Simple Digital Filters Later in the course we shall review various methods of designing frequency-selective filters satisfying prescribed specifications We now describe several low-order fr and Ir digital filters with reasonable selective frequency responses that often are satisfactory in a number of applications Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 1 Simple Digital Filters • Later in the course we shall review various methods of designing frequency-selective filters satisfying prescribed specifications • We now describe several low-order FIR and IIR digital filters with reasonable selective frequency responses that often are satisfactory in a number of applications
Simple FIR Digital Filters fiR digital filters considered here have Integer-valued impulse response coeficients These filters are employed in a number of practical applications, primarily because of their simplicity, which makes them amenable to inexpensive hardware implementations Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 2 Simple FIR Digital Filters • FIR digital filters considered here have integer-valued impulse response coefficients • These filters are employed in a number of practical applications, primarily because of their simplicity, which makes them amenable to inexpensive hardware implementations
Simple FIR Digital Filters Lowpass FIr Digital Filters The simplest lowpass fir digital filter is the 2-point moving-average filter given by 1z+1 H0(二)=(1+z-)= 2 The above transfer function has a zero at z=-l and a pole atz=0 note that here the pole vector has a unit magnitude for all values of o Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 3 Simple FIR Digital Filters Lowpass FIR Digital Filters • The simplest lowpass FIR digital filter is the 2-point moving-average filter given by • The above transfer function has a zero at and a pole at z = 0 • Note that here the pole vector has a unity magnitude for all values of w z z H z z 2 1 1 1 2 1 0 + = + = − ( ) ( ) z = −1
Simple FIR Digital Filters On the other hand as o increases fromo to T, the magnitude of the zero vector decreases from a value of 2 the diameter of the unit circle. to o Hence, the magnitude response Ho(e )l is a monotonically decreasing function of o from a=0too=兀 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 4 Simple FIR Digital Filters • On the other hand, as w increases from 0 to p, the magnitude of the zero vector decreases from a value of 2, the diameter of the unit circle, to 0 • Hence, the magnitude response is a monotonically decreasing function of w from w = 0 to w = p | ( )| 0 jw H e
Simple FIR Digital Filters The maximum value of the magnitude function is 1 at o=o and the minimum value is0ato=π,ie., (0 丌 0(e 1,|H0(e)=0 0 The frequency response of the above filter is given by 0(e e J0/2 cos(o/2) Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 5 Simple FIR Digital Filters • The maximum value of the magnitude function is 1 at w = 0, and the minimum value is 0 at w = p, i.e., • The frequency response of the above filter is given by 1 0 0 0 | 0 ( )| = , | ( )| = j jp H e H e ( ) cos( / 2) / 2 0 = w jw − jw H e e
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
