Performance of Parallel Concatenated SpaceâTime Codes
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IEEE COMMUNICATIONS LETTERS, VOL. 5, NO. 6, JUNE 2001
Performance of Parallel Concatenated Space–Time Codes Dongzhe Cui and Alexander M. Haimovich, Senior Member, IEEE
Abstract—We study a channel turbo-coding scheme that consists of parallel concatenated systematic space–time codes and is referred to as turbo space–time coded modulation (turbo-STCM). The scheme features full rate and simulation results shows that it also provides full diversity. Performance with recursive versus nonrecursive space–time constituent codes is investigated. The advantage of recursive component codes is demonstrated by simulations for a four state 4-PSK turbo-STCM scheme operating over a Rayleigh block-fading channel. It is also shown that the turbo-STCM performs better than conventional space–time codes of similar complexity. Index Terms—Channel coding, decoding, diversity methods, Rayleigh fading, transmit antennas, trellis codes.
I. INTRODUCTION
T
URBO coding combines multiple simple codes into single powerful codes that operate close to Shannon capacity [1]. While their performance resembles that of random codes, the availability of iterative decoding algorithms makes the implementation of turbo codes practical. Another class of new codes, space–time codes (STCs), combines the diversity advantage of space–time processing with the bandwidth efficiency and error correction coding of trellis-coded modulation (TCM) for multiple-input multiple-output (MIMO) multipath channels [2]. Recently, several schemes that combine space–time and turbo codes were proposed. A scheme that combines a binary turbo encoder with transmit antenna diversity was proposed in [3]. In [4], the outputs of parallel concatenated TCM modules are routed to two separate antennas. This arrangement is a direct extension of [5], where the TCM modules are connected to a single antenna through a selector. A scheme that consists of two parallel concatenated STCs was proposed in [6]. However, the scheme lacks a mechanism for puncturing the output resulting in turbo codes with reduced data rates compared to the constituent codes. Schemes for serial concatenation of STCs and turbo codes can be found in [7], [8]. Recently, we proposed a new turbo scheme referred to as turbo space–time coded modManuscript received January 29, 2001. The associate editor coordinating the review of this letter and approving it for publication was Dr. M. Fossorier. This work was supported in part by the Air Force Office of Scientific Research under Grant F49620-00-1-0107 and by the New Jersey Center for Wireless Telecommunications. D. Cui was with the Center for Communications and Signal Processing Research, Electriacl and Computer Engineering Department, New Jersey Institute of Technology, Newark, NJ 07102 USA. He is now with Wireless Networks Group, Lucent Technologies, Whippany, NJ 07981 USA (e-mail: [email protected]). A. Haimovich is with the Center for Communications and Signal Processing Research, Electriacl and Computer Engineering Department, New Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: [email protected]). Publisher Item Identifier S 1089-7798(01)05511-9.
Fig. 1.
Turbo-STCM encoder, 2 transmit antennas.
ulation (turbo-STCM) [9], [10]. The scheme consists of two parallel concatenated recursive STCs, however, unlike [6], the underlying STCs are in systematic form. Similar to the original binary turbo codes [1], this arrangement suggests a natural mechanism for achieving full coding rate by puncturing of the parity data. In the sense that the constituent codes are systematic and recursive STCs, the proposed scheme can be viewed as a true extension from the bit to the symbol level of the original turbo scheme. The basic principle of turbo coding-decoding is extended to turbo-STCM by retaining the soft-decision and iterative decoding. The encoder utilizes two transmit antennas and is designed such that one antenna transmits the systematic information, while the other transmits the parity data. Systematic constituent STCs facilitate the routing of the systematic and parity data to the different antennas. STCs available in literature are not systematic, hence we had to design our own STCs. Additionally, the constituent codes were implemented recursively. This paper extends our previous published work by studying the performance of turbo-STCM with recursive and nonrecursive constituent codes. Section II describes the turbo-STCM encoder and briefly presents the principles of the decoder. Section III contains simulations of turbo-STCM with recursive/nonrecursive constituent codes. Turbo-STCM is also compared with conventional STCs. Finally, Section IV provides the conclusions. II. TURBO-STCM ENCODER/DECODER This section briefly reviews the turbo-STCM encoder/decoder. For further details see [9], [10]. The turbo-STCM encoder consists of two STC modules in systematic form operating in a parallel concatenation structure and connected transmit antennas as shown in Fig. 1. The systematic to form for the component codes is dictated by the required separation and transmission through different antennas of the systematic and parity data. The systematic structure is further
1089–7798/01$10.00 © 2001 IEEE
CUI AND HAIMOVICH: PERFORMANCE OF PARALLEL CONCATENATED SPACE–TIME CODES
(a)
(b) Fig. 2. Recursive systematic space–time encoder for 4-state 4-PSK, 2 transmit antennas, 2 bits/s/Hz: (a) code implementation, (b) code trellis.
(a)
237
…). The component codes represented here were designed to be also systematic. Not shown here due to space limitations, these codes features full diversity, but lower coding gain than equivalent Tarokh codes. A theoretical proof of full diversity based on the rank analysis of difference matrices can be found in [11]. The initial coding gain deficit of the component codes is more than made up with the turbo iterations. For a turbo-STCM receiver utilizing an array of antennas, the MIMO channel output symbols at time are represented by vectors , where is a complex set of signal constellation points (1) is the energy per symbol. The ( ) mawhere consists of the channel coefficients. The channel is astrix sumed flat, Rayleigh and block-fading. The transmitted symbol . Additive white Gaussian noise is modeled by vector the vector . The noise is assumed complex-valued, Gaussian for each dimendistributed with zero-mean and variance sion. The turbo-STCM decoder has an iterative structure. The decoder employs two a posteriori probability (APP) computing modules concatenated in parallel; one module for each constituent code. The generic APP algorithm for nonbinary trellis and MIMO channel is based on the BCJR algorithm [12] and on [5]. The key to the iterative process of the turbo-STCM decoder is that the a priori input to each APP decoder can be derived from the output of the other decoder. Data is shared between the two decoders, and an iterative process is applied to refine the soft decisions.
(b) Fig. 3. Non-recursive systematic space–time encoder for 4-state 4-PSK, 2 transmit antennas, 2 bits/s/Hz: (a) code implementation, (b) code trellis.
motivated by the need to puncture the parity data of the turbo code such that the data rate of the overall code is the same as that of the constituent codes. The integration of STCs into turbo-STCM is similar to the integration of coded modulation modules into the turbo-TCM structure of [5]. There are some important differences between the binary turbo codes in [1] and turbo-STCM: 1) the constituent STCs need to be systematic at the symbol level rather than the bit level, 2) the interleaver operates symbol-wise (for example, with 4-PSK modulation the interleaver operates on pairs of bits). In addition, turbo-STCM is different from turbo-TCM in [5] in that the output consists of multiple streams of symbols that are being transmitted through multiple antennas. Multiple antennas may also be employed at the receiver. Recursive systematic component codes are often encountered in turbo code applications and one of the goals in the paper is to compare performance of turbo-STCM with recursive and nonrecursive component codes. The component STCs are defined by the diagrams and trellis codes shown in Figs. 2 and 3. Recursive component STCs are according to Fig. 2, while the nonrecursive form is shown in Fig. 3. The trellis code is understood to represent a mapping of one of four input symbols (groups of two input bits represented by the labels “0,” “1,” …) to two output 4-PSK symbols (also represented by the labels “0,” “1,”
III. NUMERICAL EXAMPLES Simulations were performed for frames of 130 information symbols and 4-PSK symbols transmitted over a quasistatic, flat Rayleigh fading channel. The quasistatic assumption implies that the channel was fixed during a frame, but independent frame-to-frame. Performance for turbo-STCM is shown after eight iterations of the decoder. The figures display the frame error rate (FER) curves versus the bit signal-to-noise ratio in dB per receive antenna. The interleaver used in the simulations was an “ -random” interleaver, as proposed in [13]. Following guidelines in the reference and choosing , an interleaver was obtained by generating random permutations (interleaves) without replacement, subject to the restriction that adjacent symbols are not interleaved within a distance of symbols of each other. This interleaver was used in all simulations. Fig. 4 shows the FER for two transmit-one receive antenna turbo-STCM utilizing the recursive and nonrecursive four state 4-PSK component STCs shown in Figs. 2 and 3, respectively. Curves are also provided for two transmit-one receive antenna, 4 and 32 state 4-PSK Tarokh et al. codes [2]. The 32 state Tarokh code (labeled in the figure “4p32s”) is shown since it has roughly the same decoding complexity as turbo-STCM with 4 state constituent codes, two APP decoders, and 8 iterations per APP decoder. Complexity of the decoder is evaluated as proper decoder and per iteration, where is the portional to code memory [14]. It is well known that Tarokh’s codes feature
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IEEE COMMUNICATIONS LETTERS, VOL. 5, NO. 6, JUNE 2001
recursive codes (at FER is 3.8 dB and 0.7 dB over the conventional Tarokh code with, respectively, 4 and 32 states.
IV. CONCLUSIONS
Fig. 4.
FER of turbo-STCM (2T1R) 4-state 4-PSK over fading channel.
In this letter, we studied a turbo-STCM scheme utilizing parallel concatenated systematic STCs with multilevel modulation and multiple transmit/receive antennas. The turbo-STCM encoder consists of two systematic recursive space–time component codes and it features full rate. Simulation results show that the scheme also provides full diversity. We study the performance of turbo-STCM with recursive and nonrecursive space–time constituent codes. Simulation results are provided for 4 state 4-PSK over the block-fading channel. It , and for a two transmit-one is shown that at FER receive antenna configuration, the performance with recursive codes has an advantage of about 1.3 dB over the configuration with nonrecursive codes. For the same FER and antenna configuration, turbo-STCM provides an advantage of 2.7 dB over conventional 4-state STCs and 0.6 dB over conventional 32-state space–time codes.
REFERENCES
Fig. 5.
FER of turbo-STCM (2T2R) 4 state 4-PSK over fading channel.
full diversity. Due to the puncturing operation, the turbo-STCM architecture of Fig. 1 cannot guarantee full diversity, yet as observed from the figure, both recursive and nonrecursive forms of turbo-STCM have full diversity, a fact borne out by the asymp) of the FER curves being parallel totic slopes (at high with the slopes of Tarokh’s codes. Codes are differentiated only by their coding gain. Turbo-STCM with recursive component codes has an advantage of 2.7 dB over Tarokh’s 4 state code , while turbo-STCM with non(labeled “4p4s”) at FER recursive component codes has a performance 1.3 dB below , turbo-STCM with that of the recursive form. At FER recursive codes has a 0.6 dB advantage over the 32-state conformula for complexity, ventional STC. Based on the turbo-STCM has a relative complexity of 128 (for eight iterations), while the 32 state STC has a relative complexity of 160. Comparison with a 16 state code from [2] (complexity 64) shows an advantage for the turbo code of about 1 dB. The curve labeled “outage cap” is the outage channel capacity defined as the probability that the spectral efficiency (in this case 2 bit/s/Hz) is not supported with a probability given by the ordinate [15]. It is observed that turbo-STCM with recursive codes . performs within 2.5 dB of the outage capacity at FER Fig. 5 is similar to Fig. 4, except that two antennas are used at the receiver. In this case, the advantage of turbo-STCM with
[1] C. Berrou, A. Glavieux, and P. Thititmajshima, “Near Shannon limit error-correcting coding and decoding: Turbo-codes (1),” in IEEE Int. Conf. on Communications (ICC ’93), May 1993, pp. 1064–1070. [2] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communications: Performance criterion and code construction,” IEEE Trans. Inform. Theory, vol. 44, pp. 744–765, Mar. 1998. [3] A. Stefanov and T. M. Duman, “Turbo coded modulation for systems with transmit and receive antenna diversity,” IEEE Globecom’99, pp. 2336–2340, Dec. 1999. [4] Y. Liu and M. P. Fitz, “Space-time turbo codes,” in 37th Annu. Allerton Conf. on Communication, Control and Computing, Sept. 1999. [5] P. Robertson and T. Wörz, “Bandwidth-efficient turbo trellis-coded modulation using punctured component codes,” IEEE J. Select. Areas Commun., vol. 16, pp. 206–218, Feb. 1998. [6] K. R. Narayanan, “Turbo decoding of concatenated space–time codes,” in 37th Annu. Allerton Conf. on Communication, Control and Computing, Sept. 1999. [7] G. Bauch, “Concatenation of space–time block codes and turbo-TCM,” in IEEE Int. Conf. on Communications (ICC’99), June 1999, pp. 1202–1206. [8] X. Lin and R. S. Blum, “Improved space–time codes using serial concatenation,” IEEE Commun. Lett., vol. 4, pp. 221–223, July 2000. [9] D. Cui and A. M. Haimovich, “A new bandwidth efficient antenna diversity scheme using turbo codes,” in Conf. on Information Sciences and Systems (CISS ’00), vol. 1, Princeton, NJ, Mar. 2000, pp. TA–6.24–TA–6.29. [10] , “Design and performance analysis of turbo space–time coded modulation,” IEEE Globecom 2000, vol. 3, pp. 1627–1631, Nov. 27–Dec. 1, 2000. [11] , “Parallel concatenated Turbo space–time coded modulation: Principles and performance analysis,” IEEE J. Select. Areas Commun., submitted for publication. [12] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, pp. 284–287, Mar. 1974. [13] D. Divsalar and F. Pollara, “Turbo codes for PCS applications,” in IEEE Int. Conf. on Communications (ICC ’95), vol. 1, 1995, pp. 54–59. [14] C. Berrou and A. Glavieux, “Near optimum error correcting coding and decoding: Turbo-codes,” IEEE Trans. Commun., vol. 44, pp. 1261–1271, Oct. 1996. [15] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun., pp. 6:311–335, 1998.
IEEE COMMUNICATIONS LETTERS, VOL. 5, NO. 6, JUNE 2001
Performance of Parallel Concatenated Space–Time Codes Dongzhe Cui and Alexander M. Haimovich, Senior Member, IEEE
Abstract—We study a channel turbo-coding scheme that consists of parallel concatenated systematic space–time codes and is referred to as turbo space–time coded modulation (turbo-STCM). The scheme features full rate and simulation results shows that it also provides full diversity. Performance with recursive versus nonrecursive space–time constituent codes is investigated. The advantage of recursive component codes is demonstrated by simulations for a four state 4-PSK turbo-STCM scheme operating over a Rayleigh block-fading channel. It is also shown that the turbo-STCM performs better than conventional space–time codes of similar complexity. Index Terms—Channel coding, decoding, diversity methods, Rayleigh fading, transmit antennas, trellis codes.
I. INTRODUCTION
T
URBO coding combines multiple simple codes into single powerful codes that operate close to Shannon capacity [1]. While their performance resembles that of random codes, the availability of iterative decoding algorithms makes the implementation of turbo codes practical. Another class of new codes, space–time codes (STCs), combines the diversity advantage of space–time processing with the bandwidth efficiency and error correction coding of trellis-coded modulation (TCM) for multiple-input multiple-output (MIMO) multipath channels [2]. Recently, several schemes that combine space–time and turbo codes were proposed. A scheme that combines a binary turbo encoder with transmit antenna diversity was proposed in [3]. In [4], the outputs of parallel concatenated TCM modules are routed to two separate antennas. This arrangement is a direct extension of [5], where the TCM modules are connected to a single antenna through a selector. A scheme that consists of two parallel concatenated STCs was proposed in [6]. However, the scheme lacks a mechanism for puncturing the output resulting in turbo codes with reduced data rates compared to the constituent codes. Schemes for serial concatenation of STCs and turbo codes can be found in [7], [8]. Recently, we proposed a new turbo scheme referred to as turbo space–time coded modManuscript received January 29, 2001. The associate editor coordinating the review of this letter and approving it for publication was Dr. M. Fossorier. This work was supported in part by the Air Force Office of Scientific Research under Grant F49620-00-1-0107 and by the New Jersey Center for Wireless Telecommunications. D. Cui was with the Center for Communications and Signal Processing Research, Electriacl and Computer Engineering Department, New Jersey Institute of Technology, Newark, NJ 07102 USA. He is now with Wireless Networks Group, Lucent Technologies, Whippany, NJ 07981 USA (e-mail: [email protected]). A. Haimovich is with the Center for Communications and Signal Processing Research, Electriacl and Computer Engineering Department, New Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: [email protected]). Publisher Item Identifier S 1089-7798(01)05511-9.
Fig. 1.
Turbo-STCM encoder, 2 transmit antennas.
ulation (turbo-STCM) [9], [10]. The scheme consists of two parallel concatenated recursive STCs, however, unlike [6], the underlying STCs are in systematic form. Similar to the original binary turbo codes [1], this arrangement suggests a natural mechanism for achieving full coding rate by puncturing of the parity data. In the sense that the constituent codes are systematic and recursive STCs, the proposed scheme can be viewed as a true extension from the bit to the symbol level of the original turbo scheme. The basic principle of turbo coding-decoding is extended to turbo-STCM by retaining the soft-decision and iterative decoding. The encoder utilizes two transmit antennas and is designed such that one antenna transmits the systematic information, while the other transmits the parity data. Systematic constituent STCs facilitate the routing of the systematic and parity data to the different antennas. STCs available in literature are not systematic, hence we had to design our own STCs. Additionally, the constituent codes were implemented recursively. This paper extends our previous published work by studying the performance of turbo-STCM with recursive and nonrecursive constituent codes. Section II describes the turbo-STCM encoder and briefly presents the principles of the decoder. Section III contains simulations of turbo-STCM with recursive/nonrecursive constituent codes. Turbo-STCM is also compared with conventional STCs. Finally, Section IV provides the conclusions. II. TURBO-STCM ENCODER/DECODER This section briefly reviews the turbo-STCM encoder/decoder. For further details see [9], [10]. The turbo-STCM encoder consists of two STC modules in systematic form operating in a parallel concatenation structure and connected transmit antennas as shown in Fig. 1. The systematic to form for the component codes is dictated by the required separation and transmission through different antennas of the systematic and parity data. The systematic structure is further
1089–7798/01$10.00 © 2001 IEEE
CUI AND HAIMOVICH: PERFORMANCE OF PARALLEL CONCATENATED SPACE–TIME CODES
(a)
(b) Fig. 2. Recursive systematic space–time encoder for 4-state 4-PSK, 2 transmit antennas, 2 bits/s/Hz: (a) code implementation, (b) code trellis.
(a)
237
…). The component codes represented here were designed to be also systematic. Not shown here due to space limitations, these codes features full diversity, but lower coding gain than equivalent Tarokh codes. A theoretical proof of full diversity based on the rank analysis of difference matrices can be found in [11]. The initial coding gain deficit of the component codes is more than made up with the turbo iterations. For a turbo-STCM receiver utilizing an array of antennas, the MIMO channel output symbols at time are represented by vectors , where is a complex set of signal constellation points (1) is the energy per symbol. The ( ) mawhere consists of the channel coefficients. The channel is astrix sumed flat, Rayleigh and block-fading. The transmitted symbol . Additive white Gaussian noise is modeled by vector the vector . The noise is assumed complex-valued, Gaussian for each dimendistributed with zero-mean and variance sion. The turbo-STCM decoder has an iterative structure. The decoder employs two a posteriori probability (APP) computing modules concatenated in parallel; one module for each constituent code. The generic APP algorithm for nonbinary trellis and MIMO channel is based on the BCJR algorithm [12] and on [5]. The key to the iterative process of the turbo-STCM decoder is that the a priori input to each APP decoder can be derived from the output of the other decoder. Data is shared between the two decoders, and an iterative process is applied to refine the soft decisions.
(b) Fig. 3. Non-recursive systematic space–time encoder for 4-state 4-PSK, 2 transmit antennas, 2 bits/s/Hz: (a) code implementation, (b) code trellis.
motivated by the need to puncture the parity data of the turbo code such that the data rate of the overall code is the same as that of the constituent codes. The integration of STCs into turbo-STCM is similar to the integration of coded modulation modules into the turbo-TCM structure of [5]. There are some important differences between the binary turbo codes in [1] and turbo-STCM: 1) the constituent STCs need to be systematic at the symbol level rather than the bit level, 2) the interleaver operates symbol-wise (for example, with 4-PSK modulation the interleaver operates on pairs of bits). In addition, turbo-STCM is different from turbo-TCM in [5] in that the output consists of multiple streams of symbols that are being transmitted through multiple antennas. Multiple antennas may also be employed at the receiver. Recursive systematic component codes are often encountered in turbo code applications and one of the goals in the paper is to compare performance of turbo-STCM with recursive and nonrecursive component codes. The component STCs are defined by the diagrams and trellis codes shown in Figs. 2 and 3. Recursive component STCs are according to Fig. 2, while the nonrecursive form is shown in Fig. 3. The trellis code is understood to represent a mapping of one of four input symbols (groups of two input bits represented by the labels “0,” “1,” …) to two output 4-PSK symbols (also represented by the labels “0,” “1,”
III. NUMERICAL EXAMPLES Simulations were performed for frames of 130 information symbols and 4-PSK symbols transmitted over a quasistatic, flat Rayleigh fading channel. The quasistatic assumption implies that the channel was fixed during a frame, but independent frame-to-frame. Performance for turbo-STCM is shown after eight iterations of the decoder. The figures display the frame error rate (FER) curves versus the bit signal-to-noise ratio in dB per receive antenna. The interleaver used in the simulations was an “ -random” interleaver, as proposed in [13]. Following guidelines in the reference and choosing , an interleaver was obtained by generating random permutations (interleaves) without replacement, subject to the restriction that adjacent symbols are not interleaved within a distance of symbols of each other. This interleaver was used in all simulations. Fig. 4 shows the FER for two transmit-one receive antenna turbo-STCM utilizing the recursive and nonrecursive four state 4-PSK component STCs shown in Figs. 2 and 3, respectively. Curves are also provided for two transmit-one receive antenna, 4 and 32 state 4-PSK Tarokh et al. codes [2]. The 32 state Tarokh code (labeled in the figure “4p32s”) is shown since it has roughly the same decoding complexity as turbo-STCM with 4 state constituent codes, two APP decoders, and 8 iterations per APP decoder. Complexity of the decoder is evaluated as proper decoder and per iteration, where is the portional to code memory [14]. It is well known that Tarokh’s codes feature
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IEEE COMMUNICATIONS LETTERS, VOL. 5, NO. 6, JUNE 2001
recursive codes (at FER is 3.8 dB and 0.7 dB over the conventional Tarokh code with, respectively, 4 and 32 states.
IV. CONCLUSIONS
Fig. 4.
FER of turbo-STCM (2T1R) 4-state 4-PSK over fading channel.
In this letter, we studied a turbo-STCM scheme utilizing parallel concatenated systematic STCs with multilevel modulation and multiple transmit/receive antennas. The turbo-STCM encoder consists of two systematic recursive space–time component codes and it features full rate. Simulation results show that the scheme also provides full diversity. We study the performance of turbo-STCM with recursive and nonrecursive space–time constituent codes. Simulation results are provided for 4 state 4-PSK over the block-fading channel. It , and for a two transmit-one is shown that at FER receive antenna configuration, the performance with recursive codes has an advantage of about 1.3 dB over the configuration with nonrecursive codes. For the same FER and antenna configuration, turbo-STCM provides an advantage of 2.7 dB over conventional 4-state STCs and 0.6 dB over conventional 32-state space–time codes.
REFERENCES
Fig. 5.
FER of turbo-STCM (2T2R) 4 state 4-PSK over fading channel.
full diversity. Due to the puncturing operation, the turbo-STCM architecture of Fig. 1 cannot guarantee full diversity, yet as observed from the figure, both recursive and nonrecursive forms of turbo-STCM have full diversity, a fact borne out by the asymp) of the FER curves being parallel totic slopes (at high with the slopes of Tarokh’s codes. Codes are differentiated only by their coding gain. Turbo-STCM with recursive component codes has an advantage of 2.7 dB over Tarokh’s 4 state code , while turbo-STCM with non(labeled “4p4s”) at FER recursive component codes has a performance 1.3 dB below , turbo-STCM with that of the recursive form. At FER recursive codes has a 0.6 dB advantage over the 32-state conformula for complexity, ventional STC. Based on the turbo-STCM has a relative complexity of 128 (for eight iterations), while the 32 state STC has a relative complexity of 160. Comparison with a 16 state code from [2] (complexity 64) shows an advantage for the turbo code of about 1 dB. The curve labeled “outage cap” is the outage channel capacity defined as the probability that the spectral efficiency (in this case 2 bit/s/Hz) is not supported with a probability given by the ordinate [15]. It is observed that turbo-STCM with recursive codes . performs within 2.5 dB of the outage capacity at FER Fig. 5 is similar to Fig. 4, except that two antennas are used at the receiver. In this case, the advantage of turbo-STCM with
[1] C. Berrou, A. Glavieux, and P. Thititmajshima, “Near Shannon limit error-correcting coding and decoding: Turbo-codes (1),” in IEEE Int. Conf. on Communications (ICC ’93), May 1993, pp. 1064–1070. [2] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communications: Performance criterion and code construction,” IEEE Trans. Inform. Theory, vol. 44, pp. 744–765, Mar. 1998. [3] A. Stefanov and T. M. Duman, “Turbo coded modulation for systems with transmit and receive antenna diversity,” IEEE Globecom’99, pp. 2336–2340, Dec. 1999. [4] Y. Liu and M. P. Fitz, “Space-time turbo codes,” in 37th Annu. Allerton Conf. on Communication, Control and Computing, Sept. 1999. [5] P. Robertson and T. Wörz, “Bandwidth-efficient turbo trellis-coded modulation using punctured component codes,” IEEE J. Select. Areas Commun., vol. 16, pp. 206–218, Feb. 1998. [6] K. R. Narayanan, “Turbo decoding of concatenated space–time codes,” in 37th Annu. Allerton Conf. on Communication, Control and Computing, Sept. 1999. [7] G. Bauch, “Concatenation of space–time block codes and turbo-TCM,” in IEEE Int. Conf. on Communications (ICC’99), June 1999, pp. 1202–1206. [8] X. Lin and R. S. Blum, “Improved space–time codes using serial concatenation,” IEEE Commun. Lett., vol. 4, pp. 221–223, July 2000. [9] D. Cui and A. M. Haimovich, “A new bandwidth efficient antenna diversity scheme using turbo codes,” in Conf. on Information Sciences and Systems (CISS ’00), vol. 1, Princeton, NJ, Mar. 2000, pp. TA–6.24–TA–6.29. [10] , “Design and performance analysis of turbo space–time coded modulation,” IEEE Globecom 2000, vol. 3, pp. 1627–1631, Nov. 27–Dec. 1, 2000. [11] , “Parallel concatenated Turbo space–time coded modulation: Principles and performance analysis,” IEEE J. Select. Areas Commun., submitted for publication. [12] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, pp. 284–287, Mar. 1974. [13] D. Divsalar and F. Pollara, “Turbo codes for PCS applications,” in IEEE Int. Conf. on Communications (ICC ’95), vol. 1, 1995, pp. 54–59. [14] C. Berrou and A. Glavieux, “Near optimum error correcting coding and decoding: Turbo-codes,” IEEE Trans. Commun., vol. 44, pp. 1261–1271, Oct. 1996. [15] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun., pp. 6:311–335, 1998.
