has a better stopband attenuation .  To be exact,the stopband attenuation for the Blackman is -74d
B (-0.02%), while the Hamming is only -53dB (-0.2%).  Although it cannot be seen in these graphs,the Blackman has a passband ripple  of only about 0.02%, while the Hamming is typically 0.2%.  In general, the Blackman should be your first choice; a slow roll-off is easier to handle than poor stopband attenuation.
There are other windows you might hear about, although they fall short of the Blackman and Hamming.
  The Bartlett window  is a triangle, using  straight lines for the taper.  The Hanning window, also called the raised cosine window , is given by: .  These two windows have w [i ]'0.5&0.5cos(2B i /M )about the same roll-off speed as the Hamming, but worse stopband attenuation (Bartlett: -25dB or 5.6%, Hanning -44dB or 0.63%).  You might also hear of a rectangular window .  This is the same as no  window, just a truncation of the tails (such as in Fig. 16-1c).  While the roll-off is -2.5 times faster than the Blackman, the stopband attenuation is only -21dB (8.9%).
Designing the Filter
To design a windowed-sinc, two parameters must be selected: the cutoff frequency, , and the length of the filter kernel,  M .  The cutoff frequency
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Chapter 16- Windowed-Sinc Filters
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Frequency
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A m p l i t u d e
A m p i t u d M .
4BW
EQUATION 16-3
Filter length vs. roll-off.  The length of the filter kernel, M , determines the transition bandwidth of the filter, BW.  This is only an approximation since roll-off depends on the particular window being used.
is expressed as a fraction of the sampling rate, and therefore must be between 0 and 0.5.  The value for M  sets the roll-off  according to the approximation:
where BW  is the width of the transition band, measured from where the curve just barely leaves one, to where it almost reaches zero (say, 99% to 1% of the curve).  The transition bandwidth is also expressed as a fraction of the sampling frequency, and must between 0 and 0.5.  Figure 16-3a shows an example of how this approximation is used.  The three curves shown are generated from filter kernels with: .  From Eq. 16-3, the M '20,40,and 200transition bandwidths are: , respectively.  Figure (b)BW '0.2,0.1,and 0.02shows that the shape of the frequency response does not depend on the cutoff frequency selected.
Since the time required for a convolution is proportional to the length of the signals, Eq. 16-3 expresses a trade-off between computation time  (depends on the value of M ) and filter sharpness  (the value of BW ).  For instance, the 20%slower roll-off of the Blackman window (as compared with the Hamming) can be compensated for by using a filter kernel 20% longer.  In other words, it could be said that the Blackman window is 20% slower to execute that an equivalent roll-off Hamming window.  This is important because the execution speed of windowed-sinc filters is already terribly slow.
As also shown in Fig. 16-3b, the cutoff frequency of the windowed-sinc filter is measured at the one-half amplitude  point.  Why use 0.5 instead of the

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