Low-Complexity Sphere Decoding Algorithm for ... - Semantic Scholar
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006
377
Transactions Letters________________________________________________________________ Low-Complexity Sphere Decoding Algorithm for Quasi-Orthogonal Space–Time Block Codes Alice Yen-Chi Peng, Il-Min Kim, Member, IEEE, and Shahram Yousefi, Member, IEEE
Abstract—Space–time codes can be decoded by the sphere decoding (SD) algorithm to reduce the complexity and retain maximum-likelihood (ML) performance. In this letter, the ML metric of quasi-orthogonal space–time block codes is written into two independent Euclidean norms, thus SD can be applied to each function independently. The new scheme reduces the complexity by at least 85% for systems with four or more transmit antennas, compared with the conventional SD algorithm. Index Terms—Bounded distance decoding, codes, maximumlikelihood (ML) decoding, quasi-orthogonal space–time block codes (QOSTBCs), sphere decoding (SD).
I. INTRODUCTION
P
AST research has shown that the deployment of multiple-input multiple-output (MIMO) systems along with space–time coding introduces a significant increase in information capacity [1]–[6]. Quasi-orthogonal space–time block codes (QOSTBCs) have been designed to provide the full transmission rate for complex symbols. QOSTBCs for systems with three or four transmit antennas were first designed by Jafarkhani [7]. Recently, Xian and Liu have proposed QOSTBCs for systems with up to eight transmit antennas [8]. However, for QOSTBCs, as all the symbols are not completely decoupled, the maximum-likelihood (ML) decoding complexity still increases exponentially. Thus, in a practical MIMO environment, the use of ML detection is not feasible, especially in a system with a large number of transmit antennas and/or a large signal constellation size. Sphere decoding (SD) was originally designed for lattice decoding, and lately it has been shown to be applicable to space–time coded MIMO systems, resulting in lower decoding complexity. In most cases, the exponential complexity is reduced to polynomial, while achieving ML performance [3], [9]. The complexity of a sphere decoder is independent of the constellation size. It depends on the number of total transmitted symbols. In this letter, SD for QOSTBCs is considered. As one important property of QOSTBCs, the ML metric can Paper approved by A. Lozano, the Editor for Wireless Communication of the IEEE Communications Society. Manuscript received September 27, 2004; revised July 20, 2005. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grants 283192 and 288195. This paper was presented in part at the Canada Workshop on Information Theory (CWIT 2005), Montreal, QC, Canada, June 2005. The authors are with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, ON K7L 3N6, Canada (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2006.869881
be decoupled into two independent functions [7], [8]. Our low-complexity SD algorithm is developed by rewriting the two independent functions into two square Euclidean distance forms. Hence, a sphere decoder can be applied to each function separately. When compared with the conventional SD algorithm, the modified SD scheme for QOSTBCs reduces the decoding complexity by at least 85% (for a system with four or more transmit antennas) and maintains the ML performance. The rest of this letter is organized as follows. Section II introduces a system model for wireless MIMO communication and the transmission matrices for QOSTBCs, followed by a discussion on ML decoding and the conventional SD for QOSTBCs. The low-complexity SD is proposed in Section III. We conclude the letter with simulation results in Section IV and concluding remarks in Section V. II. PRELIMINARY A. System Model transmit and receive Consider a MIMO system with antennas. The received signals during the period of time slots are given by (1) The received block is an matrix, and the th of denotes the received signal at the th receive element antenna and the th time slot. A flat Rayleigh-fading MIMO matrix , and the th channel is modeled by an of denotes the channel between the th receive element antenna and the th transmit antenna. A QOSTBC over transmit antennas and time slots is represented by , and is the additive white Gaussian noise (AWGN) matrix. The matrix symbols are written with “tilde” signs , in order to emphasize that they all contain complex elements. B. Quasi-Orthogonal Space–Time Block Codes The transmission matrix of linear space–time block codes transmitted symbols can be written as [10] (STBCs) with (2) is the real part, and where complex number.
0090-6778/$20.00 © 2006 IEEE
is the imaginary part of a
378
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006
In particular, the transmission matrix of a QOSTBC for (by Jafarkhani [7]) is
can also be expressed by (2). For , the QOSTBCs can be constructed by deleting rows of the transmission matrix. C. ML Decoding ML decoding is equivalent to minimizing the metric
Since QOSTBCs are a class of linear STBCs, pressed by (2) with
can be ex-
(5) for valid transmitted symbols ( deand notes the transpose of a matrix or vector), where denotes the constellation. As one advantage of QOSTBCs, the ML decoding can be performed by minimizing two independent ( and ) code and metrics. A system with a receive antennas is the simplest example to consider. For a , the ML metric can be written into two indepenQOSTBC dent functions as
(6)
(3)
After expanding and
(for
) and deleting the constant terms, become [7]
, the transmission matrix can be constructed simply For . That is by deleting the fourth row of
And
can also be expressed by (2) with
(7)
(8) (4)
Similarly, for a system with a QOSTBC metric can be decoupled as
[8], the ML
Also, as reported by Xian and Liu [8], the full-rate transmission is matrix of a QOSTBC for (9) For all QOSTBCs, the ML metric can only be decoupled into two functions. Since the number of related symbols in one func, each function must solve two or more symtion is exactly bols concurrently. That is, all the symbols are not entirely decoupled, and the naive ML decoding complexity is still exponential and may not be practical for a large constellation.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006
D. Conventional SD Algorithm
379
Theorem: For QOSTBCs with arbitrary
As the complexity for naive ML decoding of QOSTBCs is still exponential, the SD algorithm is considered. The idea of SD [3], [11], [12] is to perform minimization on the same metric, but the search is restricted to the points found within the sphere centered around the received signal. The SD alof radius gorithm finds the shortest vector in the sense of Euclidean norm within the above-mentioned sphere. The decoupled metric functions presented in (7) and (8) are not in the required Euclidean norm format. Therefore, the SD scheme can not be applied to decode these two decoupled functions independently. Thus, unlike ML decoding, applying a sphere decoder to a quasi-orthogonal solves only one nondespace–time coded system with coupled metric
and into square Euclidean distance forms as
and
,
can be written
(12)
(13) where
(10) Similarly, a system coded by a QOSTBC coder solves for
, a sphere de-
(11) In this letter, this decoding scheme is referred to as the conventional SD algorithm. As Damen reported, choosing initial to be the lower bound for the eigenvalues of the Gram matrix [11], the number of arithmetic operations then can be ap[3]. However, this is just a theoretical proximated to be approximation; recent enhancements to SD [13], [14] have reduced its sensitivity to initial radius selection, and decreased the actual number of operations. However, the theoretical approximation of complexity order remains the same. For a detailed analysis on the complexity, please refer to [9].
and
Furthermore,
’s and
are given by
III. LOW-COMPLEXITY SD ALGORITHM The conventional SD algorithm has been favorable for its independency of constellation size; however, the complexity is . This motivates greatly dependent on the size of vector the search for a low-complexity SD algorithm for QOSTBCs. The results presented in (7) and (8) reveal that the quasi-orthogonal property simplifies ML decoding, but this property has not been used in the conventional SD scheme. The low-complexity SD is developed by taking advantage of the quasi-orthogonal property of QOSTBCs. For any QOSTBC, the transmitted symbols can be divided into two independent sets. We denote the symbol indexes of the first set as a set of integers
(14) where and are defined using (2). Proof: Recalling the metric presented in (5), replacing by the form in (2), the metric becomes
and the second set as
In order to apply SD separately to the two independent functions and , they need to be in square Euclidean distance forms. Also, it is necessary to perform real transformations on the complex matrices. To this end, the following theorem is derived.
(15)
380
For
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006
, we define
and
as
Adding a constant to the metric in (15) and performing some mathematical manipulations, and can be rewritten as shown in the equations at the bottom of the page. In order to , we rewrite as find an expression for ,
Since for all QOSTBCs,
, it follows
where is given by (14). In order to derive an expression for , is rewritten as
Since is quasi-orthogonal, and only differ by signs in some of the columns, also in any QOSTBC is always an identity matrix or a rectangular identity matrix, and is similar to , but with some of the columns negated. Thus, it follows
where is given by (14). The usage of and ensures that the equivalent form is derived, preserving the order of elements in the original . Now we can represent and
as
Finally, it is straightforward to show that and
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006
381
can be written as (12) and (13), respectively. In the following, there are four examples to show the applicaand an arbitrary tions of the Theorem for various numbers of . number of , and Example 1: For QOSTBCs with can be written into square Euclidean distance forms as
(16) ,
, ,
where
,
, and
Fig. 1. Average number of floating point operations required by the conventional and the proposed SD algorithms for decoding one block (four symbols per block) of 16-quadrature amplitude modulation (QAM) symbols , n = 4, n = 4. for ~
X
Furthermore, ’s and are given by (14), and and are defined as in (3). , the indeExample 2: For a system with a QOSTBC and are in the same pendent metrics and forms as (16). Also, and are given by (14), with defined as in (4). , we have a similar result as for . To apply For and SD, the two independent metrics must be rewritten into square distance forms, and then they must be transformed into real matrices. The result is shown in the following example. , Example 3: For QOSTBCs with and can be expressed in square Euclidean distance forms as
(17) where
, , ,
, ,
and
Moreover, and are defined by (14). , for Example 4: For a system with a QOSTBC , the independent metrics and are in the same forms as (17). Also, and are defined by (14).
Proposed SD Algorithm: To decode , the initial and are chosen to be the lower square distances bounds for the eigenvalues of the Gram matrices generated by and , respectively. Then a sphere decoder is applied independently to and . For , the initial square distances and are chosen similarly. Then a sphere decoder is applied to and solve independently for . Lemma: The proposed algorithm has an approximate com. putational complexity of Proof: The conventional SD has an approximate comwhen solving for a transmitted vector plexity of with symbols [3]. For the new algorithm introduced, SD is applied separately to two decision metrics. For each minimization, the sphere decoder is solving for a vector with symbols. Therefore, its complexity is approximated to be . IV. SIMULATION RESULTS The complexity is measured by the average number of floating point operations for decoding one block of symbols. The sphere decoder used in the simulation adjusts the radius adaptively. When no point is found inside the sphere, the square radius is increased by 80% of the constellation bit energy for both the conventional and the proposed sphere decoders. Therefore, the conventional and proposed SD schemes provide exactly the same performance as the ML method. Figs. 1 and 2 are the complexity comparisons on both SD algorithms. For , the case of and was simulated and shown in Fig. 1. The complexity is reduced by more than 85%. , a system with and was modeled. For The complexity reduction in this case is more than 90%, as shown in Fig. 2.
382
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006
operations has decreased by at least 85% for systems with four or more transmit antennas. The computational complexity is increases. expected to be further reduced as REFERENCES
Fig. 2. Average number of floating point operations required by the conventional and the proposed SD algorithms for decoding one block (eight , n = 8, n = 8. symbols per block) of 16-QAM symbols for ~
X
V. CONCLUSION When applying SD to STBCs, the complexity is greatly dependent on the number of transmitted symbols. By the quasi-orthogonal property, the ML metric of QOSTBCs has been written into two independent functions in their Euclidean distance forms. Hence, they can be separately decoded by a sphere decoder as finding the two shortest vectors of two independent square distance functions. With the proposed SD algorithm, it has been proven that without reducing the transmission rate, or sacrificing the performance, the complexity to . As for practical applicadecreases from tions, the simulations have shown the number of floating point
[1] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: Performance results,” IEEE J. Sel. Areas Commun., vol. 17, no. 3, pp. 451–459, Mar. 1999. , “Space-time block codes from orthogonal designs,” IEEE Trans. [2] Inf. Theory, vol. 45, no. 7, pp. 1456–1466, Jul. 1999. [3] O. Damen, A. Chkeif, and J.-C. Belfiore, “Lattice code decoder for space-time codes,” IEEE Commun. Lett., vol. 4, no. 5, pp. 161–163, May 2000. [4] G. J. Foschini and R. A. Valenzuela, “Layered space-time architecture for wireless communication in a fading environment when using multiple antennas,” Bell Labs. Tech. J., vol. 1, pp. 41–59, Autumn 1996. [5] P. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela, “VBLAST: An architecture for realizing very high data rates over the rich scattering wireless channel,” in Proc. URSI Int. Symp. Signals, Syst., Electron., 1998, pp. 295–300. [6] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, pp. 585–595, Nov./Dec. 1999. [7] H. Jafarkhani, “A quasi-orthogonal space-time block code,” IEEE Trans. Commun., vol. 49, no. 1, pp. 1–4, Jan. 2001. [8] L. Xian and H. Liu, “Rate one space-time block codes with full diversity,” IEEE Trans. Commun., submitted for publication. [9] J. Jaldén and B. Ottersten, “On the complexity of sphere decoding in digital communications,” IEEE Trans. Signal Process., vol. 53, no. 4, pp. 1474–1484, Apr. 2005. [10] E. Larsson and P. Stoica, Space-Time Block Coding for Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2003. [11] E. Viterbo and J. Boutros, “A universal lattice code decoder for fading channel,” IEEE Trans. Inf. Theory, vol. 45, no. 7, pp. 1639–1642, Jul. 1999. [12] U. Fincke and M. Pohst, “Improved methods for calculating vectors of short length in a lattice, including a complexity analysis,” Math. Comput., vol. 44, pp. 463–471, Apr. 1985. [13] D. Pham, K. R. Pattipati, P. K. Willett, and J. Luo, “An improved complex sphere decoder for V-BLAST systems,” IEEE Signal Process. Lett., vol. 11, no. 9, pp. 748–751, Sep. 2004. [14] W. Zhao and G. B. Giannakis, “Sphere decoding algorithms with improved radius search,” IEEE Trans. Commun., vol. 53, no. 7, pp. 1104–1109, Jul. 2005.
377
Transactions Letters________________________________________________________________ Low-Complexity Sphere Decoding Algorithm for Quasi-Orthogonal Space–Time Block Codes Alice Yen-Chi Peng, Il-Min Kim, Member, IEEE, and Shahram Yousefi, Member, IEEE
Abstract—Space–time codes can be decoded by the sphere decoding (SD) algorithm to reduce the complexity and retain maximum-likelihood (ML) performance. In this letter, the ML metric of quasi-orthogonal space–time block codes is written into two independent Euclidean norms, thus SD can be applied to each function independently. The new scheme reduces the complexity by at least 85% for systems with four or more transmit antennas, compared with the conventional SD algorithm. Index Terms—Bounded distance decoding, codes, maximumlikelihood (ML) decoding, quasi-orthogonal space–time block codes (QOSTBCs), sphere decoding (SD).
I. INTRODUCTION
P
AST research has shown that the deployment of multiple-input multiple-output (MIMO) systems along with space–time coding introduces a significant increase in information capacity [1]–[6]. Quasi-orthogonal space–time block codes (QOSTBCs) have been designed to provide the full transmission rate for complex symbols. QOSTBCs for systems with three or four transmit antennas were first designed by Jafarkhani [7]. Recently, Xian and Liu have proposed QOSTBCs for systems with up to eight transmit antennas [8]. However, for QOSTBCs, as all the symbols are not completely decoupled, the maximum-likelihood (ML) decoding complexity still increases exponentially. Thus, in a practical MIMO environment, the use of ML detection is not feasible, especially in a system with a large number of transmit antennas and/or a large signal constellation size. Sphere decoding (SD) was originally designed for lattice decoding, and lately it has been shown to be applicable to space–time coded MIMO systems, resulting in lower decoding complexity. In most cases, the exponential complexity is reduced to polynomial, while achieving ML performance [3], [9]. The complexity of a sphere decoder is independent of the constellation size. It depends on the number of total transmitted symbols. In this letter, SD for QOSTBCs is considered. As one important property of QOSTBCs, the ML metric can Paper approved by A. Lozano, the Editor for Wireless Communication of the IEEE Communications Society. Manuscript received September 27, 2004; revised July 20, 2005. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grants 283192 and 288195. This paper was presented in part at the Canada Workshop on Information Theory (CWIT 2005), Montreal, QC, Canada, June 2005. The authors are with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, ON K7L 3N6, Canada (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2006.869881
be decoupled into two independent functions [7], [8]. Our low-complexity SD algorithm is developed by rewriting the two independent functions into two square Euclidean distance forms. Hence, a sphere decoder can be applied to each function separately. When compared with the conventional SD algorithm, the modified SD scheme for QOSTBCs reduces the decoding complexity by at least 85% (for a system with four or more transmit antennas) and maintains the ML performance. The rest of this letter is organized as follows. Section II introduces a system model for wireless MIMO communication and the transmission matrices for QOSTBCs, followed by a discussion on ML decoding and the conventional SD for QOSTBCs. The low-complexity SD is proposed in Section III. We conclude the letter with simulation results in Section IV and concluding remarks in Section V. II. PRELIMINARY A. System Model transmit and receive Consider a MIMO system with antennas. The received signals during the period of time slots are given by (1) The received block is an matrix, and the th of denotes the received signal at the th receive element antenna and the th time slot. A flat Rayleigh-fading MIMO matrix , and the th channel is modeled by an of denotes the channel between the th receive element antenna and the th transmit antenna. A QOSTBC over transmit antennas and time slots is represented by , and is the additive white Gaussian noise (AWGN) matrix. The matrix symbols are written with “tilde” signs , in order to emphasize that they all contain complex elements. B. Quasi-Orthogonal Space–Time Block Codes The transmission matrix of linear space–time block codes transmitted symbols can be written as [10] (STBCs) with (2) is the real part, and where complex number.
0090-6778/$20.00 © 2006 IEEE
is the imaginary part of a
378
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006
In particular, the transmission matrix of a QOSTBC for (by Jafarkhani [7]) is
can also be expressed by (2). For , the QOSTBCs can be constructed by deleting rows of the transmission matrix. C. ML Decoding ML decoding is equivalent to minimizing the metric
Since QOSTBCs are a class of linear STBCs, pressed by (2) with
can be ex-
(5) for valid transmitted symbols ( deand notes the transpose of a matrix or vector), where denotes the constellation. As one advantage of QOSTBCs, the ML decoding can be performed by minimizing two independent ( and ) code and metrics. A system with a receive antennas is the simplest example to consider. For a , the ML metric can be written into two indepenQOSTBC dent functions as
(6)
(3)
After expanding and
(for
) and deleting the constant terms, become [7]
, the transmission matrix can be constructed simply For . That is by deleting the fourth row of
And
can also be expressed by (2) with
(7)
(8) (4)
Similarly, for a system with a QOSTBC metric can be decoupled as
[8], the ML
Also, as reported by Xian and Liu [8], the full-rate transmission is matrix of a QOSTBC for (9) For all QOSTBCs, the ML metric can only be decoupled into two functions. Since the number of related symbols in one func, each function must solve two or more symtion is exactly bols concurrently. That is, all the symbols are not entirely decoupled, and the naive ML decoding complexity is still exponential and may not be practical for a large constellation.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006
D. Conventional SD Algorithm
379
Theorem: For QOSTBCs with arbitrary
As the complexity for naive ML decoding of QOSTBCs is still exponential, the SD algorithm is considered. The idea of SD [3], [11], [12] is to perform minimization on the same metric, but the search is restricted to the points found within the sphere centered around the received signal. The SD alof radius gorithm finds the shortest vector in the sense of Euclidean norm within the above-mentioned sphere. The decoupled metric functions presented in (7) and (8) are not in the required Euclidean norm format. Therefore, the SD scheme can not be applied to decode these two decoupled functions independently. Thus, unlike ML decoding, applying a sphere decoder to a quasi-orthogonal solves only one nondespace–time coded system with coupled metric
and into square Euclidean distance forms as
and
,
can be written
(12)
(13) where
(10) Similarly, a system coded by a QOSTBC coder solves for
, a sphere de-
(11) In this letter, this decoding scheme is referred to as the conventional SD algorithm. As Damen reported, choosing initial to be the lower bound for the eigenvalues of the Gram matrix [11], the number of arithmetic operations then can be ap[3]. However, this is just a theoretical proximated to be approximation; recent enhancements to SD [13], [14] have reduced its sensitivity to initial radius selection, and decreased the actual number of operations. However, the theoretical approximation of complexity order remains the same. For a detailed analysis on the complexity, please refer to [9].
and
Furthermore,
’s and
are given by
III. LOW-COMPLEXITY SD ALGORITHM The conventional SD algorithm has been favorable for its independency of constellation size; however, the complexity is . This motivates greatly dependent on the size of vector the search for a low-complexity SD algorithm for QOSTBCs. The results presented in (7) and (8) reveal that the quasi-orthogonal property simplifies ML decoding, but this property has not been used in the conventional SD scheme. The low-complexity SD is developed by taking advantage of the quasi-orthogonal property of QOSTBCs. For any QOSTBC, the transmitted symbols can be divided into two independent sets. We denote the symbol indexes of the first set as a set of integers
(14) where and are defined using (2). Proof: Recalling the metric presented in (5), replacing by the form in (2), the metric becomes
and the second set as
In order to apply SD separately to the two independent functions and , they need to be in square Euclidean distance forms. Also, it is necessary to perform real transformations on the complex matrices. To this end, the following theorem is derived.
(15)
380
For
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006
, we define
and
as
Adding a constant to the metric in (15) and performing some mathematical manipulations, and can be rewritten as shown in the equations at the bottom of the page. In order to , we rewrite as find an expression for ,
Since for all QOSTBCs,
, it follows
where is given by (14). In order to derive an expression for , is rewritten as
Since is quasi-orthogonal, and only differ by signs in some of the columns, also in any QOSTBC is always an identity matrix or a rectangular identity matrix, and is similar to , but with some of the columns negated. Thus, it follows
where is given by (14). The usage of and ensures that the equivalent form is derived, preserving the order of elements in the original . Now we can represent and
as
Finally, it is straightforward to show that and
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006
381
can be written as (12) and (13), respectively. In the following, there are four examples to show the applicaand an arbitrary tions of the Theorem for various numbers of . number of , and Example 1: For QOSTBCs with can be written into square Euclidean distance forms as
(16) ,
, ,
where
,
, and
Fig. 1. Average number of floating point operations required by the conventional and the proposed SD algorithms for decoding one block (four symbols per block) of 16-quadrature amplitude modulation (QAM) symbols , n = 4, n = 4. for ~
X
Furthermore, ’s and are given by (14), and and are defined as in (3). , the indeExample 2: For a system with a QOSTBC and are in the same pendent metrics and forms as (16). Also, and are given by (14), with defined as in (4). , we have a similar result as for . To apply For and SD, the two independent metrics must be rewritten into square distance forms, and then they must be transformed into real matrices. The result is shown in the following example. , Example 3: For QOSTBCs with and can be expressed in square Euclidean distance forms as
(17) where
, , ,
, ,
and
Moreover, and are defined by (14). , for Example 4: For a system with a QOSTBC , the independent metrics and are in the same forms as (17). Also, and are defined by (14).
Proposed SD Algorithm: To decode , the initial and are chosen to be the lower square distances bounds for the eigenvalues of the Gram matrices generated by and , respectively. Then a sphere decoder is applied independently to and . For , the initial square distances and are chosen similarly. Then a sphere decoder is applied to and solve independently for . Lemma: The proposed algorithm has an approximate com. putational complexity of Proof: The conventional SD has an approximate comwhen solving for a transmitted vector plexity of with symbols [3]. For the new algorithm introduced, SD is applied separately to two decision metrics. For each minimization, the sphere decoder is solving for a vector with symbols. Therefore, its complexity is approximated to be . IV. SIMULATION RESULTS The complexity is measured by the average number of floating point operations for decoding one block of symbols. The sphere decoder used in the simulation adjusts the radius adaptively. When no point is found inside the sphere, the square radius is increased by 80% of the constellation bit energy for both the conventional and the proposed sphere decoders. Therefore, the conventional and proposed SD schemes provide exactly the same performance as the ML method. Figs. 1 and 2 are the complexity comparisons on both SD algorithms. For , the case of and was simulated and shown in Fig. 1. The complexity is reduced by more than 85%. , a system with and was modeled. For The complexity reduction in this case is more than 90%, as shown in Fig. 2.
382
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006
operations has decreased by at least 85% for systems with four or more transmit antennas. The computational complexity is increases. expected to be further reduced as REFERENCES
Fig. 2. Average number of floating point operations required by the conventional and the proposed SD algorithms for decoding one block (eight , n = 8, n = 8. symbols per block) of 16-QAM symbols for ~
X
V. CONCLUSION When applying SD to STBCs, the complexity is greatly dependent on the number of transmitted symbols. By the quasi-orthogonal property, the ML metric of QOSTBCs has been written into two independent functions in their Euclidean distance forms. Hence, they can be separately decoded by a sphere decoder as finding the two shortest vectors of two independent square distance functions. With the proposed SD algorithm, it has been proven that without reducing the transmission rate, or sacrificing the performance, the complexity to . As for practical applicadecreases from tions, the simulations have shown the number of floating point
[1] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: Performance results,” IEEE J. Sel. Areas Commun., vol. 17, no. 3, pp. 451–459, Mar. 1999. , “Space-time block codes from orthogonal designs,” IEEE Trans. [2] Inf. Theory, vol. 45, no. 7, pp. 1456–1466, Jul. 1999. [3] O. Damen, A. Chkeif, and J.-C. Belfiore, “Lattice code decoder for space-time codes,” IEEE Commun. Lett., vol. 4, no. 5, pp. 161–163, May 2000. [4] G. J. Foschini and R. A. Valenzuela, “Layered space-time architecture for wireless communication in a fading environment when using multiple antennas,” Bell Labs. Tech. J., vol. 1, pp. 41–59, Autumn 1996. [5] P. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela, “VBLAST: An architecture for realizing very high data rates over the rich scattering wireless channel,” in Proc. URSI Int. Symp. Signals, Syst., Electron., 1998, pp. 295–300. [6] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, pp. 585–595, Nov./Dec. 1999. [7] H. Jafarkhani, “A quasi-orthogonal space-time block code,” IEEE Trans. Commun., vol. 49, no. 1, pp. 1–4, Jan. 2001. [8] L. Xian and H. Liu, “Rate one space-time block codes with full diversity,” IEEE Trans. Commun., submitted for publication. [9] J. Jaldén and B. Ottersten, “On the complexity of sphere decoding in digital communications,” IEEE Trans. Signal Process., vol. 53, no. 4, pp. 1474–1484, Apr. 2005. [10] E. Larsson and P. Stoica, Space-Time Block Coding for Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2003. [11] E. Viterbo and J. Boutros, “A universal lattice code decoder for fading channel,” IEEE Trans. Inf. Theory, vol. 45, no. 7, pp. 1639–1642, Jul. 1999. [12] U. Fincke and M. Pohst, “Improved methods for calculating vectors of short length in a lattice, including a complexity analysis,” Math. Comput., vol. 44, pp. 463–471, Apr. 1985. [13] D. Pham, K. R. Pattipati, P. K. Willett, and J. Luo, “An improved complex sphere decoder for V-BLAST systems,” IEEE Signal Process. Lett., vol. 11, no. 9, pp. 748–751, Sep. 2004. [14] W. Zhao and G. B. Giannakis, “Sphere decoding algorithms with improved radius search,” IEEE Trans. Commun., vol. 53, no. 7, pp. 1104–1109, Jul. 2005.
