On the Relationship Between Mutual Information and Bit Error Probability for Some Linear Dispersion Codes

Transcription

1 9 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., JANUARY 29 On the Relationship Between Mutual Information an Bit Error Probability for Some Linear Dispersion Coes Xianglan Jin, Jae-Dong Yang, Kyoung-Young Song, Jong-Seon No, Member, IEEE, an Dong-Joon Shin, Member, IEEE Abstract In this paper, we erive the relationship between the bit error probability (BEP of maximum a posteriori (MAP bit etection an the bit minimum mean square error (BMMSE. By using this result, the relationship between the mutual information an the BEP is erive for multiple-input multiple-output (MIMO communication systems with the bitlinear linear-ispersion (BLLD coes for the Gaussian channel. From the relationship, the lower an upper bouns on the mutual information can be erive. Inex Terms Bit error probability (BEP, bit-linear linearispersion (BLLD coes, maximum a posteriori (MAP bit etection, minimum mean square error (MMSE, multiple-input multiple-output (MIMO, mutual information. I. INTRODUCTION IN the analysis of communication systems, the error probability an the minimum mean square error (MMSE are very important performance criteria. The bit error probability (BEP of the multiple-input multiple-output (MIMO communication systems has been extensively stuie an many results have been obtaine. The mutual information can also be use for measuring the performance of communication systems an is wiely stuie [] [4]. Recently, Guo, Shamai, an Verú [5] erive an interesting relationship between the mutual information an the MMSE for the Gaussian channel. Lozano, Tulino, an Verú [6] obtaine an approximation form of the mutual information for the single-input single-output (SISO system with binary phase shift keying (BPSK an quarature phase shift keying (QPSK in high signal to noise ratio (SNR region. Since the relationship between the mutual information an the BEP for MIMO systems has not been foun, we erive this relationship for some linear ispersion coes. In this paper, we consier the maximum a posteriori (MAP bit etection for MIMO systems an use BEP to enote the BEP of MAP bit etection an bit MMSE (BMMSE to enote the MMSE in estimating an information bit for any coing an moulation schemes. Then, the relationship between the BEP an the BMMSE is erive. Using the result in [5], Manuscript receive January 9, 27; revise May 25, 27, October 8, 27, an April, 28; accepte April, 28. The associate eitor coorinating the review of this paper an approving it for publication was T. Duman. This work was supporte by the IT R&D program of MKE/IITA. [28-F- 7-, Wireless Communication Systems in 3 Dimensional Environment]. X. Jin, J.-D. Yang, K.-Y. Song, an J.-S. No are with the Department of Electrical Engineering an Computer Science, Seoul National University, Seoul 5-744, Korea ( {xianglan.jin, yjong, sky674}@ccl.snu.ac.kr, jsno@snu.ac.kr. D.-J. Shin is with the Department of Electronics an Computer Engineering, Hanyang University, Seoul 33-79, Korea ( jshin@hanyang.ac.kr. Digital Object Ientifier.9/T-WC /9$25. c 29 IEEE the relationship between the mutual information an the BEP for MIMO systems with bit-linear linear-ispersion (BLLD coes [7] is erive in the Gaussian channel if their ispersion matrices satisfy a given conition. From the relationship, the lower an upper bouns on the mutual information can be erive by using the BEP. The following notations will be use in this paper: capital letter enotes matrix; unerscore enotes vector; bolface letter enotes ranom object; I n enotes the n n ientity matrix; Re( an Im( mean the real an imaginary parts of a complex value, respectively; enotes the Frobenius norm of a matrix; E{ } is the expectation; the superscripts ( T, ( an ( enote transpose, complex conjugation, an complex conjugate transpose, respectively; finally, vec( an tr( represent the vectorization an trace of a matrix. II. BEP OF MAP DETECTION AND BMMSE Let L t an L r be the numbers of transmit an receive antennas in a MIMO communication system, respectively. Let x = [x, x 2,...,x Lb ] T be an information vector consisting of inepenent binary bits x i {, } an f(x a bijective function corresponing to coing an moulation schemes. We assume that the average transmitte power is ρ an the perfect channel state information is available at the receiver. Then, the output signal y is given as y = ρhf(x+n ( where H is an L r L t channel matrix with ranom entries having unit power, n is an L r column noise vector with ranom entries having unit power an being inepenent of x, anρ represents the SNR. MAP etection chooses x i to maximize the posterior probability mass function (PMF, i.e., x i =argmaxp (x i = x i y = x i y, i =, 2,...,L b. Since we assume that x i is a binary information bit, in this paper we will use MAP etection to enote MAP bit etection. Now, we efine BMMSE which is a new performance criterion. Definition : A BMMSE of x is the MMSE in estimating abitxfor a given ρ, i.e., bmmse(ρ =E{ x ˆx(y 2 }, where ˆx(y is the BMMSE estimator efine as ˆx(y = E{x y} = x {,} xp (x y. For the MIMO system efine in (, we have bmmse(ρ = L b bmmse i (ρ, Authorize license use ite to: IEEE Xplore. Downloae on March 8, 29 at 2:43 from IEEE Xplore. Restrictions apply.

2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., JANUARY 29 9 where bmmse i (ρ =E{ x i ˆx i (y 2 }. Then, the relationship between the BEP an the BMMSE for the MIMO systems can be erive as follows. Theorem : For the MIMO system efine in (, the BEP of MAP etection an the BMMSE of binary information vector x have the following relationships. 4 bmmse(ρ <P b(ρ < bmmse(ρ, (2a 2 an ρ P b(ρ = 2 ρ bmmse(ρ P b(ρ = 4 bmmse(ρ. (2b (2c Proof: Let Rj i, j {, }, be the ecision region of y satisfying P (x i = j y = y >P(x i = j y = y. Then, the BEP of x i can be written as P bi = P (x i P ( x i x i x i = x i {,} R i p(x i =, yy + R i The BMMSE of x i can be erive as p(x i =, yy. (3 bmmse i (ρ 4p(xi =, yp(x i =, y = p(x i =, y+p(x i =, y y p(x i =, yp(x i =, y =4 R i p(x i =, y+p(x i =, y y p(x i =, yp(x i =, y +4 R i p(x i =, y+p(x i =, y y p(x i =, y p(x i =, y =4 y +4 y. R i + p(x,y p(x i=,y R i + p(xi=,y p(x,y Since < p(xi= j,y < in the region Rj i, j {, }, we have the following inequality p(x i = j, y p(x i = j, yy < y 2 < + p(xi= j,y (4 p(x i = j, yy. (5 Using (3, (4, an (5, we have the inequality 4 bmmse i(ρ <P bi (ρ < 2 bmmse i(ρ, an surely 4 bmmse(ρ <P b(ρ < 2 bmmse(ρ. As ρ goes to, p(x i= j,y approaches to in the region an we have ρ P bi(ρ = 2 ρ bmmse i(ρ Fig.. The relationship between BEP an BMMSE for the SISO system with Gray coe 6QAM moulation. an also, as ρ goes to infinity, p(xi= j,y the region Rj i an we have approaches to in P bi(ρ = 4 bmmse i(ρ. Therefore, we can have (2b an (2c. An approximation similar to the proof of Theorem was use to erive the relationship between the mean square error of the maximum likelihoo estimator an the MMSE in [8]. As an example, we consier a SISO system y = ρf(x + n, wheref( is a Gray coe 6QAM mapper an n is a complex Gaussian ranom variable with CN(,. Usingthe Monte Carlo metho, the BEP an the BMMSE values are plotte in Fig. which shows the relationship in Theorem. III. RELATIONSHIP BETWEEN MUTUAL INFORMATION AND BEP OF MAP DETECTION FOR BLLD CODES In this section, for BLLD coes, the lower an upper bouns on the mutual information are erive using the BEP. Especially, for the homogeneous orthogonal space time block coes(ostbcs in the Rayleigh faing channel, these bouns can be erive in close form. A. The Case of BLLD Coes Let X C Lt T an Y C Lr T be the transmit an receive signal matrices, respectively, where C enotes the set of complex numbers an T enotes the number of symbol urations. Let H C Lr Lt be the channel matrix which is known to the receiver only. H oes not change within T symbol urations an changes inepenently from block to block. Then, the MIMO system in the Gaussian channel can be expresse as Y = ρ HX + N (6 where the elements of N are inepenent an ientically istribute (i.i.. circularly symmetric complex Gaussian ranom variables with mean zero an variance.5 per imension, i.e., N CN(,I LrT. Authorize license use ite to: IEEE Xplore. Downloae on March 8, 29 at 2:43 from IEEE Xplore. Restrictions apply.

3 92 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., JANUARY 29 C = A BLLD coe C efine in [7] is given as { X : X = k= x k A k, x k {, }, k=, 2,...,L b } (7 where A k C Lt T are ispersion matrices an x,x 2,...,x Lb are the information bits. Let H =[vec(ha, vec(ha 2,...,vec(HA Lb ] an x =[x, x 2,...,x Lb ] T. Then, the MIMO system in (6 using BLLD coe can be rewritten as y = ρ Fx + n (8 [ ] [ ] Re(vec(Y Re(vec(N where y =, n =,anf = [ Im(vec(Y Im(vec(N Re(H ] Im(H. Clearly, this satisfies Theorem. In (8, for the fixe F, the MMSE of F x is E{ F (x ˆx(y 2 } an the mutual information I(ρ F = F of x an y is a function of ρ. Then, the mutual information an the MMSE of F x satisfy the following relationship for any input statistics [5] ρ I( ρ F = F = E{ F (x ˆx(y 2 } log 2 e. (9 Using Theorem an (9, we erive the following relationship. Theorem 2: Let X = L b k= x ka k be a BLLD coe where L b enotes the number of information bits uring T symbol urations. Suppose that A k s satisfy the conition A i A j + A ja i =, i<j L b. ( Then, for the MIMO system in (6 the relationship between the mutual information an the BEP of MAP etection of x i can be erive as 2log 2 e HA i 2 ( P bi ρ H = H < ρ I( ρ H = H < 4log 2 e HA i 2 ( P bi ρ H = H, ρ ρ I( ρ H = H =2log 2 e HA i 2 ( P bi ρ H = H, ρ ρ I( ρ H = H =4 log 2 e HA i 2 P bi (a (b ( ρ H = H. (c Proof: Using the previously efine F an x, the MMSE of F x can be given as If F satisfies E{ F (x ˆx 2 } = E { (x ˆx T F T F (x ˆx }. F T F = iag(λ,λ 2,...,λ Lb (2 where λ i > an iag( enotes the iagonal matrix, we have E{ F (x ˆx 2 } = L b λ ie{ x i ˆx i 2 } = Lb λ ibmmse i (ρ H = H. Then, we have the relationship between the mutual information an the BEP in (a, (b, an (c. Thus, it is enough to prove that (2 hols if ( is satisfie. The element at the jth row an ith column of F T F is equal to Re{tr(HA i A j H }. Since Re{tr(HA i A j H } = 2 tr{h(a ia j +A ja i H }, i<j L b, from (, F T F = iag( HA 2,..., HA Lb 2. Thus, the theorem is prove. Several examples satisfying Theorem 2 are given as follows. Example (A single transmit antenna system with BPSK or QPSK: For BPSK, A =an AA =, an for QPSK, A =/ 2, A 2 = j/ 2,anA A 2 + A 2A =. Therefore, the ispersion matrices satisfy the conition ( in Theorem 2. Example 2 (Generalize linear complex OSTBCs: The generalize linear complex OSTBCs [9] can be written as { } X : X = x i A i,x i R,A i C Lt T, i =, 2,...,L b an have the property XX = iag It is equivalent to ( Lb l,i x 2 i, l 2,i x 2 i,..., l Lt,ix 2 i A i A i = iag(l,i,l 2,i,...,l Lt,i, i L b, A i A j + A ja i =, i<j L b where l k,i,k =, 2,...,L t,i =, 2,...,L b, are positive numbers etermine by the type of the coe. Therefore, when BPSK or QPSK is use, the coes become the BLLD coes satisfying Theorem 2. Example 3 (Pseuo OSTBCs: Pseuo OSTBC, propose by Jafarkhani [], is efine as an L t T matrix X with entries that are linear combinations of the ineterminate variables s k S k,k =, 2,...,K, an their conjugates such that XX = c( s 2 + s s K 2 I Lt, where c is a constant an S k,k =, 2,...,K, are arbitrary subset of C. Whens k can be escribe as a binary signal form, the pseuo OSTBCs satisfy Theorem 2. For example, when s,s 2 {, }, s 3,s 4 { j, j}, the following pseuo OSTBC satisfies Theorem 2. s s 2 s 3 s 4 X = s 2 s s 4 s 3 s 3 s 4 s s 2. s 4 s 3 s 2 s To confirm Theorem 2, we compare the erivative of the mutual information an our bouns for Alamouti space-time coe [] with QPSK. We assume N CN(,I LrT an then for the fixe H = H, Fig. 2 shows that the inequalities in (a are satisfie an the lower an upper bouns are quite tight (within.3 B in the high SNR region.. Authorize license use ite to: IEEE Xplore. Downloae on March 8, 29 at 2:43 from IEEE Xplore. Restrictions apply.

4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., JANUARY Fig. 2. The relationship between the mutual information an BEP for the Alamouti space-time coe with QPSK moulation. Fig. 3. Comparison of the bouns in (7, the approximation using (8, an the capacity for the Alamouti space-time coe with QPSK moulation. Integrating the terms in (a with respect to ρ, we obtain ρ 2log 2 e HA i 2 ( ( P bi γ H = H γ < I ρ H = H ρ < 4log 2 e HA i 2 ( P bi γ H = H γ. (3 Theorem 2 can be applie not only to the fixe H but also to the ranom H in (3 by taking the expectation as given in the following corollary. Corollary : The average mutual information of X an Y of the MIMO system in (6 with the BLLD coes satisfying ( has the following lower an upper bouns ρ 2log 2 e E { HA i 2 ( } P bi γ H γ <I(ρ ρ < 4log 2 e E { HA i 2 ( P bi γ H } γ. (4 B. Calculation of Mutual Information for Homogeneous OS- TBCs In this subsection, we will simplify the bouns in (4 when the homogeneous OSTBCs efine in [2] are use with BPSK or QPSK in the Rayleigh faing channel. The homogeneous OSTBCs satisfy the conitions A i A i =ci L t, i L b, A i A j + A ja i =, i<j L b. (5 Thus, we can use (4 to fin the bouns on the mutual information. We assume that the information bits x j {, },j =, 2,...,L b, are equiprobable. From (5, we have F T F = c tr(hh I Lb = c H 2 I Lb as in the proof of Theorem 2, an ˇx = ρ (F T F F T y = F T y c ρ H 2 with ˇx N(x, 2c H 2 ρ I L b. Then, each information bit can be etecte separately an thus the BEP for the fixe H is equal to Q( 2c H 2 ρ. Hence we obtain the lower an upper bouns on the mutual information in (4 as ρ 2L b log 2 e E {c H 2 Q ( 2c H 2 γ } γ <I(ρ ρ < 4L b log 2 e E {c H 2 Q ( 2c H 2 γ } γ. (6 Using the result in [3], we can transform the expectation value in (6 as ρ E {c H 2 π/2 ( } exp c H 2 γ π sin 2 θγ θ = c ρ π/2 { ( } E H 2 exp c H 2 γ π sin 2 θγ. θ Let p = H 2 = i,j h i,j 2 an s = cγ/sin 2 θ,where h ij CN(,. Then, the probability ensity function of y = h i,j 2 is p(y = y =e y, y >, an E{p exp(sp} = s E{exp(sp} = L t L r ( s LtLr. Then, the lower an upper bouns on the mutual information in (4 are erive as L b log 2 e 2L b log 2 e π/2 ( sin 2 sin 2 θ LtLr θ θ < I(ρ 2 π sin 2 θ + cρ <L b log 2 e 4L b log 2 e π/2 ( sin 2 sin 2 θ LtLr θ θ. (7 π sin 2 θ + cρ For BPSK an QPSK, Lozano, Tulino, an Verú [6] obtaine the approximation of the mutual information for SISO systems in high SNR region as follows. I(ρ log 2 m ( e 2 4 ρ log 2 e /( ρ/π /2 (8 Authorize license use ite to: IEEE Xplore. Downloae on March 8, 29 at 2:43 from IEEE Xplore. Restrictions apply.

5 94 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., JANUARY 29 where m = 2 an = 2 for BPSK, an m = 4 an = 2 for QPSK. Since ˇx N ( x, 2c H 2 ρ I L b, the homogeneous OSTBCs can be ecouple into several parallel SISO channels. Thus, the approximation of the mutual information for the homogeneous OSTBCs can be obtaine using (8. The approximation of the mutual information given in [4] may also be use. In Fig. 3, Inorm(ρ enotes the mutual information normalize by T = 2, which is obtaine by Monte Carlo simulation, C enotes the capacity, Ia,norm enotes the approximation of the mutual information using (8 normalize by T = 2, an LBnorm an UBnorm enote the lower an upper bouns in (7 normalize by T =2, respectively. From Fig. 3, we can see that although the approximation of the mutual information using (8 is very accurate in the high SNR region, it cannot be use in the low SNR range (< B. Note that the capacity can be use as an upper boun on the mutual information. Finally, although our upper boun is not tight, it can be use in the whole SNR range an, especially, the lower boun is tight in the low SNR region. IV. CONCLUSION In this paper, BMMSE is efine for the MIMO systems with any coing an moulation schemes an the relationship between the BEP an the BMMSE is erive. Using this result, for the MIMO systems with BLLD coes in the Gaussian channel, the lower an upper bouns on the mutual information are erive by using BEP when their ispersion matrices satisfy a given conition. Especially, the lower an upper bouns on the mutual information for the MIMO systems with the homogeneous OSTBCs are erive in close form. REFERENCES [] E. Telatar, Capacity of multi-antenna Gaussian channels," AT&T Bell Labs Tech. Rep., June 995. [2] G. J. Foschini an M. J. Gans, On its of wireless communications in a faing environment when using multiple antennas," Wireless Pers. Commun., vol. 6, pp , Mar [3] M. Dohler an H. Aghvami, On the approximation of MIMO capacity," IEEE Trans. Wireless Commun., vol. 4, no., pp. 3-34, Jan. 25. [4] M. Kang an M.-S. Alouini, Capacity of MIMO Rician channels," IEEE Trans. Wireless Commun., vol. 5, no., pp. 2-22, Jan. 26. [5] D. Guo, S. Shamai (Shitz, an S. Verú, Mutual information an minimum mean-square error in Gaussian channels," IEEE Trans. Inform. Theory, vol. 5, no. 4, pp , Apr. 25. [6] A. Lozano, A. M. Tulino, an S. Verú, Optimum power allocation for parallel Gaussian channels with arbitrary input istributions, IEEE Trans. Inform. Theory, vol. 52, no. 7, pp , July 26. [7] Y. Jiang, R. Koetter, an A. C. Singer, On the separability of emoulation an ecoing for communications over multiple-antenna block faing channels," IEEE Trans. Inform. Theory, vol. 49, no., pp , Oct. 23. [8] N. Chayat an S. Shamai, Bouns on the capacity of intertransitiontime-restricte binary signaling over an AWGN channel," IEEE Trans. Inform. Theory, vol. 45, no. 6, pp , Sept [9] W. Su an X.-G. Xia, On space-time block coes from complex orthogonal esigns," Wireless Pers. Commun., vol. 25, no., pp. -26, Apr. 23. [] H. Jafarkhani, Space-Time Coing Theory an Practice. Cambrige, U.K.: Cambrige Univ. Press, 25. [] S. Alamouti, A simple transmit iversity technique for wireless communications," IEEE J. Select. Areas Commun., vol. 6, no. 8, pp , Oct [2] S.-H. Kim, I.-S. Kang, an J.-S. No, Exact bit error probability of orthogonal space-time block coes with quarature amplitue moulation," in Proc. ISIT 3, p. 63, 23. [3] J. W. Craig, A new, simple an exact result for calculating the probability of error for two-imensional signal constellations," in Proc. IEEE MILCOM 9, pp , 99. [4] S. ten Brink, G. Kramer, an A. Ashikhmin, Design of low-ensity parity-check coes for moulation an etection," IEEE Trans. Commun., vol. 52, no. 4, pp , Apr. 24. Authorize license use ite to: IEEE Xplore. Downloae on March 8, 29 at 2:43 from IEEE Xplore. Restrictions apply.