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FIR Filters - design frequency sampling method

The magnitude and phase response of the designed filter is shown in Figure (1). The figure shows both reference filter response and response of the filter after coefficients have been quantized to fixed point representation. Order of the filter comes out to be 161 i.e. total 162 coefficients. Design of such a higher order filter is difficult task | Chapter FIR Filter Design Frequency Sampling Method 12.1 Introduction In Chap. 11 the desired frequency response for an FIR filter was specified in the continuous-frequency domain and the discrete-time impulse response coefficients were obtained via the Fourier series. We can modify this procedure so that the desired frequency response is specified in the discretefrequency domain and then use the inverse discrete Fourier transform DFT to obtain the corresponding discrete-time impulse response. Example 12.1 Consider the case of a 21-tap lowpass filter with a normalized cutoff frequency of 3x17. The sampled magnitude response for positive frequencies is shown in Fig. 12.1. The normalized cutoff frequency z 7 falls midway between n 4 and n 5 and the normalized folding frequency of z n falls midway between n 10 and n 11. Note that 45 10.5 3 7. We assume that Hd n Hd n and use the inverse DFT to obtain the filter coefficients listed in Table 12.1. The actual continuous-frequency Hd n n 1 2 3 4 5 6 7 8 9 10 11 X Figure 12.1 Desired discrete-frequency magnitude response for a lowpass filter with .v 3n 7. 211 212 Chapter Twelve TABLE 12.1 Coefficients for the 21-tap Filter of Example 12.1 A 0 20 0.037334 1 19 -0.021192 2 18 -0.049873 3 17 0.000000 4 16 0.059380 5 15 0.030376 6 14 -0.066090 7 13 -0.085807 i 12 0.070096 9 11 0.311490 10 0.428571 response of an FIR filter having these coefficients is shown in Figs. 12.2 and 12.3. Figure 12.2 is plotted against a linear ordinate and dots are placed at points corresponding to the discrete-frequencies specified in Fig. 12.1. Figure 12.3 is included to provide a convenient baseline for comparison of subsequent plots that will have to be plotted against decibel ordinates in order to show low stop-band levels. The ripple performance in both the pass-band and stop-band responses can be improved by specifying one or more transition-band samples at values somewhere between the pass-band value of Hd m 1 and the stop-band Figure 12.2 .