A flatness-based generalized optimization approach to spectral estimation

Signal P ocessing 19 (1990) 311-320 Else ie 311 A FLATNESS-BASED GENERALIZED OPTIMIZATION APPROACH TO SPECTRAL ESTIMATION Climen NADEU Dep . de Teo ia del Senyal i Comunicacions, Uni e si a Po/i ecnica de Ca alunya, Ba celona, Spain Miquel BERTRAN Dep . de Ma emii ica Aplicada i Telemii ica, Uni e si a Poli ecnica de Ca alunya, Ba celona, Spain Recei ed 12 Ap il 1988 Re ised 10 July 1989 Abs ac . In he op imiza ion o mula ion, once he known au oco ela ions a e ixed as cons ain s, he objec i e unc ional comple ely cha ac e izes he co esponding me hod o spec al es ima ion. A e obse ing ha his unc ional in ol es a so o spec al la ness maximiza ion, he ela ionship be ween he o m o i s in eg and and he salien ends o he a ious me hods is poin ed ou . This se es as a basis o p opose a gene alized op imiza ion app oach ha encompasses some classical es ima o s and o igina es new ones. Zusammen assung. In de Op imie ungs-Besch eibung cha ak e isie das Funk ional die jeweilige Me hode zu Spek alschal- zung olls andig, wenn on es en Au oko ela ionswe en ausgegangen wi d. Nach dem Au zeigen de Ta sache, daB dieses Funk ional eine A on Maximie ung de Flachhei des Spek ums beinhal e , wi d die Beziehung zwischen de Fo m des In eg anden und den he aus agenden Rich ungen de e schiedenen Me hdden he ausges ell . Au diese G undlage wi d ein e allgemeine e Op imie ungs-Ansa z o geschlagen, de einige klassische Scha ze einschlieB und neue he o b ing . SchlieBlich wi d die B auchba kei de Vo s ellung de spek alen Flachhei ii die Spek alscha zung disku ie . Resume. Dans un con ex e d'op imisa ion,la onc ion objec i ca ac e ise comple emen la me hode co espondan e d'es ima- ion spec ale des que les aleu s d'au oco ela ion connues sen ixees comme con ain es. Obse an que ce e onc ion objec i implique une ce aine maximisa ion du ca ac e e pla du spec e (en anglais spec al la ness), la ela ion en e la o me de !'in eg a ion e les ai s saillan s de di e en es me hodes es mise en e idence. Ceci se de base a une app oche gene alisee de l'op imisa ion qui ecou e ce ains es ima eu s classiques e pe me d'en c ee de nou eaux. Keywo ds. Spec al analysis, spec al modeling, maximum la ness. 1. In oduc ion The op imiza ion o a ia ional o mula i9n is an in e es ing s a ing poin o spec al es ima ion because a numbe o meaning ul es ima o s can be de i ed om i [1-3, 6, 8, 9, 12-14]. Succinc ly, he app oach is as ollows. On he one hand, a numbe o measu emen s ob ained om he signal samples ca y in o ma ion abou he spec al densi y unc ion S(w) o he unde lying andom This wo k was suppo ed by he PRONTIC g an numbe 105/88. 0165-1684/90/$3.50 © 1990, Else ie Science Publishe s B.V. p ocess; in mos cases, hey a e he au oco ela ion unc ion alues n iom lag 0 o M. In he de e - minis ic app oach [11], hese alues a e assumed o be exac ly known, so hey ac ually ac as con- s ain s o he op imiza ion p oblem. On he o he hand, a cos measu e J ha , as i will be shown in he ollowing, is conce ned wi h he deg ee o emphasis gi en o each ype o spec- al shape is de ined. The aim is o minimize he unc ional 1 "' J =- F[S( w )] dw, 2'lT _, (1) l 312 C. Nadeu, M. Be an I A la ness-based op imiza ion o spec al es ima ion subjec o he au oco ela ion cons ain s 1 " . . - S(w) eJWn dw = n, 2'IT _, n =0, ±1, ... , ±M, (2) whe e he a ea o S(w) will be no malized he e o uni y, i.e., 0 = 1, wi hou loss o gene ali y. Ob iously, once he cons ain s (2) a e speci ied, he pe o mance o he spec al es ima ion me hod a ising om he abo e app oach is comple ely cha ac e ized by F(S), so we ha e o ace wi h he p oblem o designing a p ope cos unc ion F(S). We may ega d his p oblem as ha o measu ing he dissimila i y be ween he shape o he spec um es ima e S(w) and he la spec um S0 (w) = 1. In ac , among all he spec a ha ma ch he gi en au oco ela ions (2), he maximally la o smoo h spec um could be a sensible choice because i would be maximally close o he whi e noise powe densi y unc ion [3]. On he o he hand, i he spec al a ea 0 is he only cons ain , i.e. M= 0, S(w) = 1 wha e e F(S) is [9]. The e o e, maximum la ness seems o be a common endency o he me hods a ising om he op imiza ion app oach which is only bounded by he con- s ain s. Fo his eason, i was exploi ed in [1, 2] as a uni ying p inciple o spec al es ima ion. Fi s o all, conside he well-known maximum en opy me hod [3] (he ea e we will call i MEM1) ha maximizes he en opy measu e gi en by 1 " -log S(w) dw, 2T _, (3) which belongs o he gene al class o unc ionals J o (1) when F(S) =-logS. (4) I has o en been ega ded as a maximum la ness me hod [3] mainly due o he ac ha i maximizes he en opy o he associa ed Gaussian p ocess wi h he idea o app oaching he whi e noise p o- cess and, consequen ly, he whi e noise ( la ) spec- um. Signal P ocessing The e exis s an al e na i e e sion o he maxi- mum en opy p inciple (we will e e o i as MEM2) which s a s om a di e en la ness measu e [9]. Since a powe spec al densi y ac ually is a p obabili y densi y unc ion o a an- dom a iable 2T , whe e makes sense in e ms o he ins an aneous equency o he associa ed andom (no necessa ily Gaussian) p ocess [14], we can also use he ela i e en opy measu e be ween S(w) and S0 (w) [7], namely 1 " S(w) E(S, S0) =- S(w) log-S ( ) dw. 2'IT _, 0 w (5) Since (5) is a measu e o closeness be ween he wo densi y unc ions S(w) and S0(w), a la ness measu e is ob ained by se ing S0( w) = 1. Thus, he MEM2 a ises om he op imiza ion app oach o F(S) =SlogS. (6) Con e sely o he maximum en opy app oach, we can a oid any e e ence o he en opy o he p ocess and jus no ice he spec um i sel , so ha we me ely hink o la ness om a geome ical poin o iew. Then, he Euclidean measu e o sepa a ion om he cons an spec um (7) appea s as a sensible unc ional J, leading o he classical Blackman-Tukey me hod wi h ec- angula window which ex apola es wi h ze oes he au oco ela ion unc ion beyond M. Ob iously, i s co esponding cos unc ion F(S) is F(S) = (S -1?. (8) The o egoing spec al es ima ion me hods a e h ee di e en ways o aiming a maximum la ness es ima es. Howe e , he e a e many o he possibili ies. The pu pose o his pape is no only he explo a ion o new me hods, bu mainly o poin ou , in a a he quali a i e manne , he in luence o he o m o he unc ion F(S) on he pe o mance o he me hods om he iewpoin o la ness in o de o a i e a a gene alized app oach. C. Nadeu, M. Be an I A la ness-based op imiza ion o spec al es ima ion 313 The pape is o ganized as ollows. In Sec ion 2, he op imiza ion p oblem is sol ed o ind he spec um es ima e associa ed o each unc ion F(S) o a gi en se o cons ain s n, n = 0, ±1, ... , ±M. In Sec ion 3 he second de i a i e o F(S) is p esen ed as a sui able ool o explain- ing he salien de e minis ic ea u es o he co e- sponding es ima o . This ac is used in Sec ion 4 o p opose an unbounded amily o me hods ha includes he h ee abo e men ioned BTM, MEM1 and MEM2. Some illus a i e examples a e gi en in Sec ion 5 and, inally, a gene alized app oach is p esen ed in Sec ion 6, along wi h an algo i hm o de e mine he disc e e spec al es ima e. 2. Sol ing he op imiza ion p oblem The minimiza ion o he unc ional (1) wi h con- s ain s (2) can be ca ied ou by means o Lag ange mul iplie s An, n = 0, ±1, ... , ±M, by minimizing he in eg al 2 ~ J:"' { F[S(w)] + n=~M An[ n- S(w) ej"'"J} dw. (9) I he de i a i e o he in eg and wi h espec o S(w) is made equal o ze o, an ex emal o (9) is ob ained. The co esponding spec um e i ies he equali y M F'[S(w)]= L Anej"'"=P(w), (10) n=-M whe e F' is he de i a i e o F wi h espec o S and P( w) is a eal and e en igonome ic poly- nomial o o de M. I can be shown [3] ha , i he e is a solu ion o he cons ained minimiza ion p oblem, i is unique and gi en by S(w) = G[P(w )], (11) whe e G is he in e se unc ion o F'. No e ha , in gene al, G is no linea and he model is no a ional. In o de o ind he spec um es ima e S(w) we should subs i u e (11) in (2), ob aining, in gene al, a sys em o M+ 1 non-linea equa ions. The M+ 1 a iables An can be de e mined by means o an i e a i e algo i hm. Then, S(w) is compu ed om P(w) wi h (11). Thus, he co esponding au oco ela ion unc ion R ( n) ag ees wi h i s known alues n up o M and ex apola es hem up o in ini y. 3. Analy ical compa ison o me hods Gi en a pa icula se o cons ain s n, n = 0, ±1, ... , ±M, he spec al es ima ion me hods esul ing om he abo e app oach only di e by hei cos unc ion F(S). Howe e , cons ain s could be di e en om au oco ela ions ( o example, ceps al coe icien s [8]), so, wi h he objec o compa ing me hods, he ype o con- s ain s should be included. This can be accom- plished by using as subjec o compa ison he spec al model since i is a consequence o bo h F(S) and he ela ionship be ween he unc ion o which co espond he cons ain s, i.e. he au oco ela ion unc ion in ou case, and he spec- um. Fo example, i ceps al coe icien s we e used, F'(S) in (10) should be subs i u ed by SF'(S) o ob ain he spec al model whe eas F(S) would be le unchanged. spec al model is jus de e mined by he i s de i a i e o he cos unc ion acco ding o (10). Consequen ly, each me hod o spec al es ima ion a ising om he op imiza ion app oach is cha ac- e ized by F'(S). Howe e F'(S) + K1 leads o he same spec al model as F'(S) because he cons an K1 may be included as pa o he coe icien A0 in (10). Thus, due o he ac ha bo h unc ions F'(S) and F'(S) + K1 ha e he same de i a i e wi h espec o S, we can choose (S), he second de i a i e o F(S) wi h espec o S, as he unc ion ha bes cha ac e izes he spec al model. Equa ion (10) can hen be ew i en in he ollow- ing way: J S(w) (x) dx=P(w), XJ (12) whe e he cons an x1 is a bi a y. Vol. 19, No. 4, Ap il 1990 314 C. Nadeu, M. Be an I A la ness-based op imiza ion o spec al es ima ion The ollowing example is illus a i e. Conside wo di e en unc ions Fa(S) = Kz(S -1) 2, Fb(S) = Kz(S 2 -1). (13) (14) Thei i s de i a i es di e by an addi i e cons an o alue - 2K 2• Howe e , hei second de i a i es a e iden ical, as well as hei spec al models (which co espond o he BTM). The e o e, assuming he cons ain s ", n = 0, ±1, ... , ±M, he way in which he cos unc ion F(S) a ou s a spec al shape wi h espec o ano he one is de e mined by i s cu e con exi y. In ac , a ze o alue o i s second de i a i e (S) o all S implies a linea unc ion F(S) so ha he unc ional J in (1) only depends on 0, he a ea o S(w ). Hence, when (S) = 0 o all S, S(w) may show any shape consis en wi h he cons ain s. Mo eo e , when (S) has a cons an non-ze o alue K3, we can choose x1 such ha (15) esul ing in F(1 +AS)= F(l- AS), o AS :s;; 1. (16) Since he mean alue o S(w) is 0 = 1, (16) means ha spec al peaks (AS> 0) and alleys (AS< 0) a e iden ically ea ed by he spec al model as long as 0 :s;; S :s;; 2. A his poin , once he signi icance o he second de i a i e ( S) has been shown, i is wo h compa - ing di e en spec al es ima o s om he poin o iew o hei associa ed unc ions. Since he spec- al models emain una ec ed when a cons an is added o F(S), F'(S) o bo h, o when (S) is mul iplied by a cons an , a no maliza ion is needed o emo e he a bi a i y o he compa ison. Fo his eason, we will ix he alues o he unc ions F, F' and a he mos cha ac e is ic poin , namely S = 1. The imposed condi ions a e F(1) =0, (17) F'(1) =0 (18) (l) = 1. (19) Signal P ocessing Condi ion (17) o ces he cos alue J in (1) o be ze o when S( w) = 1 o all w ( la spec um). Con- di ion (18) ensu es ha F(S) has an ex emal a S = 1, which has o be a minimum in o de o weigh nega i ely any spec al de ia ion om uni y. Finally, condi ion (19) makes a con exi y no maliza ion ha will allow us o compa e he ea men o peaks wi h espec o alleys and ice e sa. Le us show he e ec o using he abo e condi- ions on he h ee unc ions F(S) in (4), (6) and (8). The esul ing unc ions a e shown in Table 1, along wi h hei i s and second de i a i es. The unc ional J co esponding o he MEM1 is exac ly he I aku a-Sai o measu e o sepa a ion be ween wo spec a [5] when one o hem is S( w) and he o he is he cons an spec um S(w) = 1. Fu he - mo e, i can be shown ha all F(S) a e nonnega i e unc ions. Also no ice ha he second de i a i es show a egula o m o all me hods, namely, Sk(w ), k = 0, -1 and -2. The unc ions F(S) and (S) co esponding o he h ee spec al es ima ion me hods a e plo ed in Fig. 1. Obse e ha he MEM1 weigh s alleys (S < 1) in he cos measu e mo e han he BTM and he opposi e occu s a peaks. The MEM2 lies be ween he o he wo, consis en ly wi h esul s epo ed in [9]. The second de i a i es a e mono onic unc ions and hey a e such ha , gi en a alue o S, a g ea e con exi y implies a g ea e weigh ing in he cos unc ion. No only he ela i e ea men o peaks and alleys cha ac e is ic o e e y spec al es ima o can be an icipa ed om he unc ions used in he Table 1 No malized cos unc ions o he op imiza ion app oach co e- sponding o he h ee p e iously known spec al es ima ion me hods, and hei i s " wo de i a i es. MEM1 MEM2 BTM F(S) -logS+(S-1) Slog S-(S-1) !(S -1) 2 F'(S) -(1/S)+1 JogS S-1 (S)= F"(S) 1/Sz 1/S 1 C. Nadeu, M. Be an I A la ness-based op imiza ion o spec al es ima ion 315 2 ~----------------~ -- BlM ----- MEM2 ....... MEM1 §:1 LL . ·._ . '' _ .. -···· .."''./;::.<>/ 0 ~--~~~----~--~ 0 2 3 (a) 2 ,-~--------------, ,. ·. · ,_·. -- BlM ----· MEM2 ....... MEM1 (J)j ;. - - '· ·-::.-·., · .. :::::···-··- ............... _____ ,_ ····-~ ···- ......... . 0 1------ ----------~ 0 2 3 (b) Fig. 1. (a) No malized cos unc ions F(S) co esponding !' he BTM (-), MEM2 (----)and MEMl(· ···);(b) hei second de i a i es (S). op imiza ion app oach, bu also o he ea u es ela ed wi h he geome ical shape o spec a. Le us show wo examples. O he examples will appea in Sec ion 5. Fi s o all conside he well-known possibili y o he BTM o p oduce spec al es ima es wi h nega i e alues. This e ec has an explana ion in e ms o i s associa ed unc ion since F'(O) is ini e. In ac , when he slope o F(S) a S = 0 is in ini e he spec um is d i en away om ze o by he minimiza ion p ocess [3]. As i was poin ed ou in [6, 9] MEM2 spec a can exhibi e y deep alleys when he e exis p ominen peaks. This e ec can also be explained by obse ing he MEM2 unc ions. On he one hand, F'(O) is in ini e so, unlike in he BTM, nega- i e o ze o alues o he spec um a e no allowed. On he o he , unlike in he MEM1, F(O) is ini e and i is e y close o uni y inside he in e al [ 0, s], whe e s « 1, so S ( w) can app oach he ze o alue a some equency bands wi hou inc easing no iceably he unc ional J ha has o be minim- ized. This is specially ue when he au oco ela ion cons ain s o ce he spec um es ima e o ha e p ominen peaks since in his case J has a high alue ha is only sligh ly a ec ed by he deg ee o dep h o spec al alleys. 4. A amily o spec al es ima ion me hods The egula o m o he second de i a i es (S) shown in Table 1 sugges s he possibili y o de ining a amily o spec al es ima o s cha ac e - ized by a eal cons an g and he ollowing simple exp ession o he second de i a i e (20) which e i ies (19) and encompasses he abo e conside ed BTM, MEM2 and MEM1, espec i ely, o g=O, -1 and -2. Pe o ming a double in eg a ion and imposing (17), (18), he amily o cos unc ions (s) 1 (Sg+2 ) Fg (g+1)(g+2) -1 1 --(S-1) (21) g+1 ollows. They a e alid o all eal alues o g excep o g = -1 and g = -2 which a e singula poin s in he amily ( hei co esponding unc ions a e shown in Table 1). In a s ic sense, his amily should comp ise only hose es ima o s ha gua an ee non-nega i i y o spec a. Since a su icien condi ion is o show an in ini e alue o F~(O), we should es ic he amily o alues g,;:;; -1. Howe e , i will also be wo h s udying he beha iou o he me hods co e- sponding o g > -1. Vol. 19, No. 4, Ap il 1990 316 C. Nadeu, M. Be an I A la ness-based op imiza ion o spec al es ima ion Figu e 2 shows Fg(S) and g(S) o se e al in ege alues o g. All he cos unc ions a e necessa ily non-nega i e since hey a e con ex and i s minimum alue is ze o. F om he sequence o cu es i is appa en ha spec al peaks a e mo e a ou ed o lowe alues o g and he same occu s wi h alleys o g ea e alues o g. I F(S) in (1) is subs i u ed by Fg(S) o (21), a la ness measu e o S(w) is ob ained. To make mo e appa en ha i eally measu es sepa a ion om he la spec um, we obse e ha Fg(S) is equi alen o 1 [( s)g+ 2 J Fg(S, S0) = (g + 1 )(g + 2) So -1 __ 1 (~-1) g+1 S0 (22) 0 2 3 (a) 0 2 3 (b) Fig. 2. (a) Plo o Fg(S) o se e al alues o g. (b) Plo o /g(S) o he same alues o g. Signal P ocessing when S0(w) = 1 o all w. This exp ession also shows ha peaks (S > S0) and alleys (S <So) a e di e en ly weigh ed by he measu e, he ype o weigh ing depending on he alue o g. Mo eo e , no ice ha , i a p io spec al es ima e S0( w) di e en om he cons an spec um exis s [12], i is eadily inco po a ed in o he op imiza ion app oach using Fg(S, S0) ins ead o Fg(S). No e ha he unc ion J co esponding o g = -2 is he I aku a-Sai o dis ance be ween S and S0 since he limi o he i s e m when g ~ -2 is -log(S/ S0 ). Acco ding o (11), he spec al models a ising om he op imiza ion app oach and Fg(S) a e S(w) = [(g + 1)P(w) + 1F 1(g+l g;C-1. (23) The spec al model o he MEM2 (g = -1) can be ob ained om j_ 1(S) = s-1 o aking he limi when g ~ -1 in (23). As i can be obse ed om Table 1 and conside ing (10), i co esponds o a polynomial modeling o he log spec um. Table 1 also shows he models associa ed o he BTM and he MEM1, which a e a ional models. Fo he BTM, he ob ained au oco ela ion unc ion R(n) is ze o beyond M and ma ches he gi en alues n om 0 o M. In he MEM1, an i e a i e algo i hm is no equi ed o ind he ex apola ed au oco ela ions o , equi alen ly, he spec um, because he e exis s he e icien Le inson-Du bin algo i hm [3]. Any o he eal g di e en om 0 and -2 gi es ise o a non a ional model and needs an i e a i e algo i hm o ind S ( w). No e om (10) ha he polynomial coe icien s An a e he Fou ie 's se ies coe icien s o F'[S(w )], which a e ze o o In I> M. Hence, e e y me hod eme ging om he op imiza ion app oach is equi alen o mul iplying he Fou ie se ies coe icien s o he i s de i a i e o he exac spec- um by a window o leng h 2M + 1. The window depends on he gi en au oco ela ions n, since he esul ing coe icien s mus p ese e hese da a. The BTM is an excep ion since, in his case, he ( ec- angula ) window is di ec ly applied o he au oco ela ion unc ion. C. Nadeu, M. Be an I A la ness-based op imiza ion o spec al es ima ion 317 5. Examples Now we will illus a e he s a emen s o he p e ious sec ions wi h some examples. Pa - icula ly, we desi e o show he ela ionship be ween he pa ame e g o he abo e-men ioned amily o me hods and he pe o mance o he co esponding me hods. All he esul s o his sec ion we e ob ained by means o an i e a i e algo i hm based on he New on-Raphson app oach [9], excep o he cases g = 0 and -2 which ha e simple special algo i hms. Fo small alues o lgl, h ee i e a ions usually su ice o sol e he non-linea sys em o equa ions (2) accu a ely, when he ini ial solu ion is compu ed om he da a n wi h he p ocedu e desc ibed in [9]. Howe e , he con e gence becomes mo e di icul when lgl g ows, specially i he spec um shows a la ge ampli ude ange. Fi s o all, le us conside he spec um plo ed in Fig. 3(a), which is o med adding wo Gaussian shapes o he same a ea and wi h opposi e sign o uni y. We assume exac ly known he ou i s alues o i s au oco ela ion unc ion and ex apo- la e hem o se e al in ege alues o g. The esul - ing es ima es a e shown in Figs. 3(b) and 3(c). These esul s clea ly show he abo e asse ions abou he signi icance gi en by each me hod o high and low alues o S(w ). I g1 is lowe han g2, he me hod co esponding o g1 has a sha pe peak and a la e alley han he me hod co e- sponding o g2• Th ee pa icula obse a ions can be no iced om he esul s. Fi s ly, S ( w) becomes nega i e o some w when g > 0, a ac ha is qui e possible o g > -1, in gene al, as shown in he p e ious sec ion. Secondly, he ampli ude ange is minimal o g = 0 and g ows when lgl inc eases. Thi dly, he lowe g is he highe he esolu ion capabili y o he me hod is, due o i s g ea e pe missi eness a peaks. This claim is illus a ed in Fig. 4, using a spec um wi h wo e y close Gaussian peaks. While he MEMl (g = -2) can no esol e he peaks, a smalle alue o g (g = -4) p oduces a spec um ha can sepa a e hem. 2,---------~~--~ Exac spec um 0'------"'=""---------------l 0 1 (a) 0 1 (b) Fig. 3. (a) Exac spec um; (b), (c) spec um es ima es, o M=3, ob ained, wi h g=3, 2, 1, 0, and g=O, -1, -2, -3, espec i ely. To comple e his se o examples, we shall con- side a new spec um which is plo ed in Fig. 5(a). As he spec um in Fig. 3(a), i consis s o wo Gaussian unc ions o opposi e sign; howe e , in his case, he ampli udes o he peak and he alley a e loga i hmically equi alen , so he peak is no iceably highe . Figu e 5(b) shows he spec a co esponding o g = 0 and g = -5 ob ained using he i s wen y au oco ela ions as cons ain s. In his example, we can obse e a clea e ec o leakage ha is mo e accen ua ed o g = 0 o such an ex en ha he alley is no be e cha ac e ized Vol. 19, No. 4, Ap il !990 318 C. Nadeu, M. Be an I A la ness-based op imiza ion o spec al es ima ion 5 Exac 4 spec um 3 2 V 0 0 1 (a) 3,----------, -g=-4 ---- g=-2 (MEM1) 2 0~-------~ 0 1 (b) Fig. 4. (a) Exac spec um; (b) spec um es ima es, o M= 10, ob ained wi h g = -4 (-)and g = -2 (-- -). han o g = -5. These di e ences o leakage can also be explained om F(S) due o he ac ha he luc ua ions essen ially co espond o spec al alues lowe han 1, so hey a e mo e a ou ed by high alues o g (i.e., g = 0) han by low alues (i.e. g = -5). 6. Gene alized op imiza ion app oach Un il now we es ic ed ou sel o he amily o unc ions Fg(S). This amily shows e y in e es ing p ope ies. Fi s o all, i de ines a ow o in ini e me hods including h ee basic app oaches o spec- um es ima ion (BTM, MEMl and MEM2) among hem. Mo eo e , all he unc ions a e Signal P ocessing dB 10 Exac spec um 5 0 -5 ·1 0 0 1 (a) dB 10 -g=O (BTM) -- --9=-5 5 0 -~. -· .. "). -5 0 1 (b) Fig. 5. (a) Exac spec um; (b) spec um es ima es, o M= 19, ob ained wi h g = 0 (-) and g = -5 (- - - ). mono onic o S(w) ;;;.1 o S(w):;;:; 1 and he e a e no c ossings be ween hem excep o S(w) = 1. These egula cha ac e is ics lead o well speci ied changes in he beha iou o he es ima es when g is a ied. Howe e , we may de ine o he unc ions F(S) di e en om Fg(S) and use hem o ob ain new me hods wi h gi en cha ac e is ics. In any case, we should impose wo su icien condi ions on hei de i a i es, namely (1) F'(S) ~ oo, S->0 (2) (S);;;. 0 o all S. Condi ion (1) gua an ees he posi i i y o S(w) and condi ion (2) ensu es ha , i an ex emal o he cons ained minimiza ion p oblem exis s, i is unique (as shown in [3]). C. Nadeu, M. Be an / A la ness-based op imiza ion o spec al es ima ion 319 Fo ins ance, we could use he unc ion Fs(S) =![F_ 1(S) + F_iS)] =!(S -1) logS. (24) No e ha (24) can also be ob ained a e aging he en opy unc ions (4) and (6). Mo eo e , i s unc- ional J esul s om he symme iza ion o he ela i e en opy measu e (5), i.e., !(E(S, S0 )+E(S 0, S)) by equa ing S0 o 1. Suppose also he ollowing unc ion 4 "'T S-1 F(S)=-- g--+S-1 "'T 2 S+1 ' (25) which e i ies he abo e wo condi ions in addi ion o (17)-(19). Figu e 6 shows he es ima e ob ained wi h his unc ion o he spec um depic ed in Fig. 3(a) and, as be o e, M= 3. We can obse e ha he esul ing spec um es ima e lies be ween hose co esponding o g = -2 (MEMl) and g = -3; his ac can be explained conside ing ha ,(S) is a mono onic unc ion like /g(S) and he slope o ,(S) in he cen al poin S = 1 lies be ween hose co esponding o j_ 2 (S) and _ 3(S), since j;(l) = -2.74 and ~(l) =g. In ac , he spec um ob ained by Fg(S) o g = -2.74 isually coincides wi h he spec um o F1(S) in Fig. 6. Now we desi e o go u he in o he a emp on gene alizing he me hodology o spec al es ima- ion wi hin he amewo k gi en by he op imiz- 0 1 Fig. 6. Spec um es ima e ob ained om F,(S) using he same cons ain s as hose o es ima es shown in Fig. 3. a ion app oach. In ac , i is possible o selec any kind o unc ion F(S) e i ying he o egoing con- di ions o o selec (S) and o in eg a e i wice in o de o ob ain F'(S) and F(S). Mo eo e , he unc ions may be de ined bo h analy ically o nume ically. This app oach is qui e di ec ; i does no need any in e p e a ion o he unc ional ( 1) in e ms o en opy o o he concep s, i only equi es he design o a unc ion acco ding o he ype o ea men desi ed o he di e en spec al shapes. Un o una ely, he algo i hm o New on- Raphson may no be use ul in his gene al app oach because unc ion Gin (11) may no be known. Ne e heless, we can always use a nume i- cal p ocedu e by disc e izing he a iable w so ha he alues S(k) o he spec um in he N + 1 equencies wk equally dis ibu ed be ween 0 and "'T a e he a iables o be de e mined by means o a cons ained op imiza ion algo i hm. The aim is hen o minimize N I F[S(k)], (26) k=-N subjec o he au oco ela ion cons ain s __ 1_ £ S(k) ej[2'1Tkn/(2N+l)] = 2N +1 k=-N n• n=0,±1, ... ,±M (27) and he posi i i y cons ain (i equi ed) S(k);;;.O, k=O, ±1, ... , ±N. (28) In a p e ious in es iga ion [10], he same p ob- lem was sol ed wi h a sequen ial quad a ic p o- g amming algo i hm [ 4] which uses he g adien o (26). Fo example, he spec um in Fig. 6 was compu ed using his algo i hm wi h N = 128. Un o una ely, he algo i hm does no exhibi a con e gence as good as he New on-Raphson's one. Howe e , o he algo i hms which a e mo e speci ic and show a be e con e gence pe o m- ance ha e ecen ly been de eloped [15]. Vol. 19, No. 4, Ap il 1990 l'l I I