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Abstract—In this letter, we present a unified mathematical expression for the decoding algorithm of space-time block codes. (STBC). Based on the unified ...
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 3, MAY 2005

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Transactions Letters________________________________________________________________ On Decoding Algorithm and Performance of Space-Time Block Codes Changjiang Xu and Kyung Sup Kwak, Member, IEEE

Abstract—In this letter, we present a unified mathematical expression for the decoding algorithm of space-time block codes (STBC). Based on the unified expression, we make a comparison of the STBC transmit diversity and the maximal-ratio combining MRC receive diversity, and analyze the symbol error probability for the STBC transmissions. The effects of the channel correlation and the number of transmit and receive antennas on the performance of the STBC are discussed. Index Terms—Multiple-input multiple-output (MIMO), spacetime block codes, transmit diversity.

I. INTRODUCTION

T

HE SPATIAL diversity using multiple transmit and/or receive antennas can mitigate channel fading without necessarily sacrificing bandwidth resources and increases the capacity significantly over single-antenna systems [1], [2]. The space-time coding is an effective transmit diversity technique to combat fading in wireless communications. It was originally presented in the form of convolutional codes, called space-time trellis codes (STTC) [3]. STTC requires computationally complex trellis search algorithms at the receiver. To reduce decoding complexity, space-time block codes (STBC) with 2 transmit antennas were first introduced in [4] and later generalized to an arbitrary number of transmit antennas in [5]. An attractive property of STBC is that maximum-likelihood (ML) decoding can be performed using only linear processing at little loss of performance. However, the ML decoding algorithm of STBC described in [5] is inconvenient for calculation. In [6], the decoding algorithms were enumerated one by one for some distinct STBCs. In this letter, we present a unified mathematic expression for the decoding algorithm of a general STBC and then analyze the performance of STBC. We first represent the STBC transmission as an orthogonal vector-channel model. Based on the orthogonal model, the unified expression for the decoding algorithms of STBCs is directly derived. It can be seen from this unified expression that the STBC multiple-input multiple-output

Manuscript received December 22. 2001; revised May 14, 2002, March 1, 2003, and November 11, 2003; accepted March 8, 2004. The editor coordinating the review of this paper and approving it for publication is J. V. Krogmeier. This work was supported in part by the Jiangsu National Science Foundation of China under Grant 01KJB510002 and in part by the University IT Research Center Project (Inha UWB-ITRC), Korea. C. Xu is with the Department of Telecommunication Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China (e-mail: [email protected]). K. S. Kwak is with the UWB Wireless Communications Research Center, School of Information and Communication Engineering, Inha University, Inchon 402-752, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2005.847098

(MIMO) channel is equivalent to a single-input single-output (SISO) Gaussian channel. Thus, the performance analysis can be done as in Gaussian channel. The rest of this letter is organized as follows. Section II introduces the encoding scheme of STBC and the STBC transmission model. In Section III, the ML decoding algorithm for a general STBC is derived. The further discussions on the performance of the STBCs are given in Section IV. Section V contains the conclusions. Notations: In this letter, boldfaced lower-case and upper-case letters denote column vectors and matrices, respectively. The superscripts , , and stand for transpose, complex conjugate, and conjugate transpose, respectively. denotes identity matrix of size . Boldfaced zero “ ” denotes zero matrix or vector with corresponding dimension. II. ENCODING SCHEME AND TRANSMISSION MODEL A. Encoding Scheme transmit antennas and generalized linear

A space-time block encoder with rate is defined by a orthogonal design matrix

.. .

.. .

..

.

.. .

(1)

the entries of which are linear combinations of the information symbols and their complex conjugates such that is a diagonal matrix with the th diagonal element of the form (2) where all the coefficients are strictly positive numbers, each symbol belongs to a complex signal constellation1 , is the number of the information symbols and is the number of time slots. The coding matrix can be represented as (3) where

and

are

matrices. Let be diagonal matrix. We have the following proposition (The detailed proof is given in Appendix). 1In this paper, we deal with complex-valued signals only. The extension to the real-valued signals is straightforward.

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Proposition 1: The above-defined matrix is a generalized linear complex orthogonal design if and only if

where and is the Kronecker delta. Note if , then above-defined matrix is called a linear complex orthogonal design (COD). Most of the proposed STBCs are based on the COD.

III. DECODING ALGORITHM In this Section, we consider the decoding algorithm of STBC. The maximum-likelihood (ML) decoding amounts to minimizing the decision metric

(9) Using the orthogonal property (8), it can be calculated that

B. Transmission Model Consider a wireless communication system equipped with transmit antennas and receive antennas. At the transmitter, a block of information symbols is mapped to the code matrix by space-time block encoder. At each time slot , the code symbols , are transmitted simultaneously from the transmit antennas. Assume the channels between pairs of transmit and receive antennas are quasi-static flat faded. Then the channel from th transmit antenna to th receive antenna can be modeled as complex constant, denoted as , within each block. At the th receive antenna, the received signal at th time slot is given by

(10) where

(4) (11)

denotes the average energy of the code symbol and where are independent samples of a zero-mean complex Gaussian random variable with variance . The above equation can be rewritten in a matrix form as (5) , and . Define the vectors . By use of the relationship (3), the received signal model is transferred to

(12) Then omitting the terms that are independent of the symbols, the ML decision metric (10) is equivalent to

where

(13) Thus the ML decision rule is reduced to minimizing symbols separately, i.e., for .

(6) where

and

(14) where and are defined by (11) and (12), respectively. Note that the decision variables (11) can be rewritten in the matrix form as

(7) It can be readily verified from Proposition 1 that , are orthogonal, i.e., for , it holds that

,

, ,

(8) where

and

is the Kronecker delta.

(15) where

, and . Thus the decision variables (11) or (15) can be calculated by use of the pair of matrices and , which are determined by the coding matrix , i.e., matrices and . For some distinct STBCs, the expressions of the decision variables were derived one by one in [6]. Note that in [6], the signs of the first and fifth terms in the decision variable

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for decoding of the STBC positive instead of negative.

are incorrect and should be

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C. Probability of Symbol Error Uncoded STBC Systems: Using the equivalent SISO Gaussian model (16) and the similar methods as in [7], we can derive the probability of symbol error for a general STBC as follows

IV. FURTHER DISCUSSIONS A. Equivalent SISO Gaussian Channel From orthogonal property (8), the decision variables (11) can be expressed as

(21)

(16) where is defined as (12) and is a zero-mean complex Gaussian random variable with variance . Thus the STBC MIMO channel is transferred to an equivalent SISO Gaussian channel. The simple equivalent SISO channel model is very useful for analytical evaluation of STBC. Next we use this model to analyze the performance of STBC.

and are the average number of nearest neighbors where and minimum distance of separation of the underlying constellation, respectively. Moreover the average error probability and outage property can be calculated as (22) (23)

B. Comparison With MRC From the equivalent SISO model (16), we have the output signal-to-noise ratio of STBC . In order to constrain the total transmission power, we have , where is the average energy of the information symbol and is the rate of STBC. Thus (17) is the signal-to-noise ratio per symbol. where Consider the wireless systems where is sent from one transmit antenna and received using receive antennas. Denote , as the path gains from the transmit antenna to the receive antennas. The received signal at the th antenna is

where and stands for the probability density function (pdf) and cumulative distribution function (cdf) of , respectively. Assume the channels are Rayleight fading, i.e., are zero-mean Gaussian random variables with the covariance matrix , where and . Moreover, we assume is nonsingular. Let be the Kronecker product and with . Denote . We have , and

(24)

(18) where is a zero-mean independent complex Gaussian noise with variance . The output signal-to-noise ratio of the maximal-ratio combining (MRC) is (19) . where For brevity, we consider the COD-based STBCs, i.e., or . Assume that the channel gains between transmit and receive antennas are independent and identically distributed (i.i.d) random variables. Thus and are identically distributed random variables. Then it holds that in the mean sense

and are statiswhere tically independent and distributed according to a chi-squared contribution with two degrees of freedom. The characteristic function of is (25) where of is

. And then the characteristic function

(26) (20) It is seen from (20) that the STBC transmit diversity has dB loss in performance compared to the same level MRC. Note if each transmit antenna in the STBC transmit diversity radiates the same energy as the single transmit antenna in the MRC receive diversity, i.e., , then the STBC transmit diversity achieves the same performance as the MRC.

Thus the PDF of can be calculated from the characteristic function . For the sake of simplicity, we consider the case of . In this case, the PDF of is [8] (27)

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where and error probability is

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, and the average

where denotes the number of signal sequences with distance . The pairwise error probability can be computed as in the uncoded systems [9]. The detail is not further discussed herein.

V. CONCLUSION

(28) . When where ability can be approximated as [8]

, the average error prob-

(29) Thus the error probability decreases inversely with the th power of . Notice that and . Particularly, the parameter represents the correlation between the channels. Hence it can be known from (29) which parameters affect the performance of STBC. Coded STBC Systems: In the wireless systems, the forward error correction (FEC) coding is required. From the equivalent Gaussian SISO model (16), the same FEC codes as in single antenna systems should be used in the STBC MIMO channel. For the concatenated systems where a convolutional code is followed by a STBC, the first-event error probability is upperbounded by [8] (30)

In this letter, the STBC transmissions have been represented as an orthogonal vector-channel model, and a unified mathematical expression for the decoding algorithms of STBCs has been given. Using an equivalent SISO channel model, we have analyzed the performance of STBCs. It has been shown that the STBC transmit diversity has dB loss in performance compared to the same level MRC. By analyzing the probability of symbol error, we have illustrated how the channel correlation and the number of antennas affect on the performance of STBCs.

APPENDIX PROOF OF PROPOSITION 1 Let

,

. Then and

(31), shown at the bottom of the page. Assume the matrix is a generalized linear complex orthogonal design. It holds

(32)

(31)

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(33)

(35)

(36)

Let , in (31) and (32). Then (33), shown at the top of the page. Comparing two sides of above identity yields

(34) for . From (31), (32), and (33), we further have (35), shown at the top of the next page.Comparing two sides of the above identity yields (36), shown at the top of the next page, for . Thus the necessary conditions can be derived directly from (34) and (36). The sufficiency can be readily verified. The detailed process is omitted here.

ACKNOWLEDGMENT The authors wish to thank the anonymous reviewers for the comments and suggestions, which are helpful to improve this paper.

REFERENCES [1] J. H. Winters, J. Salz, and R. D. Gitlin, “The impact of antenna diversity on the capacity of wireless communication systems,” IEEE Trans. Commun., vol. 42, no. 4, pp. 1740–1751, Apr. 1994. [2] G. J. Foschini and M. J. Gans, “On limits of wireless communication in a fading environment when using multiple antennas,” Wireless Personal Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998. [3] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-Time codes for high data rates wireless communications: Performance criterion and code construction,” IEEE Trans. Inf. Theory, vol. 44, no. 3, pp. 744–765, Mar. 1998. [4] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas in Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [5] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-Time block codes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, no. 7, pp. 1456–1467, Jul. 1999. , “Space-Time block coding for wireless communications: Perfor[6] mance results,” IEEE J. Sel. Areas Commun., vol. 17, no. 3, pp. 450–460, Mar. 1999. [7] R. U. Nabar, H. Bolcskei, and A. J. Paulraj, “Outage properties of spacetime block codes in correlated rayleigh or ricean fading environments,” in Proc. IEEE Int. Conf. Acoustics, Speech and Signal Processing, vol. 3, 2002, pp. 2381–2384. [8] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2001. [9] G. Bauch and J. Hagenauer, “Analytical evaluation of space-time transmit diversity with FEC-coding,” in Proc. IEEE Global Telecommunications Conf., vol. 1, 2001, pp. 435–439.