Simple FIR Digital Filters The 3-dB cutoff frequency of a cascade of M sections is given 0=2c52(212M) For M=3. the above yields o=0.302T Thus the cascade of first-order sections yields a sharper magnitude response but at the expense of a decrease in the width of the passband Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 11 Simple FIR Digital Filters • The 3-dB cutoff frequency of a cascade of M sections is given by • For M = 3, the above yields • Thus, the cascade of first-order sections yields a sharper magnitude response but at the expense of a decrease in the width of the passband 2cos (2 ) 1 1/ 2M c − − w = wc = 0.302p
Simple FIR Digital Filters a better approximation to the ideal lowpass filter is given by a higher-order moving- average filter Signals with rapid fluctuations in sample values are generally associated with higl frequency components These high-frequency components are essentially removed by an moving-average filter resulting in a smoother output wavefo 12 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 12 Simple FIR Digital Filters • A better approximation to the ideal lowpass filter is given by a higher-order movingaverage filter • Signals with rapid fluctuations in sample values are generally associated with highfrequency components • These high-frequency components are essentially removed by an moving-average filter resulting in a smoother output waveform
Simple FIR Digital Filters Highpass FIR Digital Filters The simplest highpass Fir filter is obtained from the simplest lowpass FIR filter by replacing z with -z This results in H1(=)=1(1-z-) 13 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 13 Simple FIR Digital Filters Highpass FIR Digital Filters • The simplest highpass FIR filter is obtained from the simplest lowpass FIR filter by replacing z with • This results in ( ) ( ) 1 2 1 1 1 − H z = − z − z
Simple FIR Digital Filters Corresponding frequency response is given io/2 e )=je sin(0/2) whose magnitude response is plotted below First-order Fir highpass filter 0.8 30.6 0.4 0.2 0.2 0.4 0.8 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 14 Simple FIR Digital Filters • Corresponding frequency response is given by whose magnitude response is plotted below ( ) sin( / 2) / 2 1 = w jw − jw H e j e 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 w/p Magnitude First-order FIR highpass filter
Simple FIR Digital Filters The monotonically increasing behavior of the magnitude function can again be demonstrated by examining the pole-zero pattern of the transfer function H1(z) The highpass transfer function Hi(z) has a zero at z=1 or o=o which is in the stopband of the filter 15 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 15 Simple FIR Digital Filters • The monotonically increasing behavior of the magnitude function can again be demonstrated by examining the pole-zero pattern of the transfer function • The highpass transfer function has a zero at z = 1 or w = 0 which is in the stopband of the filter H (z) 1 H (z) 1