Simple FIR Digital Filters The magnitude response Ho(eo)=cos(o/2) can be seen to be a monotonical decreasing function of o First-order FIr lowpass filter 0.8 06 0.4 0 0.4 0.6 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 6 Simple FIR Digital Filters • The magnitude response can be seen to be a monotonically decreasing function of w | ( )| cos( / 2) 0 = w jw H 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 lowpass filter
Simple FIR Digital Filters The frequency (=0 at which 0(e0)= 0 0 is of practical interest since here the gang(Oc) in dB is given by G(Oc)=20log1o H(e/0c) 2010g10 (e70)-20log0y2全-3dB since the dc gain G(0)=201og. H(ej0)=o 7 10 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 7 Simple FIR Digital Filters • The frequency at which is of practical interest since here the gain in dB is given by since the dc gain w= wc ( ) 2 1 ( ) 0 0 0 j j H e H e c = w ( ) G wc 20log ( ) 20log10 2 3 dB 0 = 10 − − j H e ( ) G wc 20log ( ) 10 c j H e w = 0 20 0 0 10 ( ) = log ( ) = j G H e
Simple FIR Digital Filters Thus, the gain G(O)at @=o is approximately 3 db less than the gain at o=0 As a result, @. is called the 3-dB cutoff frequency To determine the value of o. we set Hole cos(oc/2)=2 which yields Oc=T/2 8 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 8 Simple FIR Digital Filters • Thus, the gain G(w) at is approximately 3 dB less than the gain at w = 0 • As a result, is called the 3-dB cutoff frequency • To determine the value of we set which yields w= wc wc wc wc = p/ 2 2 2 2 1 0 | ( )| = cos (w / 2) = w c j c H e
Simple FIR Digital Filters The 3-dB cutoff frequency @c can be considered as the passband edge trequency As a result, for the filter Ho(z) the passband width ly兀/2 Is approximately The stopband is from T /2 to T Note Ho(z) has a zero at z=-I oro=T which is in the stopband of the filter Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 9 Simple FIR Digital Filters • The 3-dB cutoff frequency can be considered as the passband edge frequency • As a result, for the filter the passband width is approximately p/2 • The stopband is from p/2 to p • Note: has a zero at or w = p, which is in the stopband of the filter wc H (z) 0 H (z) 0 z = −1
Simple FIR Digital Filters A cascade of the simple fir filters 0 2(+) results in an improved lowpass frequency response as illustrated below for a cascade of 3 sections First-order FIR lowpass filter cascade 80: 0.4 0.4 0.6 lI Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 10 Simple FIR Digital Filters • A cascade of the simple FIR filters results in an improved lowpass frequency response as illustrated below for a cascade of 3 sections ( ) ( ) 1 2 1 0 1 − H z = + z 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 lowpass filter cascade