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Unformatted text preview: 285 CHAPTER 16 h [ i ] sin(2 B f C i ) i B WindowedSinc Filters Windowedsinc 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, windowedsinc 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 WindowedSinc Figure 161 illustrates the idea behind the windowedsinc filter. In (a), the frequency response of the ideal lowpass filter is shown. All frequencies below the cutoff frequency, , are passed with unity amplitude, while all higher f C 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. 114), 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 lowpass 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 . The Scientist and Engineer's Guide to Digital Signal Processing 286 w [ i ] 0.54 &amp; 0.46 cos(2 B i / M ) EQUATION 161 The Hamming window. These windows run from to M , i for a total of points. M % 1 w [ i ] 0.42 &amp; 0.5 cos(2 B i / M ) % 0.08 cos(4 B i / M ) EQUATION 162 The Blackman window. FIGURE 161 (facing page) Derivation of the windowedsinc filter kernel. The frequency response of the ideal lowpass 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 truncatedsinc with a smooth window, (e), resulting in the windowedsinc filter kernel, (f). The frequency response of the windowedsinc, (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 % 1 even number. All samples outside these points are set to zero, or simply M % 1 ignored. 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...
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 Summer '09
 Frequency

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