2. Finite Impulse Response Filters (FIR)
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1 .. Mthos for FIR filtrs implmntation. Finit Impuls Rspons Filtrs (FIR. Th winow mtho.. Frquncy charactristic uniform sampling. 3. Maximum rror minimizing. 4. Last-squars rror minimizing.. Mthos for FIR filtrs implmntation... Th winow mtho... Th winow mtho Lt s consir th sir (ial transfr function ( It is a prioical function ovr with a prio. From th Fourir sris composition: ( = h n h = ( n A stanar FIR filtr has th transfr function: may consir th FIR cofficints to b: hn ( = h,,..., ( = h n w kp th first cofficints of th ial h (n. Also, in orr to hav a linar phas FIR filtr w must apply th rstrictions prsnt in th prvious chaptr
2 Frquncy charactristic rstrictions Frquncy charactristic rstrictions Tip ( ( în = a' cos( n '( b n cos n 3 c' sin( n 4 '( fără constrângri fără constrângri ( în = fără constrângri n sin n Zrouri obligatorii fără constrângri z = fără constrângri z = şi z = z = S pot proicta FTJ FTS FTB FOB FTJ FTB FTB tr. ilbrt ifrnţiator FTS FTB tr. ilbrt ifrnţiator Typ : vn function an symmtrical P = a'cosn - 3 Typ : vn function an asymmtrical P = b'cos( n - 3 Frquncy charactristic rstrictions... Th winow mtho Typ 3: o function an asymmtrical P = c'sinn Typ 4: o function an asymmtrical P = 'sin( n Stp. Choos a sir (ial zro phas transfr function an a th linar phas trm. For th typs an, is an vn function: [ ] = (,, Th corrsponing linar phas trm: ( =
3 ... Th winow mtho Stp. Choos a sir (ial zro phas transfr function an a th linar phas trm. For th typs 3 an 4, is an o function: (, (, ] (, [, = (, = = Th corrsponing linar phas trm: ( =... Th winow mtho Stp. trmin th Fourir sris (infinit trms composition of th ( n h = ( ( = (, =,, h n h n n Stp 3. Establish th ral filtr cofficints... Th winow mtho Kping a finit numbr of trms is quivalnt with multiplying with a tmporal winow w(n: hn ( = h( nwn ( wn ( =, pntru n, Th ffct in th Z transform omain is: { h n } = z ( z = Z { w } Z ( ( z hn ( = h( nwn ( ( ( z = ( z... Th winow mtho An in th frquncy omain: u ( u ( = ( ( = ( ( u In orr to maintain th linar phas charactristic, th winowing squnc must b symmtrical: wn ( = w ( n ( =
4 ... Th winow mtho Th amplitu-frquncy charactristics for th usual winows hav a main lob cntr in = an a numbr of sconary lobs with a crasing lvls. ( Th Gibbs ffct A frquncy transition ban btwn th passban an th stopban. It pns on th main lob of th winow charactristic Th amplitu charactristic has rippls both in th passban an in th stopban. ( ( Thy pn on th amplitu of th sconary lobs of th winow charactristic Th rctangular winow Th rctangular winow frquncy charactristic Is fin by: w ( n n, n, w =, in rst n z ( z = z = z ( Th spctrum is: ( = = sin sin (
5 First sconary lob amplitu Th maximum is for: sin =± sin ( = sin Th first frquncy th maximum is obtain: 3 = + ci =. First sconary lob amplitu ( ( B -3B -5 3 ( (, 3 = 3 = 5 or -3B frcvnţă normată Th rctangular winow In th ral filtr charactristics, thr will rsult rippls with a maximum lvl of -B in th stopban. Th amming winows Ar fin by: α ( α cos ( n+, n, w =, ls ( -B Thy ar pning on paramtr α. Forα=.54 w gt th fault amming winow. Forα=.5 w gt th ann winow.
6 Th amming winows α ( α cos ( n+, n, w =, ls Th amming winows α ( α cos ( n+, n, w =, ls Th first an th last winow sampls: w( = w( = α ( αcos For larg thy ar almost for α.5 w ( = w ( α Compar to th rctangular winow (with an abrupt transition at th n of th winow, th amming winows introuc a smooth transition. Th amming winows α ( α cos ( n+, n, w =, ls α n w = αw w α ( ( = α( ( whr ( is th rctangular winow spctrum: ( = α n w sin = sin α ( + ( Th amming winows Thrfor: with: α ( ( = ( = α ( α α = α ( ( α + + = ( =
7 Th amming winows ( α α α = ( α ( Th amming winows - - ( ( B frcvnţă normată Th amming winows Th Blackman winow gt a oubl with 8 main lob than th rctangular winow th sconary lob lvl is much lowr than th rctangular winow ( 4B. Th rsult filtr charactristic will hav lowr rippls, lss than -54B. w B 4, 4,5 cos n+,8 cos n, + + = n,, in rst Th amplitu-frquncy charactristic will hav: a 3r tim largr main lob than th rctangular winow charactristic. th sconary lobs will hav an amplitu lowr than 58 B of th main lob. Thrfor, th amplitu-frquncy charactristic s rippls of th filtr will b lss than 74 B.
8 Th Bartltt winow Is a triunghiular winow fin by: n+, n wbt = n+, n, inrst Th amplitu-frquncy charactristic will hav th sam main lob with lik th amming winow with highr sconary lobs amplitu ( 5 B, an th filtr charactristics rippls of 6 B. Frastra Kaisr w K I α n+ =, n, I ( α, in rst whr I (α is th moifi Bssl function of orr. hav two austing paramtrs, α şi : Changing α w can aust th sconary lobs lvl an, thrfor th filtr charactristics. Choosing w can aust th transition banwith... Frquncy charactristic sampling mtho ( Lt it b th sir charactristic an th sir lngth for th filtr. impos that for qually spac frquncis = k k =,..., k th sir charactristic an th sign filtr on ar qual: k k ( ( = k =,...,.. Frquncy charactristic sampling mtho That givs a quation systm = h n k with unknowns h kn ( k =,..., { } n =,, Also, th prvious rlation is th iscrt Fourir Transform (k of th squnc h(n: k = FT = { ( }( ( h n k k k =,...,
9 .. Frquncy charactristic sampling mtho { },, Thrfor h n can b comput from th = Invrs transform: h n = n k ( IFT (,..., ( k kn h n =,..., k = ( ( z = h n z n k kn = z k= n.. Frquncy charactristic sampling mtho k k ( z = z k= n = k z ( z n k n z z = = z z k k k z k = k = (z function is actually a polynomial in z, th z k xprssions ar iviing z. z.. Frquncy charactristic sampling mtho ( ( k k ( = (..3. Mtoa şantionării caractristicii frcvnţă Onulaţiil pot fi rus: accptân o bană tranziţi mai largă, sau impunân un număr şantionar cu valori intrmiar într şi în acastă bană, sau lăsân nprcizat valoril câtorva şantioan în zona tranziţi, gral librtat rămas fiin utilizat pntru minimizara onulaţiilor.
10 ..3. Mtoa şantionării caractristicii frcvnţă În cazul cân s orşt obţinra unui filtru cu coficinţi rali { h } R n, caractristica =,..., impusă trbui să fi o funcţi circular conugat simtrică: k ( k = k =,..., acă acastă coniţi nu st rspctată, vor rzulta coficinţi complcşi.
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