Error probability analysis of bit-interleaved coded modulation

Transcription

1 Error robability analyi of bit-interleaved coded modulation Citation for ublihed verion (APA): Martinez, A., Guillén i Fàbrega, A., & Caire, G. (006). Error robability analyi of bit-interleaved coded modulation. IEEE Tranaction on Information Theory, 5(), 6-7. DOI: 0.09/TIT DOI: 0.09/TIT Document tatu and date: Publihed: 0/0/006 Document Verion: Publiher PDF, alo known a Verion of Record (include final age, iue and volume number) Pleae check the document verion of thi ublication: A ubmitted manucrit i the verion of the article uon ubmiion and before eer-review. There can be imortant difference between the ubmitted verion and the official ublihed verion of record. Peole intereted in the reearch are advied to contact the author for the final verion of the ublication, or viit the DOI to the ubliher' webite. The final author verion and the galley roof are verion of the ublication after eer review. The final ublihed verion feature the final layout of the aer including the volume, iue and age number. Link to ublication General right Coyright and moral right for the ublication made acceible in the ublic ortal are retained by the author and/or other coyright owner and it i a condition of acceing ublication that uer recognie and abide by the legal requirement aociated with thee right. Uer may download and rint one coy of any ublication from the ublic ortal for the uroe of rivate tudy or reearch. You may not further ditribute the material or ue it for any rofit-making activity or commercial gain You may freely ditribute the URL identifying the ublication in the ublic ortal. If the ublication i ditributed under the term of Article 5fa of the Dutch Coyright Act, indicated by the Taverne licene above, leae follow below link for the End Uer Agreement: Take down olicy If you believe that thi document breache coyright leae contact u at: oenacce@tue.nl roviding detail and we will invetigate your claim. Download date:. Ar. 09

2 6 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., JANUARY 006 Error Probability Analyi of Bit-Interleaved Coded Modulation Alfono Martinez, Member, IEEE, AlbertGuillén i Fàbrega, Member, IEEE, and Giuee Caire, Fellow, IEEE Abtract Thi correondence reent a imle method to accurately comute the error robability of bit-interleaved coded modulation (BICM). Thank to the binary-inut outut-ymmetric (BIOS) nature of the channel, the airwie error robability (PEP) i equal to the tail robability of a um of random variable with a articular ditribution. Thi robability i in turn comuted with a addleoint aroximation. It reciion i numerically validated for coded tranmiion over tandard Gauian noie and fully interleaved fading channel for both convolutional and turbo-like code. Index Term Additive white Gauian noie (AWGN) channel, bit-interleaved coded modulation (BICM), error robability, addleoint aroximation, Gauian aroximation, fading channel. I. INTRODUCTION Bit-interleaved coded modulation (BICM) wa introduced by Zehavi [] a a ragmatic coding cheme for ectrally efficient modulation. Under the aumtion of ufficient bit interleaving at the encoder outut, it wa later extenively tudied by Caire et al. [], who uggeted that the ytem eentially behave a a memoryle binaryinut outut-ymmetric (BIOS) channel. Thi conideration allow for an eay calculation of channel caacity (average mutual information) and cutoff rate for arbitrary modulation alhabet and ymbol labeling. However, the analyi of error robabilitie in [] wa either not tight or exceedingly comlex to comute. In thi correondence, we elaborate on their method and obtain a imle and very accurate method to etimate the error robability. II. ERROR PROBABILITY ANALYSIS A. Channel Model We tudy coded modulation over Gauian noie channel. The dicrete-time received ignal can be exreed a yk = hkxk + zk; k =;...;L () where yk i the (comlex-valued, i.e., yk C) kth received amle, hk C i the kth fading attenuation, xk C i the tranmitted ignal Manucritreceived November 30, 004; revied Augut, 005. Thi work wa uorted in art by the ANTIPODE roject of the French Telecommunication Reearch Council RNRT, and by Intitut Eurécom indutrial artner: Bouygue Télécom, Fondation d Entrerie Groue Cégétel, Fondation Haler, France Télécom, Hitachi, STMicroelectronic, Swicom, Texa Intrument, and Thale. The material in thi correondence wa reented in art at the 004 Conference on Information Science and Sytem, Princeton Univerity, Princeton, NJ, March 004, and at the 004 International Symoium on Information Theory and It Alication, Parma, Italy, October 004. A. Martinez i with the Deartment of Electrical Engineering, Techniche Univeriteit Eindhoven, 5600 MB Eindhoven, The Netherland ( alfono.martinez@ieee.org). A. Guillén i Fàbrega i with the Intitute for Telecommunication Reearch, Univerity of South Autralia, Mawon Lake SA 5095, Autralia ( albert.guillen@unia.edu.au). G. Caire wa with Intitut Eurécom, Sohia-Antioli, France. He i now with the Electrical Engineering Deartment, Univerity of Southern California, Lo Angele, CA USA ( caire@uc.edu). Communicated by Ø. Ytrehu, Aociate Editor for Coding Technique. Digital Object Identifier 0.09/TIT attime k, and zk C i the kth noie amle, aumed to be comlex Gauian indeendent and identically ditributed (i.i.d.) NC(0; ). BICM codeword x = (x;...; xl) are obtained by bit interleaving the codeword c =(c;...;cn) of the code C, each of dimenion K information bit and length N, and maing over the ignal contellation X with the labeling rule : f0; g M!X;M = log jx j. The correonding tranmiion rate i R = KM N bit er channel ue. The average received ignal-to-noie ratio i. We denote the vector of received ymbol by y =(y;...; yl). The tandard additive white Gauian noie (AWGN) and fully interleaved Rayleigh-fading channel are obtained from () by imly letting hk = and hk NC(0; ), reectively. The oeration i deicted in Fig.. B. Error Probability Under ML Decoding For maximum-likelihood (ML) decoding, the error robability of linear binary code over BIOS channel i accurately given by the union bound in the region above the cutoff rate [3]. Let Ad denote the number of codeword in C with Hamming weight d. In the region above the cutoff rate, the codeword error robability i very cloely uer-bounded by Pe d AdPEP(d; ; X ; ) () where PEP(d; ; X ; ) i the airwie error robability (PEP) for two codeword differing in d bit. Etimating the error robability reduce therefore to comuting the PEP. Auming that codeword c wa tranmitted, the robability of chooing a candidate codeword c 0 athamming ditance d from c i given by PEP(d; ; X ; ) = Pr(Pr(c 0 j y) > Pr(c j y) j c) =Pr log Pr(c0 i j yk(i)) Pr(ci j yk(i)) i > 0 j c where we have ued that the ith bit deend only it correonding channel outut yk(i). The random element in the channel outut include the noie and fading realization z and h, reectively, the articular modulation ymbol x, and the bit oition in the binary label m. In order to avoid cumberome notation, we grou them in a vector V = (z; h; x; m); V deend on the modulation alhabet X, the labeling, and. Taking into account that only the bit oition for which c 0 i 6= ci mutbe conidered, the PEP i given by PEP(d; ; X ; ) = Pr d j= 3j > 0 (3) where we have defined a new random variable, denoted by 3, thea oteriori log-likelihood ratio, a 3=log Pr(^c =c jv) Pr(^c = c jv) : (4) Thank to the reence of the interleaver [], the variable 3 can be conidered, to a ractical extent, i.i.d. Furthermore, due to the ym- We aume erfect channel tate information (CSI) at the receiver. However, the extenion of technique decribed here to the nonerfect CSI cae i traightforward. Similarly, the bit-error robability P i given by the right-hand ide of () with A relaced by ~ A = A ;A being the number of codeword in C with outut Hamming weight d and inutweighti /$ IEEE Authorized licened ue ited to: Eindhoven Univerity of Technology. Downloaded on October 8, 009 at 09:54 from IEEE Xlore. Retriction aly.

3 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., JANUARY Fig.. Channel interface: tandard nonbinary ymbol at channel level, or at demodulator level, with binary ymbol. metry of the channel outut, 3 their ditribution doe not deend on the value of c, and we can afely aume that the all-zero codeword ha been tranmitted. It hould be noted that thi formulation i imly a retatement of the reult in [] with a different notation. In articular, the exact deendence of the error robability on the modulation ymbol or the bit index i droed, or rather conidered another random variable imilar to the noie or fading realization. Fig. how the location of 3 in the communication channel, after the demodulator. The a oteriori robabilitie ued in the comutation of 3 are given by Pr(^c = c jv) =Pr(^c = c j z; h; x; m) / ex(0jy 0 hxj ) (5) xx where Xc m i the ubet of ignal contellation oint with mth binary label oition equal to c. In [], three alternative method were given to comute PEP(d;; X ; ): the Bhattacharyya-union bound (B-UB), the BICM bound, and the exurgated BICM union bound (ex-ub). Of thee, the B-UB will be analyzed later. The BICM bound wa ued a a mean to derive the tighter exurgated bound and, therefore, we do not analyze it further. It i intereting to note that a careful examination of the exreion for the exurgated bound in [] reveal that it i equal to (4) retricting the um in (5) to one ingle term, the nearet neighbor. Proceeding directly from the aumtion of a memoryle BIOS channel, their derivation can be ignificantly hortened. Furthermore, for non-gray labeling, the effect of the other neighbor i not negligible, and thu the ex-ub may not be accurate []. C. Log-Likelihood Ratio Ditribution For ome BIOS channel, the ratio 3 ha a known and eaily manageable ditribution. For examle, for the binary-ymmetric channel (BSC) 3 i a binomial random variable. For the binary-inut AWGN channel with ignal-to-noie ratio, 3 i normally ditributed N (04; 8). A little algebra how that for binary-inut Rayleigh-fading channel, the denity i two-ided exonential f 3 (3) = 4 (+) ex ign(3) : (6) Even though a cloed-form exreion for the denity of 3 for BICM eem difficult to obtain, it i neverthele imle to evaluate it by comuter imulation if required. 3 For ignal contellation X that lead to a BICM channel which i not ymmetric, the channel can be rendered BIOS by uing the maing and it comlement with robability = []. In etimate of tail robabilitie, the cumulant tranform () (or cumulant generating function) of a random variable 3 i a more convenient rereentation than the denity. The tranform i given by () = log E[e 3 ] (7) with C [4]. Uing the definition of 3, we rewrite () a Pr(^c =jv) () =loge V (8) Pr(^c =0jV) where the ubcrit V indicate that the exectation i taken with reect to all nuiance arameter V = (z; x; h; m). Thi exectation can be eaily evaluated by numerical integration uing the Gau Hermite (for the AWGN channel) and a combination of the Gau Hermite and Gau Laguerre (for the fading channel) quadrature rule, which are tabulated in [5]. It will alo rove convenient to define the addleoint ^ a the value for which 0 (^) = 0. It can be hown that thi oint exit and i unique [6]. For BIOS channel, ymmetry dictate that the addleoint i laced at ^ ==, with no need to carry an exlicit numerical minimization te [7]. Fig. how the comuter-imulated denity of 3 for 6-QAM over an AWGN channel and 8-PSK over a Rayleigh-fading channel with = db and = 7 db, reectively. In both cae, the labeling i Gray. For the ake of comarion, Fig. alo how the ditribution of a Gauian random variable with ditribution N (04; 8), with = 0(^). It hould be noted that thi Gauian aroximation i valid in the tail of the ditribution, rather at the mean a would be the cae for the tandard Gauian aroximation N (E[3]; E[3 ] 0 E[3] ). It i remarkable how cloe the tail are to the tail of a Gauian random variable for the cae of AWGN. For the Rayleigh fading, the denity inherit the exonential behavior of the binary-inut cae, and the Gauian aroximation to the tail i omewhat le accurate. D. Gauian Aroximation The receding dicuion ugget aroximating the PEP by PEP(d; ; X ; ) ' Q( 0d(^)) (9) a reult which wa heuritically introduced in [8]. The aroximation in (9) correond a well to the zeroth-order term in the Lugannani Rice formula [9] (ee alo [0]). E. Bhattacharyya Union Bound The Bhattacharyya bound [7] can be ued to uerbound the PEP a PEP(d; ; X ; ) e d(^) (0) Pr(^c =jv) = E V : () Pr(^c =0jV) Notice that thi coincide with the Chernoff bound a ^ ==. Uing thi in () we obtain the B-UB rooed in []. d Authorized licened ue ited to: Eindhoven Univerity of Technology. Downloaded on October 8, 009 at 09:54 from IEEE Xlore. Retriction aly.

4 64 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., JANUARY 006 Fig.. Denity of the a oteriori log-likelihood ratio 3: emirical ditribution (olid line, comuter imulated) and Gauian aroximation to the tail (dah-dotted) for 6-QAM/8-PSK, Gray maing, and AWGN/Rayleigh fading. F. Saddleoint Aroximation In the Aendix I, we reent the derivation of the addleoint aroximation and of an etimate of the aroximation error to the PEP. Even though the derivation in the Aendix i uniformly valid for all value of the addleoint ^, including mall value of ^, in our cae thi i notrequired a ^ = =. Keeing only the firt-order term in the aymtotic erie, the PEP can be aroximated by PEP(d; ; X ; ) = ed(^) +O(d 00 (^)) 0 d 00 (^)^ () where the term O(d 00 (^)) 0 decay fata a ower of (d 00 (^)) 0. The effect of the correction i found to be negligible in ractical calculation, which imlie that we need not um over any more term in the aymtotic erie and we may then dro the O( )term. The exonent i the ame a for the Bhattacharyya bound, in accordance to the aymtotic otimality of the latter, and coincide a well with the exonential decay of the Gauian aroximation. Note that efficient comutation of the econd derivative 00 (^) 00 (^) = E[3 e^3 ] E[e^3 ] = E[e^3 ] E V log Pr(^c =jv) Pr(^c =0jV) Pr(^c =jv) Pr(^c =0jV) (3) can again be erformed uing Gauian quadrature rule. It i worthwhile remarking that the method advocated in [] to comute thi robability for the exurgated union bound (UB) wa the ue of integration in the comlex lane. It can be een 4 that the addleoint 4 With the caveat indicated at the end of Section II-B on the metric (5). method i an alternative to the comlex-lane integration. Intead of directly comuting the integral, it value i very accurately aroximated with a method of ignificantly lower comlexity. III. NUMERICAL RESULTS AND DISCUSSION In thi ection, we how ome numerical reult that illutrate the accuracy of the rooed method a well a it aymtotic behavior. In articular, we how the following: the B-UB, the addleoint aroximation () union bound (SP-UB), the Gauian aroximation tangential-here bound (GA-TSB) [8], 5 and the imulation of the bit-error rate (BER im). For every block of information bit a different bit interleaver i randomly generated. A. AWGN Channel Fig. 3 and 4 how the bit-error robability a a function of E b =N 0 = =R for the aforementioned method and for convolutional and reeat accumulate (RA) code with 6-QAM in the AWGN channel with no fading. In Fig. 3, we ue the otimum 64-tate and rate-= convolutional code with Gray and et artitioning maing and in Fig. 4 an RA code [] of rate =4 with Gray maing. The erformance at medium-to-high ignal-to-noie ratio i very well aroximated by both the Gauian and the addleoint aroximation, for all conidered labeling and code. Note that the erformance etimate in the cae of et-artitioning labeling remarkably imrove the bound reented in []. In eence, thi can be traced back to the accuracy of the Gauian aroximation to the tail of the log-likelihood ratio 3, already dicued in Section II-C. The B-UB yield the correct decay of the bit error curve but it remain at a fixed ga from the true bit error robability. The accuracy of 5 Thi i the tandard tangential here bound [] for a binary-inut AWGN channel with = 0(^). Authorized licened ue ited to: Eindhoven Univerity of Technology. Downloaded on October 8, 009 at 09:54 from IEEE Xlore. Retriction aly.

5 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., JANUARY Fig. 3. Comarion of imulation and addleoint and Gauian aroximation on the BER of BICM with a 64-tate, rate-= convolutional code with 6-QAM modulation with Gray and et artitioning maing. Fig. 4. Comarion of imulation and addleoint and Gauian aroximation on the BER of BICM with a RA code of rate =4 with 6-QAM modulation and Gray maing, K = 04 information bit, 0 iteration of belief roagation decoding the AWGN channel. the union bound-baed aroximation for the RA code enemble aear only in the error floor region, ince the union bound i not tight for random-like code for below the correonding cutoff rate. Neverthele, the GA-TSB yield a fairly good etimate of the waterfall behavior of the error curve alo for low. In all cae, the decay of the bit error for increaing ignal-to-noie ratio eem to be of exonential nature. Aendix III rove the aymtotic validity of thi conjecture and how that! (^) = 0 d min 4 (4) where dmin i the minimum Euclidean ditance of the contellation. A outlined in the roof, at large, BICM behave a a binary modulation with ditance dmin, regardle of the maing. Thi reult confirm that BICM reerve the roertie of the underlying binary code and that for large the error robability decay exonentially with a 0 e d. In thi line, Fig. 5 how 0 (^) for 6-QAM with Gray and et artitioning maing in the AWGN channel. The aymtotic value i d =0:, a etablihed by the receding reult. 4 In the Gauian aroximation, the quantity 0 (^) can be interreted a the caling with reect to when uing BICM [8] and Authorized licened ue ited to: Eindhoven Univerity of Technology. Downloaded on October 8, 009 at 09:54 from IEEE Xlore. Retriction aly.

6 66 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., JANUARY 006 Fig. 5. Cumulantit: 0 and for 6-QAM modulation with Gray and et artitioning maing in the AWGN channel and (^) for 6-QAM and 8-PSK with Gray maing in the Rayleigh-fading channel. thu, the aymtotic caling deend only on the ignal contellation X (through it minimum ditance) and not on the labeling. A hown in Aendix I, the econd-order cumulant evaluated at the addleoint 00 (^) lay an imortant role in aeing the error of the aroximation. Aendix III alo how that 00 (^)! =d min : (5) (^) Fig. 5 alo how for 6-QAM with Gray and et artitioning maing in the AWGN channel. The it coincide with the above reult, and imlie that, in the AWGN channel, the addleoint aroximation become more and more accurate a grow. B. Fully Interleaved Rayleigh Fading Channel Fig. 6 and 7 how the etimate of the bit error robability for convolutional and RA code, reectively, in a fully interleaved AWGN channel with Rayleigh fading. Fig. 6 how two cae, a rate-=3, 8-tate otimum code over 8-PSK, and the rate-=, 64-tate otimum code over 6-QAM both with Gray maing. Fig. 7 how the erformance of an RA code of rate =4 with Gray maing and 6-QAM modulation. Similarly to the AWGN cae, the three aroximation to the error rate give the correct loe of the decay with atmedium-to-high ignal-to-noie ratio, while the horizontal hift of the curve i different. All aroximation are cloe to the imulated value, but now only the addleoint aroximation give an accurate etimate. A we aw in Section II-C, the tail of the log-likelihood ratio 3 in the fading channel i aroximately exonential, rather than Gauian, and thi hae i not correctly tracked by the Gauian aroximation. On the contrary, the addleoint aroximation i able to learn the hae of the variable. A evidenced by the reult of 6-QAM with the 64-tate convolutional code, thi effect become le aarent for more owerful code with large minimum ditance, ince the um in (3) contain more term and it tail i cloer to a Gauian. Again, the GA-TSB yield the mot accurate etimate of the error robability in the low- region. The accuracy of the UB-baed aroximation for the RA code enemble i accurate in the error floor region. Note alo that BICM reerve the roertie of the underlying binary code for fully interleaved Rayleigh-fading channel a well, a the error robability decay a an invere ower of. Aendix III how that in the it for large the rate of decay varie a and that! (^) = 0 (6) log! 00 (^) =8 (7) confirming that BICM indeed behave a a binary modulation and thu, the aymtotic erformance deend on the Hamming ditance of the code rather than on the Euclidean ditance. Fig. 5 alo how 00 (^) a a function of for 6-QAM and 8-PSK with Gray maing in the fully interleaved Rayleigh-fading channel. A exected, the it value i 8, and doe not deend on the modulation. IV. CONCLUSION In thi correondence, we have reented a imle method to comute a tight aroximation to the error robability of BICM. Thi robability i found to correond in a natural way to the tail robability of a um of indeendentrandom variable, which i calculated uing the addleoint aroximation. The exact form of the aroximation Authorized licened ue ited to: Eindhoven Univerity of Technology. Downloaded on October 8, 009 at 09:54 from IEEE Xlore. Retriction aly.

7 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., JANUARY Fig. 6. Comarion of imulation and addleoint and Gauian aroximation on the bit error rate of BICM with a 8-tate, rate-=3 convolutional code with 8-PSK modulation and a 64-tate, rate-= convolutional code with 6-QAM, both with Gray maing. Fig. 7. Comarion of imulation and addleoint and Gauian aroximation on the bit error rate of BICM with a Reeat-and-Accumulate code of rate =4 with 6-QAM modulation and Gray maing, K = 5 information bit, 0 iteration of belief roagation decoding the fully-interleaved Rayleigh fading channel. i new ince, a ooed to the uual formula, it i uniformly valid for all value of the addleoint. The rooed method benefit from imle numerical integration uing Gauian quadrature for noie and fading averaging. We have verified the validity of the aroximation for both, convolutional and turbo-like code enemble with BICM, over AWGN and fully interleaved Rayleigh-fading channel. In both cae, the aymtotic behavior of BICM mimic that of binary modulation. Thi imle technique contitute a owerful tool to the analyi of finite-length BICM. Furthermore, being imler and tighter than the original bound in [], it how a wide range of ractical alication. Authorized licened ue ited to: Eindhoven Univerity of Technology. Downloaded on October 8, 009 at 09:54 from IEEE Xlore. Retriction aly.

8 68 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., JANUARY 006 APPENDIX I DERIVATION OF THE SADDLEPOINT APPROXIMATION We wih to etimate the tail robability of Z, a continuou random variable with denity f Z (z). ToZ we aociate it cumulant tranform (or cumulant generating function) (), defined a () = loge[e Z ]; C. We hall be concerned with the cae of M i= X i.for Z being the um of M random variable X i ;Z = indeendent X i, it i immediate that the total cumulant tranform i the um of the tranform for each comonent. In thi cae, the denity of Z and it tail robability (or equivalently it ditribution) can be recovered from () by Fourier inverion [6] f Z(z) = j Pr(Z >z)= j j e ()0z d (8) =0j j e ()0z d =0j : (9) In the following, we tudy the tail robability only and aume, without lo of generality, that z>e[z]. An alication of Cauchy integral theorem allow u to move the integration ath to the right, from the imaginary axi to a line L = (^ 0 j; ^ + j) that croe the real axi at another oint ^ [3]. It i motconvenientto chooe ^ o that 0 (^) =z; thi oint i called a addleoint, a comlex-variable analytic function do not reach extreme oint in their domain of analyticity [3]. Thi oint exit and i unique due to the convexity of () [6]. Along the integration ath =^+j; 0 << and (0^) = j. Uing thi new variable of integration, we now exand the argument of the exonential term in a Taylor erie around ^ () 0 z = (^) 0 ^z + 00 (^) (j) + R () (0)! where we have ued that the firt derivative i zero and R () i a horthand for the remaining term in the exanion around ^ (`) (^) R () = (j)`: () `! `=3 In the following, we hall inditinctly refer to the `th-order derivative a the `th-order cumulant. Equation (9) can be rewritten a Pr(Z >z)= e(^)0^z = e(^)0^z e 0 0 e 0 e R () d ^ + j e R () ^ 0 j ^ + d () where we have multilied numerator and denominator time a factor ^ 0 j. Uing the Taylor exanion for the exonential, e z = m! zm, we have now (^ 0 j)e R () (`) (^) =(^ 0 j) (j)` m! `! `=3 m = ~ m(j) = m m m where we have groued the term with common factor (j) m, and called the correonding coefficient ~ m ; imilarly for m, which incororate the ower of j into the coefficient. The ymmetry of the integrand (ee ()) imlie that the integral of the term with odd m i zero; we need thu conider only the even value of m. The firtfew term are 0 =^; =0 4 = 0 (3) (^) 3! 6 = (5) (^) 5! +^ (4) (^) 4! 0 ^ (6) (^) 6! 0 ^! (3) (^) At thi oint, we normalize the cumulant. A the cumulant are all linear term in m, the number of random variable contributing to Z, we getrid of thi deendence on m by dividing all cumulant by 00 (^) and denote the normalized cumulant by ~ (`) (^). The coefficient m become now a olynomial of 00 (^) 4 = 0 ~(3) (^) 3! 6 = ~(5) (^) 5! 00 (^) +^ ~(4) (^) 00 (^) 4! 00 (^) 0 ^ ~(6) (^) 00 (^) 0 ^ 6!! 3! ~ (3) (^) 3! : ( 00 (^)) : The degree of the olynomial will rove ueful when tracking the variou term in the final exanion. The next roblem i the evaluation of integral of the form I(m) = m + ex (^) m d (3) ^ + where m i an even number. The value of thi integral i given in (34), in Aendix II. Setting = 00 (^) and =^ we obtain I(m) = m 3 (m 0 ) ^ ( 00 (^)) m+ f +O((00 (^)) 0 )g: (4) In articular, for m =0and dicarding the O() term, we recover the claical addleointaroximation Pr(Z >z) ' 00 (^)^ e(^)0^z : (5) Note that even though thi equation loe it validity for mall ^, we may ue the original (33) and how that the robability tend to = for ^! 0 0 ^ erfc ^ 00 (^) ex ^ 00 (^) = erfc ^ 00 (^) ex ^ 00 (^) : (6) Thi yield an aroximation which i uniformly valid for all value of the addleoint [6]. If required, higher order term may be obtained by extending the outlined rocedure. The following term i given by m =4, which give an extra term with leading coefficient ( 00 (^)) 0.A 4 i a degree- olynomial of 00 (^), the term grow rather like ( 00 (^)) 0. A careful analyi of the remaining term how that there i only one additional term with the ame factor, namely, the one correonding to the quared Authorized licened ue ited to: Eindhoven Univerity of Technology. Downloaded on October 8, 009 at 09:54 from IEEE Xlore. Retriction aly.

9 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., JANUARY third cumulant in 6. Putting all thi information back together, the econd-order addleointaroximation i given by Pr(Z >z) ' e(^)0^x + 00 (^)^ 00 (^) + (4) (^) 8 00 (^) (3) (^) 00 (^) 0 ^ 0 ~(3) (^) ^ 00 (^) +O(( 00 (^)) 0 ) : (7) Thi additional term in the exanion alo erve a an etimate of the error made by the aroximation. In general, the firt term of the exanion give a very good aroximation to the real tail robability, with no need of conidering extra term. APPENDIX II SOME INTEGRALS AND EXPANSIONS OF INTEREST The error comlementary function i defined a erfc(x) = e 0t dt: x It aymtotic erie i derived by integration by art [3] and give erfc(x) = e0x x (0) m 3 (m 0 ) m x m (8) = e0x x 0 x + 3 x x 6 + : (9) Here the abolute error committed by truncating may be hown to be maller than the firt neglected term. For large value of the arameter, the aroximation erfc(x) ' x e 0x i valid. More reciely, for value of x larger than, the relative error in aroximating erfc(x)ex(x ) by x i maller than 5% (obtained by evaluation of the formula, not with an etimate of the error). An integral that aear often in our calculation i the following [3]: + ex(0 x ) 0 + x dx = erfc()ex( ): (30) Note that we may eaily aly the aymtotic exanion for erfc(x). We will alo evaluate integral of the more general form + ex(0 x x n ) 0 dx (3) + x where n i an integer. Their value i calculated a follow. Firt exand the fraction in the integrand x n n + x = (0) m0 (m0) x (n0m) +(0) n n + x ; m= n0 = (0) n0m0 (n0m0) x m +(0) n n + x (3) and then integrate term by term. Each term i een to be the th momentof a normal random variable with zero mean and variance ( ) 0, whoe value i 3 ( 0 )( ) 0 [4]. Combining thee value back into (3) and after ome algebraic maniulation we get n0 + (0) n0m0 (n0m0) ex(0 x )x m dx 0 +(0) n n + ex(0 x ) 0 + x dx n0 = (0) n0m0 (n0m0) 3 (m 0 )( ) 0m +(0) n n erfc()ex( ) (33) n0 = (0) n0m0 (n0m0) 3 (m 0 )( ) 0m +(0) n n e 0 (0) m 3 (m 0 ) m m m ex( ) n0 = (0) n0m0 (n0m0) 3 (m 0 )( ) 0m + (0) n+m (n0) = (0) n+m (n0m0) m=n 3 (m 0 ) m m m 3 (m 0 ) m m : (34) In the lat te, we exloit that the firt n 0 term in both ummation exactly cancel each other. A it i derived from the aymtotic exanion of erfc(x), the formula inherit the former bound on error, that i, the error by truncating the erie i uer-bounded by the abolute value of the following term. APPENDIX III CUMULANT TRANSFORM ASYMPTOTIC ANALYSIS In thi aendix, we how that in the it for large, BICM behave a a binary modulation with quared Euclidean ditance d min In articular, we have that and for the AWGN channel, and and = min x;x X d (x; x 0 )= min x;x X jx 0 x0 j : (^)! = 0 d min 4 (35) 00 (^)! =d min (36)! (^) = 0 (37) log! 00 (^) =8 (38) for the fully interleaved Rayleigh-fading channel. In thi aendix, and without lo of generality, aume that i real. Authorized licened ue ited to: Eindhoven Univerity of Technology. Downloaded on October 8, 009 at 09:54 from IEEE Xlore. Retriction aly.

10 70 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., JANUARY 006 A. AWGN Channel Conider firt the AWGN channel without fading. Then ()! =! x X e0j (x0x )+zj log E z;x;m x X e0j (x0x )+zj We can uer-bound the term inide the exectation by uer-bounding the um at the numerator by jxj and lower-bounding the um at the denominator by e 0jzj. Then and ince x X x X e0j (x0x )+zj e0j (x0x )+zj E e jzj < ; jx j e jzj (39) if <, we can ue the dominated convergence theorem [4]. Note that ince ^ = for BIOS channel, thi retriction oe no ractical itation to the validity of the reult. We can then take the dominant term in the um and write that ()! =! log E z;x;m e 0j (x0x )+zj e 0jzj where x 0 denote the ignal contellation ymbol cloet to x in the comlementary et X m. Then we have that ()! =! log E z;x;m e 0j (x0x )+zj +jzj =! 0d (x;x )0Ref (x0x )z g log E z;x;m e =! =! = 0d min( 0 ) (x;x )(0 ) log Ex;m e0d (0 ) log Ke0d where K may deend on the actual maing rule. Note, however, that the reult doe not. By letting =^ =, we then obtain that (^)! = 0 d min 4 : Furthermore, at large, the econd-order cumulant behave a 00 ()! =d min which again mimic the behavior of a binary modulation with quared minimum ditance d min. : B. Fully Interleaved Rayleigh-Fading Channel In the cae of the fully interleaved fading channel, we have! () log =! log x X e0j h(x0x )+zj log E z;h;x;m e0j h(x0x )+zj x X The uer bound in (39) alie here a well and, then, for <, the dominated convergence theorem lead to ()! log =! log log E e 0j h(x0x )+zj z;h;x;m e 0jzj =! log log E 0d (x;x )(0 ) ;x;m e where = jhj i the fading ower and x 0 denote again the cloet ointto x in the et X m. Averaging over the fading we get that ()! log =! log log E x;m +d (x; x 0 )(0 ) =! log log =0: Therefore, ince atlarge! () =! log K +d min (0 ) K +d min ( 0 ) we obtain that the econd-order cumulant behave a! 00 () dmin =! +dmin ( 0 ) + dmin( 0 ) +dmin ( 0 ) By letting =^ =, it i eay to verify that at the addleoint a in the binary cae.! 00 (^) =8 (40) REFERENCES [] E. Zehavi, 8-PSK trelli code for a rayleigh channel, IEEE Tran. Commun., vol. 40, no. 5, , May 99. [] G. Caire, G. Taricco, and E. Biglieri, Bit-interleaved coded modulation, IEEE Tran. Inf. Theory, vol. 44, no. 3, , May 998. [3] A. J. Viterbi and J. K. Omura, Princile of Digital Communication and Coding. New York: McGraw-Hill, 979. [4] R. Durrett, Probability: Theory and Examle. Belmont, CA: Duxbury, 996. [5] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Function with Formula, Grah and Mathematical Table. New York: Dover, 97. [6] J. L. Jenen, Saddleoint Aroximation. Oxford, U.K.: Clarendon, 995. [7] R. G. Gallager, Information Theory and Reliable Communication. New York: Wiley, 968. [8] A. Guillén i Fàbrega, A. Martinez, and G. Caire, Error robability of bit-interleaved coded modulation uing the gauian aroximation, in Proc. Conf. Information Science and Sytem, Princeton, NJ, Mar [9] R. Lugannani and S. O. Rice, Saddle oint aroximation for the ditribution of the um of indeendent random variable, Adv. Al. Probab., vol., , 980. : : Authorized licened ue ited to: Eindhoven Univerity of Technology. Downloaded on October 8, 009 at 09:54 from IEEE Xlore. Retriction aly.

11 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., JANUARY [0] A. Martinez, A. Guillén i Fàbrega, and G. Caire, New imle evaluation of the error robability of bit-interleaved coded modulation uing the addleoint aroximation, in Proc. 004 Int. Sym. Information Theory and it Alication, Parma, Italy, Oct. 004, [] G. Poltyrev, Bound on the decoding error robability of linear code via their ectra, IEEE Tran. Inf. Theory, vol. 40, no. 4,. 84 9, Jul [] D. Divalar, H. Jin, and R. J. McEliece, Coding theorem for turbolike code, in Proc. 36th Allerton Conf. Communication, Control, and Comuting, Allerton Houe, Monticello, IL, Se. 998, [3] F. W. J. Olver, Aymtotic and Secial Function. New York: Academic, 974. On the Ditribution of SINR for the MMSE MIMO Receiver and Performance Analyi Ping Li, Debahi Paul, Ravi Naraimhan, Member, IEEE, and John Cioffi, Fellow, IEEE Abtract Thi correondence tudie the tatitical ditribution of the ignal-to-interference-lu-noie ratio (SINR) for the minimum mean-quare error (MMSE) receiver in multile-inut multile-outut (MIMO) wirele communication. The channel model i aumed to be (tranmit) correlated Rayleigh flat-fading with unequal ower. The SINR can be decomoed into two indeendent random variable: SINR = SINR +, where SINR correond to the SINR for a zero-forcing (ZF) receiver and ha an exact Gamma ditribution. Thi correondence focue on characterizing the tatitical roertie of uing the reult from random matrix theory. Firt three aymtotic moment of are derived for uncorrelated channel and channel with equicorrelation. For general correlated channel, ome iting uer bound for the firt three moment are alo rovided. For uncorrelated channel and correlated channel atifying certain condition, it i roved that converge to a Normal random variable. A Gamma ditribution and a generalized Gamma ditribution are rooed a aroximation to the finite amle ditribution of. Simulation ugget that thee aroximate ditribution can be ued to etimate accurately the robability of error even for very mall dimenion (e.g., two tranmit antenna). Index Term Aymtotic ditribution, channel correlation, error robability, Gamma aroximation, minimum mean quare error (MMSE) receiver, multile-inut multile-outut (MIMO) ytem, random matrix, ignal-to-interference-lu-noie ratio (SINR). I. INTRODUCTION Thi tudy conider the following ignal and channel model in a multile-inut multile-outut (MIMO) ytem: y r = m H W R t P x t + n c = m Hx t + n c () Manucritreceived January 0, 005; revied Setember 9, 005. P. Li i with the Deartment of Statitic, Stanford Univerity, Stanford, CA USA ( ingli@tat.tanford.edu). D. Paul i with the Deartment of Statitic, Univerity of California, Davi, Davi, CA 9566 USA ( debahi@wald.ucdavi.edu). R. Naraimhan i with the Deartment of Electrical Engineering, Univerity of California, Santa Cruz, Santa Cruz, CA USA ( ravi@oe.ucc.edu). J. Cioffi i with the Deartment of Electrical Engineering, Stanford Univerity, Stanford, CA USA ( cioffi@tanford.edu). Communicated by R. R. Müller, Aociate Editor for Communication. Digital Object Identifier 0.09/TIT where x t i the (normalized) tranmitted ignal vector and y r m i the received ignal vector. Here i the number of tranmit antenna and m i the number of receive antenna. H W m conit of indeendent and odentically ditributed (i.i.d.) tandard comlex Normal entrie. R t i the tranmitter correlation matrix. P = diag[~c ; ~c ;...; ~c ] m ; ~c k = c k, where c k i the ignal-to-noie ratio () for the kth atial tream. Thi definition of i conitent with [, Sec. 7.4]. H = H W R t P m i treated a the channel matrix. n c m i the comlex noie vector and i aumed to have zero mean and identity covariance. Note that the ower matrix P ha term involving the variance of the noie. The correlation matrix R t and ower matrix P are aumed to be nonrandom. Alo, we retrict our attention to m. We conider the oular linear minimum mean-quare error (MMSE) receiver. Conditional on the channel matrix H, the ignal-tointerference-lu-noie ratio (SINR) on the kth atial tream can be exreed a (e.g., [] [6]) SINR k = MMSE k 0 = I + m Hy H 0 kk 0 () where I i a identity matrix, and H y i the Hermitian tranoe of H. Note that (), in the ame form a equation (7.49) of [], i derived baed on the econd-order tatitic of the inut ignal, not retricted to binary ignal. For binary inut, Verdú [4, eq. (6.47)] rovide the exact formula for comuting the bit-error rate (BER) (alo ee [7]). Conditional on H, thi BER formula require comuting 0 Q-function. To comute BER unconditionally, we need to amle H enough time (e.g., 0 5 ) to get a reliable etimate. When 3 (or 64), the comutation become intractable [4], [8]. Recently, tudy of the aymtotic roertie of multiuer receiver (e.g., [] [4], [6], [8] []) ha received a lot of attention. Work that relate directly to the content of thi correondence include Te and Hanly [] and Verdú and Shamai [6], who indeendently derived the aymtotic firt moment of SINR for uncorrelated channel. Te and Zeitouni [3] roved the aymtotic Normality of SINR for the equal ower cae, and commented on the oibility of extending the reult to the unequal ower cenario. Zhang et al. [] roved the aymtotic Normality of the multile-acce interference (MAI), which i cloely related to SINR. Guo et al. [8] roved the aymtotic Normality of the deciion tatitic for a variety of linear multiuer receiver. [8] conidered a general ower ditribution and correonding unconditional aymtotic behavior. Baed on the aymtotic Normality reult, Poor and Verdú [] (alo in [4], [8]) rooed uing the iting BER (denoted by BER ) for binary modulation, which i a ingle Q-function BER = Q( E(SINR k )) = E(SINR ) e 0t = dt (3) where E(SINR k ) denote the aymtotic firt moment of SINR k. Equation (3) i convenient and accurate for large dimenion. However, it accuracy for mall dimenion i of ome concern. For intance, [8] comared the aymtotic BER with imulation reult, which howed that even with =64there exited ignificant dicreancie. In general, (3) will underetimate the true BER. For examle, in our imulation, when m =6; =8; = 5 db, the aymtotic BER given by (3) i roughly of the exact BER. In 0000 current ractice, code-diviion multile-acce (CDMA) channel with m; between 3 and 64 are tyical and in multile-antenna ytem array of 4 antenna are tyical but array with 8 to 6 antenna would be feaible in the near future [9]. Therefore, it would be ueful if /$ IEEE Authorized licened ue ited to: Eindhoven Univerity of Technology. Downloaded on October 8, 009 at 09:54 from IEEE Xlore. Retriction aly.