Lecture 19: Lowpass, bandpass and highpass filters
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1 Leure 9: Lowpass, bandpass and higass filers UHTXHQF\6HOHFLYH LOHUV Ideal frequeny-seleive filers are filers ha le frequeny omponens over a given frequeny band (he passband pass hrough undisored, while omponens a oher frequenies (he sopband are ompleely u off. The usual senario where filering is needed is when a noise n ( is added o a signal, bu he noise has mos of is energy a frequenies ouside of he bandwidh of he signal. e wan o reover he original signal from is noisy measuremen. n ( x ( ~ + x( H( j + x( filer X( j N( j Here he noise sperum is assumed o have all of is energy a higher frequenies han he bandwidh of he signal. Thus, an ideal lowpass filer would perfely reover he signal, i.e., ~ x ( x (. X ( j H( j ~ X( j X( j /RZSDVV LOHUV An ideal lowpass filer us off frequenies higher han is uoff frequeny. The frequeny response of his filer is given by: H lp % &, ( j ' 0, < >. Hlp ( j
2 The orresponding impulse response was found o be a sin funion of ime in Seion 6.4 (i is even and real, as expeed. hlp π π sin (6.70 ( π h ( lp π π Reall ha he onvoluion propery of he Fourier ransform is he basis for filering: he oupu of FT an LTI sysem wih impulse response h ( H( j subjeed o an inpu signal FT x ( X( j is given by y ( x ( h ( Y( j H( j X( j. FT Even hough he above filer is ermed "ideal" in referene o is frequeny response, i may no be so desirable in he ime-domain for some appliaions beause of he ripples in is impulse response. For example, he sep response of he ideal lowpass filer is he running inegral of he sin funion. Is plo indiaes ha here are poenially undesirable osillaions before (beause he impulse response is nonausal and afer he disoninuiy. s ( lp As menioned before, he impulse response of he ideal lowpass filer is no realizable as i is nonausal. Approximaions o he ideal lowpass filer ha an be realized by ausal LTI differenial equaions have been proposed in he pas, and some of hem have found widespread use. One of hese filers is he uerworh filer. The magniude of he frequeny response of an N h -order uerworh filer wih uoff frequeny is given by:
3 H ( j + N (6.7 Remarks: The d gain H ( j0 The aenuaion a uoff is H( j (or -3d for any order N. Here is a plo of he magniude for N (seond-order ompared o he ideal "brik wall" magniude. The ransiion band is he frequeny band around where he magniude "rolls off". The higher he order, he narrower he ransiion band ges. H ( j The seond-order uerworh filer is defined by is haraerisi polynomial jπ 4 jπ 4 p( s ( s+ e ( s+ e s+ + j s j! " + $#! " $# s + s+ (6.7 Therefore he differenial equaion relaing he inpu and oupu signals of his filer mus have he form d y( dy( + + y ( x (. (6.73 d d Noe ha i is neessary o add a gain of muliplying he inpu signal o obain a d gain of for he filer. Reall ha he d gain is he gain of he filer when he inpu and he oupu signals are onsan, whih means ha he derivaives of y ( are zero. Le's hek ha he frequeny response of his differenial equaion has he magniude of Equaion (6.7. The frequeny response of he seond-order uerworh filer is given by: 3
4 and he magniude is Y( j H ( j X( j H ( j ( j + j + j + ( + ( +! (! ( + ( 4 j j " $# + " $# + (6.74 (6.75 The impulse response of a buerworh filer is given by (following a parial fraion expansion of he frequeny response: h( e sin( u(. (6.76 This impulse response does no osillae muh even hough i is a damped sinusoid. The deay rae is fas enough o damp ou he osillaions. For example, if we plo he sep response of his seond-order uerworh filer, we obain s ( 4
5 +LJKSDVV LOHUV An ideal higass filer us off frequenies lower han is uoff frequeny. The frequeny response of his filer is given by: H % &, ( j ' 0, < >. H( j Noie ha he frequeny response of an ideal higass filer an be wrien as he differene beween and he frequeny response of an ideal lowpass filer. H ( j H ( j (6.77 The resuling impulse response is simply h ( δ ( h (. (6.78 This suggess one possible approah o obain a realizable higass filer. Firsly, design a lowpass filer wih uoff frequeny and desirable haraerisis in he ransiion band and he sopband. Seondly, form he frequeny response of he higass filer using Equaion (6.77. lp lp Example: e an design a higass filer using he seond-order lowpass uerworh filer of he above example. H ( j H ( j ( j + j + ( j + j + ( j + j + ( j + j ( j + j + (6.79 H ( j 5
6 The orresponding ausal LTI differenial equaion is d y( dy( d x( dx( + + y ( +. (6.80 d d d d and he impulse response is as given by Equaion (6.78: h( δ( e sin( u( (6.8 %DQGSDVV LOHUV An ideal bandpass filer us off frequenies lower han is firs uoff frequeny and higher han is seond uoff frequeny. The frequeny response of suh a filer is given by: H bp % &, < < ( j ' 0, oherwise. Hbp( j The frequeny response of an ideal bandpass filer an be wrien as he produ of he frequeny responses of ideal overlapping lowpass and higass filers. The higass filer should have a uoff frequeny of and he lowpass filer. H ( j H ( j H ( j. (6.8 bp lp This is one approah o design bandpass filers. 6
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