The Schur-Cohn Algorithm
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1 Modelng, Estmton nd Otml Flterng n Sgnl Processng Mohmed Njm Coyrght 8, ISTE Ltd. Aendx F The Schur-Cohn Algorthm In ths endx, our m s to resent the Schur-Cohn lgorthm [] whch s often used s crteron for testng the stblty of bounded-nut bounded-outut systems []. To smlfy the descrton of ths lgorthm, we frst te u the nlyss of the stblty domn of nd -order trnsfer functon. Ths rtculr cse leds to smlfcton of the stblty crter mosed on the denomntor of the trnsfer functon. Unfortuntely, t cnnot be led to trnsfer functons of n order greter thn. We lso resent the Schur-Cohn stblty lgorthm bsed on the trnsfer functon of n ll-ss flter, llowng us to estblsh equvlence relton between the Schur coeffcents nd the reflecton coeffcents. Let there be second-order trnsfer functon defned s follows: b b N [F.] The oles of b b re equl to: 4 nd 4 [F.]
2 36 Modelng, Estmton nd Otml Flterng n Sgnl Processng nd ts eros re defned s follows: b b 4b nd b b 4b [F.3] eendng on the vlues ten by nd, the oles cn be rel or comlex. For exmle, when 4, the oles re comlex conjugtes of ech other. Otherwse, they re rel. To ensure stblty, the oles of the trnsfer functon must be locted wthn the unt crcle n the -lne,.e.: nd [F.4] Ths constrnt mles tht the followng two nequltes re stsfed: [F.5] nd: [F.6] Reltons [F.5] nd [F.6] me t ossble to defne trngle n the, lne where the flter s stble nd whch s clled the stblty trngle. Ths trngle dected n Fgure F.. s smle tool for testng the stblty s t s bsed on the vlues of the flter s coeffcents.
3 Aendces 363 Fgure F.. The stblty trngle Alcton of the stblty trngle Let there be th -order trnsfer functon defned s follows: N [F.7] where. The frst condton requred for the stblty s exressed n terms of : snce,.., [F.8] In the rest of ths stblty test, we wll te nd ssume tht the frst condton [F.8] s stsfed. Let us develo the trnsfer functon of th -order ll-ss flter usng. [F.9]
4 364 Modelng, Estmton nd Otml Flterng n Sgnl Processng Furthermore, we defne s follows: [F.] Note : We note tht s lso n ll-ss flter of order -, nd ts exresson cn be smlfed by mosng:,.., [F.] Thus, we hve: [F.] From equton [F.], we note tht the oles of,, re such tht: [F.3] Thus, tng equton [F.8] nto ccount, we obtn: [F.4] We now show tht stsfyng the ssertons s the trnsfer functon of stble ll-ss flter nd s equvlent to syng tht s stble. To do ths, we frst show tht f s the trnsfer functon of stble llss flter nd, then s the trnsfer functon of stble ll-ss flter.
5 We cn esly show tht ny gven ll-ss functon roertes: G G G f f f Aendces 365 G stsfes the followng [F.5] Consequently, f s n fct the trnsfer functon of stble ll-ss flter, when. owever, from equton [F.4], we see tht. Therefore, the oles of le nsde the unt crcle n the -lne nd s stble.. Let us te u equton [F.] nd exress Let us now ssume tht nd tht. We thus obtn: If s ole of s the trnsfer functon of stble ll-ss flter s functon of [F.6], t must verfy: [F.7] As, we get: [F.8] Snce gves us: s the trnsfer functon of stble ll-ss flter, equton [F.5] [F.9] f
6 366 Modelng, Estmton nd Otml Flterng n Sgnl Processng [F.] f [F.] f Condton [F.8] contrdcts [F.9], but s n greement wth [F.]. Consequently, s stble. nd Usng the sme develoment s resented bove, we cn defne n terms of, then defne 3 n terms of, nd so on, untl we obtn. At ech successve ste, we test the vlue of, then, nd so on. The Schur stblty crteron sttes tht s stble f for ll vlues of. Let us now loo t the corresondence between Schur coeffcents nd reflecton coeffcents. Let us te equton [F.]:,.., P [F.] nd wrte t n mtrx form, tng nto ccount tht nd : [F.] Let us comre ths mtrx wth equton [.9], obtned usng the Levnson lgorthm, nto whch we ntegrte :
7 Aendces 367 [F.3] The followng olynoml s ssocted wth the vector T : Thus, we cn ssocte the followng olynoml wth the vector T : The Schur-Cohn lgorthm s wrtten s follows: [F.4] whle the exresson for the Levnson lgorthm stsfes: [F.5] We note tht the mtrces n equtons [F.4] nd [F.5] re the nverses of one nother. Thus, we hve:
8 368 Modelng, Estmton nd Otml Flterng n Sgnl Processng [F.6] so: [F.7] Ths roves the equvlence between the Schur coeffcents nd the reflecton coeffcents. References [] T. Klth, A Theorem of I. Schur nd ts Imct on Modern Sgnl Processng, Oertor Theory: Advnces nd Alctons I. Schur Methods n Oertor Theory nd Sgnl Processng, 8,. 9-3, Brhuser, 986. [] S. K. Mtr, gtl Sgnl Processng Comuter Bsed Aroch, 3 rd edton, McGrw-ll, 6.
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