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

References 1 OSHIBA, s., MATOBA, A., HORIKAWA, H., KAWAI, Y., and SAKUTA, M.: ‘Reliability of 1.3pm V-grooved inner-stripe laser diodes under high-power operation’, Electron. Lett., 1986, 22, (8), p. 428 2 IMANAKA, K., HORIKAWA, H., MATOBA, A., KAWAI, Y., and SAKUTA, M.: ‘High power output, low threshold, inner stripe GaInAsP laser diode on a ptype InP substrate’, Appl. Phys. Lett., 1984, 45, (3), p. 282 inner-stripe laser diodes on a p-type substrate operating over IOOmW at 1-5pm wavelength’, Appl. Phys. Lefters, 1987, 50, (7), 3 HORIKAWA, H., OSHIBA, S., MATOBA, A., and KAWAI, Y.: ‘V-grooved 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 power operalion of InPflnGaAsP double-channel planar buried-heterostructure lasers with asymmetric facet coatings’, Appl. Phys. Lett., 1987, 51, (26). p. 2219 A. Y.: ‘High power output over 200mW of GaInAsP/InP VIPSLD. 10th IEEE Int. Semiconductor Laser Conf., 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-substrate buried-crescent lasers with semi-insulating Fepi-doped InP current blocking layers’, Electron. Lett., 1988, 24, (II), p. 695 BAUM, A.. WOLF, o., RENNER, D., HESS, K. L., and ZEHR. s. w.: ‘1.3pm InGaAsP buried crescent lasers with cobalt-doped semi-insulating current blocking layers grown by metalorganic chemical vapor deposition’, Appl. Phys. Lett., 1988, 53, (14), p. 1257 8 OLSHANSKY, R., FQWAZINK, w., HILL, P., LANZISERA, v., and LAUER, R. B.: ’InGaAsP buried heterostructure laser with 22GHz bandwidth and high modulation efficiency’, Electron. Lett., 1987, 23, (Wt6), p. 839 KAWAI, Y.: ‘High-power and high-speed semi-insulating blocked V-grooved inner-stnpe laser at 1.3 pm wavelength fabricated on p-InP substrates’, Appl. Phys. Lett., 1989,54, (12), p. 1077 SUSAKI, w., IK~DA, K., and NIIKAWA, K.: ‘InCaAsP/lnP buried crescent laser diode emitting at 1.3pm wavelength’, IEEE J. Quantum Electronics, 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., APPEL9 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 terms: Speech recognition, Spectral analysis Spectral estimators based on finite-length cepstrum modelling are useful in several applications. The Cramer-Rao lower bound of the asymptotic variance of their logarithmic spectral estimates can be obtained in explicit form. This does not depend on the specific underlying spectrum. Introduction: Parametric modelling of stationary random process is widely used in many engineering and statistics applications. Especially popular are rational models as the all-pole or autoregressive (AR) model and the general polezero or autoregressive moving-average (ARMA) model.’ There also exist nonrational models of interest in a number of applications. Particularly, the model S(w) = 8‘”’ (1) where P(w) is a trigonometric polynomial of the angular frequency 4-n 10 I n) M-1 P(w) = C c,e-jok k=-M+I appears in various fields, e.g., radar and sonar applications that make use of Gaussian spectra,’ or the absorption spectra encountered in interferometric ~pectroscopy.~ From a theoretical point of view, this type of model arises from the maxiELECTRONICS LETTERS 5th July 1990 Vol. 26 No. 14 misation of an entropy measure of the underlying random process assuming an accurate set of autocorrelation values is given.“ Since the coeficients ck of P(w) are the cepstral coeficients of S(w),’ this type of spectral model is equivalent to consider a finite-length cepstrum. Finite-length cepstrum spectral estimators obtained from AR (or LPC) modelling are very successful in speech processing whenever a distance between spectra has to be evaluated.6 Since the AR spectral modelling involves an infinite cepstral sequence, the model underlying these distances (of Euclidean type) is no longer of AR type. Actually, it is a finite-length cepstrum model, as indicated by eqn. 1. In fact, whenever a spectral estimator involves a cepstrum windowing, it will be based on this type of modelling (see Reference 7 for another example). In spite of the use that is being made of spectral estimators based on this type of model, the statistical eficiency of them can not be easily evaluated due to the lack of a theoretical explicit lower bound for the variance. In this letter a general expression for the asymptotic Cramer-Rao lower bound of the log spectrum is given. As will be shown, it does not depend on the specific underlying spectrum. Before finding this Cramer-Rao bound we will calculate the corresponding Fisher information matrix. Asymptotic Fisher information matrix of finite-length models: Let x(n) be a zero-mean Gaussian stationary time series. Assume that measurements are available in the range 1 I n I N where N + 00, and the cepstral parameters cy, k = 0, 1, . . . , M - 1, are estimated from x(n). The Fisher information matrix associated with this estimation is given by Whittle’s formula (expression (3.3 in Reference 8): dw (3) F,, = 5 j log S(4 a log S(4 4n ac, ac, -I where Fij are the elements of the Fisher matrix F, where i,j=O,l,_.., M1. According to the spectral model eqns. 1 and 2 log s(w) = co + 1 ck(Ba’ + ,-jot) MI k=l so that the derivatives involved in eqn. 3 are d log S(W) 8% ~- -1 (4) The Fisher matrix takes the simple form F,, = NS(i - j) if i, j # 0 N F,, = - 2 F=N(: 1/2 0 0 (7) which does not depend on the parameter values ck. Crame-Rao lower_ bound of the asymptotic log spectrum variance: Assume C = [fo, fl, . . . , ZM - is an asymptotically unbiased estimator of the vector of M cepstral parameters of the model. The spectral estimate log g(w) obtained through eqns. 1 and 2 is also asymptotically unbiased and its asymptotic variance is bounded by the corresponding Cramer-Rao bound? If the estimator is eficient, this variance will approach the Cramer-Rao lower bound as N, the number of data points, tends to infinity. If it is not efficient, then var {log &a)} will be strictly greater than the bound as N + 00. 987 The asymptotic Cramer-Rao lower bound asserts that, if the Fisher information matrix is not singular, the asymptotic variance of log 9(w) verifies’ var {log g(w)} t DJ(w)F~‘D(w) (8) where T denotes transpose and D(w) is the vector of partial derivatives a log s(w) a log s(~) a log s(w) From the non-singular Fisher matrix given by eqn. 7 and the elements of the vector o(w) indicated in eqn. 5, it readily follows from eqn. 8 that which is independent of S(o), being dependent only on N and M. Adding up the geometrical exponential series involved in eqn. 10, the asymptotic Cramer-Rao lower bound of log $(U) can be expressed in the following more closed form: Notice that if M + m 4M N 2w=O,n In this case, the Cramer-Rao lower bound exactly coincides with the asymptotic variance (M + cc and N + CO) of the efficient AR spectral estimator.” This is a consistent result since when the number of parameters of the model tends to infinity the finite-length cepstrum model and the AR model become identical. To summarise the performance of the spectral estimator with a scalar measure, we can compute the lower bound of the average variance, which is 2M L = 1 var {log $w)} dw 2 - 271 N -I In Reference 9 was shown that eqn. 13 is an universal result, independent of both the specific parametric model and the underlying spectrum. Observe that L is actually the integrated mean square estimation error for the log spectrum since the estimator is unbiased. Conclusions: Spectral estimators based on finite-length cepstrum modelling are useful in several applications. In this paper, general explicit expressions for the asymptotic Fisher information matrix and the asymptotic Cramer-Rao lower bound of the log spectrum variance have been easily obtained. None of them depends on the specific underlying spectrum, a result that is characteristic of the finite-length cepstrum model due to the particular form of eqns. 3 and 9. Acknowledgment: This work was supported by a PRONTIC grant number lO5jSS. C. NADEU E. LLEIDA Department Teoria del Senyal i Comunicacions Uniuersitat Politecnica de Caralunya Ap. 30002,08080 Barcelona, Spain 988 3rd May 1990 References 1 KAY, s.: ‘Modern spectral estimation’ (Prentice-Hall, 1987) 2 VAN TREES, H. L.: ‘Estimation and modulation theory, Part 111’ (Wiley, New York, 1971) 3 GINGRAS, D. I., and BOEHME, 1. F.: ‘Mathematical modelling and parametric estimation in the cepstrum domain of absorption spectral encountered in interferometric spectroscopy’. Proc. 3rd ASSP Workshop on Spectrum Estimation and Modelling, Boston, Nov. 1986, pp. 7%73 4 NADEU, c., BERTRAN, M., and SOLE, J.: ‘Spectral estimation with rational modelling of the log spectrum’, Signal Processing, 1986, 10, (I), pp. 7-18 5 OPPENHEIM, A., and SCHAFER, R.: ‘Discrete-time signal processing’ (Prentice-Hall, 1989) 6 TOKHWRA, Y.: ‘A weighted cepstral distortion measure for speech recognition’. Proc. Int. Conf. Acoust. Speech and Signal Process., Tokyo (Japan), April 1986, pp. 761-764 7 WAHBA, G.. ‘Automatic smoothing of the log periodogram’, J. Am. Statist. Assoc., 1980,75 8 WHITTLE, P.: ‘The analysis of multiple stationary time series’, J. Royal Statist. Soc., 1953, 15, pp. 125139 9 FRIEDLANDER, B., and PORAT, 8.: ‘A general lower bound for parametric spectrum estimation’, IEEE Trans., 1984, ASSP-32, (4), pp. 728-733 IO KROMER, R. E.: ‘Asymptotic properties of the autoregressive spectral estimator’. Ph.D. Dissertation, Standford University, 1970 ADAPTIVE ALGORITHM FOR VARIABLE TELETRAFFIC DEMAND IN HIGHWAY MICROCELLS Indexing terms: Telecommunications, Algorithms I An adaptive algorithm is presented which ensures that the probability of a mobile radio call in progress being forced to terminate during handover in highway microcells is always small, even in the presence of high new call request rates. As the teletraffic demand increases for mobile radio communications it becomes necessary to operate with small cells that are configured to the localised teletrafic load. By opting for smaller cells, often called microcells or picocells,i,z the spectral efficiency is dramatically increased. A particular type of microcell that is of commercial interest is the highway microcell. The mobile stations (MSs) are mounted in vehicles that are restrained to the tight confines of the highway lanes. The drivers make calls while in motion using transceivers having a hands off mode of operation. A highway microcell covers a segment of the highway containing a small base station (BS) mounted at lamp post elevation, say 15m, radiating mobile radio signals with a cigar shape beam along the high~ay.~ Contigious microcellular BSs form a cluster, with the BSs connected either by an optical LAN, or by point-topoint radio links to the mobile switching centre (MSC). The clusters are also contigious over the length of the highway. As a vehicular MS, in the process of making a call, travels along the highway it requests from the BS in the next microcell a channel in order to continue its communication. Should no channel be available the call is terminated. The probability of this happening is P,. We require that P, be significantly lower than the probability P, of a newly requested call being blocked. To ensure that P, P, we have devised a number of scheme^.^ One is to arrange for a BS to have a set number of channels N, exclusively available for handovers. If the BS can support N channels, the remaining N, = (N - Nh) channels may be used for either handover or new calls. The teletraffc performance may be improved by deploying an overlaying macrocell which provjdes No channels to assist those BSs in the microcellular cluster that currently have no channels available for handover to requesting MSs. Thus a MS entering a microcell whose BS does not have a channel available to allow the MS to continue its call is assigned one from the macrocell BS. When a channel becomes available at FLECTRONICS LETTERS 5th July 1990 Vol. 26 No. 14