On Asymptotic BER Performance of the Optimal Spatial-Temporal Power Adaptation Scheme with Imperfect CSI

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1 SCC 203, January 2 24, 203 in Munich, Germany On Asymptotic BER Performance of the Optimal Spatial-Temporal Power Adaptation Scheme with Imperfect CSI Quan Kuang Institute of Telecommunications University of Stuttgart Stuttgart, Germany quan.kuang@inue.uni-stuttgart.de Shu-Hung Leung Department of Electronic Engineering City University of Hong Kong Kowloon, Hong Kong SAR eeeugshl@cityu.edu.hk Astract This paper studies the asymptotic it-error-rate BER performance of the joint spatio-temporal S-T power adaptation which has recently een proposed for eamforming space-time-coded multiantenna systems in the presence of imperfect channel state information at the transmitter CSIT. We derive a closed-form asymptotic solution of the power adaptation, which is instrumental in characterizing the asymptotic BER performance. We show that with imperfect CSIT, the optimal S-T scheme gives exponential diversity gain in the low and medium SNR regions when the quality of CSIT is relatively good, ut cannot increase the diversity order of the space-time-coded MIMO link at very high SNR. Compared to the previous design y treating imperfect CSIT as perfect to adjust the temporal power, the optimal S-T is more roust in retaining the exponential decrease in BER over a larger range of SNR. The results sharpen our understanding of the ehavior of the optimal S-T strategy. I. INTRODUCTION The time-varying fading features of wireless channels have posed an enormous challenge for the system design. In an additive white Gaussian noise AWGN channel, the it-errorrate BER decreases exponentially with the signal-to-noise ratio SNR. However, the average BER in a Rayleigh-fading channel decreases only inversely with SNR i.e. SNR [], [2], at least at high SNR. This indicates a severe degradation in BER performance over Rayleigh fading channels. This degradation can e partially mitigated y installing multiple antennas at the transmitter and/or receiver. One popular signaling scheme to otain the enefits of multiple antennas is the space-time lock codes STBCs [3], [4], the transmit diversity can e achieved without the knowledge of channel state information CSI at the transmitter CSIT. By using STBC, the average BER decays with SNR MN, M and N are the antenna numers at the transmitter and receiver respectively, and MN is called the diversity order. If partial/imperfect CSIT is availale, the BER performance of STBC systems can e further improved via precoding [5] [0]. The STBCs exploit the space diversity of multiple-input and multiple-output MIMO channels as the precoding exploits the eigen-property of the CSIT. This STBC and precoding comination is roust to channel fading and provides an efficient way to exploit the availale CSIT []. For many common forms of imperfect/partial CSIT, such as the channel mean [8], [0], channel correlation [9], compound channel model [2], the optimal precoder consists of adaptive power allocation and multiple eigen-eamforming, the outputs of the STBC are power-loaded and then transmitted along eigen directions of the autocorrelation matrix of the spatial channel estimate. Among these transmitters, various techniques and criteria have een used to derive the power adaptation strategies. However, most of the existing power adaptation schemes perform power allocation only in the spatial domain among eigeneams. It has een shown that the diversity order remains MN for precoded STBC systems with spatial-only power allocation [8]. Recently, power allocation oth in space and time has een proposed in [3] and [4] for precoded STBC systems. In [3], it shows that with perfect CSI, the BER decease exponentially with SNR i.e. infinite diversity order, as in an AWGN channel. However, when the CSIT is imperfect, the diversity order reduces to MN at high SNR. Since the power adaptation approach adopted in [3] treated the imperfect CSIT as perfect to control the temporal power, it is interesting to investigate whether the optimal design in [4] y explicitly considering the imperfection of CSIT can improve the diversity at large SNR. However, the asymptotic analysis is missing in [4]. This paper studies the asymptotic performance of the optimal spatio-temporal S-T power adaptation scheme proposed y [4]. We derive an asymptotic solution of the power adaptation. Compare to the optimal design in [4] which requires numerical search, the closed-form asymptotic solution is simpler to implement. More important, the asymptotic solution enales us to study the diversity gain analytically. We show that with imperfect CSIT, the optimal S-T scheme gives exponential diversity gain in the low and medium SNR regions when the quality of CSIT is relatively good, ut cannot increase the diversity order of the space-time coded MIMO link at very high SNR. Although the diversity order cannot e improved at very high SNR, the optimal S-T is more roust than the previous design in [3] in retaining the exponential VDE VERLAG GMBH Berlin Offenach

2 SCC 203, January 2 24, 203 in Munich, Germany Input data Spacetime coding Power adaptation P P M Fig.. Beam forming Tx TxM Feedack channel System diagram Rx RxN Spacetime decoding Channel estimation Output data decrease in the average BER over a larger SNR region. The remainder of this paper is organized as follows. In Section II, we outline our system model and the joint spatiotemporal power adaptation scheme of [4]. In section III, the asymptotic solution is derived. Afterwards, the diversity analysis is provided in Section IV, followed y the numerical results in Section V. Finally, we conclude the paper in Section VI. II. SYSTEM MODEL We consider a wireless multi-antenna communication system with M transmit antennas and N receive antennas operating over a flat and quasi-static Rayleigh fading channel as depicted in Fig.. The space-time encoder, which is represented y an M T orthogonal STBC OSTBC transmission codeword matrix D [4], is used to encode K data symols into an M-dimensional vector sequence of T time slots with code rate r = K/T. The OSTBC vectors are sent along the M eigen-directions of the autocorrelation matrix of the spatial channel estimate at the transmitter with power allocation in space and time. The channel is represented y an N M matrix H = {h nm }, h nm denotes the channel gain from the mth transmit antenna to the nth receive antenna. It is assumed that h nm remains constant over an OSTBC frame and varies from frame to frame, and {h nm } are modeled as independent identically distriuted i.i.d. complex Gaussian random variales r.v.s with zero-mean and variance 0.5 per dimension. At the transmitter, only an imperfect channel estimate Ĥ is availale for the current frame, modeled as Ĥ = H + E [5], [6], E is the channel error matrix independent of H. The elements of E are assumed to e i.i.d. complex Gaussian r.v.s with zero mean and variance σe. 2 Let h = vech, ĥ = vecĥ, and e = vece e the column vectors constructed y stacking the columns of H, Ĥ, and E respectively. Based on the Bayesian Linear Model and Theorem 0.3 in [7], the mean and covariance matrices of h given ĥ are given as E[h ĥ] =C hc h + C e ĥ =+σe 2 ĥ, C h ĥ = C h C h C h + C e C h = σe 2 + σe 2 I NM 2 E[ ] denotes the expectation, C e = σ 2 ei NM and C h = I NM are the covariance matrices of e and h respectively, I NM is the NM NM identity matrix. Hence, conditioned on Ĥ, the elements {h nm } of H ecome complex Gaussian r.v.s with mean + σ 2 e ĥ nm and variance σ 2 e + σ 2 e. The received signals of the system can e expressed as Y = SHÛPD + Z = S HPD + Z 3 H HÛ, Û = {û ij,i,j =,,M} is an M M unitary matrix containing the M-eigenvectors of Ĥ H Ĥ corresponding to the eigenvalues {ˆζ m } sorted in decreasing order the superscript H stands for conjugate transpose, D is the OSTBC codeword matrix with normalized energy as E[trDD H ]/T =, Z is an N T received noise matrix with i.i.d. entries modeled as complex Gaussian r.v.s with zero mean and variance σn, 2 S is the total transmit power radiated from the M transmit antennas, Y is the N T received signal matrix, and P = diag P, P 2,, P M denotes a diagonal power allocation matrix which satisfies M P m = 4 m= P m 0, m =,,M. 5 It is assumed that the receiver perfectly knows the CSI. After space-time decoding, the instantaneous received SNR per symol at the receiver is expressed as [4] ρ = S HP 2 F = S M P m β m, 6 rσ 2 n rσn 2 m= F denotes the Froenius norm, β m is the mth eigenchannel power gain defined as N N M β m = h nm 2 = h ni û im 2. 7 n= n= A. Optimal Spatio-Temporal Power Adaptation Scheme of [4] In the scheme of [4], the total power S is suject to the long term time average constraint and can e varied from one OSTBC lock to another according to the updated CSIT, which is referred to as temporal power allocation. {P m } s are the spatial power allocation parameters. The optimal algorithm is derived from the minimization of a tight BER approximation as follows. We consider quadrature amplitude modulation QAM of size Q and Gray mapping. The following unified BER approximation for all QAM constellations has een used in [4]: i= BER ρ 0.2exp gρ 8 ρ is the received SNR and {.5 Q for square QAM g = for rectangular QAM. 6 5Q 4 This approximation is also tight for BPSK y regarding it as a special case of rectangular QAM with Q =2. With 6, the BER approximation of 8 can e written as M BER ρ 0.2exp γ P m β m 0 m= 9 VDE VERLAG GMBH Berlin Offenach

3 SCC 203, January 2 24, 203 in Munich, Germany γ gs rσn 2. Since {β m } in 0 are not availale at the transmitter, the transmitter can rely on the following conditional average BER given Ĥ to adapt the power [4]: P = E Ĥ β[ber ρ Ĥ] M =0.2 + γσ 2 P m N exp γ β m P m +γσ 2 P m m= σ2 e n= i= 2 σ 2 = +σe 2 3 N M ˆζ β m = + σe 2 2 ĥ ni û im 2 m = + σe 2 2,4 and β β 2 β M ecause the eigenvalues satisfy ˆζ ˆζ 2 ˆζ M. The optimization prolem of minimizing the average BER y joint spatio-temporal power adaptation can e formulated as minimize E β[p γ Ĥ β,p m β, β] γ β,{p m β} s.t. E β[γ β] = g S γ,γ β 0 rσ 2 n M P m β =, P m β 0 m m= 5a 5 5c P is given in 2, Ĥ β = β,, β M denotes the availale CSIT defined as 4, and S is the power udget. This prolem can e solved optimally y an inner-outer formulation introduced in [4]. The solution is γ β = M0 σ 2 I m λ M 0 6 m= { } Pm β =max 0, σ 2 γ β I mλ 7 N 2 σ 4 +4 β m λ + Nσ 2 I m λ = 8 2λ and λ is uniquely given y the following equation M 0 βm 0.2 I m λ N exp I m λσ 2 β m σ 2 λ = ξ 9 m= M 0 is the numer of eigeneams allocated with nonzero power, ξ is the Lagrange multiplier whose value is determined such that the average power constraint in 5 is satisfied. Note that ξ is a constant determined y the statistical distriution of β, as λ is a function of the realization of β as implicitly given y 9. Please see [4] for the method to determine the value of M 0 and ξ. III. ASYMPTOTIC SOLUTION The optimal solution given in 6 to 9 requires numerical search. In this section, we derive the closed-form asymptotic solution to simplify the calculation. Besides, the asymptotic solution enales us to study the asymptotic diversity gain analytically in Section IV. At high SNR, the optimal spatial power allocation strategy is to distriute the power equally among all the eigeneams [6], [0]. Considering P = P2 = = PM = M,we neglect βm λ in the square root in 8 since λ is generally small for high SNR case. Then 8 ecomes I m λ Nσ2 λ = λ, m =,,M 20 λ λ/nσ 2. After we sustitute 20 to 9 and consider M 0 = M, 9 is simplified into λ MN+ e λ = 5ξ Nσ 2 e 2 M m= m σ The solution to 2 can e expressed in terms of W-function [8] as λ = MN + W MN + 5ξe Nσ 2 MN+ 23 W denotes the principal ranch of the Lamert W function, whose value can e accurately calculated [8]. The Lamert function W x is a monotonically increasing function of x for x 0. Thus the solution of λ is unique. With λ, the temporal power is otained y sustituting 20 into 6, resulting in γ β = M σ 2 λ. 24 In order to otain positive value of γ from 24, the value of the Lagrange multiplier ξ should satisfy 0 <ξ<0.2nσ Interestingly, once 25 is satisfied, λ is always less than for all channel realizations as shown in 26, ensuring that γ otained from 24 is always positive λ < MN + W MN + e i= MN+ =. 26 Equations 23 and 24 provide the closed-form formulae to calculate the temporal power parameter γ for high SNR. The value of ξ in 23 can e determined offline from the average power constraint. We propose to adopt Newton s method as follows. First we express the average power constraint in 5 using Gauss-Laguerre numerical integration formula [9] as N p γ = γf d w i L! zl i γ i 27 VDE VERLAG GMBH Berlin Offenach

4 SCC 203, January 2 24, 203 in Munich, Germany f is the pdf of defined in 22, L = M N, {w i } denote the weights associated with zeros {z i } of the one-dimensional N p th order Laguerre polynomial [9], γ i is otained from 24 in which the λ is calculated y sustituting i = z i /[ + σeσ 2 2 ] into the parameter of 23. The detail of the derivation of 27 can e found in Appendix. From the constraint 27, we let N p Φξ = γ w i L! zl i γ i 28 i= The value of ξ is the root of the monotonic Φξ, which can e calculated y Newton s method as ξ n+ = ξ n Φξ n /Φ ξ n 29 ξ n is the value of ξ at the nth iteration of the algorithm. The first order derivative of Φξ with respect to ξ is with Φ ξ = M σ 2 N p i= w i L! z L i λ λ i 30 2 i λ i = W y i 3 [ + W y i ] ξ i y i = i MN + 5ξe i Nσ 2 MN+ 32 The initial value of ξ for Newton s method can e derived from the condition of 25. Since ξ should e within the range of 0, 0.2Nσ 2, we set the initial value ξ 0 =0.Nσ 2. Now we complete the derivation of the asymptotic solution. As will e shown in section V, its performance agrees with the optimal solution of [4] perfectly in high SNR regions. IV. DIVERSITY GAIN In this section, we analyze the diversity gain of the optimal spatio-temporal power adaptation y using the asymptotic solution we derived previously. For very large SNR, it is noted that the Lagrange multiplier ξ in 23 of the asymptotic solution is very small. Thus, the argument of the Lamert function in 23 ecomes small. The Taylor s series expansion of the principle ranch of the Lamert function around zero is [8] W x =x x x3. We can use the following function gx to approximate the Lamert function, which is a ounded function whose Taylor s expansion has the first two terms identical to the Lamert function: gx = x +x. Using gx, the λ in 23 is approximated y λ 5ξe MN+ Nσ 2 + 5ξe MN+ MN+ Nσ 2 +ɛ 5ξe Nσ 2 MN+ 5ξe MN+ Nσ 2 33 ɛ is a constant to make λ less than, whose value can e determined experimentally. As a result, according to 24, γ is guaranteed to e positvie and expressed as [ γ M 5ξe ] MN+ σ 2 Nσ 2 + ɛ. 34 Sustituting the γ in 34 into the average power constraint gives [ M 5ξe ] MN+ γ= 0 σ 2 Nσ 2 + ɛ f d [ = M 5ξ MN+ MN σ 2 Nσ 2 + ɛ ] 35 κ f is the pdf of defined in 22, which has een derived in Appendix, κ =+ +σe 2σ2 MN+. From 35, we can otain the Lagrange multiplier ξ as 5ξ MN+ MN M = Nσ 2 γσ M ɛ Note that in order to have positive ξ, ɛ must satisfy ɛ< + γσ2 M. Equation 36 verifies that ξ ecomes small as γ increases. Sustituting 36 into 33 and 34 gives Mκ MN λ [ γσ 2 + M ɛ]e MN+ γ γσ2 + M ɛ σ 2 κ MN e MN+ κ + ɛmκ MN 37 M + ɛ 38 σ2 Equations 37 and 38 can e regarded as simplified modifications of the original asymptotic solution 23 and 24. Now we apply the aove results to analyze the asymptotic ehavior of the average BER. For large SNR, the spatial power allocation P m approximately equals /M. Hence, the conditional BER given Ĥ is expressed as P Ĥ 0.2 M M + γσ 2 =0.2 λ MN e λ MN e γσ2 M+γσ 2 39 Based on 2, the conditional BER given Ĥ in 39 is simplified as P Ĥ = ξ λnσ Based on 40 together with 37, the average BER for large SNR is P = 0 P Ĥ f d ξ Nσ 2 0 λ f d = ξ [ γσ 2 + M ɛ Nσ 2 M ] + ɛ. 4 VDE VERLAG GMBH Berlin Offenach

5 SCC 203, January 2 24, 203 in Munich, Germany 0 Optimal S T Asymptotic solution Sharma S T D eamforimg Optimal S T Asym solution Sharma S T D eamforming Average BER 0 3 Average BER SNRdB SNRdB Fig. 2. Asymptotic performance of the optimal S-T power adaptation, M =2,N =,σe 2 =0.4, and BPSK. Fig. 3. Asymptotic performance of the optimal S-T power adaptation, M =2,N =,σ 2 e =0.0, and QPSK. Sustituting ξ in 36 into 4 otains MN P =0.2κ MNMN+ M γσ 2 + M ɛ MN+ Mκ MN +0.2ɛ γσ M ɛ For large SNR, γ is large. Thus, 42 can e approximated as MN M P 0.2 σ 2 κ MN+ γ MN. 43 which shows that the diversity gain is MN. V. NUMERICAL RESULTS In this section, we provide numerical results to validate the asymptotic solution and the diversity result we derived. In the evaluation, G 2 code Alamouti code with r =, and H 3 with r = 3/4 are used for illustration. The specific code matrices can e found in [4]. We denote a system with M transmit and N receive antennas as M N system. In the following figures, SNR is defined as S/σ n. 2 We refer to the existing spatio-temporal scheme in [3] as Sharma S-T, and the optimal solution in [4] as Optimal S-T. For comparison, we also give the results of using conventional eamforming without STBC the signal with constant power is transmitted only along the eigen direction with the largest eigenvalue named as -D eamforming. In our performance evaluation, we use the following formula to determine the average BER: P = EĤ[P Ĥ ] 44 P Ĥ has een given in 2, and the expectation is calculated y Gauss-Laguerre numerical integration. Figs.2 and 3 show the performance comparison. The system eing used for evaluation is a 2 system with BPSK and QPSK, respectively. We consider moderate estimation error σ 2 e = 0.4 in Fig.2 and small estimation σ 2 e = 0.0 in Fig.3. It is verified in the figures that the asymptotic solution is valid, resulting in the same performance as the optimal S-T solution asymptotically in high SNR regions. It is also oserved from Fig.3 that for small estimation error variance the optimal S-T has exponential diversity in the low and medium SNR regions. This result makes the optimal S-T outperform Sharma S-T scheme significantly for small estimation error variance. However, the diversity gain asymptotically reduces to MN=2 at high SNR. This asymptotic diversity gain will e further illustrated in Fig.4. Constant-power -D eamforming lacks the feasiility of adapting the power either in space or in time, leading to the worst performance. The comparison of the asymptotic BER for very large SNR in 43 with the BER performance is shown in Fig.4. The 2 and 3 systems are considered. The modulation is QPSK and the estimation error σ 2 e is set equal to 0.05 and 0.0. As shown in the figure, the closed-form asymptotic BER of 43 is in good agreement with the BER performance curves at high SNR. As derived in Section IV, the temporal power adaptation cannot improve the diversity gain at high SNR when the CSIT is imperfect. The reason is that the BER performance is mainly limited y the additive noise at low SNR and at medium SNR, ut largely affected y the errors in the CSIT at high SNR. Through adapting the temporal power to comat fading effects, we can achieve exponential decrease in BER yielding similar performance ehavior as in AWGN channel. At very high SNR, however, the diversity gain we can otain is MN, just like the space-time coding methods with no knowledge of CSIT. This result shows that the imperfection of CSIT at very high SNR causes the feedack knowledge of CSIT is useless. Although the diversity gain cannot e improved at high SNR y the power adaptation, it is clearly shown from Fig.2 and Fig.3 that the optimal S-T scheme outperforms the existing Sharma S-T approach since it is more roust than the existing scheme in retaining the exponential decrease in BER over a VDE VERLAG GMBH Berlin Offenach

6 SCC 203, January 2 24, 203 in Munich, Germany Average BER M=2,N=,σ e 2 =0.05 M=3,N=,σ e 2 =0.0 Optimal S T Closed form P for high SNR SNRdB Fig. 4. Diversity analysis for Optimal S-T power adaptation under different system configurations. larger SNR region for different σ 2 e. VI. CONCLUSION In this paper, we studied the asymptotic performance of the optimal spatio-temporal power adaptation scheme. The average BER is characterized y using an unified tight approximation for QAM constellations, including BPSK as a special case. A closed-form asymptotic solution of the power adaptation has een derived, ased on which the asymptotic diversity gain has een analyzed. The results show that y adapting the temporal power to comat fading effect, we can achieve exponential decrease in average BER at low and medium SNR, yielding the performance ehavior similar as in AWGN channel. At very high SNR, however, the diversity gain we can otain is the same as that of space-time coding without the knowledge of CSIT, since the errors in the CSIT ecomes a dominant factor making the CSIT less and less useful. Although the diversity order cannot e improved at very high SNR, the optimal design y explicitly considering the CSIT imperfection achieves the exponential decrease in BER over a larger range of SNR under different CSIT uncertainties, in comparison with the previous design. These results provide us etter understanding of the ehavior of the optimal S-T design with imperfect CSIT. APPENDIX DERIVATION OF THE NUMERICAL INTEGRATION OF 27 According to 22, 4 and our channel estimation model, we have = M i= ˆζ i + σ 2 e 2 σ 2 = H 2 F + σ 2 eσ 2 = t + σ 2 eσ 2 we define t H 2 F, which is central chi-square distriuted with the degree of freedom of 2L, and L = M N. The pdf of t is given as f t t = L! tl e t. According to the Gauss-Laguerre formula [9], the expectation of γ is E β[γ β]= γ βf β βd β which gives 27. = γf d 0 t = γ 0 + σeσ 2 2 N p z i w i γ + σeσ 2 2 i= REFERENCES f t tdt L! zl i [] D.Tse and P.Vishwanath, Fundamentals of Wireless Communications. Camridge, U.K.: Camridge University Press, [2] A.Goldsmith, Wireless Communications. Camridge, U.K.: Camridge University Press, [3] E. G. Larrsson and P. Stoica, Space-Time Block Coding for Wireless Communications. Camridge, U.K.: Camridge University Press, [4] V. Tarokh, H. Jafarkhani, and A. R. Calderank, Space-time lock coding for wireless communications: performance results, IEEE J. Sel. Areas Commun., vol. 7, no. 3, pp , March 999. [5] A. Pascual-Iserte, D. P. Palomar, A. I. Perez-Neira, and M. A. Lagunas, A roust maximin approach for MIMO communications with imperfect channel state information ased on convex optimization, IEEE Trans. Signal Process., vol. 54, no., pp , Jan [6] G. Jongren, M. Skoglund, and B. Ottersten, Comining eamforming and orthogonal space-time lock coding, IEEE Trans. Inf. Theory, vol. 48, no. 3, pp , March [7] M. Vu and A. Paulraj, Optimal linear precoders for MIMO wireless correlated channels with nonzero mean in space-time coded systems, IEEE Trans. Signal Process., vol. 54, no. 6, pp , June [8] J. W. Huang, E. K. S. Au, and V. K. N. Lau, Precoder design for spacetime coded MIMO systems with imperfect channel state information, IEEE Trans. Wireless Commun., vol. 7, no. 6, pp , June [9] S. Zhou and G. B. Giannakis, Optimal transmitter eigen-eamforming and space-time lock coding ased on channel correlations, IEEE Trans. Inf. Theory, vol. 49, no. 7, pp , July [0], Optimal transmitter eigen-eamforming and space-time lock coding ased on channel mean feedack, IEEE Trans. Signal Process., vol. 50, no. 0, pp , Oct [] M. Vu and A. Paulraj, MIMO wireless linear precoding, IEEE Signal. Proc. Mag., vol. 24, no. 5, pp , [2] J. Wang and D. P. Palomar, Worst-case roust MIMO transmission with imperfect channel knowledge, IEEE Trans. Signal Process., vol. 57, no. 8, pp , [3] V. Sharma, K. Premkumar, and R. N. Swamy, Exponential diversity achieving spatio-temporal power allocation scheme for fading channels, IEEE Trans. Inf. Theory, vol. 54, no., pp , Jan [4] Q. Kuang, S.-H. Leung, and X. Yu, Optimal joint spatial and temporal power adaptation for space-time-coded systems with imperfect CSI, IEEE Trans. Signal Process., vol. 60, no. 6, pp , 202. [5] E. G. Larsson, Diversity and channel estimation errors, IEEE Trans. Commun., vol. 52, no. 2, pp , Fe [6] Z. Shi and H. Lei, Transmit antenna selected V-BLAST systems with power allocation, IEEE Trans. Veh. Technol., vol. 57, no. 4, pp , July [7] S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theroy. Englewood Cliffs: NJ: Prentice-Hall, 993. [8] R. Corless, G. Gonnet, D. Hare, D. Jeffrey, and D. Knuth, On the Lamert W function, Advances in Computational Mathematics, vol. 5, pp , 996. [9] P. J. Davis and P. Rainowitz, Methods of Numerical Integration. New York: Academic Press, 975. VDE VERLAG GMBH Berlin Offenach