58 Chapter 6 rectangular window determines the length of the impulse response of the filter. This is called a rectangular window because the window coefficients W i are all 1 within the window and 0 outside. With the rectangular window, we abruptly truncate the impulse response of the filter. Obviously, for realistic filter implementations, we have to limit the impulse response at some point because each tap or coefficient requires a multiplication operation to compute each filter output. But perhaps we can get a more desirable response by reducing the coefficient values gradually at either end of the impulse response before we reach the point of impulse response truncation. 6.2 Tapering of Coefficients Reducing the coefficient values has led to efforts to develop other window functions besides the default rectangular window. Window design and analysis involve a fair bit of mathematics. But after the rigors of the preceding chapter, you may not mind too much if we skip over this. Actually, many filter designers do not know the details of the various window functions offered by their filter design software but work iteratively instead. That is, designers experiment with moving the frequency cutoff point slightly, and playing the allowable number of taps, the various window options, and sometimes the numerical precision (number of bits) of the input data and coefficients. By observing the computer-generated frequency plots, designers can iterate to find an optimum combination of these parameters to meet application requirements. Often, the requirements are a certain degree of filter rejection or attenuation at one or more specific frequency points, a maximum amount of ripple or variance in the passband region of the frequency response, and a specified region of the frequency response. (See Figure 6.1.) Most windows are named after their inventors. They include Hanning, Hamming, von Hann, Kaiser, Blackman, Bartlett, and others. The window coefficients are not equal to 1, as in the rectangular window, but will gradually transition from 1 to 0 in some fashion near the edges of the window. The form of this transition, or tapering off, of the coefficients defines the window properties. Note that this is a different function from filter design, which produces the original and ideal set of coefficients. Windowing is used to avoid the abrupt truncation of the filter coefficient set, required to allow implementation of the filter with a finite number of multiply-add operations. In general, a window cannot increase the steepness of the transition region, but it can be used to reduce either the passband or stopband ripple in the frequency response. Most filter design programs offer several window options. The following figures show the frequency www.newnespress.com