Lattice Realization Here we will study an other FIR filter structure called the LATTICE FILTER or lattice realization.

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James Hall
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1 Lattice Realiati Here we will study a ther IR ilter structure called the LTTICE ILTER r lattice realiati. Lattice ilters are etesively used i speech prcessi ad i the ipleetati adaptive ilters. Let us csider a sequece IR ilters with syste uctis: here [] H is the plyial : [] [],,,...,( M ) [] ( ) ad [] The uit ipulse respse the th ilter is h () ad h () () r,,...,. Here subscript als detes the deree the plyial. It is desirable t view IR ilters as liear predictrs sice the iput sequece [-], [-],...,[-] ca be used t predict the value the sial []. The liearly predicted value [] equals: [] [][ ] ˆ where, {- [] } represet the predicti ceiciets. The utput sequece y[] ay be epressed as: y[] []- ˆ [] [] [ ] [ ] () Hece the IR ilter utput ive by equati () ay be iterpreted as the errr betwee the ˆ. true sial value [] ad the predicted value [ ] Tw direct r realiati the IR predicti ilter are shw belw:

2 [] () () () (-) () y[] i. y[] [] () - () - () - () i. Suppse that we have a ilter r which. The the utput this ilter is: [] [] [ ] [ ] y () This utput ca als be btaied r the irst rder r sile-stae lattice ilter shw belw: [] [] []y[] [] - [-] [] Sile stae lattice ilter.

3 [] [] [ ] [] [] [ ] [] [ ] [] [] [ ] [] [ ] I we select [] the [] will be equal t equati () Paraeter i the lattice ilter is called the relecti ceiciet. Nw csider a IR ilter r which. I this case the utput r a direct-r structure is: y[] [] [] [ ] [ ] [ ] [ ] [ ] [ ] y[] [] [] [ ] [ ] [ ] () Tw stae Lattice ilter Output r the irst stae is: [] [ ] [ ] [] [] [ ] the utput r the secd stae is: [] [ ] [ ] [] [] [ ] r equivaletly usi the results stae- [] [] [ ] [ ] [ ] [ ] [] ( ) [ ] [ ] (4) [] [] [] []y[] [ ] [] []

4 Nte here that equati (4) is idetical t equati () i we equate: [], [ ] ( ) Or equivaletly : [ ] [] [] hece the relecti ceiciets ad the lattice ilter ca be btaied r the ceiciets { [] } the direct r realiati. y ctiui this prcess e ca easily destrate by iducti the equivalece betwee a th der direct-r IR ilter ad a -stae lattice ilter. The lattice ilter is eerally described by the llwi set rder-recursive equatis. [] [ ] [ ] [] [] [ ],,..., M [] [] [ ],,..., M The utput the (M-)th stae crrespds t utput a (M-) rder IR ilter. i.e. y[] [ ] M s a csequece the equivalece betwee a IR ilter ad a lattice ilter the utput [ ] a -stae lattice ilter ca be writte as: (5) [] [][ ] where, [] Sice eq.(5) is a cvluti su it llws that the Z-trasr is a siple ultiplicati. r equivaletly [] [] X [ ] [] X [ ] [] [ ] []

5 The secd utput [] the lattice ca als be epressed as a cvluti su by usi a ther set ceiciets {β ()}. The utput [] r a -stae lattice ilter ay be epressed by the cvluti su the r: [] [][ ] where, β [] β (6) where the ilter ceiciets {β ()} are assciated with a ilter that prduces [] y[ ] but perates i reverse rder. Suppsi [], [-],.., [-] is used t liearly predict the sial value [-]. The predicted value equals: [ ] [ ] [ ] ˆ β (7) where the ceiciets β () i the predicti ilter are siply the ceiciets { [] i reverse rder. β [] [ ],,..., } tae Nw i we csider the sae i Z-dai, the trasr eq. (6) beces: [] [] X [ ] [] X [] [] here [] represets the syste ucti the IR ilter with ceiciets β (). Sice β () (-) [] β [] [] [ ] l [] l l [] l l [ ] l

6 This iplies that ers the IR ilter with syste ucti [ ] are siply the reciprcal the ers []. Hece [] is called the reciprcal plyial []. Nw that this relatiship is shw lets et bac t recursive lattice equatis ad traser the t -dai. [] [] [ ] [] [] [] [] [] [],,...,,,..., X I we divide each equati by X[] [ ] [ ] [] [] [] [] [] [],,...,,,..., Hece the lattice stae is described i the -dai by the atri equati: [] [] [ ] [] The lattice ceiciets { i } ca be cverted t direct r ceiciets{ [] } as shw belw: [ ] [ ] [] [] [] [] [ ],,...,,,..., Sluti is btaied recursively beii with. Thus we btai a sequece (M-) IR ilters e r each value. Eaple: ive a three-stae lattice ilter with ceiciets /4, /, /, deterie the IR ilter ceiciets r the direct r structure. We ca slve this prble recursively. Let us bei with. Thus we have : [] [ ] [ ] 4

7 Hece the ceiciets a IR ilter crrespdi t the sile-stae lattice are: ( ), ( ). Sice [] is the reverse plyial [] we have : 4 [] Net we add the secd stae t the lattice. r 4 [ ] [ ] [ ] 8 Hece the IR ilter paraeters crrespdi t the tw-stae lattice are ( ), ( ), ( ). ls 8 8 [] ially the additi the third stae t the lattice results i the plyial [] [] [ ] Csequetly, the desired direct-r IR ilter is characteried by the ceiciets 4 ( ), ( ), ( ), () 5 8 MTL r calculati relecti ceiciets M-uctis: plyrc rcply >> [/4 / / ].5.5. >> rcply()

8 as >> OR >> a[ /4 5/8 /] a >> plyrc(a) as >>.5.5.

BINOMIAL MODELS FOR INTEREST-RATE DERIVATIVES. We shall discuss the construction of simple binomial models for pricing interest-rate

BINOMIAL MODELS FOR INTEREST-RATE DERIVATIVES. We shall discuss the construction of simple binomial models for pricing interest-rate BINOMIAL MODLS FOR INTRST-RAT DRIVATIVS We shall discuss the cstructi f simple bimial mdels fr pricig iterest-rate derivatives The mai feature f these mdels is that the \risk-free" iterest rate ca vary