DIGITAL SIGNAL PROCESSING (DSP) 211
Figuring out the coefficient c(n) from this formula might involve some difficult calculus with an integral over a range of 2p. This is the case for a general-purpose (custom) frequency response, but if the frequency response curve is like the low-pass filter, the calculations are simpler. The gain is flat at a value of 1 and then drops off completely (in the ideal math equation). Taking advantage of the simplified filter shape, and with a few other mathematical manipulations, the integral reduces to a closed math solution as follows: c(n) (sin (nv)/np) Using the math identity sinc (x) sin(x)/x, c(n) v sinc (nv)/p The sinc function is well known as the spectral envelope of a train of pulses. Figure 8-13 shows the shape of the sinc function. One of the difficulties of the Fourier method is that it produces an infinite set of coefficients. This presents a problem because we cannot have an infinite number of taps in the FIR filter. If we simply eliminate some taps, the filter won’t work as designed or simulated. Instead, various techniques are used to minimize the taps to a conveniently small number. These techniques create a window value for every coefficient in the infinite series. All the coefficients are multiplied by the window during the FIR filter computations. All these windows limit the number of coefficients to the desired number of taps because the window has a value of zero for taps outside the range of the window.
SINC (x) = Sin (x) / x 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 1 -0.4
FIGURE 8-13 The sinc function