On the cramer-rao bound of finite-length cepstrum spectral estimators

Re e ences 1 OSHIBA, s., MATOBA, A., HORIKAWA, H., KAWAI, Y., and SAKUTA, M.: ‘Reliabili y o 1.3pm V-g oo ed inne -s ipe lase diodes unde high-powe ope a ion’, Elec on. Le ., 1986, 22, (8), p. 428 2 IMANAKA, K., HORIKAWA, H., MATOBA, A., KAWAI, Y., and SAKUTA, M.: ‘High powe ou pu , low h eshold, inne s ipe GaInAsP lase diode on a p ype InP subs a e’, Appl. Phys. Le ., 1984, 45, (3), p. 282 inne -s ipe lase diodes on a p- ype subs a e ope a ing o e IOOmW a 1-5pm wa eleng h’, Appl. Phys. Le e s, 1987, 50, (7), 3 HORIKAWA, H., OSHIBA, S., MATOBA, A., and KAWAI, Y.: ‘V-g oo ed p. 374 4 KOSZI, L. A., TEMKIN, H., PRYZBYLEK, G. I., SEGNER, B. P., IVAPHOLTZ, s. G., BOGDANOWIZ, C. M., and DUTTA, N. K.: ‘High powe ope alion o InP lnGaAsP double-channel plana bu ied-he e os uc u e lase s wi h asymme ic ace coa ings’, Appl. Phys. Le ., 1987, 51, (26). p. 2219 A. Y.: ‘High powe ou pu o e 200mW o GaInAsP/InP VIPS- LD. 10 h IEEE In . Semiconduc o Lase Con ., Kanazawa, Tokyo, 1986, pp. 148-149 6 ZAH, C. E., CANEAU, C., MENOCAL, S. G., FAVIRE, F., and LEE, T. P: ‘High-speed 1.3 pm GalnAsP p-subs a e bu ied-c escen lase s wi h semi-insula ing Fepi-doped InP cu en blocking laye s’, Elec on. Le ., 1988, 24, (II), p. 695 BAUM, A.. WOLF, o., RENNER, D., HESS, K. L., and ZEHR. s. w.: ‘1.3pm InGaAsP bu ied c escen lase s wi h cobal -doped semi-insula ing cu en blocking laye s g own by me alo ganic chemical apo deposi ion’, Appl. Phys. Le ., 1988, 53, (14), p. 1257 8 OLSHANSKY, R., FQWAZINK, w., HILL, P., LANZISERA, ., and LAUER, R. B.: ’InGaAsP bu ied he e os uc u e lase wi h 22GHz band- wid h and high modula ion e iciency’, Elec on. Le ., 1987, 23, (W 6), p. 839 KAWAI, Y.: ‘High-powe and high-speed semi-insula ing blocked V-g oo ed inne -s npe lase a 1.3 pm wa eleng h ab ica ed on p-InP subs a es’, Appl. Phys. Le ., 1989,54, (12), p. 1077 SUSAKI, w., IK~DA, K., and NIIKAWA, K.: ‘InCaAsP/lnP bu ied c escen lase diode emi ing a 1.3pm wa eleng h’, IEEE J. Quan um Elec onics, 1989, QE-20, (8), pp, 806-814 5 OSHIBA, S., HORIKAWA. H., MATOBA, A., KAWAHARA, M., and KAWAI, 7 CHENG, W. H., eOOLADDU, I., HUANG, S. Y., BUEHRING, K. D., APPEL- 9 HORIKAWA, H., WADA, H., MATWI, Y., YAMADA, T., OGAWA, Y., and 10 OOMURA, E., HIGUCHI, H., SAKAKIBAKA, Y., HIRANO, R., NAMIZAKI, H., ON THE CRAMER-RA0 BOUND OF FINITE-LENGTH CEPSTRUM SPECTRAL EST1 MATORS Indexing e ms: Speech ecogni ion, Spec al analysis Spec al es ima o s based on ini e-leng h ceps um model- ling a e use ul in se e al applica ions. The C ame -Rao lowe bound o he asymp o ic a iance o hei loga i hmic spec al es ima es can be ob ained in explici o m. This does no depend on he speci ic unde lying spec um. In oduc ion: Pa ame ic modelling o s a iona y andom p ocess is widely used in many enginee ing and s a is ics applica ions. Especially popula a e a ional models as he all-pole o au o eg essi e (AR) model and he gene al pole- ze o o au o eg essi e mo ing-a e age (ARMA) model.’ The e also exis non a ional models o in e es in a numbe o applica ions. Pa icula ly, he model S(w) = 8‘”’ (1) whe e P(w) is a igonome ic polynomial o he angula e- quency 4-n 10 I n) M-1 P(w) = C c,e-jok k=-M+I appea s in a ious ields, e.g., ada and sona applica ions ha make use o Gaussian spec a,’ o he abso p ion spec a encoun e ed in in e e ome ic ~pec oscopy.~ F om a heo- e ical poin o iew, his ype o model a ises om he maxi- ELECTRONICS LETTERS 5 h July 1990 Vol. 26 No. 14 misa ion o an en opy measu e o he unde lying andom p ocess assuming an accu a e se o au oco ela ion alues is gi en.“ Since he coe icien s ck o P(w) a e he ceps al coe i- cien s o S(w),’ his ype o spec al model is equi alen o conside a ini e-leng h ceps um. Fini e-leng h ceps um spec al es ima o s ob ained om AR (o LPC) modelling a e e y success ul in speech p o- cessing whene e a dis ance be ween spec a has o be e alu- a ed.6 Since he AR spec al modelling in ol es an in ini e ceps al sequence, he model unde lying hese dis ances (o Euclidean ype) is no longe o AR ype. Ac ually, i is a ini e-leng h ceps um model, as indica ed by eqn. 1. In ac , whene e a spec al es ima o in ol es a ceps um windowing, i will be based on his ype o modelling (see Re e ence 7 o ano he example). In spi e o he use ha is being made o spec al es ima o s based on his ype o model, he s a is ical e iciency o hem can no be easily e alua ed due o he lack o a heo e ical explici lowe bound o he a iance. In his le e a gene al exp ession o he asymp o ic C ame -Rao lowe bound o he log spec um is gi en. As will be shown, i does no depend on he speci ic unde lying spec um. Be o e inding his C ame -Rao bound we will calcula e he co esponding Fishe in o ma ion ma ix. Asymp o ic Fishe in o ma ion ma ix o ini e-leng h models: Le x(n) be a ze o-mean Gaussian s a iona y ime se ies. Assume ha measu emen s a e a ailable in he ange 1 I n I N whe e N + 00, and he ceps al pa ame e s cy, k = 0, 1, . . . , M - 1, a e es ima ed om x(n). The Fishe in o - ma ion ma ix associa ed wi h his es ima ion is gi en by Whi le’s o mula (exp ession (3.3 in Re e ence 8): dw (3) F,, = 5 j log S(4 a log S(4 4n ac, ac, -I whe e Fij a e he elemen s o he Fishe ma ix F, whe e i,j=O,l,_.., M- 1. Acco ding o he spec al model eqns. 1 and 2 log s(w) = co + 1 ck(Ba’ + ,-jo ) M- I k=l so ha he de i a i es in ol ed in eqn. 3 a e d log S(W) 8% ~- -1 (4) The Fishe ma ix akes he simple o m F,, = NS(i - j) i i, j # 0 N F,, = - 2 F=N(: 1/2 0 0 (7) which does no depend on he pa ame e alues ck. C ame-Rao lowe _ bound o he asymp o ic log spec um a iance: Assume C = [ o, l, . . . , ZM - is an asymp o ically unbiased es ima o o he ec o o M ceps al pa ame e s o he model. The spec al es ima e log g(w) ob ained h ough eqns. 1 and 2 is also asymp o ically unbiased and i s asymp- o ic a iance is bounded by he co esponding C ame -Rao bound? I he es ima o is e icien , his a iance will app oach he C ame -Rao lowe bound as N, he numbe o da a poin s, ends o in ini y. I i is no e icien , hen a {log &a)} will be s ic ly g ea e han he bound as N + 00. 987 The asymp o ic C ame -Rao lowe bound asse s ha , i he Fishe in o ma ion ma ix is no singula , he asymp o ic a iance o log 9(w) e i ies’ a {log g(w)} DJ(w)F~‘D(w) (8) whe e T deno es anspose and D(w) is he ec o o pa ial de i a i es a log s(w) a log s(~) a log s(w) F om he non-singula Fishe ma ix gi en by eqn. 7 and he elemen s o he ec o o(w) indica ed in eqn. 5, i eadily ollows om eqn. 8 ha which is independen o S(o), being dependen only on N and M. Adding up he geome ical exponen ial se ies in ol ed in eqn. 10, he asymp o ic C ame -Rao lowe bound o log $(U) can be exp essed in he ollowing mo e closed o m: No ice ha i M + m 4M N 2- w=O,n In his case, he C ame -Rao lowe bound exac ly coincides wi h he asymp o ic a iance (M + cc and N + CO) o he e icien AR spec al es ima o .” This is a consis en esul since when he numbe o pa ame e s o he model ends o in ini y he ini e-leng h ceps um model and he AR model become iden ical. To summa ise he pe o mance o he spec al es ima o wi h a scala measu e, we can compu e he lowe bound o he a e age a iance, which is 2M L = 1 a {log $w)} dw 2 - 271 N -I In Re e ence 9 was shown ha eqn. 13 is an uni e sal esul , independen o bo h he speci ic pa ame ic model and he unde lying spec um. Obse e ha L is ac ually he in eg a ed mean squa e es ima ion e o o he log spec um since he es ima o is unbiased. Conclusions: Spec al es ima o s based on ini e-leng h ceps- um modelling a e use ul in se e al applica ions. In his pape , gene al explici exp essions o he asymp o ic Fishe in o ma ion ma ix and he asymp o ic C ame -Rao lowe bound o he log spec um a iance ha e been easily ob ained. None o hem depends on he speci ic unde lying spec um, a esul ha is cha ac e is ic o he ini e-leng h ceps um model due o he pa icula o m o eqns. 3 and 9. Acknowledgmen : This wo k was suppo ed by a PRONTIC g an numbe lO5jSS. C. NADEU E. LLEIDA Depa men Teo ia del Senyal i Comunicacions Uniue si a Poli ecnica de Ca alunya Ap. 30002,08080 Ba celona, Spain 988 3 d May 1990 Re e ences 1 KAY, s.: ‘Mode n spec al es ima ion’ (P en ice-Hall, 1987) 2 VAN TREES, H. L.: ‘Es ima ion and modula ion heo y, Pa 111’ (Wiley, New Yo k, 1971) 3 GINGRAS, D. I., and BOEHME, 1. F.: ‘Ma hema ical modelling and pa ame ic es ima ion in he ceps um domain o abso p ion spec- al encoun e ed in in e e ome ic spec oscopy’. P oc. 3 d ASSP Wo kshop on Spec um Es ima ion and Modelling, Bos on, No . 1986, pp. 7%73 4 NADEU, c., BERTRAN, M., and SOLE, J.: ‘Spec al es ima ion wi h a ional modelling o he log spec um’, Signal P ocessing, 1986, 10, (I), pp. 7-18 5 OPPENHEIM, A., and SCHAFER, R.: ‘Disc e e- ime signal p ocessing’ (P en ice-Hall, 1989) 6 TOKHWRA, Y.: ‘A weigh ed ceps al dis o ion measu e o speech ecogni ion’. P oc. In . Con . Acous . Speech and Signal P ocess., Tokyo (Japan), Ap il 1986, pp. 761-764 7 WAHBA, G.. ‘Au oma ic smoo hing o he log pe iodog am’, J. Am. S a is . Assoc., 1980,75 8 WHITTLE, P.: ‘The analysis o mul iple s a iona y ime se ies’, J. Royal S a is . Soc., 1953, 15, pp. 125139 9 FRIEDLANDER, B., and PORAT, 8.: ‘A gene al lowe bound o pa a- me ic spec um es ima ion’, IEEE T ans., 1984, ASSP-32, (4), pp. 728-733 IO KROMER, R. E.: ‘Asymp o ic p ope ies o he au o eg essi e spec al es ima o ’. Ph.D. Disse a ion, S and o d Uni e si y, 1970 ADAPTIVE ALGORITHM FOR VARIABLE TELETRAFFIC DEMAND IN HIGHWAY MICROCELLS Indexing e ms: Telecommunica ions, Algo i hms I An adap i e algo i hm is p esen ed which ensu es ha he p obabili y o a mobile adio call in p og ess being o ced o e mina e du ing hando e in highway mic ocells is always small, e en in he p esence o high new call eques a es. As he ele a ic demand inc eases o mobile adio communi- ca ions i becomes necessa y o ope a e wi h small cells ha a e con igu ed o he localised ele a ic load. By op ing o smalle cells, o en called mic ocells o picocells,i,z he spec- al e iciency is d ama ically inc eased. A pa icula ype o mic ocell ha is o comme cial in e es is he highway mic o- cell. The mobile s a ions (MSs) a e moun ed in ehicles ha a e es ained o he igh con ines o he highway lanes. The d i e s make calls while in mo ion using anscei e s ha ing a hands o mode o ope a ion. A highway mic ocell co e s a segmen o he highway con aining a small base s a ion (BS) moun ed a lamp pos ele a ion, say 15m, adi- a ing mobile adio signals wi h a ciga shape beam along he high~ay.~ Con igious mic ocellula BSs o m a clus e , wi h he BSs connec ed ei he by an op ical LAN, o by poin - o- poin adio links o he mobile swi ching cen e (MSC). The clus e s a e also con igious o e he leng h o he highway. As a ehicula MS, in he p ocess o making a call, a els along he highway i eques s om he BS in he nex mic ocell a channel in o de o con inue i s communica ion. Should no channel be a ailable he call is e mina ed. The p obabili y o his happening is P,. We equi e ha P, be signi ican ly lowe han he p obabili y P, o a newly eques ed call being blocked. To ensu e ha P, P, we ha e de ised a numbe o scheme^.^ One is o a ange o a BS o ha e a se numbe o channels N, exclusi ely a ailable o hando e s. I he BS can suppo N channels, he emaining N, = (N - Nh) chan- nels may be used o ei he hando e o new calls. The ele- a c pe o mance may be imp o ed by deploying an o e laying mac ocell which p o jdes No channels o assis hose BSs in he mic ocellula clus e ha cu en ly ha e no channels a ailable o hando e o eques ing MSs. Thus a MS en e ing a mic ocell whose BS does no ha e a channel a ailable o allow he MS o con inue i s call is assigned one om he mac ocell BS. When a channel becomes a ailable a FLECTRONICS LETTERS 5 h July 1990 Vol. 26 No. 14