IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 3, MARCH Transactions Letters

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1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 3, MARCH Transactons Letters BER Analyss of QAM on Fadng Channels wth Transmt Dversty M. Surendra Raju, Ramesh Annavajjala, Student Member, IEEE, and A. Chocalngam, Senor Member, IEEE Abstract In ths letter, we derve analytcal expressons for the bt error rate BER of space-tme bloc codes STBC from complex orthogonal desgns COD usng quadrature ampltude modulaton QAM on Raylegh fadng channels. We tae a bt log-lelhood rato LLR based approach to derve the BER expressons. The approach presented here can be used n the BER analyss of any STBC from COD wth lnear processng for any value of M n an M-QAM system. Here, we present the BER analyss and results for a 6-QAM system wth -Tx,L-Rx antennas usng Alamout code rate- STBC, 3-Tx, L-Rx antennas usng a rate-/ STBC, and 5-Tx, L-Rx antennas usng a rate-7/ STBC. In addton to beng used n the BER analyss, the LLRs derved can also be used as soft nputs to decoders for varous coded QAM schemes, ncludng turbo coded QAM wth space-tme codng as n hgh speed downln pacet access HSDPA n 3G. Index Terms Space-tme bloc codes, transmt dversty, QAM, bt log-lelhood rato. I. INTRODUCTION THE potental capacty gans acheved by usng multple antenna systems have led to consderable attenton n the area of space-tme codng []. Space-tme bloc codes STBC from complex orthogonal desgns COD are of nterest as they can be used for complex constellatons such as quadrature ampltude modulaton QAM to acheve hgher data rates n wreless communcaton systems [],[3]. Recent wors have reported analytcal expressons for the symbol error rate SER and the bt error rate BER of orthogonal STBCs. In [4], Shn and Lee derved expressons for the SER of orthogonal STBCs on Raylegh fadng channels. They derved the SER by convertng the multple nput multple output MIMO system model to an equvalent sngle nput sngle output SISO model. Recently, Smon n [5], and Tarcco and Bgler n Manuscrpt receved September, 003; revsed January 0, 005; accepted October 3, 005. The assocate edtor coordnatng the revew of ths letter and approvng t for publcaton was L. Vandendorpe. Ths wor n part was presented n IEEE GLOBECOM 003, San Francsco, December 003. Ths wor was supported n part by the Swarnajayant Fellowshp, Department of Scence and Technology, New Delh, Government of Inda, under Project Ref: No.6/3/00-S.F. M. Surendra Raju s wth Ianos Communcatons Inda Prvate Lmted, Bangalore 56005, Inda e-mal: mraju@anos.com. Ramesh Annavajjala s wth the Department of Electrcal and Computer Engneerng, Unversty of Calforna San Dego, La Jolla, CA 9093 USA. e-mal: ramesh@cwc.ucsd.edu. A. Chocalngam s wth the Department of Electrcal Communcaton Engneerng, Indan Insttute of Scence, Bangalore 5600, Inda e-mal: achocal@ece.sc.ernet.n. Dgtal Object Identfer 0.09/TWC [6], have reported exact expressons for the parwse error probablty PEP as well as approxmate expressons for the BER for space-tme codes. In ths letter, we derve analytcal expressons for the BER for lnear STBCs from COD usng QAM on Raylegh fadng channels. We adopt a bt log-lelhood rato LLR based approach, where we frst derve expressons for the LLRs of the ndvdual bts formng the QAM symbol, and then use these LLRs to obtan the BER expressons. We pont out that ths approach can be used n the BER analyss of any STBC from COD wth lnear processng for any value of M n an M-QAM system. Here, we present the BER analyss and results for a 6-QAM system wth -Tx,L- Rx antennas usng the rate- Alamout code, 3-Tx,L- Rx antennas usng a rate-/ code, and 5-Tx,L-Rx antennas usng a rate-7/ code. Another major usefulness of ths contrbuton s that the derved LLRs provde a soft metrc for each bt n the mappng, whch can be used as soft nputs to decoders for varous coded QAM schemes wth spacetme codng. Examples of such schemes nclude turbo coded QAM wth transmt dversty n hgh speed downln pacet access HSDPA n 3G, and convolutonally coded QAM wth orthogonal frequency dvson multplexng OFDM n dgtal vdeo broadcastng DVB and IEEE 80.. II. SYSTEM MODEL We consder a wreless communcaton system wth L t transmt and L r receve antennas. We consder space-tme bloc codes, where each codeword s a matrx wth P rows and L t columns, wth complex valued symbols as ts entres. Here, P s the number of tme slots requred to transmt one codeword. For some K nformaton symbols, s,s,,s K,whch are selected from the 6-QAM constellaton see Fg., the entres of the codeword X = {x t,t=,,,p; =,,,L t } are a lnear combnaton of the nformaton symbols s,,,,k, and ther complex conjugates. At tme slot t, t =,,,P,thet th row of the codeword X.e., x t,x t,,x Lt t s transmtted smultaneously from L t antennas. The symbol transmsson rate, R, sdefnedas the number of nformaton symbols transmtted per tme slot,.e., R = K/P. The channel fade coeffcents are assumed to /06$0.00 c 006 IEEE Four bts, r,r,r 3,r 4 are mapped on to a complex symbol s = s I + js Q. The horzontal/vertcal lne peces n Fg. denote that all bts under these lnes tae the value, and the rest tae the value 0.

2 48 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 3, MARCH 006 Fg.. 6-QAM Constellaton. reman constant over P tme slots. The receved codeword, Y, can be wrtten as Y = XH+ N, where Y = {y j t : t =,,,P; j =,,,L r } s a matrx of sze P L r, whose entry y j t s the sgnal receved at antenna j at tme slot t; H = {h,j } s the channel matrx of sze L t L r, whose entry h,j s the complex channel coeffcent from the transmt antenna to the receve antenna j. The random varables h,j s are assumed to be..d Raylegh dstrbuted wth E h,j =Ω. N = {n j t} s the nose matrx of sze P L r, whose entres are..d complex Gaussan nose wth zero mean and varance σ. Let C. be a mappng from a K-tuple complex message vector s =s,s,,s K to the columnwse orthogonal P L t codeword X = Cs. Due to the columnwse orthogonalty of the lnear orthogonal space-tme bloc codes consdered, the L t L t matrx Cs H Cs s gven by { K } K Cs H Cs =dag g, s,, g,lt s, where. H denotes the Hermtan operator, and G = {g m,n } s a matrx of sze K L t whose entres can tae nonnegatve nteger values for example, for the Alamout code [7] g m,n =, m, n. Assumng perfect nowledge of the channel coeffcents at the recever, the combned sgnal output for the symbol s s gven by ŝ =Δ s + ζ, 3 where L r [ Δ = g, h,j + g, h,j + + g,lt h Lt,j ], 4 and ζ s a complex Gaussan random varable wth zero mean and varance Δ σ. III. BIT LOG-LIKELIHOOD RATIOS We defne the LLR for the bt r, =,, 3, 4, of symbol s, =,,,K,as Pr r = Y, H LLR s r = log Pr r =0 Y, H Pr r = ŝ, H = log Pr r =0 ŝ, H. 5 Assumng that all the symbols are equally lely and that the fadng s ndependent of the transmtted symbols, usng Bayes rule, we have α S LLR s r =log fŝ H,s ŝ H,s = α β S 0 fŝ H,s ŝ H,s = β. 6 Snce fŝ H,s ŝ H,s = α = exp πˆσ ˆσ ŝ Δ α where ˆσ =Δ σ, 6 can be wrtten as exp ŝ α S ˆσ LLR s r =log Δ α exp ŝ β S 0 ˆσ Δ β. 7 Usng the approxmaton log j exp X j mn j X j, LLR s r can be approxmated as LLR s r = ˆσ mn β S 0 mn ŝ Δ α α S ŝ Δ β Defne complex varables, ẑ, =,,,K,as. 8 ŝ ẑ =. 9 Δ Usng 9 n 8 and normalzng by 4/ˆσ, LLR s r s wrtten as LLR s r = Δ 4 mn β S 0 ẑ β mn α S ẑ α. 0 Note that the set parttons S and S 0 are delmted by horzontal or vertcal boundares. As a consequence, two symbols n dfferent sets closest to the receved symbol always le ether on the same row f the delmtng boundares are vertcal or on the same column f the delmtng boundares are horzontal. Usng the above fact, the LLRs for each of the bts formng the symbol, s, are obtaned as LLR s r = dẑ I Δ, dd ẑ I Δ, dd + ẑ I Δ, dẑ Q Δ, LLR s r = dd ẑ Q Δ, dd + ẑ Q Δ, ẑ I d ẑ I > d ẑ I < d ẑ Q d ẑ Q > d ẑ Q < d As we wll see n Sec. V, the analytcal BER evaluated usng ths approxmate LLR s almost the same as the BER evaluated through smulatons wthout ths approxmaton.

3 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 3, MARCH LLR s r 3 =d ẑ I dδ 3 LLR s r 4 =d ẑ Q dδ. 4 In the above equatons, ẑ I and ẑ Q are the real and magnary parts of ẑ, respectvely, and d s the mnmum dstance between pars of sgnal ponts. We note that, lewse, the LLR expressons for other values of M n M-QAM can be derved. For example, we have derved the bt LLR expressons for the 3-QAM constellaton n Fg. 4 of [0] as well as the 64-QAM constellaton n Fg. 4 of [] and presented them n Table I. These LLR expressons can be used to derve the BER expressons for M-QAM as llustrated n the followng secton. IV. DERIVATION OF BER In ths secton, we derve the probablty of error for the bt r, =,, 3, 4, formng a 6-QAM symbol. The probablty of error for bt r n symbol s, Pb, can be wrtten as Pb = P b s I = d Pr s I = d+p b s I = 3d Pr s I = 3d + P b s I =d Pr s I = d+p b s I =3d Pr s I =3d, 5 where s I represents the real part of s. Let us frst consder Pb s I = d, whch s gven by Pb s I = d = Pb s I = d,h 6 where the overlne ndcates averagng over the complex random varables {h,j }. Pb s I = d,h can be wrtten as Pb s I = d,h LLR = Pr s r < 0 s I = d, H ζi d Δ = Pr d = Q, 7 Δ σ I where σi = σ /. Let us defne ξ = L t K g,. P 8 = We then have d σ I = 4Eb R 5N ol r ξ,wheree b s the energy per bt per transmt antenna and R s the rate of the STBC used. From the above, we can wrte P b s I = d,h = Q 4E b R Δ 5N o L r ξ. 9 To obtan Pb s I = d, we need to uncondton P b s I = d,h w.r.t Δ, whch s gven by L r Δ = g, h,j + g, h,j + + g,lt h Lt,j Lr = g, h,j Lr + + g,lt h Lt,j.0 Let us defne θ n = L r h n,j,n =,,,L t.snce h,j are..d exponental wth mean Ω, the random varables θ n are..d Gamma random varables wth densty functon f θn x = exp x x Lr, Lr ΓL r Ω Ω and the moment generatng functon MGF s gven by Lr M θn s =. +sω Snce Δ = L t n= g,nθ n, ts MGF, M Δ,sgvenby L Lr t M Δ =. 3 +sωg n=,n Usng the above and Crag s formula [8], we can show that Pb s 4E b R Δ I = d = Q 5N o L r ξ = π L t sn Lr φ π φ=0 sn dφ, 4 φ + μ n= g,n where μ = γ b R 5L and γ rξ b = ΩE b N o. Smlarly, the condtonal error probablty Pb s I = 3d,H s gven by Pb s I = 3d,H LLR = Pr s r < 0 s I = 3d, H ζi 36E b R Δ = Pr 3d = Q.5 Δ 5N o L r ξ Uncondtonng Pb s I = 3d,H w.r.t Δ, t can be shown that Pb s 36E b R Δ I = d = Q 5N o L r ξ = π L t sn Lr φ π φ=0 sn dφ, 6 φ + μ n= g,n where μ = 8 γ b R 5L. It can further be shown that P rξ b s I = d = Pb s I =d and P b s I = 3d = P b s I =3d. Moreover, for the 6- QAM constellaton consdered, t can be shown that Pb = P b and Pb3 = P b4. Wth the above, the BER expressons for the bts r, r, r 3, r 4 of the symbol s can be wrtten as Pb = Pb = P + P 7 Pb3 = P b4 = P + P P 3, 8 where Pj,,, 3, aregvenby Pj = π L t sn Lr φ π φ=0 sn dφ, 9 φ + μ n= j g,n and μ = γ b R 5L r ξ, μ = 8 γ b R 5L r ξ, μ 3 = 0 γ b R L r ξ. 30 Note that for STBCs where g,n = g,, n, the ntegral n 9 has a closed-form expresson gven by [9] LrL t L Pj = λ rl j t Lr L t + +λ j., =0 3

4 484 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 3, MARCH 006 TABLE I BIT LLR EXPRESSIONS FOR THE 3-QAM CONSTELLATION IN FIG.4OF [0] AND THE 64-QAM CONSTELLATION IN FIG.4OF [] IN RAYLEIGH FADING.FOR 3-QAM, THE MAPPING OF BITS r STOBITS j S AND q j SINFIG.4OF [0] IS AS FOLLOWS: r =,r = q,r 3 =,r 4 = q,r 5 = 3. Bt LLR Expressons for 3-QAM Expressons for 64-QAM r LLR s r = dẑ ji Δ, ẑ ji d dẑ ji Δ, ẑ ji d dd ẑ ji Δ, d <ẑ ji 4d dd ẑ ji Δ, d <ẑ ji 4d 3dd ẑ ji Δ, 4d <ẑ ji 6d 3dd ẑ ji Δ, 4d <ẑ ji 6d 4d3d ẑ ji Δ, ẑ ji > 6d 4d3d ẑ ji Δ, ẑ ji > 6d dd + ẑ ji Δ, 4d ẑ ji < d dd + ẑ ji Δ, 4d ẑ ji < d 3dd + ẑ ji Δ, 6d ẑ ji < 4d 3dd + ẑ ji Δ, 6d ẑ ji < 4d 4d3d + ẑ ji Δ, ẑ ji < 6d 4d3d + ẑ ji Δ, ẑ ji < 6d r LLR s r = dẑ jq Δ, ẑ jq d dẑ jq Δ, ẑ jq d dd ẑ jq Δ, ẑ jq > d dd ẑ jq Δ, d <ẑ jq 4d dd + ẑ jq Δ, ẑ jq < d 3dd ẑ jq Δ, 4d <ẑ jq 6d 4d3d ẑ jq Δ, ẑ jq > 6d dd + ẑ jq Δ, 4d ẑ jq < d 3dd + ẑ jq Δ, 6d ẑ jq < 4d 4d3d + ẑ jq Δ, ẑ jq < 6d r 3 LLR s r 3 = d 3d + ẑ ji Δ, ẑ ji d d 3d + ẑ ji Δ, ẑ ji d d 4d + ẑ ji Δ, d < ẑ ji 6d d 4d + ẑ ji Δ, d < ẑ ji 6d d 5d + ẑ ji Δ, d < ẑ ji > 6d d 5d + ẑ ji Δ, d < ẑ ji > 6d r 4 LLR s r 4 = d ẑ jq dδ d 3d + ẑ jq Δ, ẑ jq d d 4d + ẑ jq Δ, d < ẑ jq 6d d 5d + ẑ jq Δ, r 5 LLR s r 5 = dd ẑ ji Δ, ẑ ji 4d dd ẑ ji Δ, ẑ ji 4d d 6d + ẑ ji Δ, ẑ ji > 4d d 6d + ẑ ji Δ, ẑ ji > 4d d < ẑ jq > 6d r 6 LLR s r 6 = dd ẑ jq Δ, ẑ jq 4d d 6d + ẑ jq Δ, ẑ jq > 4d where λ j = gμj +gμ j. It s noted that, for STBCs ncludng rate- Alamout code C gven n the next secton and rate- / STBC C gven n the next secton, g,n are constants g = for C and g = for C, and hence the closedform expresson n 3 can be used to compute the BER for these STBCs. For STBCs where g,n s not a constant e.g., rate-7/ STBC C 3 gven n the next secton, 9 can be evaluated numercally and accurately usng the Gauss- Chebyshev Quadrature rule. The average BER for symbol s,,,,k, Pb, s then gven by Pb = P 4 b + Pb + P b3 + P b4. 3 Fnally, the average BER of the system, P b,sgvenby P b = K K Pb. 33 The BER expressons for other values of M n M-QAM can be derved lewse. V. RESULTS AND DISCUSSIONS We computed the BER performance of 6-QAM on Raylegh fadng channels as a functon of average SNR for the followng space tme bloc codes: s s s 3 s s s 4 s 3 s 4 s s s C = s s, C = s 4 s 3 s s s s, 3 s s s 4 s 3 s 4 s s 4 s 3 s and C 3 = s s s 3 0 s 4 s s 0 s 3 s 5 s 3 0 s s s 6 0 s 3 s s s 7 s s 7 s 0 s 4 0 s 6 s 0 0 s 4 s 5 s 3 0 s 5 s 6 0 s s 5 0 s 7 0 s s 6 s s 3 s 7 s 6 s 5 s 4 0. C s the well nown Alamout code wth parameters P = K = L t =, R =,andc H C s a dagonal matrx wth the, th dagonal element, D,, of the form s. C s a rate-/ STBC wth parameters P =8,K =4,L t =3, R =/,andc H C s a 3 3 dagonal matrx wth the, th

5 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 3, MARCH Average Probablty of Bt Error Smulated BER usng True LLR s, Tx, Rx antennas Analytcal BER usng apprx. LLRs, Tx, Rx antennas Smulated BER usng True LLRs, Tx, Rx antennas Analytcal BER usng apprx. LLRs, Tx, Rx antennas Average Probablty of Bt Error No Dversty 5Tx, Rx 5Tx, Rx 5Tx, 4Rx 5Tx, 0Rx Average SNR per bt db γ b db Fg.. Comparson of the analytcal BER evaluated usng approxmate LLRs vs the smulated BER usng the LLRs wthout approxmaton. 6-QAM wth rate- STBC Alamout code n Raylegh fadng. -Tx/-Rx and -Tx/-Rx antennas. Fg. 3. BER performance of 6-QAM wth 5 transmt antennas and L r =,, 4, 0 receve antennas usng rate-7/ STBC n Raylegh fadng. 0 0 dagonal element, D,, of the form 4 s. C 3 s a rate-7/ STBC wth parameters P =,K =7,L t =5, R =7/, andc H 3 C 3 s a 5 5 dagonal matrx wth the, th dagonal element, D,, of the form D, = D, = D3, 3 = D4, 4 = D5, 5 = 3. s + 7 s,34 7 s. 35 =3 In Fg., we compare the analytcal BER evaluated usng the approxmate LLRs derved versus the smulated BER usng the LLRs wthout approxmaton for rate- STBC Alamout code usng 6-QAM for -Tx/-Rx and -Tx/-Rx antennas. It s observed that the analytcally computed BER s almost the same as the smulated BER, ndcatng that the approxmaton to the LLRs results n nsgnfcant dfference between the analytcally computed BER and the true BER. We would le to pont out that the BER obtaned usng the approxmate LLR expresson s the same as that of the tradtonal BER results for M-QAM as publshed, for example, n the paper by Cho and Yoon [0]. The reason for ths observaton s that the decson statstc for each bt formng the QAM symbol wth Gray codng and approxmate LLR s the same as that of the conventonal symbol-to-bt demappng approach. In other words, wthout the approxmaton, the average BER performance for M-QAM wll be slghtly better than the conventonal symbol-to-bt demappng approach. In [], t s shown that for all practcal values of the bt SNR ths mprovement can be neglgble. We further pont out that [3] presents BER results for Gray-coded M-QAM by dvdng the SER by the number of bts per symbol. However, ths result s only approxmate, as the exact BER analyss requres evaluatng the number of bt errors occurrng for each possble transmtted symbol. We also note that the approxmate BER Average Probablty of Bt Error No dversty Tx, Rx Tx, Rx, Alamout code 3Tx, Rx Scheme rate / 5Tx, Rx Scheme rate 7/ AWGN γ db b Fg. 4. BER performance of 6-QAM wth dfferent STBCs n Raylegh fadng; Tx antennas usng rate- STBC Alamout code, 3 Tx antennas usng rate-/ STBC, 5 Tx antennas usng rate-7/ STBC. Number of receve antennas, L r =. results n [3] match the exact BER only at large-enough SNR values. In Fg. 3, we present the analytcal results of the average BER performance as a functon of the average SNR, γ b,for the rate-7/ STBC, C 3. The number of receve antennas consdered nclude L r =,, 4, 0. Fgure 4 presents the comparatve BER performance of the dfferent STBCs C, C and C 3 when the number of receve antennas L r =. The performance n AWGN s also shown for comparson. As we ponted out earler, n addton to beng used n the BER analyss, the derved LLRs for the ndvdual bts n the QAM symbols can be used as soft nputs to the decoders n varous coded QAM schemes. As an example, we employed the LLRs as soft nputs to the turbo decoder n a rate-/3 turbo

6 486 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 3, MARCH 006 Average Probablty of Bt Error AWGN Fadng No Dversty Tx, Rx Tx, Rx Average SNR per bt db Fg. 5. BER performance of rate-/3 turbo coded 6-QAM wth two transmt antennas and L r =, receve antennas usng rate- STBC Alamout code n Raylegh fadng. LLRs of bts n QAM symbols used as soft nputs to the turbo decoder. coded 6-QAM scheme on Raylegh fadng wthout and wth transmt dversty usng Alamout code C. Fgure 5 shows the smulated BER performance of the turbo coded 6-QAM system usng the derved LLRs as soft nputs to the decoder. The turbo code used n the smulatons s the one specfed n the 3GPP standard. Lewse, the LLRs can be used as soft nputs to decoders n DVB and IEEE 80.a, where convolutonally coded QAM wth OFDM s used. REFERENCES [] G. J. Foschn and M. J. Gans, On lmts of wreless communcatons n a fadng envronment when usng multple antennas, Wreless Personal Commun., vol. 6, pp , 998. [] V. Taroh, H. Jafarhan, and A. R. Calderban, Space-tme bloc codng for wreless communcatons: performance results, IEEE Jl. Sel. Areas n Commun., vol. 7, no. 3, pp , Mar [3] W. Su and X.-G. Xa, On space-tme bloc codes from complex orthogonal desgns, Wreless Pers. Commun., vol. 5, no., pp.-6, Apr [4] H. Shn and J. H. Lee, Exact symbol error probablty of orthogonal space-tme bloc codes, Proc. IEEE GLOBECOM 00, vol., pp. 97-0, Nov. 00. [5] M. K. Smon, Evaluaton of average bt error probablty for space tme codng based on a smpler exact evaluaton of parwse error probablty, J. Commun. Networs, vol. 3, no. 3, pp , Sept. 00. [6] G. Tarcco and E. Bgler, Exact parwse error probablty of space-tme codes, IEEE Trans. Inform. Theory, vol. 48, pp , Feb. 00. [7] S. M. Alamout, A smple transmt dversty technque for wreless communcatons, IEEE J. Sel. Areas n Commun., vol. 6, no. 8, pp , Oct [8] J. W. Crag, A new, smple and exact result for calculatng the probablty of error for two-dmensonal sgnal constellatons, Proc. IEEE MILCOM 9, pp , 99. [9] M. K Smon and M.-S. Aloun, Dgtal Communcatons Over Fadng Channels: A Unfed Approach to Performance Analyss, Wley Seres, July 000. [0] K. Cho and D. Yoon, On the general BER expresson of one and two dmensonal ampltude modulatons, IEEE Trans. Commun., vol. 50, no. 7, pp , July 00. [] X. Tang, M.-S. Aloun, and A. J. Goldsmth, Effect of channel estmaton error on M-QAM BER performance n Raylegh fadng, IEEE Trans. Commun., vol. 47, no., pp , Dec [] M. K. Smon and R. Annavajjala, On the optmalty of bt detecton of certan dgtal modulatons, IEEE Trans. Commun., vol. 53, no., pp , Feb [3] C.-J. Km, Y.-S. Km, G.-Y. Jeong, and H.-J. Lee, BER analyss of QAM wth MRC space dversty n Raylegh fadng channel, Proc. PIMRC 95, pp , Sept. 995.