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In next step impulse signal is shifted by one time unit i.e. ‘sample1’ is assigned a value of one and remaining all sample values are made zero. The corresponding obtained output is same as second coefficient value (-13). Similarly impulse signal is moved through all the input samples and corresponding outputs are observed. Simulation results clearly show all the coefficient values coming out as ‘filt_out’ as and when impulse signal traverses through input samples. Thus designed filter is verified by verifying the impulse response of the system | Chapter Butterworth Filters Butterworth lowpass filters LPF are designed to have an amplitude response characteristic that is as flat as possible at low frequencies and that is monotonically decreasing with increasing frequency. 3.1 Transfer Function The general expression for the transfer function of an nth-order Butterworth lowpass filter is given by n 1 s-S s -SjXs -s2 s -sn where s ejni 21 n 1 2n cosfn n _ _ j sjnin 3.2 2n J 2n J Example 3.1 Determine the transfer function for a lowpass third-order Butterworth filter. solution The third-order transfer function will have the form W s ------------------------ s -sx s -s2 s s3 The values for s1 s2 and s3 are obtained from Eq. 3.2 Í 2 r 2tt Sí cosl I j sinl 0.5 0.866 V 3 3 s2 eJn cos n j sin rc 1 s3 cosf j sin 0.5 0.866J 3 3 65 66 Chapter Three US S s 0.5 - 0.866 s l s 0.5 0.866 1 s3 2s2 2s 1 The form of Eq. 3.1 indicates that an nth-order Butterworth filter will always have n poles and no finite zeros. Also true but not quite so obvious is the fact that these poles lie at equally spaced points on the left half of a circle in the s plane. As shown in Fig. 3.1 for the third-order case any odd-order Butterworth LPF will have one real pole at s 1 and all remaining poles will occur in complex conjugate pairs. As shown in Fig. 3.2 for the fourth-order case the poles of any even-order Butterworth LPF will all occur in complex conjugate pairs. Pole values for orders 2 through 8 are listed in Table 3.1. 3.2 Frequency Response A C function butterworthFreqResponse for generating Butterworth frequency response data is provided in Listing 3.1. Figures 3.3 through 3.5 ju -1 Figure 3.1 Pole locations for a third-order Butterworth LPF. Figure 3.2 Pole locations for a fourth-order Butterworth LPF. Butterworth Filters 67 table 3.1 Poles of Lowpass Butterworth Filters n Pole values 2 0.707107 0.707107 3 -1.0 -0.5 0.866025 4 -0.382683 0.923880 -0.923880 0.382683 5 -1.0 -0.809017 0.587785J -0.309017 0.951057 6 -0.258819 0.965926 .
