2. Finite Impulse Response Filters (FIR)

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1 ..3.3aximum error minimizing method. Finite Imule Reone Filter (FIR)..3 aximum error minimizing method he zero hae tranfer function N H a' n con tye n N H b n con n tye ' the lat relation can be exreed lie: N H b n co n co n..3.3aximum error minimizing method For any of the 4 th tye of linear hae filter: where: r H Q P co P n n n, tiul co, tiul Q in, tiul3 in, tiul4..3.3aximum error minimizing method Let conider we imoe the deired zero hae function H d he weighted error function: E W H H d W i a real function having bigger value in the frequency interval of interet.

2 ..3.3aximum error minimizing method E i a real function that can be exreed: where: E W Hd Q P W ˆ H ˆ P d not not H d Q Wˆ W Q Hˆ H d..3.3aximum error minimizing method r P n co n We want to find the function hat mean to determine the coefficient α(n) o that we minimize the maximum of error function E over the frequency domain A, min max E n A he aroximation will be an equirile one of Cebaev tye. he olution i given by the alternating theorem. n he alternating ati theorem e r A function P n co n n give the bet aroximation of Cebaev ene of a continuou function H ˆ d over a comact interval A, if and only if the weighted error function E ha at leat r + extreme frequency oint in A, the extreme oint having the ame magnitude and alternative ign...3.3aximum error minimizing method herefore it mut exit a grou of frequencie... r o that: E,,,, i E i i r E max E, i,,, r i A A ractical method i given by the witching algorithm rooed by Remez.

3 he Remez ez algorithm. Chooe an initial et of r + frequencie ω i and imoe that the error function ha alternative value, ±δ in thi oint, δ being unecified. ˆ ˆ i W,,,, i Hd P i r co co cor Wˆ ˆ H d co co cor ˆ Hˆ W d ˆ r H d r cor co r cor r ˆ W r he Remez ez algorithm. By olving the reviou ytem we obtain α(n) and δ. 3. Evaluate E() ( ) over the entire frequency domain. If E at every oint, it mean that thee are the deired coefficient. Ele, we tart a new iteration, chooing a grou of r + where the error ha maximum value. he Remez algorithm wa imlemented for FIR deign oftware (lie ALAB) by cclellan, ll Par and Rabiner. Filter deign ecification When deigning a digital filter we may imoe: - the uer limit of the banda, ω ; - the maximum rile (error) in the banda, δ ; - the lower limit of the bandto ω ; - the maximum rile (error) in the toband, δ. He ( ) He ( ) for, He ( ) for, Filter deign ecification We may ecify, intead of δ, the maximum variation of the attenuation inthe banda in db: a lg lg lg (db) And in the ame way, intead of δ we may ecify the minimum attenuation in the bandto: a lg (db) ω

4 Filter deign ecification For examle, in cae of a LPF deigned uing the Remez algorithm (Par-cClellan), the amlitude-frequency characteritic i etablih by ω, ω, δ, δ. H ( ) d e Filter deign ecification Remez ezalgorithm If we have an imoed N, we cannot imoe both δ, and δ. We can imoe their ratio by the weighted error function ;, W ;, hen, in the bandto, the error will be t and, in the anda t b b ω Fie funcţia dorită, e care o vom reuune de fază liniară şi cea realizată d H e e H d d H e e H Caracteritica de fază fiind realizată în mod exact, în cazul filtrului de fază liniară rămâne de aroximat funcţia de fază nulă. Fie un et de frecvenţe ω, =,...,. Se oate defini eroarea onderată la frecvenţa ω e E W Hd H,,, unde W Definim vectorul eroare: e e, e,, e şi eroarea ătratică: ete o funcţie de ondere reală. E e ee

5 d E W H H În funcţie de tiul filtrului, e oate exrima rin coeficienţii a, b, c, d. H e Vom lua dret exemlu filtrul de tiul : Definim vectorii: a P H a' n co n n a, a,, a co,co,,co P P aşa încât H a şi e d W unde a d W H d Vom introduce în continuare vectorul d d d, d,, d matricea diagonală W a onderilor, de dimeniuni W şi matricea S, de dimeniuni diag W, W,, W S,,, P Rezultă Vom nota şi deci e d WS a X WS Problema e reduce la minimizarea formei ătratice E ee dxa edxa unde. emnifică norma euclidiană a unui vector, în funcţie de vectorul a. În rinciiu, eroarea ar utea fi nulă dacă Xa = d. Sitemul aceta are P + necunocute şi ecuaţii. El oate fi rezolvat exact numai dacă = P +. În mod uzual > P + (numărul de condiţii ete mai mare ca al gradelor de libertate), atfel încât itemul it ldevine uradeterminat. dt t El oate fi rezolvat aroximativ, rin minimizarea erorii în enul celor mai mici ătrate.

6 Pentru aceata mai utem crie: d ax d Xa dd axxa axd dxa E Aceată exreie trebuie minimizată în raort cu a. Pentru aceata, e egalează cu gradientul exreiei E E a E a ae E ap Pentru imlificarea crierii vom nota: R X X r X d R ete o matrice imetrică de dimeniuni P P şi e verifică uşor că ete ozitiv emidefinită. r ete un vector P. E devine deci: E ddaraarra Să evaluăm derivatele termenilor ce comun exreia: P P ara ara i i a a i aşa încât: a P P P a R ar ar i i i i i i ara Ra Aoi: deci P ar ra ar i i a a a i a r a ar ra r Condiţia E a e reduce deci la Ra r numită şi ecuaţia normală, cu oluţia: a R r

7 Aceatã valoare a lui a conduce realmente la un minim al formei ătratice E deoarece Heianul tranformării ete E R, matrice ozitiv emidefinită. Vectorul a e mai crie atricea a X X X d X X X X ete eudo invera matricei idretunghiulare X şi oluţia roblemei e oate crie: a X d Cele frecvenţe ot fi eventual luate echiditante,,..., În cazul = N e obţine, ca un caz articular, metoda eşantionării în frecvenţă. etoda aceata oate fi rivită deci ca o generalizare a metodei eşantionării ii în frecvenţă.

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