Exact pairwise error probability of space~time codes - Information ...

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 2, FEBRUARY 2002

combinatorial search. Novel programming paradigms for DNA computation would be desirable. The quest for a “killer application” goes on [11]. This is a relevant application that can be performed much faster on a DNA computer than on a conventional computer. ACKNOWLEDGMENT The author wishes to thank the referees for helpful comments.

Exact Pairwise Error Probability of Space–Time Codes Giorgio Taricco, Member, IEEE, and Ezio Biglieri, Fellow, IEEE

Abstract—We describe a simple technique for the numerical calculation, within any desired degree of accuracy, of the pairwise error probability (PEP) of space–time codes. for any nonThis method applies also to the calculation of negative random variable whose moment-generating function is known.

[exp(

REFERENCES [1] L. Adleman, “Molecular computation of solutions to combinatorial problems,” Science, vol. 266, pp. 1021–1024, Nov. 1994. [2] L. Adleman, P. W. K. Rothemund, S. Roweis, and E. Winfree, “On applying molecular computation to the Data Encryption Standard,” in DNA Based Computers, Proc. of the 2nd Annu. Meet., E. B. Baum et al., Eds., Princeton, NJ, 1996, pp. 28–48. [3] E. B. Baum and D. Boneh, “Running dynamic programming algorithms on a DNA computer,” in DNA Based Computers, Proc. 2nd Annu. Meet., E. B. Baum et al., Eds., Princeton, NJ, 1996, pp. 122–127. [4] R. J. Britten and D. E. Kohnle, “Repeated sequences in DNA,” Science, vol. 161, pp. 529–540, 1968. [5] D. Boneh, C. Dunworth, and R. Lipton, “Breaking DES using a molecular computer,” in DNA Based Computers, Proc. DIMACS Workshop, E. B. Baum and R. Lipton, Eds, 1996, pp. 37–65. [6] K. Drlica, Understanding DNA and Gene Cloning. New York: Wiley, 1992. [7] R. P. Feynman, Miniaturization, D. H. Gilbert, Ed. New York: Reinhold, 1961, pp. 282–296. [8] J. Hartmanis, “On the weight of computations,” Bull. Europ. Assoc. Theor. Comput. Sci., vol. 55, pp. 136–138, 1995. [9] T. Head, “Formal language theory and DNA: An analysis of the generative capacity of specific recombinant behaviors,” Bull. Math. Biol., vol. 49, pp. 737–759, 1987. [10] N. Jonoska and S. Karl, “A molecular computation of the road color problem,” in DNA Based Computers, Proc. 2nd Annu. Meet., E. B. Baum et al., Eds., Princeton, NJ, 1996, pp. 148–158. [11] L. Kari, “DNA computing: Arrival of biological mathematics,” Math. Intelligencer, vol. 19, no. 2, pp. 9–22, 1997. [12] L. Kari, G. Paun, G. Rozenberg, A. Salomaa, and S. Yu, “DNA computing, sticker systems, and universality,” Acta Inform., vol. 35, no. 5, pp. 401–420, 1998. [13] J. Kendrew et al., Ed., The Encyclopedia of Molecular Biology. Oxford, U.K.: Blackwell, 1994. [14] T. Lai and K.-H. Zimmermann, “A software platform for the sticker model,” Dept. Comput. Eng., TU Hamburg, Harburg, Germany, Tech. Rep. 2001.2, 2001. [15] R. J. Lipton, “DNA solution of hard computational problems,” Science, vol. 268, pp. 542–545, Apr. 1995. , Using DNA to solve NP complete problems. Preprint. [Online]. [16] Available: www.cs.princeton.edu/r˜jl [17] J. MacWilliams and N. Sloane, The Theory of Error Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1988. [18] G. Praun, G. Rozenberg, and A. Salomaa, DNA Computing. Berlin, Germany: Springe-Verlag, 1998. [19] S. Roweis, E. Winfree, R. Burgoyne, N. Chelyapov, M. Goodman, P. Rothemund, and L. Adleman, “A sticker based architecture for DNA computation,” in DNA Based Computers, Proc. 2nd Annu. Meet., E. B. Baum et al., Eds., Princeton, NJ, 1996, pp. 1–27.

[ ( )]

)]

8( )=

Index Terms—Multiple antennas, pairwise error probability (PEP), space–time codes.

I. INTRODUCTION The application of multiple-antenna architectures for high-rate transmission requires the development of effective tools for the calculation of error probabilities of space–time coding schemes. In this correspondence, we derive a simple technique for the numerical calculation, within any desired degree of accuracy, of the pairwise error probability (PEP). It is well known (see, e.g., [2]) that the PEP is the basic building block for the derivation of union bounds to the error probability of a coding scheme. The exact calculation of the PEP over fading channels is comprehensively addressed in [7]. More specifically, its calculation through the method of residues was recently advocated in [10], which generalizes [4] for single-input single-output fading channels. Now, this method is easily applicable when a certain complex function (to be described later) exhibits simple poles, but becomes long and intricate when multiple poles or essential singularities are present. A different approach to the exact calculation of the PEP, and one which guarantees arbitrarily high accuracy irrespective of the structure of the above function, is presented here. The use of Gauss–Chebyshev quadrature to calculate the PEP has been proposed in [3] and, subsequently, in [9] in a more specific context. This approach is also reported in [7, Appendix 9B.2]. The results presented in this correspondence share with [3] the use of Gauss–Chebyshev quadrature and the moment-generating function (MGF) approach. However,phere we provide a new general method for the calculation of [Q(  )] for a nonnegative random variable  which is based on its MGF 8 (s) = [exp(0s )]. A related method has been proposed in [1]. II. GENERAL RESULT Here we first derive a general result on the PEP. Although our derivation is done in the framework of space–time codes, we believe it to be widely applicable to other areas of error performance analysis. Consider the computation of the expectation

P

=1

Q



(1)

where Q(x) = ( > x), with  real Gaussian random variable with mean zero and unit variance, and  is a nonnegative random variable Manuscript received July 9, 2001; revised September 12, 2001. The work of E. Biglieri was performed under a consulting arrangement with Lucent Technologies. The authors are with the Dipartimento di Elettronica, Politecnico di Torino, 10129 Torino, Italy (e-mail: [email protected]; [email protected]). Communicated by G. Caire, Associate Editor for Communications. Publisher Item Identifier S 0018-9448(02)00308-5. 0018–9448/02$17.00 © 2002 IEEE

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 2, FEBRUARY 2002

(independent of  ). We advocate a simple method to obtain the value of P based on numerical integration. Assume that the MGF of 

8 (s) =1 [exp(0s)]

(2)

is known. In this case, we have [3]

Q where



1 = (1 < 0) = 2j

+ c0j 1

c j1

81 (s) ds s

1 =1  0  2 . It is straightforward to obtain 81 (s) = [exp(0s1)] =8 (s)8 (0s) =8 (s)(1 0 2s)01=2 :

(3)

2) Block fading (BF): we assume that the transmitted symbols in a codeword are affected by the same fading realization. We show that, in both cases, the PEP can be expressed as in (1) by suitably defining the random variable  . We assume t transmit and r receive antennas, and a code with block length N . A. Independent Fading Channel The discrete-time low-pass equivalent channel equation can be written as

yi

(4)

Hence,

1 = 2j

511

+

c j1

= H i xi + z i ;

i

= 1; . . . ; N

(9)

where H i 2 r2t is the ith channel gain matrix, x i 2 t21 is the ith transmitted symbol vector (each entry transmitted from a different antenna), y i 2 r21 is the ith received sample vector (each entry received from a different antenna), and z i 2 r21 is the ith received noise sample vector (each entry received from a different antenna). We assume that the channel gain matrices H i are element-wise independent and independent of each other with H i jk  Nc ; , i.e., each element is circularly Gaussian distributed with mean zero and variance j H i jk j2 . Also, the noise samples are independent with z i  Nc ; N0 . It is straightforward to obtain the PEP as follows:

8 (s) (1 0 2s)01=2 ds: (5) (0 1) [ ] 2s c0j 1 [[ ] ] =1 Here we assume that c is in the region of convergence (ROC) of 81 (s). (0 ) This is given by the intersection of the ROC of 8 (s) and the ROC of [ ] 0 1 =2 8 (0s) = (1 0 2s) . Thus, the ROC of 81 (s) includes the complex region defined by f0 < Re(s) < 1=2g. Therefore, we can safely ^ P X !X assume 0 < c < 1=2 and a good choice is c = 1=4 corresponding to N an integration line in the middle of the minimal ROC of 81 (s). = ky i 0 H ix^i k2 0 ky i 0 H ixi k2 < 0 Then, expanding the real and imaginary parts in (5), we have the Q



=1

i

following result:

Q



1 = 2j = 21 = 21

+

c j1 c0j 1

= 81 (s) ds

1 81 (c + j!) d! 01 c + j! 1 cRef81 (c + j!)g + !Imf81 (c + j!)g d!: c2 + ! 2 01 p The change of variables ! = c 1 0 x2 =x yields

1 Re 01 p

1 2

=

s

= 21

p

dx

1 0 x2 :

= tan((

1 2)

) = 64

where k k 0 = = and E numerical results we assumed  .

0

N

2 2N0 i=1 kH i (xi 0 x^i ) k

:

(10)

Here we assume that the channel gain matrices H i coincide and are equal to H : under this assumption, the channel equation can be written compactly as follows:

(7)

[H ]ij  Nc (0; 1) ( [j[H ]ij j2 ] = 1) and i.i.d. [Z ]ij  Nc (0; N0 ). We obtain, after straightforward calcuX

(8)

! 0 as  ! 1. In our

(11)

t2N where H 2 r2t , X , Y 2 r2N , and r2N Z2 . We assume independent and identically distributed (i.i.d.) entries

lations

fRe[81 (c(1+ jk ))]+ k Im[81(c(1+ jk ))]g + E

=1

1

Q