A Study on the Series Expansion of Gaussian Quadratic Forms

Transcription

1 c WISRL, KAIST, APRIL, 202 A Study on the Seres Expanson of Gaussan Quadratc Forms Techncal Report WISRL-202-APR- KAIST Juho Park, Youngchul Sung, Donggun Km, and H. Vncent Poor All rghts reserved c WISRL, KAIST Jan. 4, 200

2 c WISRL, KAIST, APRIL, Ths report s a supplementary document to the paper Outage Probablty and Outage-Based Robust Beamformng for MIMO Interference Channels wth Imperfect Channel State Informaton, by J. Park, Y. Sung, D. Km and H. V. Poor [Parket2TWC], submtted to IEEE Transactons on Wreless Communcatons. I. Dstrbuton of a Non-Central Gaussan Quadratc Form A. Prevous work and lterature survey There exst extensve lterature about the probablty dstrbuton and statstcal propertes of a quadratc form of non-central complex Gaussan random varables n the communcatons area and the probablty and statstcs communty. Through a lterature survey, we found that the man technque to compute the dstrbuton of a central or a non-central Gaussan quadratc form s based on seres fttng, whch was concretely unfed and developed by S. Kotz [Kotz-67a, Kotz-67b], and most of other works are ts varants, e.g., [Nabar-05]. Frst, we brefly explan ths seres fttng method here. Consder a Gaussan quadratc form x H Qx, where x CNµ,Σ wth sze n and Q = QH. The frst step of the seres fttng method s to convert the non-central Gaussan quadratc form nto a lnear combnaton of ch-square random varables: n x H Qx = z + δ 2 = = n [Rez + δ 2 + Imz + δ 2 ], = ndependent where z CN0, 2 for =,, n, and {δ, } are constants determned by Q, µ and Σ. Note that Rez N0, and Rez N0,. Thus, the non-central Gaussan quadratc form s equvalent to a weghted sum of non-central Ch-square random varables of whch moment generatng functon MGF s known. The MGF of a weghted sum of n ndependent non-central χ 2 random varables wth degrees of freedom 2m and non-centralty parameter µ 2 s gven by Φs = exp{ 2 n µ = n = µ 2 n 2 s } 2 s m. 2 = Note here that Φ s s nothng but the Laplace transform of the probablty densty functon PDF of x H Qx or equvalently n = z + δ 2. Now, the seres fttng method expresses the PDF as an nfnte seres composed of a set of known bass functons and tres to fnd the lnear combnaton coeffcents so that the Laplace transform of ths seres s the same as the known Jan. 4, 200

3 c WISRL, KAIST, APRIL, Fg.. Computaton of the dstrbuton of a Gaussan quadratc form Φ s. Specfcally, let the PDF be g n Q, µ,σ; y = c k h k y, 3 where {h k y, k = 0,, } s the set of known bass functons and {c k, k = 0,, } s the set of lnear combnaton coeffcents to be determned. Here, to make the problem tractable, n most cases, the followng condtons are mposed. Frst, the sequence {h k y} of bass functons s chosen among measurable complex-valued functons on [0, ] such that c k h k y Ae by, y [0, ] almost everywhere, 4 where A and b are real constants. Second, the Laplace transform ĥks of h k y has a specal form: ĥ k s = ξsη k s, 5 where ξs s a non-vanshng, analytc functon for Res > b, and ηs s analytc for Res > b and has an nverse functon. The frst condton s for the exstence of Laplace transform and the second condton s to make the problem tractable. Fnally, wth the pre-determned {h k y} Jan. 4, 200

4 c WISRL, KAIST, APRIL, wth the condtons, the coeffcents {c k } are computed so that Lg n Q, µ,σ; y = c k ĥ k s = Φ s, 6 where L denote the Laplace transform of a functon. Wdely used {h k y} for the seres expanson of the PDF of a quadratc form of non-central Gaussan random varables s as follows. [Kotz-67a, Kotz-67b]. Power seres: h k y = k y/2 n/2+k 2Γn/2+k. 2. Laguerre polynomals: Γn/2 h k y = gn; y/β[k! βγn/2 + k ]Ln/2 k y/2β, 7 where gn; y s the central χ 2 densty wth n degrees of freedom and L n/2 k x s the generalzed Laguerre polynomal defned by Rodrges formula for a > and a postve control parameter β. L n/2 k x = k! ex n/2 dk x dx k e x x k+ For the detal computaton of {c k }, please refer to [Kotz-67a, Kotz-67b, Matha-92]. The whole procedure s summarzed n Fg.. Reference group [Kotz-67a] S. Kotz, N. L. Johnson, and D. W. Boyd, Seres representaton of dstrbutons of quadratc forms n normal varables. I. Central Case, Ann. Math. Statst., vol, 38, pp , Jun [Kotz-67b] S. Kotz, N. L. Johnson, and D. W. Boyd, Seres representaton of dstrbutons of quadratc forms n normal varables. II. Non-central Case, Ann. Math. Statst., vol. 38, pp , Jun [Matha-92] A. M. Matha and S. B. Provost, Quadratc forms n random varables: Theory and applcatons, New York:M. Dekker, 992. [Nabar-05] R. Nabar, H. Bolcske, and A. Paulraj, Dversty and Outage Performance of Space- Tme Block Coded Rcean MIMO Channels, IEEE Trans. on Wreless Commun., vol. 4, no. 5, Sept Reference group 2 Jan. 4, 200

5 c WISRL, KAIST, APRIL, [Pachares-55] J. Pachares, Note on the dstrbuton of a defnte quadratc form, Ann. Math. Statst., vol. 26, pp. 28 3, Mar Power seres representaton of quadratc form of central Gaussan random varables. [Shah-6] B. K. Shah and C. G. Khatr, Dstrbuton of a defnte quadratc form for noncentral normal varates, Ann. Math. Statst., vol. 32, pp , Sep. 96. Power seres representaton of quadratc form of non-central Gaussan random varables. [Shah-63] B. K. Shah, Dstrbuton of defnte and of ndefnte quadratc forms from a noncentral normal dstrbuton, Ann. Math. Stats., vol. 34, pp , Mar Extends [Gurland-55] to derve a representaton of quadratc form of non-central Gaussan random vector wth Laguerre polynomal. Double seres of Laguerre polynomals s requred. [Gurland-55] J. Gurland, Dstrbuton of defnte and ndefnte quadratc forms, Ann. Math. Statst., vol. 26, pp , Jan Provdes a smple representaton of quadratc form of central Gaussan random vector n Laguerre polynomal. [Gurland-56] J. Gurland, Quadratc forms n normally dstrbuted random varables, Sankhya: The Indan Journal of Statstcs vol. 7, pp , Jan CDF for the ndefnte quadratc form of central random varable. [Ruben-63] H. Ruben, A new result on the dstrbuton of quadratc forms, Ann. Math. Statst., vol. 34, pp , Dec Represents the CDF of quadratc form of central and non-central Gaussan random vector wth central/non-central χ 2 dstrbuton functon. [Tku-65] M. L. Tku, Laguerre seres forms of non central χ 2 and F dstrbutons, Bometrka, vol. 52, pp , Dec Another seres representaon wth Laguerre polynomals. [Davs-77] A. W. Davs, A dfferental equaton approach to lnear combnatons of ndependent ch-squares, J. of the Ame. Statst. Assoc. vol. 72, pp , Mar Provdes another seres representaton wth power seres. [Imhof-6] J. P. Imhof, Computng the dstrbuton of quadratc forms n normal varables, Bometrka vol. 48, pp , Dec. 96. Provdes a numercal method of computng the dstrbuton [Rce-80] S. O. Rce, Dstrbuton of quadratc forms n normal varables - Evaluaton by numercal ntegraton, SIAM J. Scent. Statst. Comput., vol., no. 4, pp , 980. Another numercal method of computng dstrbuton. [Byar-93] K. H. Byar and W. C. Lndsey, Statstcal dstrbuton of Hermtan quadratc forms Jan. 4, 200

6 c WISRL, KAIST, APRIL, n complex Gaussan varables, IEEE Trans. Inform. Theory, vol. 39, pp , Mar Seres expanson of mult-varate complex Gaussan random varables. Ths paper deals wth the case that the Hermtan matrx n the quadratc form s a specal block-dagonal matrx. Reference group 3 [Raphael-96] D. Raphael, Dstrbuton of noncentral ndefnte quadratc forms n complex normal varables, IEEE Trans. Inf. Theory, vol. 42, pp , May 996. [Al-Naffour-09] T. Al-Naffour and B. Hassb, On the dstrbuton of ndefnte quadratc forms n Gaussan random varables, n Proc. of IEEE Int. Symp. Inform. Theory, Seoul, Korea, Jun. Jul B. The dfference of our work from the prevous works Frst, let us remnd our outage event n MIMO nterference channels. From equatons 5, 6 and 7 n [Parket2TWC], we have K d Pr{outage} = Pr where X mj k = j= X mjh k X mj k umh k Ĥ kk v m k 2 2 Rm k σ 2 =: τ, 8 s a non zero-mean Gaussan random varable. Note that the outage probablty s an upper tal probablty of the dstrbuton of the Gaussan quadratc form d j= XmjH k X mj k. However, as seen n Fg. 2, the most wdely-used seres fttng method explaned n the prevous subsecton yelds a good approxmaton of the dstrbuton at the lower tal not at the upper tal. The dscrepancy between the seres and the true PDF s large at the upper tal for a truncated seres. On the other hand, our approach yelds a good approxmaton to the true dstrbuton at the upper tal. Thus, the proposed seres s more relevant to our problem than the seres fttng method. Our approach to the upper tal approxmaton s based on the recent works by Raphael [Raphael-96] and by Al-Naffour and Hassb [Al-Naffour-09]. Frst, let us explan Raphael s method. The procedure n Fg. up to obtanng the MGF of the Gaussan quadratc form s common to both the sequence fttng method and Raphael s method. However, Raphael s method obtans the PDF by drect nverse Laplace transform of the MGF Φs. Typcally, the In the case of the problem consdered n [Nabar-05], the outage defned n [Nabar-05] s assocated wth the lower tal of the dstrbuton and thus the seres fttng method s well suted to that case. However, our system setup and consdered problem are dfferent from those n [Nabar-05]. Jan. 4, 200

7 c WISRL, KAIST, APRIL, [Proposed] 5 [Proposed] 0 [Laguerre] 5 [Laguerre] [Proposed] 5 [Proposed] 30 [Laguerre] 25 [Laguerre] 20 [Laguerre] Pr{Y y} [Proposed] Pr{Y y} [Proposed] Emprcal dstrbuton 20 Proposed expresson [Proposed, Laguerre] Laguerre polynomal y Emprcal dstrbuton 20 Proposed expresson [Proposed] Laguerre polynomal y a b Fg. 2. Seres fttng method versus drect nverse Laplace transform method: number of varables = 4, µ = 0.5, Q = [,0.5,0,0;0.5,,0,0;0,0,,0;0,0,0,], and Σ = 0.3I. a β = and b β = 2. β s the control parameter for the Laguerre polynomals n 7. Note that the convergent speed of the seres fttng method based on the Laguerre polynomals depends much on β. In the case of β = 2, the seres fttng method based on the Laguerre polynomals yelds large errors at the upper tal. It s not smple how to choose β and an effcent method s not known. One cannot run smulatons for emprcal dstrbutons for all cases. The seres fttng method based on the power seres shows bad performance, and t cannot be used n practce. nverse Laplace transform of the MGF s represented as a complex contour ntegral and then the complex contour ntegral s computed as an nfnte seres by the resdue theorem. However, to obtan the cumulatve dstrbuton functon CDF, whch s actually necessary to compute the tal probablty, Raphael s method requres one more step, the ntegraton of the PDF, to obtan the CDF snce the MGF Φs s the Laplace transform of the PDF. In [Parket2TWC], to obtan the CDF of a general Gaussan quadratc form, we dd not use the MGF Φs, whch s a bt complcated and requres an addtonal step, lke Raphael, but nstead we drectly used a smple contour ntegral for the CDF, eq. 2 n [Parket2TWC], obtaned by Al-Naffour and Hassb [Al-Naffour-09]. Then, the contour ntegral was computed as an In [Al-Naffour-09], Al-Naffour and Hassb obtaned the contour ntegral, eq. 2 n [Parket2TWC] for the CDF of a Gaussan quadratc form. However, they dd not obtan closed-form seres expressons for the contour ntegral n general cases except a few smple cases. The man goal of [Al-Naffour-09] was to derve a nce and smple contour ntegral form for the CDF. Jan. 4, 200

8 c WISRL, KAIST, APRIL, nfnte seres by the resdue theorem. Usng the resdue theorem s borrowed from Raphael s work. Thus, our result s smpler than Raphael s approach and does not requre the ntegraton of a PDF for the CDF. As mentoned already, the seres expanson n [Parket2TWC] has a partcular advantage over the seres fttng method consdered n [Nabar-05] for the outage event defned n [Parket2TWC]; The seres n [Parket2TWC] fts the upper tal of the dstrbuton well wth a few number of terms. We shall provde a detaled proof for ths n a specal case n the next subsecton. Thus, our seres expressons for outage probablty n MIMO nterference channels are meanngful and relevant. II. Computatonal Issues and Convergence of the Obtaned Seres A. Computng hgher order dervatves Recall the general outage expresson n Theorem n [Parket2TWC]: Pr{outage} = Pr{log 2 + SINR m k R m κ = = e τ + κ j= χj 2 k } n=κ n! gn 0 n κ +! κ j= χj 2 n κ + 9, where g s = eτs exp s / To compute 9, we need to compute s / p κp p +s / p q= χq p 2 κp. 0 + s / p p { } the egenvalues of the Kd Kd covarance matrx Σ = ΨΛΨ H, {χ j } the elements of Kd vector χ = Λ /2 Ψ H µ, where µ s the mean vector of the Gaussan dstrbuton, and the hgher order dervatves of g s. The computaton of { } and {χ j } s smple snce the szes of the mean vector and the covarance matrx are Kd and Kd Kd, respectvely. Furthermore, the hgher order dervatves of g s can also be computed effcently based on recurson [Matha-92],[Raphael-96]. Note that g s = Jan. 4, 200

9 c WISRL, KAIST, APRIL, e log g s. Thus, the dervatve of g s can be wrtten as g s = g s[log g s], g 2 s = g s[log g s] + g s[log g s] 2,. g n s = n n g l s[log g s] n l, n l l=0 where g l s and [log g s] l denote the l-th dervatves of g s and log g s, respectvely. Here, [log g s] n can be computed from 0 as [log g s] n n! n = τδ n s / n p n! n n p + p s / n+ κ p q= χ q p 2 n! n κ p n p + p s / n p where δ n s Kronecker delta functon. Thus, for gven g s and [log g s] l, we can compute g l s effcently n a recursve way, as shown n. B. Convergence analyss In ths subsecton, we provde some convergence analyss on the derved seres expanson n [Parket2TWC]. Consder the general result n Theorem of [Parket2TWC] for the CDF of a Gaussan quadratc form: Pr{Y y} = + where κ = e y + κ j= χj 2 g s, y = esy s n=κ exp n! gn 0, y n κ +! s / p κp p +s / p q= χq p 2 κp. + s / p p κ j= χj 2 n κ + Here, we explctly use the varable y as an nput parameter of the functon g s for later explanaton. g n s, y denotes the n-th partal dervatve of g s, y wth respect to s. Here, κ s the number of dstnct egenvalues of the Kd Kd covarance matrx Σ and κ s the geometrc order of egenvalue. κ = κ = Kd. The resdual error caused by truncatng the nfnte seres after the frst N terms s gven by 2 R N y = κ = e y + κ j= χj 2 n! gn 0, y n κ +! κ j= χj 2 n κ +, 3 Jan. 4, 200

10 c WISRL, KAIST, APRIL, and we have Pr{Y y; nfnte sum} = Pr{Y y; truncaton at N} + R N y. The truncaton error R N y can be expressed as κ R N y = RNy, 4 where R Ny = e y + κ j= χj 2 = n! gn 0, y n κ +! κ j= χj 2 n κ + for each κ. Then, the magntude of each term RN y n the truncaton error s bounded as RNy { y exp + κ j= χ j 2 } g n 0, y n! 5 κ j= χj 2 n κ +. n κ +! As seen n Fg. 2, our seres expanson fts the upper tal dstrbuton frst. Now, to assess the overall convergence speed of our seres, for the same step as n Fg. 2, we ran some smulatons to obtan an emprcal dstrbuton, and computed the overall mean square error MSE between the truncated seres and the emprcal dstrbuton over 0 y 0 as CDF MSE = Pr{Y y ; N, type of seres} Pr{Y y ; emprcal} 2, = where {y } are the unform samples of [0, 0]. Fg. 3 shows the CDF MSE of the three methods n Fg. 2: the proposed seres, the seres fttng method wth β = and the seres fttng method wth β = 2. It s seen n Fg. 3 that the overall convergence of the proposed seres can be worse than the seres fttng method at the small values for the number of summaton terms for the settng n Fg. 2. The bad overall convergence s due to worse fttng at the lower tal of the dstrbuton, but the bad lower tal approxmaton s not mportant to our outage computaton. Please see Fg. 2. Fg. 4 shows another case. In ths case, the proposed seres outperforms the seres fttng method both n the overall convergence and n the upper tal convergence. It s seen numercally that the proposed seres fts the upper tal dstrbuton frst. Now, we shall prove ths property of the proposed seres. However, t s a dffcult problem to prove ths property n general cases. Thus, n the next subsecton, we provde a proof of ths property when the number of dstnct egenvalues of the covarance matrx Σ s one, e.g., n the..d. case. 6 Jan. 4, 200

11 c WISRL, KAIST, APRIL, CDF MSE Proposed expresson Laguerre polynomal β= Laguerre polynomal β= Order Fg. 3. CDF MSE of the CDFs n Fg [Laguerre] 0 0 Proposed expresson Laguerre polynomal β= [Laguerre] Pr{Y y} [Proposed] CDF MSE [Proposed] Emprcal dstrbuton 0.4 Proposed expresson Laguerre polynomal y Order a b Fg. 4. number of varables = 4, µ = 0.5, Q = I, and Σ = [ ; ; ; ]. In ths case egenvalues are.0000, 0.638, 0.258, and wth β =. a CDF, b CDF MSE. Unform sample of y s taken over [0,5.9]. B. The dentty covarance matrx case Suppose that there s only one egenvalue, > 0, wth multplcty κ for the covarance matrx Σ. Ths case corresponds to Corollary 4 n [Parket2TWC], and the outage probablty Jan. 4, 200

12 c WISRL, KAIST, APRIL, s gven by where Pr{Y y} = + exp η2 exp y g n 0, y η2 / n κ+ n!n κ +!, 7 n=κ gs, y = eys s 8 and η 2 = κ j= χj 2. The resdual error caused by truncatng the nfnte seres after the frst N terms s gven by R N y = exp η2 exp y g n 0, y η2 / n κ+ n!n κ +!. 9 Before we proceed, we frst obtan the n-th dervatve of gs, y at s = 0, whch s gven n the followng lemma. Lemma : For n 0, g n 0, y = n n! n k! k y n k. 20 Proof: Proof s gven by nducton. The valdty of the clam for n = 0, and 2 s shown by drecton computaton: g 0 0, y = = yeys 0 0! s / = = s=0 0 k! k y 0 k, g 0, y = = yeys s / e ys! s / 2 = y + = s=0 k! k y k, g 2 0, y = yeys ys y/ + e ys y s 2 2e ys ys y/ s s / 4 2 = y 2 + 2y ! = 2 k! k y 2 k. Now, suppose that 20 holds up to the n -th dervatve of gs, y. From the recursve formula n, g n 0, y s obtaned as g n 0,y n n = k n = 0 g k 0,ylog g0,y n k g 0 0,ylog g0,y n + n g 0,ylog g0,y n + + s=0 n g n 0,ylog g0,y. n 2 Jan. 4, 200

13 c WISRL, KAIST, APRIL, Snce [log gs] = ys logs /, we can easly see that [log g0] = y + and [log g0] n = n! n for n 2. Therefore, 2 can be rewrtten as n g n 0, y =n!g0, y n + n g 0, yn 2! n + g 2 0, yn 3! n n g n 2 0, y 2 + g n 0, yy + =n!g0, y n + n!g 0, y n + + n g n 2 0, y 2 + g n 0, y + yg n 0, y [ n l n n! l! l! l k! k y l k n l +y l=0 m=0 [ n l n a = = l=0 n! l! l! l k! k y l k n l + n! g 2 0, y n m=0 n! n m! m y n m n! n m! m y n m ] ] 22 where a holds snce 20 holds for all g 0 0, y,, g n 0, y by the nducton assumpton. Here, consder the coeffcent of each y n 22 for = 0,, n. y n s obtaned only when m = 0. The coeffcent of y n from 22 s therefore gven by. It corresponds to the coeffcent of y n n 20. For 0 < p n, the coeffcent of y n p s obtaned by consderng all l, k that satsfes l k = n p due to the frst term n the rght-hand sde RHS of 22, and m = p due to the second term of the RHS of 22. In the frst case, we obtan y n p wth the followng pars l, k = n, p, n 2, p 2,, n p, 0. For these l, k pars, we have n l=n p n! l! l! n p! l n+p y n p n l = In the second case of m = p, we have n l=n p n! n p! p y n p n! = p n p! p y n p. 23 n! n p! p y n p. 24 Fnally, the coeffcent of y n q s gven by addng 23 and 24: n!! + pn p y n p n p! n p! n! = + p p y n p n p! n p n! = n p! p y n p, Jan. 4, 200

14 c WISRL, KAIST, APRIL, whch s equvalent to the coeffcent for y n p n 20 0 < p n. Thus, 20 holds for g n 0, y. Note that g n 0, y < 0 for all n 0 from 20. Therefore, R N y 0 for all N and y and g n 0, y = g n 0, y. Now, consder the resdual error term R N y n 9. The magntude of the resdual error can be upper bounded as follows: R N y = exp η2 = exp η2 = exp µ2 = exp η2 exp y g n 0, y η2 / n κ+ n!n κ +! exp y g n 0, y η2 / n κ+ n!n κ +! exp y exp y = 2 κ exp η2 exp g n 0, y η2 / n κ+ n!n κ +! n2η 2 n κ+ 2 κ n! gn 0, y 2 n κ +! y a 2 κ exp η2 exp y = 2κ expη2 exp y b 2κ expη2 exp y c = 2κ expη2 exp y g d = 2κ expη2 exp y = 2 κ expη 2 exp y 2 n 2η 2 n κ+ n! gn 0, y 2 n κ +! n n! gn 0, y exp2η 2 2 n! gn 0, y 2 n! gn 0, y 2 n=0 2, y expy/2 /2 n n 25 where a s from γk k! expγ = p=0 γp /p! for any γ > 0, b s from the fact that summand s negatve, c s by usng the Taylor seres expanson, and d s from 8. Snce η s a fxed constant, from 25, for any N 0 lm R Ny = y Thus, t s clear that the proposed seres converges from the upper tal dstrbuton! Jan. 4, 200

15 c WISRL, KAIST, APRIL, Now, let us consder the resdual error magntude as a functon of y for gven N. From 20, we have g n 0, y y Dfferentatng R N y wth respect to y yelds R N y y = exp η2 + exp η2 = exp η2 = ng n 0, y. 27 exp y g n 0, y η2 / n κ+ n!n κ +! exp y g n 0, y y exp y Furthermore, from 20 we have η 2 / n κ+ n!n κ +! η 2 / n κ+ n!n κ +! gn 0, y + ng n 0, y. 28 gn 0, y + ng n 0, y = y n. 29 By substtutng 29 nto 28, we have R N y y = exp η2 exp y η 2 / n κ+ y n n!n κ +!, 30 whch s postve. Snce R N y 0, lm y R N y = 0 and R Ny y > 0, the resdual error magntude monotoncally decreases as y ncreases and the maxmum error occurs at y = 0 for any gven N. Now, let us compute the worst truncaton error R N 0, whch s gven by R N 0 = exp η2 From 20, we have g n 0, 0 = n! n+. Therefore, R N 0 = exp η2 = exp η2 = exp η2 = exp η 2 g n 0, 0 η2 / n κ+ n!n κ +!. 3 n! n+ η2 / n κ+ n!n κ +! n+η2 / n κ+ n κ +! η 2 n κ+ n κ +! κ η 2 n κ+ n κ +!. 32 Jan. 4, 200

16 c WISRL, KAIST, APRIL, From 7, N κ 2. For general N κ 2, let m = n κ +. Then, R N 0 = exp η 2 m=n κ+2 η 2 m. m! Note that η 2 m m=n κ+2 m! s the resdual error of the Taylor seres expanson of expx after the frst N κ + terms. By the Taylor theorem, m=n κ+2 η 2 m m! = η2 N κ+2 N κ + 2! expαη2 33 where some α [0, ]. Therefore, the worst truncaton error s gven by R N 0 = exp α η 2 η2 N κ+2 N κ + 2! η2 N κ+2 N κ + 2!, 34 where the nequalty holds snce expα η 2 for 0 α. Furthermore, the resdual error magntude s a strctly decreasng functon of N for any y, Ths can be shown easly as follows. R N y = exp η2 = exp η2 =R N+ y + exp η2 R N y > R N+ y. 35 exp y g n 0, y η2 / n κ+ n!n κ +! exp y { g n 0, y η2 / n κ+ n!n κ +! + gn+ 0, y n=n+2 exp y g N+ η 2 / N κ+2 0, y N +!N κ + 2!. } η 2 / N κ+2 N +!N κ + 2! Snce R N y < 0 and g N+ y < 0 for all y 0 and N, we have 35. Now, based on 34 and 35, wth gven χ k and σh 2, we can compute the requred number N of terms n the seres to acheve the desred level of accuracy snce η 2 s known. Fnally, consder the worst case of N = κ 2 and y = 0: R κ 2 0 = exp η 2 η 2 n κ+ n κ +! = exp η2 n=κ m=0 η 2 m m! =, where the second equalty s by replacng m = n κ +. It s easy to see that the worst case error s - n the dentty covarance matrx case. Fg. 5 shows the performance of the proposed seres expanson n the case of the dentty covarance matrx. The numercal results well match our theoretcal analyss n ths subsecton. From the fgure, t seems reasonable to choose N for accurate outage probablty computaton. Jan. 4, 200

17 c WISRL, KAIST, APRIL, N=5 0 N=20 Pr{Y y} N=0 N=5 0.2 Emprcal dstrbuton N=20 Proposed expresson y RNy 0.2 N= N=0 0.8 N=5 0 2 y 3 4 a CDF b Resdual error Fg. 5. number of varables = 4, µ = 0.5, Q = I, and Σ = 0.I. Jan. 4, 200