New Bounds for the Marcum

Transcription

1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO., NOVEMBER 33 Then the Vanik Chervonenkis dimension of A N is uer-ounded as V ( A N ) 4N (d +)ln(n(d + )): Proof: We have A A, and A N fa \ B: A A;B A N g; for N : A well-known roerty of shatter coefficients (see, e.g., []) imlies that for N and k A (k) A(k) A (k): Thus, A (k) A(k)N y induction. Define D fa c : A Ag.It follows immediately from the definition that A(k) D (k). Hence, we otain A (k) D(k)N : (4) Since D is the collection of all closed alls in d,wehavev (D) d+ y a result of Dudley [3]. Next, we use a well-known consequence of Sauer s lemma which states that for any class of sets B and all integers k V (B) B(k) ke V (B) V (B) [] R. M. Gray and D. L. Neuhoff, Quantization, IEEE Trans. Inform. Theory, (Secial Commemorative Issue), vol. 44, , Oct [3] T. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 99. [4] A. Gersho and R. M. Gray, Vector Quantization and Signal Comression. Boston, MA: Kluwer, 99. [5] A. György and T. Linder, A note on the existence of otimal entroyconstrained vector quantizers, in Proc. IEEE Int. Sym. Information Theory, Lausanne, Switzerland,. [6] P. Paantoni-Kazakos and R. M. Gray, Roustness of estimators on stationary oservations, Ann. Proa., vol. 7,. 989, 979. [7] T. Linder, V. Tarokh, and K. Zeger, Existence of otimal refix codes for infinite source alhaets, IEEE Trans. Inform. Theory, vol. 43,. 6 8, Nov [8] T. Linder, On the training distortion of vector quantizers, IEEE Trans. Inform. Theory, vol. 46, , July. [9] P. A. Chou and B. J. Betts, When otimal entroy-constrained quantizers have only a finite numer of codewords, in Proc. IEEE Int. Sym. Information Theory, Camridge, MA, Aug. 6, 998,. 97. [] L. Devroye and G. Lugosi, Cominatorial Methods in Density Estimation. New York: Sringer-Verlag,. [] V. N. Vanik, Statistical Learning Theory. New York: Wiley, 998. [] M. Anthony and P. L. Bartlett, Neural Network Learning: Theoretical Foundations. Camridge, U.K.: Camridge Univ. Press, 999. [3] R. Dudley, Balls in R do not cut all susets of k +oints, Adv. Math., vol. 3, no. 3, , 979. (see, e.g., [, Corollary 4.]). This and (4) imly that for all k d+ NV (D) N(d+) A (k) ke ke : (5) V (D) d + An uer ound to V ( A N ) can now e otained y finding a k for which the right-hand side is less than k. It is easy to check that if d, then k 4N (d + ) ln(n (d + )) satisfies this requirement. Since for d we oviously have V ( A N ) N, we otain that for all N; d, V ( A N ) 4N (d + ) ln(n (d + )): REFERENCES [] D. Pollard, Quantization and the method of k-means, IEEE Trans. Inform. Theory, vol. IT-8,. 99 5, Mar. 98. [] T. Linder, G. Lugosi, and K. Zeger, Rates of convergence in the source coding theorem, in emirical quantizer design, and in universal lossy source coding, IEEE Trans. Inform. Theory, vol. 4, , Nov [3] P. Bartlett, T. Linder, and G. Lugosi, The minimax distortion redundancy in emirical quantizer design, IEEE Trans. Inform. Theory, vol. 44,. 8 83, Set [4] D. Pollard, Strong consistency of k-means clustering, Ann. Statist., vol. 9, no.,. 35 4, 98. [5], A central limit theorem for k-means clustering, Ann. Proa., vol., no. 4, , 98. [6] E. A. Aaya and G. L. Wise, On the existence of otimal quantizers, IEEE Trans. Inform. Theory, vol. IT-8, , Nov. 98. [7] A. J. Zeevi, On the erformance of vector quantizers emirically designed from deendent sources, in Proc. Data Comression Conf. (DCC 98), J. Storer and M. Cohn, Eds. Los Alamitos, CA: IEEE Comuter Society Press, 998, [8] S. Graf and H. Luschgy, Foundations of Quantization for Proaility Distriutions. Berlin, Heidelerg, Germany: Sringer-Verlag,. [9] A. György and T. Linder, On the structure of otimal entroy-constrained scalar quantizers, IEEE Trans. Inform. Theory, vol. 48, , Fe.. [] P. A. Chou, T. Lookaaugh, and R. M. Gray, Entroy-constrained vector quantization, IEEE Trans. Acoust. Seech, Signal Processing, vol. 37,. 3 4, Jan [] R. M. Gray, T. Linder, and J. Li, A Lagrangian formulation of Zador s entroy-constrained quantization theorem, IEEE Trans. Inform. Theory, vol. 48, , Mar.. New Bounds for the Marcum -Function Giovanni E. Corazza, Memer, IEEE, and Gianluigi Ferrari, Student Memer, IEEE Astract New ounds are roosed for the Marcum -function, which is defined y an integral exression where the th-order modified Bessel function aears. The roosed ounds are derived y suitale aroximations of the th-order modified Bessel function in the integration region of the Marcum -function. They rove to e very tight and outerform ounds reviously roosed in the literature. In articular, the roosed ounds are noticealy good for large values of the arameters of the Marcum -function, where reviously introduced ounds fail and where exact comutation of the function ecomes critical due to numerical rolems. Index Terms Marcum -function, modified Bessel function of the first kind, uer and lower ounds. I. INTRODUCTION Calculation of the generalized Marcum Q-function of order M, usually referred to as QM(a; ), and articularly the oular case (M ) indicated as Marcum Q-function Q(a; ), is imortant in many rolems of signal detection [], []. Immediate examles are the comutation of error roaility in transmission over fading channels or detection roaility for code acquisition in a direct-sequence code-division multile-access (DS-CDMA) system. Several algorithms have Manuscrit received Setemer 5, ; revised May 6,. G. E. Corazza is with the Deartment of Electronics, Comuter Science and Systems (D.E.I.S.), University of Bologna, 436 Bologna, Italy ( gecorazza@deis.unio.it). G. Ferrari is with the Deartment of Information Engineering, University of Parma, 43 Parma, Italy ( gferrari@tlc.unir.it). Communicated y A. Kav cić, Associate Editor for Detection and Estimation. Digital Oject Identifier.9/TIT /$7. IEEE

2 34 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO., NOVEMBER een devised in order to numerically evaluate this function [3] [5], ut it is sometimes useful to have simle ounds, in order to gain immediate hysical insight and avoid tedious and difficult numerical comutations. Recently, new ounds have een roosed, ased on suitale reresentations of the Marcum Q-function and generalized Marcum Q-function [6] [8], []. However, one difficulty lies in the fact that the tightness of the ounds is strongly deendent on the values of the arguments. In articular, oth uer and lower ounds are usually tight for a, ut they are generally loose for < a. Moreover, the tightness for a may not e sufficient for some alications. In this corresondence, we roose new ounds for the Marcum Q-function, which solve all of the aforementioned rolems. Namely, they are tight also for <a, and they imrove significantly on the tightness for a. These satisfying results are otained with the use of ad hoc ounds for the th-order modified Bessel function valid only in the integration region of the Marcum Q-function. The resulting ounds require the comutation of the th-order modified Bessel function only. The corresondence is structured as follows. In Section II, we recall the Marcum Q-function. In Section III, we roose new ounds for the Marcum Q-function, and numerical results are resented in Section IV. A few alications are mentioned in Section V and conclusions are drawn in Section VI. II. MARCUM Q-FUNCTION The generalized Marcum Q-function of order M is defined y the integral QM (a; ) x x a M ex x + a I M (ax) () where I M () is the (M )th-order modified Bessel function of the first kind. The arameters a and are ositive real. It can e shown that [9] where Q M (a; ) Q (a; ) +e (a + ) M k a k I k (a) () Q (a; ) Q(a; ) x ex x + a I (ax) (3) is usually referred to as the Marcum Q-function. By oserving the receding equations, it is evident that it is ossile to focus our attention on finding ounds for the Marcum Q-function. In fact, we will ound the Marcum Q-function in terms of the th-order modified Bessel function. The generalized Marcum Q-function of order M is then automatically ounded in terms of modified Bessel functions, from th to (M )th order. The sirit of the roosed ounds for the Marcum Q-function is that of simly considering a suitale integral function (the th-order modified Bessel function), thus avoiding further integration. In [6], [7], the authors claim that numerical rolems can e encountered in evaluating (), due to the semi-infinite integration region, and then they roose alternative integral exressions for the generalized Marcum Q-function with integration over a finite interval. In oth cases, they consider suitale integral exressions for the th-order modified Bessel function which aears inside the integral of the Marcum Q-function. In this corresondence, we take a different aroach. Instead of transforming the integration region, we work directly on the classic exression (3) and we introduce ounds for the th-order modified Bessel function which are extremely tight over the integration region. III. BOUNDS FOR THE MARCUM Q-FUNCTION The starting oint of the roosed ounding rocedure is an oservation aout the integral () defining the Marcum Q-function. In fact, we are integrating in the semi-infinite region [; ). The integrand function can e seen as a Rice roaility density function (df) with its arameter (see [9]). This function is unimodal, and it is ossile to show that its mode can e aroximated y the arameter a of the Marcum Q-function []. Assuming > a, it is evident that the integrand is monotonic decreasing. In this case, it is ossile to find a very tight uer ound for the integrand adding the constraint that the ounding function assumes the same value of the integrand function in. On the other hand, if < a, then the integrand is a comlicated function which is first increasing and then decreasing. In this case, it is extremely difficult to find a ounding function that is also easy to integrate and tight. Therefore, it is exedient to evaluate Q(a; ), which essentially translates into integrating the Rice df in [; ]. Again, theintegrand is now monotonic, ut increasing. With some effort, it is ossile to find tight ounding functions for this integrand. In the following, since Q(a; ) [; ], it is taken for granted that in the case of the roosed uer ounds one should consider minf; uer oundg, while in the case of the lower ounds one should consider maxf; lower oundg. In fact, if the ounds are out of the interval [; ], they ecome meaningless. A. First Case: a ) Uer Bound for a: A known uer ound for the modified Bessel function is the following [9]: I (ax) ex(ax); 8 x : (4) Based on this inequality, it is easy to derive the following uer ound for the Marcum Q-function: Q(a; ) x ex x + a e ax (x a) x ex (y + a) ex y a dy y ex y a dy + a ex y a ( a) ex + a dy erfc a : (5) The key idea is the following. The uer ound can e significantly imroved y oserving that we are interested in ounding I (ax) only in the interval [; ). Therefore, we may find a tighter aroximation in this interval. We make the following oservation. Since it follows that e x I (x) e x I (x) I (x) I (x) > ; 8 x I (x) e x I () ; 8 x : (6) ex() In Fig., the functions e x, I (x), and e x I () e (for 5) are comared. Note that the uer ound e x I () e for I (x) is extremely tight,

3 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO., NOVEMBER 35 Fig.. Comarison etween I (x) and e, for 5. Fig.. Comarison etween I (x) and, for 4. TABLE I UPPER BOUNDS IN THE REGION a TABLE II LOWER BOUNDS IN THE REGION a ut holds exclusively for x >, which luckily is the interval of interest. Hence, we can derive the following imroved uer ound y multilying the right-hand side of (5) y I (a), otaining ex(a) Q(a; ) I (a) ex(a) ex (a) +a erfc a : (7) This uer ound is referred to as UB. In Tale I, the exression of the new uer ound is shown, together with ounds that reviously aeared in the literature. More recisely, the ound indicated as UBS is roosed in [6], the ound UBC is roosed in [7], and UBMG is roosed in [8]. ) Lower Bound for a: In order to find a lower ound for the Marcum Q-function, the following inequality on the modified Bessel function I(x) can e verified: I (x) I() e x e ; 8 x : (8) x The functions involved in (8) in the lower ound are shown in Fig. for 4. Again, the ound is tight and valid exclusively for x>. Based on this ound, we may derive the following lower ound on the Marcum Q-function: Q(a; ) x ex x + a I (a)a e a I (a) (x a) ex e a I(a) e a I (a) erfc e a a ax e ax (x a) ex : (9) This lower ound will e referred to as LB. In Tale II, the new lower ound is shown and comared to ounds roosed in [6] [8] and indicated as LBS, LBC, and LBMG, resectively. B. Second Case: <a ) Uer Bound for <a: In this case, as reviously mentioned, we are integrating the Rice df over a region where the sloe of the integrand is first ositive and then negative. From the considerations reviously made, to derive uer and lower ounds we should not use the inequalities (6) and (8) that we used in the receding aragrah,

4 36 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO., NOVEMBER ecause they yield extremely loose ounds. Instead, it is convenient to exress the Marcum Q-function as follows: TABLE III LOWER BOUNDS IN THE REGION <a Q(a; ) x ex x + a I (ax) : () In this case, to find an uer ound for the Marcum Q-function we need to find a lower ound for the Bessel function in the interval [; ]. It is now ossile to reuse inequality (6) (which is indeed a lower ound for x<), suitaly rewritten as I (ax) I (a) e ax ; x [; ]: () e a By sustituting () into (), the following uer ound on the Marcum Q-function can e derived: Q(a; ) x ex x + a I (ax) I (a) e a I(a) ex a e a +a erfc (x a) x ex a ( a) ex erfc a : () This uer ound is referred to as UB. It is imortant to remark that, to the est of the authors knowledge, no uer ound can e found in the literature for <a. ) Lower Bound for <a: Starting again from (), to find a lower ound for the Marcum Q-function we need to find an uer ound for the Bessel function in the interval [; ]. To this end, we make the following consideration. Letting x log I (a), we end and stretch a the exonential curve such that we ring the oint (x ;I (a)) in the oint (; I (a)), i.e., we consider the function ex Defining log I In fact, since and (a) x log I(a) ax ex x :, it is ossile to rove that I (ax) ex(x); 8 x [; ]: (3) d I (ax) a I (ax) +I (ax) > ; 8 x d ex(x) ex(x) > it follows that I (ax) and ex(x) are oth concave. Since and I () ex() I (a) ex() ex[log I (a)] di (ax) I () x d ex(x) > x we conclude that I (ax) < ex(x), for x (; ). By sustituting (3) into (), it is ossile to derive the following lower ound for the Marcum Q-function, which is referred to as LB: Q(a; ) x ex x + a x ex x + a ex a ex a ex + I (ax) ex(x) ( ) ex erfc (x ) x ex erfc : (4) It is interesting to note the formal similarity etween the uer ound () and the lower ound (4). The latter ound is reorted in Tale III, together with the ounds LBS, LBaS, and LBC roosed in [6], [8], and [7], resectively. IV. NUMERICAL RESULTS In this section, we comare the Marcum Q-function with the otained ounds and with some ounds reviously roosed in the literature [6] [8] and considered in Section III. It is worth remarking that simlified ounds can e derived y using aroximations for the error function and the Bessel function []. Examles of simlified ounds are shown in []. Numerical results are resented as follows. We consider the two cases a and <a. For each of these cases, we resent the uer ounds and the lower ounds. We consider several fixed values of the arameter a and, for each case, we consider the ehavior of the Marcum Q-function and the considered ounds as a function of the remaining arameter. An extensive analysis of the erformance of the roosed ounds (esecially for larger values of the arameters a and ) can e found in [].

5 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO., NOVEMBER 37 TABLE IV UPPER BOUNDS COMPARISON: a AND a Fig. 3. Uer ounds comarison (linear scale): a and a. Fig. 4. Lower ounds comarison (logarithmic scale): a and a. A. First Case: a ) Uer Bounds for a: We comare the roosed ound UB to ounds reviously roosed in the literature. In Tale IV, we resent a comarison among the considered ounds for a and several values of. For each air (a; ), we resent the exact value Q(a; ) of the Marcum Q-function, while for each of the considered ounds we show the exact value and the relative error, with resect to the Marcum Q-function, exressed as and indicated as "%. The oundq(a; ) Q(a; ) roosed ound UB is the est one. The imrovement with resect to the second est ound (UBC), in terms of the relative error "%, can e as large as two orders of magnitude for large. As can e seen, the ound UBS is greater than for <. The ound is in this case meaningless, hence the corresonding entries in Tale IV are filled with. In Fig. 3, the case for a is considered and the ounds are comared in linear scale. The imrovement of the roosed ound over the others for large is now etween three and four orders of magnitude []. ) Lower Bounds for a: In Fig. 4, we consider the lower ounds for a in linear scale. As one can see, the roosed ound LB erforms very well. The only ound reviously roosed which has a similar erformance is the ound indicated as LBC [7]. For increasing values of a, the erformance imrovement of the roosed ound is even more ronounced []. B. Second Case: <a ) Uer Bounds for <a: Since no uer ound was found in the literature for the case <a, we consider the roosed ound UB and comare it to the Marcum Q-function only. The two curves are shown in Fig. 5. As can e seen, the ound UB ecomes less tight for increasing values of. Similar ehavior is oserved for almost all values of the arameter a. ) Lower Bounds for <a: In Fig. 6, we comare several lower ounds for a in linear scale. As is evident, the roosed ound LB is very tight, while the simle ound LBS is loose. The maximum relative error with the roosed ound is around.4%, and it is remarkaly less than with the other ounds []. In this case, the only lower ound comarale to the roosed ound LB is the ound LBaS. As shown in Fig. 6, the ound LBaS has a strange ehavior, since for increasing values of a sort of floor aears. This situation

6 38 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO., NOVEMBER the instantaneous signal-to-noise ratio of the received signal [8], []. Hence, the roosed ounds may e successfully emloyed to ound this error roaility. Fig. 5. Uer ound UB versus the Marcum Q-function: a and <a. VI. CONCLUSION AND DISCUSSION In this corresondence, we resented new ounds for the Marcum Q-function. The roosed ounds, esecially when comared to ounds reviously introduced in the literature, have shown to e extremely tight. In articular, the following secific remarks can e made. For large values of the arameters a and, the comutation of the Marcum Q-function according to any definition ecomes critical, ecause of numerical rolems in the integration region. The roosed ounds are valid for large values of a and without suffering any numerical rolem. Looking at the tightness tales relative to all ossile cominations of the arameters a and [], the following remark can e made. Considering a value of the Marcum Q-function around 5, for increasing a the roosed ounds imrove, while the ounds reviously roosed in the literature worsen. Moreover, in the case a, the erformance, i.e., the tightness, imroves quickly for increasing. As far as we know, no uer ound for the Marcum Q-function has een ever introduced for <a. Hence, the roosed uer ounds may e very useful in the comutation of a Rice cumulative density function for very small values of this function, i.e., for very low roailities. Fig. 6. Lower ounds comarison: a and < a. is even more ronounced for increasing values of a, where the ound LB is still the tightest one []. V. APPLICATIONS It can e shown that the cumulative distriution function of a noncentral chi-square random variale with M degrees of freedom can e exressed in terms of the generalized Marcum Q-function of order M []. When characterizing the erformance of differentially coherent and noncoherent digital communications, the generic form of the exression for the error roaility tyically involves the Marcum Q-function, the arguments of which are roortional to the square root of REFERENCES [] M. K. Simon and M.-S. Alouini, A unified aroach to the roaility of error for noncoherent and differentially coherent modulations over generalized fading channels, IEEE Trans. Commun., vol. 46, , Dec [], Digital Communication Over Generalized Fading Channels: A Unified Aroach to the Performance Analysis. New York: Wiley,. [3] S. Parl, A new method of calculating the generalized Q function, IEEE Trans. Inform. Theory, vol. IT-6,. 4, Jan. 98. [4] P. E. Cantrell and A. K. Ojha, Comarison of generalized Q-function algorithms, IEEE Trans. Inform. Theory, vol. IT-33, , July 987. [5] C. W. Helstrom, Comuting the generalized Marcum Q-function, IEEE Trans. Inform. Theory, vol. 38,. 4 48, July 99. [6] M. K. Simon, A new twist on the Marcum Q-function and its alication, IEEE Commun. Lett., vol.,. 39 4, Fe [7] M. Chiani, Integral reresentation and ounds for Marcum Q-function, IEE Electron. Lett., vol. 35, , Mar. 8, 999. [8] M. K. Simon and M.-S. Alouini, Exonential-tye ounds on the generalized Marcum Q-function with alication to error roaility analysis over fading channels, IEEE Trans. Commun., vol. 48, , Mar.. [9] J. G. Proakis, Digital Communications, 4th ed. New York: McGraw- Hill,. [] G. E. Corazza and G. Ferrari, New ounds for the Marcum Q-function, Communication Sciences Institute, USC, Los Angeles, CA, Tech. Re. CSI--8-, Aug.. [] M. Aramowitz and I. A. Stegun, Eds., Handook of Mathematical Functions. New York: Dover, 97. [] R. L. Peterson, R. E. Ziemer, and P. E. Borth, Introduction to Sread-Sctrum Communications. Uer Saddle River, NJ: Prentice-Hall, 995.