Performance and Design of Space–Time Coding in Fading Channels

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 12, DECEMBER 2003

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Performance and Design of Space–Time Coding in Fading Channels Jinhong Yuan, Member, IEEE, Zhuo Chen, Student Member, IEEE, Branka Vucetic, Fellow, IEEE, and Welly Firmanto, Student Member, IEEE

Abstract—The pairwise-error probability upper bounds of space–time codes (STCs) in independent Rician fading channels are derived in this letter. Based on the performance analysis, novel code design criteria for slow and fast Rayleigh fading channels are developed. It is found that in fading channels, STC design criterion depends on the value of the possible diversity gain of the system. In slow fading channels, when the diversity gain is smaller than four, the code error performance is dominated by the minimum rank and the minimum determinant of the codeword distance matrix. However, when the diversity gain is larger than or equal to four, the performance is dominated by the minimum squared Euclidean distance. Based on the proposed design criteria, new codes are designed and evaluated by simulation. Index Terms—Diversity, fading channels, space–time trellis code (STTC).

parameter for the code performance. New STTCs are designed based on the proposed design criteria and evaluated by simulation. II. SYSTEM MODEL We consider a baseband mobile communication system with transmit and receive antennas. The information data are parallel encoded by a space–time encoder, which generates parallel outputs data sequences. At each time instant , the are simultaneously transmitted by antennas, , is transmitted by antenna . whereby symbol , At time instant , the received signal at antenna , , is given by (1)

I. INTRODUCTION

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PACE–TIME coding is a powerful technique to improve the error performance of wireless communications systems by using multiple transmit antennas. In the first performance investigation of the space–time trellis codes (STTCs) [1], analytical bounds and design criteria were proposed for slow and fast fading channels. It was pointed out that in slow fading channels, the critical parameters are the rank and determinant of the codeword distance matrix, while in fast fading channels, the important parameters are the symbol-wise Hamming distance and the product distance. Based on these criteria, new 4and 8-phase-shift keying (PSK) STTCs have been reported in [2]–[7] for slow fading channels, and in [8] for fast fading channels. In this letter, we develop a more detailed performance evaluation for space–time codes (STCs) by deriving analytical bounds. It is observed that in fading channels, the code construction criterion depends on the value of the possible diversity gain of the system. In slow fading channels, when the diversity gain is small, the rank and determinant criteria are valid for code design. On the other hand, when the diversity gain is reasonably large, the trace of the codeword distance matrix, or, equivalently, the minimum squared Euclidean distance, will be the dominant Paper approved by A. F. Naguib, the Editor for Wireless Communication of the IEEE Communications Society. Manuscript received February 10, 2001; revised May 13, 2003. This paper was presented in part at the IEEE International Symposium on Information Theory, Washington, DC, June 2001 and in part at the IEEE International Conference on Communications, Helsinki, Finland, June 2001. J. Yuan is with the School of Electrical Engineering and Telecommunications, University of New South Wales, NSW 2052, Australia (e-mail: [email protected]). Z. Chen, B. Vucetic, and W. Firmanto are with the School of Electrical and Information Engineering, University of Sydney, Sydney, NSW 2006, Australia (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2003.820741

is the energy per symbol at each transmit antenna and where is the additive white Gaussian noise (AWGN) at receive antenna at time , which has a zero mean and power spectral . The coefficient is the fading coefficient bedensity tween transmit antenna and receive antenna at time . are indepenIt is assumed that the fading coefficients and dent complex Gaussian random variables with mean variance 1/2 per dimension. In this letter, we consider both slow and fast fading. For slow fading, it is assumed that the fading coefficients are constant during a frame and vary from one frame to another. In a fast fading channel, the fading coefficients are constant within each symbol period and vary from one symbol to another. III. PERFORMANCE ANALYSIS Assume that a codeword , given by was transmitted, where is the frame length. A maximum-likelihood (ML) receiver might decide erroneously in favor of another codeword Assuming that ideal channel state information (CSI) is available at the receiver, the conditional pairwise-error probability (PEP) is given by [1]

(2) where

is the

0090-6778/03$17.00 © 2003 IEEE

channel matrix and .

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 12, DECEMBER 2003

A. Error Probability in Slow Fading Channels Following the derivation in [1], we define a codeword difference matrix as .. .

..

.

.. .

where is the probability density function (pdf) of the Gaussian random variable . Using

(3) (9)

and a codeword distance matrix as , where denotes the Hermitian operation. Since is nonnegative definite Hermitian, the eigenvectors , , form a complete orthonormal of -dimensional vector space, and the eigenvalues basis of an , , of are nonnegative real numbers. and . Let The conditional PEP can be rewritten as [1]

the upper bound in (8) can be further expressed as

(10) Let us now consider the special case for Rayleigh fading, in and thus, . The PEP can be written as which

(4) Inequality (4) is an upper bound of the conditional PEP ex, which is contingent upon . pressed as a function of are independent complex Gaussian It is obvious that random variables with variance 1/2 per dimension and mean , where denotes the expectation. Let , then follows a Rician distribution with [1]. a Rician factor In order to get an upper bound on the unconditional PEP, we need to average (4) with respect to independent Rician distribu. Let denote the rank of matrix . In this tion of analysis, we will distinguish two cases, depending on the value . of : Since 1) PEP Upper Bound for Large Values of follows a Rician distribution, has a noncentral chi-squared distribution with two degrees of freedom (DOFs) . The mean and noncentrality parameter value and the variance of the noncentral chi-square-distributed are given by random variables

(11) : When the 2) PEP Upper Bound for Small Values of is small, the Gaussian number of independent subchannels assumption is no longer valid and the PEP can be obtained by term by term. The PEP is then given by averaging (4) over [1]

(12) In the case of Rayleigh fading, the PEP at high signal-to-noise ratios (SNRs) can be simplified as [1]

(5)

(13)

(6) respectively. On the right-hand side (RHS) of inequality (4), there are independent chi-square-distributed random variables. For , which corresponds to a large number of a large value of independent subchannels, according to the general central limit theorem [9] for independent random variables with unequal distributions, the expression (7) with mean approaches a Gaussian random variable and variance . The PEP can then be upper bounded by (8)

are the nonzero eigenvalues of mawhere . Note that the results (12) and (13) are the trix is called the diversity gain and same as in [1], where is called the coding gain. B. Error Probability in Fast Fading Channels The analysis for slow fading channels in the previous section can be directly applied to fast fading channels. At each time , we define a space–time symbol difference vector as (14) matrix . and an , matrix has only one nonzero eigenIf eigenvalues are zero. Let be the value, and the other . The eigenvector of nonzero eigenvalue of corresponding to the nonzero eigenvalue is denoted by .

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 12, DECEMBER 2003

Let

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PEP can be obtained by averaging (15) over term. The PEP upper bound becomes [1]

and . The conditional PEP can be given by

(15) denotes the set of time instances , where , and are Rician distributed with such that . a Rician factor The conditional PEP can be averaged over independent . If we define as the Rician-distributed variables number of space–time symbols, in which the two codewords and differ, then at the RHS of inequality (15), there are independent random variables. As before, we will distinguish . The two cases in the analysis, depending on the value of is also called the space–time symbol-wise Hamming term distance between two codewords. : Provided that the 1) PEP Upper Bound for Large is large, according to the central limit theorem, value of the expression (16) can be approximated by a Gaussian random variable with mean (17)

term by

(23) are Rayleigh distributed, the For a special case where upper bound of the PEP at high SNRs becomes [1]

(24) is the product of the squared Euclidean distances bewhere tween the two space–time symbol sequences and is given by (25) The PEP bound (24) shows that it is possible to achieve a diverand a coding gain of in fast Rayleigh sity gain of fading channels [1]. IV. STC DESIGN CRITERIA A. Design Criteria for Slow Rayleigh Fading Channels

and variance (18) By averaging (15) over the Gaussian random variable, the PEP can be upper bounded by

(19) For Rayleigh fading channels, the PEP upper bound can be approximated by

(20) where (21) is the accumulated squared Euclidean distance between the two space–time symbol sequences and (22) : When the value of 2) PEP Upper Bound for Small is small, the central limit theorem does not apply and the

As the error performance upper bounds (11) and (13) indicate, the design criteria for slow Rayleigh fading channels will . The maximum possible value of depend on the value of is . For small values of , corresponding to a small number of independent subchannels, the error probability over is dominated by the minimum rank of matrix all possible codeword pairs. The product of the minimum rank , is called the minand the number of receive antennas, imum diversity. In addition, in order to minimize the error probability, the minimum product of the nonzero eigenvalues of maalong the pairs of codewords with the minimum trix is rank should be maximized. Therefore, if the value of small, the STC design criteria for slow Rayleigh fading channels can be summarized as follows [1]. Design Criteria Set 1: over all 1) Maximize the minimum rank of matrix pairs of distinct codewords. of matrix 2) Maximize the minimum product along the pairs of distinct codewords with the minimum rank. This set of design criteria is referred to as the rank and determinant criteria. corresponding to a large number of For large values of independent subchannels, the PEP is upper bounded by (11). In order to get an insight into the code design for systems of practical interest, we assume that the STC operates at a reasonably high SNR, which corresponds to (26)

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 12, DECEMBER 2003

Note that the value of is usually small. For example, its value for the four-state 4-PSK STTCs in [1], [2], and [10] is 0.5, 0.19, and 0.11, respectively. The bound (11) can be further approximated by (27) From (27), it can be seen that in order to minimize the error probability, the minimum of the sum of all eigenvalues of matrix among all the pairs of distinct codewords should be maximized. For a square matrix, the sum of all the eigenvalues is equal to the sum of all the elements on the main diagonal, or . It can be expressed as the trace of matrix (28) where

are the elements on the main diagonal of matrix . The trace of matrix can be expressed as (29)

is Equation (29) indicates that the trace of matrix equivalent to the squared Euclidean distance between the pair of codewords and . In other words, the PEP is minimized if the Euclidean distance is maximized. This design criterion is referred to as the trace criterion. It should be pointed out that (27) is valid for a large number of independent subchannels under the condition that the minimum is high. In this case, the STC design criteria for value of slow fading channels can be summarized as follows. Design Criteria Set 2: 1) Make sure that the minimum rank of matrix over all pairs of distinct codewords is large enough, such . that of matrix 2) Maximize the minimum trace among all pairs of distinct codewords. It is important to note that this design is consistent with the trellis-code design in fading channels with a large number of diversity branches. A large number of diversity branches reduces the effect of fading and consequently, the channel approaches an AWGN model [11]. Therefore, the trellis-code design criteria for AWGN channels apply to fading channels with a large number of diversity. In a similar way, in STC design, when the is large, the channel number of independent subchannels converges to an AWGN channel. Thus, the code design is the same as that for AWGN channels. between the two design criteria The boundary value of sets was chosen to be four. This boundary is determined by in (7) to satisfy the required number of random variables the central limit theorem. In general, for random variables with smooth pdfs, the central limit theorem can be applied if the number of random variables in the sum is larger than four [9]. In the application of the central limit theorem in (7), the choice of four as the boundary has been further justified by the code design and performance simulation, as it was found that as long as , the best codes based on the trace criterion outperform the best codes based on the rank and determinant criteria [10].

B. Design Criteria for Fast Rayleigh Fading Channels As the error performance upper bounds (20) and (24) indicate, the code design criteria for fast Rayleigh fading channels will depend on the value of . For small values of , the error probability is dominated by the minimum space–time over all distinct codeword symbol-wise Hamming distance pairs. In addition, in order to minimize the error probability, the minimum product distance, , along the path of the pairs of codewords with the minimum symbol-wise Hamming distance should be maximized. Therefore, if the value of is small, the STC design criteria for fast fading channels can be summarized as follows [1]. Design Criteria Set 3: 1) Maximize the minimum space–time symbol-wise Hambetween all pairs of distinct codewords. ming distance 2) Maximize the minimum product distance, , along the path with the minimum symbol-wise Hamming distance . , the PEP is upper bounded by (20). For large values of As before, we assume that the STC works at a reasonably high SNR, which corresponds to (30) and are given by (21) and (22), respectively. where The bound (20) can be further approximated by (31) (32) From (32), it is clear that the PEP is dominated by the squared . To minimize the error probability in Euclidean distance fading channels, the codes should satisfy the following. Design Criteria Set 4: 1) Make sure that the product of the minimum space–time symbol-wise Hamming distance and the number of re, is large . ceive antennas, 2) Maximize the minimum Euclidean distance between all pairs of distinct codewords. It is interesting to note that this set of design criteria is the same as the trace criterion for STC in slow fading channels if is large. It is also consistent with the design the value of criteria for trellis-coded modulation in fading channels if the effective code length or the symbol-wise Hamming distance is large [12]. Based on the previous discussion, we can conclude that code design in fading channels depends on the possible diversity gain of the STC system. For codes on slow fading channels, the total , and the diversity is the product of the receive diversity, transmit diversity provided by the coding scheme, . On the other hand, for codes on fast fading channels, the total diver, and the time disity is the product of the receive diversity, . If the total diverversity achieved by the coding scheme, sity is small, in the code design for slow fading channels, one should attempt to maximize the diversity and the coding gain

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 12, DECEMBER 2003

Fig. 1.

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Trellis structures for four-state 4-PSK STTCs with two antennas.

TABLE I NEW 8-PSK CODES BASED ON TRACE CRITERION

Fig. 2. FER performance of the four-state STTC 4-PSK with two transmit antennas.

by choosing a code with the largest minimum rank and the determinant; while for fast fading channels, one should attempt to choose a code with the largest minimum symbol-wise Hamming distance and the product distance. However, when the total diversity is getting larger, increasing the diversity gain cannot achieve a substantial performance improvement. Since a large diversity drives the fading channel toward an AWGN channel [12], the error probability is dominated by the minimum Euclidean distance. Thus, the code design criterion for AWGN channels, which is maximizing the minimum Euclidean distance, is valid for both slow and fast fading channels, provided that the total diversity is large. C. Example To illustrate the design criteria, we compare three four-state 4-PSK STTCs for two transmit antennas. The trellis diagrams of the three STTCs are shown in Fig. 1, together with the corresponding minimum rank, determinant, and trace. The frameerror rate (FER) performance in slow Rayleigh fading channels is obtained by simulation and shown in Fig. 2. The frame length is .

From Fig. 2, it can be observed that Codes A and B outperform Code C for a single receive antenna. When the number of is small, the minimum rank of independent subchannels the code dominates the code performance. Since both Codes A and B are of full rank and Code C is not, Codes A and B achieve a better performance relative to Code C. However, for four receieve antennas, Code C outperforms , although Code C does Code A by 1.3 dB at a FER of not have a full rank. This is because when the diversity gain is large, the performance is determined by the code minimum trace. Code C has a much larger minimum trace value. In addition, it is observed that Code B is about 0.8 dB better than , although they have the same minCode C at a FER of imum trace. This is due to the fact that Code B is of full rank and Code C is not. Therefore, Code B can achieve a larger diversity. From this example, it is obvious that in the code design for slow fading channels with a large number of independent subchannels, we should first choose the code with a large minimum trace. Among the codes with the same minimum trace, we choose a code with a largest minimum rank. This is consistent with the previous performance analysis and code design criteria. D. Code Design Tables I and II [4] present some 8-PSK STTCs for two transmit antennas, based on the trace criterion and the rank

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 12, DECEMBER 2003

TABLE II NEW 8-PSK CODES BASED ON RANK AND DETERMINANT CRITERIA

More STTCs, based on the trace criterion for slow fading channels, are designed by computer search in [3]–[5] and [13]–[15]. V. CONCLUSION We presented new performance bounds and new design criteria for STTCs in both slow and fast fading channels. In order to obtain good performance in slow fading channels, codes should be designed to maximize the minimum trace of the codeword distance matrices or maximize the minimum squared Euclidean distance of the code for systems with diversity gain larger than or equal to four. Several codes based on the new design criteria are also presented. REFERENCES

Fig. 3. Performance comparison of the eight-state 8-PSK STTCs in slow fading channels.

and determinant criteria, respectively. In the table, denotes is the generator matrix of STTC the number of states, denotes the minimum rank, det denotes defined in [2], denotes the minimum the minimum determinant, and squared Euclidean distance. Compared with the corresponding Tarokh–Seshadri–Calderbank (TSC) codes in [1], the STTCs in but smaller det, except for the eight-state Table I have larger codes, for which both have the same det value. Fig. 3 shows the performance of the eight-state 8-PSK codes. The new code, presented in Table I, is superior to the TSC code by 0.6, 1.2, and 1.7 dB for two, three, and four receive antennas, respectively. When one receive antenna is employed, the new and the TSC codes have approximately the same performance. This clearly substantiates the design criteria proposed.

[1] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space–time codes for high-data-rate wireless communication: Performance criterion and code construction,” IEEE Trans. Inform. Theory, vol. 44, pp. 744–765, Mar. 1998. [2] S. Baro, G. Bauch, and A. Hansmann, “Improved codes for space–time trellis-coded modulation,” IEEE Commun. Lett., vol. 4, pp. 20–22, Jan. 2000. [3] Z. Chen, B. S. Vucetic, J. Yuan, and K. L. Lo, “Space–time trellis codes for 4-PSK with three and four transmit antennas in quasi-static flat fading channels,” IEEE Commun. Lett., vol. 6, pp. 67–69, Feb. 2002. [4] Z. Chen, B. Vucetic, J. Yuan, and K. L. Lo, “Space–time trellis codes for 8-PSK with two, three and four transmit antennas in quasi-static flat fading channels,” Electron. Lett., vol. 38, pp. 462–464, May 2002. , “Space–time trellis codes with two, three and four transmit an[5] tennas in quasi-static flat fading channels,” in Proc. IEEE Int. Conf. Communications, New York, NY, May 2002, pp. 1589–1595. [6] Q. Yan and R. S. Blum, “Improved space–time convolutional codes for quasi-static slow fading channels,” IEEE Trans. Wireless Commun., vol. 1, pp. 563–571, Oct. 2002. [7] R. S. Blum, “Some analytical tools for the design of space–time convolutional codes,” IEEE Trans. Commun., vol. 50, pp. 1593–1599, Oct. 2002. [8] W. Firmanto, B. S. Vucetic, and J. Yuan, “Space–time TCM with improved performance on fast fading channels,” IEEE Commun. Lett., vol. 5, pp. 154–156, Apr. 2001. [9] A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York: McGraw-Hill, 1991. [10] Z. Chen, J. Yuan, and B. Vucetic, “An improved space–time trellis coded modulation scheme on slow Rayleigh fading channels,” in Proc. IEEE Int. Conf. Communications, Helsinki, Finland, June 2001, pp. 1110–1116. [11] J. Ventura-Traveset, G. Caire, E. Biglieri, and G. Taricco, “Impact of diversity reception on fading channels with coded modulation—Part I: Coherent detection,” IEEE Trans. Commun., vol. 45, pp. 563–572, May 1997. [12] B. Vucetic and J. Nicolas, “Performance of 8PSK trellis codes over nonlinear fading mobile satellite channels,” Inst. Elect. Eng. Proc. I, vol. 139, pp. 462–471, Aug. 1992. [13] Z. Chen, J. Yuan, and B. Vucetic, “Improved space–time trellis coded modulation scheme on slow Rayleigh fading channels,” Electron. Lett., vol. 37, pp. 440–441, Mar. 2001. [14] J. Yuan, B. Vucetic, Z. Chen, and W. Firmanto, “Performance of space–time coding on fading channels,” in Proc. IEEE ISIT’01, Washington, DC, June 2001, p. 153. [15] B. Vucetic and J. Yuan, Space–Time Coding. New York: Wiley, 2003.