On the Design of Minimum BER Linear Space-Time Block Codes for ...

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 8, AUGUST 2006

3147

On the Design of Minimum BER Linear Space-Time Block Codes for MIMO Systems Equipped With MMSE Receivers Jing Liu, Member, IEEE, Jian-Kang Zhang, Member, IEEE, and Kon Max Wong, Fellow, IEEE

Abstract—In this paper, we consider the design of a full-rate linear space-time block code for coherent multiple-input multiple-output (MIMO) communication systems under a quasi-static Rayleigh flat-fading environment. Our design targets specifically at the use of a linear minimum mean-square error (MMSE) receiver that minimizes the asymptotic average bit error rate (BER) when the transmitted signal is selected from a 4-QAM constellation. This optimization problem is solved in two main stages: 1) a lower bound on the BER is first minimized, and 2) how this minimized lower bound can be achieved is then shown. By exploiting a rigorous convex optimization technique without any assumption on the code, we prove that individual unitary and trace-orthogonal structures are the necessary and sufficient conditions to assure the minimum asymptotic average BER with an MMSE detector. An algorithm is provided for an efficient generation of our codes, and simulation results confirm that our optimally designed codes are indeed superior in performance compared to some other commonly used codes. Index Terms—Bit error rate (BER), minimum mean-square error (MMSE), multiple-input multiple-output (MIMO), unitary trace-orthogonal.

I. INTRODUCTION HE explosive expansion in wireless communications in recent years encounters severe technical challenges caused by the demands for concurrently transmitting speech, data and video at high rates in a scatter-rich environment together with limitations on transmission power and bandwidth. Multiple-input multiple-output (MIMO) wireless links are important recent developments in wireless communication systems due to their enormous potential in meeting these transmitter anchallenges. MIMO designs rely on the use of tennas and receiver antennas that enables the communication system to exploit the high performance provided by the space diversity available, and/or the high data rate provided by the capacity obtainable in the MIMO channels [1], [2]. Full diversity offered is achieved when the total degree of freedom in the multiantenna system is utilized. This will ensure a good performance in terms of probability of error for detecting the transmitted symbols at high signal-to-noise ratio (SNR) when a maximum-likelihood (ML) detector is employed [3]. Full

T

Manuscript received October 14, 2004; revised August 29, 2005. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Martin Haardt. The authors are with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON L8S 4K1, Canada. Digital Object Identifier 10.1109/TSP.2006.874766

symbol rate, on the other hand, is achieved when, on average, one symbol is transmitted by each of the multiple transmitter antennas per time slot (often called a “channel use”). In the transmitter antennas, we will have an average of case of symbols per channel use (pcu) at full rate. One early technique trying to take advantage of the merits of MIMO systems was Bell Laboratories layered space-times (BLAST, also known as D-BLAST) scheme [4] for which diversity gain is high but the data transmission rate is rather poor. Vertical BLAST (V-BLAST) [5] was subsequently presented to achieve full transmission rate, however this gain is acquired at the loss of diversity. Other approaches aiming at performance include the orthogonal space-time block codes (OSTBC) [6]–[11], which ensure full diversity with a linear processing ML detector. However, these schemes generally suffer from low transmission data rates [9], [11]–[13] and thus do not achieve full ergodic channel capacity in general MIMO systems [14]. Another approach to high-rate transmission is by the design of linear dispersive codes [15] maximizing the ergodic capacity of a virtual MIMO system (i.e., the complete MIMO system including both the precoder and the detector). However, full rate and full diversity cannot be assured in this case either. Much advancement in the development of full-rate, full-diversity STBC has been made since then [16]–[22]. More specifically, space-time codes have been developed that simultaneously provide both full diversity and full rate [21], [22] and therefore have fully exploited the advantages of MIMO systems. The good error performance achieved in these designs, however, depends on the ML detector which, in practice, is computationally prohibitive, especially when the number of transmitter antennas or the constellation size is large. While sphere decoding [23] can be utilized to simplify the implementation of the ML detector, there still exists the problem of the choice of the covering radius of the lattice, and the complexity can be still quite high if the number of transmitter antennas is large. On the other hand, a linear MMSE receiver not only involves much simpler implementation, but also makes it possible to perform symbol by symbol detection. In addition, the MMSE equalizer has been exploited for an iterative soft algorithm [24] to be implemented on the equalized outer-coded signals which ensures competitive error performance in comparison to the ML detector while having much lower complexity. The importance of the linear MMSE receiver is further enlightened in [25] and [26] where it has been shown that the MMSE estimator plays a key role in achieving the capacity of the additive white Gaussian

1053-587X/$20.00 © 2006 IEEE

3148

noise channel for digital communications [26]. (Very recently, a new formula has been obtained revealing the connection between the input-output mutual information and the mean-square error (MSE) achieved by linear MMSE receivers which holds for Gaussian channels regardless of the input distribution [27].) For all the above reasons, we will focus our attention hereafter to the use of a linear MMSE equalizer at the receiver. In this paper, we consider a coherent MIMO system, emphasizing on the simplicity of implementation at both the code generator and the receiver. Specifically, we consider the transmission of a space-time block code (STBC) that is a linear combination of coding matrices weighted by the information symbols through a quasi-static Rayleigh flat-fading channel and received by a linear MMSE detector. Our target is the design of a code which, while maintaining full data-transmission rate, minimizes the asymptotic average (over all the random channel coefficients) bit error probability of an MMSE detector. To this symbols pcu is end, we first ensure that the full data rate of maintained, and then, based on the BER for 4-QAM signals, we derive the conditions for optimal codes and establish a code structure that minimizes the asymptotic average bit error probability. The resulting optimum code structure requires the individual coding matrices to be unitary as well as to be mutually orthogonal when vectorized. These two optimum structural characteristics of the coding matrices are described respectively as unitary and trace-orthogonal in this paper. For other signal constellations, the same mathematical techniques presented in this paper can be employed to design STBC using an MSE criterion [28]. In addition, the optimality of the combined unitary trace-orthogonal structure is not only applicable to STBC received by a linear MMSE detector, but can also be extended to provide a guideline for ML detection by confining the consideration only to the codes satisfying these structures, thereby dramatically reducing the parameter space [29]. In recent years, several authors have noted either one or both of the unitary and the trace-orthogonal structures as important properties of space-time codes for good performance [9], [15], [17], [30]–[33]. Specifically, in developing the linear dispersion (LD) codes in [15], the coding matrices have been assumed to be individually unitary. That all the vectorized coding matrices must be mutually orthogonal (trace-orthogonality) for optimal ergodic capacity has also been pointed out in [15], [17], and [30]. On the other hand, in developing STBC to achieve full rate and full diversity employing an ML receiver, both Gamal and Damen [21] and Ma and Giannakis [22] have arrived at codes whose structures comply with the unitary and trace-orthogonal requirements. In spite of the recent advance, our approach to the design problem in this paper is taken from a rigorous convex optimization point of view without any assumptions of the code structures. We prove that the joint unitary and trace-orthogonal properties are both necessary and sufficient to achieve minimum asymptotic average bit error probability for a MIMO system fitted with the MMSE detector. Through this theoretic analysis, we also show that the trace-orthogonal structure minimizes the average total MSE for the resulting virtual space-time-coded MIMO channel, while the unitary structure of the individual code matrix renders the total MSE evenly distributed over each subchannel. These two main properties together minimize the


asymptotic average bit error probability of the MMSE detector. We further provide an algorithm to generate such rectangular codes efficiently. This is followed by the presentation of some simulation results showing the superior performance of codes that possess the unitary and trace-orthogonal properties over those that do not. II. LINEAR SPACE-TIME BLOCK CODING FOR A MIMO SYSTEMS transConsider a MIMO communication system having mitter antennas and receiver antennas. The symbol stream to , be transmitted in time slots is given by and is divided into substreams each having symbols. Thus, altogether, we have symbols to be transmitted. These symbols are selected from a given constellation with zero mean and unity covariance and are, in general, complex. To facilitate antennas in the transmission of these symbols through the the time slots, each symbol is processed by an coding and then summed together, resulting in matrix STBC matrix given by an

(1)

th element of represents the coded symbol where the antenna at the to be transmitted from the th, th, , time slot. The total power assigned to all the coding matrices is constrained to a constant , i.e.,

(2) where “tr” denotes the trace of a square matrix. It is our objective to satisfy certain optimal to design such coding matrices criterion and constraints. This mode of MIMO communication symbols pcu and is therefore transmitting data at transmits full rate. th transmitter antenna, A coded symbol from the , is transmitted to the th receiver antenna, , through a transmission path which is assumed to . These random transmission be flat-fading with coefficient coefficients are modeled to be of normalized independent and identically distributed (i.i.d.) complex circular Gaussian distributions with zero-mean, and remain invariant for the time slots, but may change to another independent state after that. channel matrix can now be formed by having the An as its elements. At the receiver transmission coefficients additive space-time noise antennas, we also have the that is assumed to be spatially and temporally white matrix and is of complex circular Gaussian distribution . Thus, the received space-time signal, denoted by the matrix , can be expressed by

(3)

LIU et al.: ON THE DESIGN OF MINIMUM BER LINEAR STBCs FOR MIMO SYSTEMS

Fig. 1. Block diagram of MIMO channel model with linear STBC.

where is the SNR per receiver antenna. The system model described in (3) is depicted in Fig. 1. In the system model and the transmitted signal of (3), the design parameter are embedded in the space-time signal matrix as indicated in (1). To facilitate the design of a code, we can and by vectorizing . From (1), , separate where is an matrix and . Thus, (3) can be rewritten in an equivalent vectorized form [17], [19] as

interference and additive noise components of (5) have distributions that are symmetric with respect to the real and imaginary axes of the complex plane (and hence they have zero mean), the optimal disjoint (scalar) detector for the th element of the received data block is in fact the standard threshold detector for 4-QAM signaling [35]. To determine the probability of error for has a Gaussian distribution that detector, we observe that (because is jointly Gaussian). Also, since there are altogether possible interfering symbols, hence the interference comdiscrete values. These values, ponent can take on up to which we will denote by , can be calculated by substituting the th permutation of the interfering symbols into the interference component. In fact, by applying the standard procedures for the calculation of the probability of error for symbol-by-symbol detection of a (scalar, unit energy) 4-QAM symbol in the presence of interference [35] to the model in (5), we have that

(7)

(4) , both of dimension , are defined as , and with being identity matrix and denoting Kronecker product. the Equation (4) will be used as the system model throughout this paper. where

and

III. MMSE EQUALIZATION: ASYMPTOTIC AVERAGE BIT ERROR RATE At the receiver of the MIMO system modeled in (4), the transmitted signal has to be equalized and detected. As mentioned in Section I, we like to employ a linear MMSE receiver for the purpose. Our objective is to arrive at a space-time code structure that minimizes the bit error probability while limiting the transmission power to a fixed amount. To derive an expression for the average asymptotic BER of the space-time-coded signals for such a system, we must specify the mode of modulation for the transmitted signal which, in this paper, has been stipulated to the 4-QAM. From the equivalent system model (4), we see that by facilisignal vector tating a linear equalizer , the equalized can be written as

(5) where for an MMSE equalizer [34]

(6)

3149

, and and denote the real and imaginary parts, respectively, of a complex quantity. Unfortunately, the number of terms in the summation in (7) is exponential in , and hence exact computation of quickly becomes computationally infeasible as the number of transmitted data grows. A standard approach to reduce the complexity of evaluating the exact analytic expression for the symbol error probability of a system with interference is to come up with a simple approximate statistical model for the interference. By adopting the results in [36]–[38] to the MIMO system considered in this paper, it can be shown that the distribution of the interference in each element of converges almost surely to a circular complex Gaussian distribution as increases [39]. Therefore, the th equalized symbol in (5) can be approximated by where

(8) is a zero-mean, circular complex Gaussian random where variable with independent real and imaginary parts each having the same variance. Equation (8) is the asymptotic model for the signal and noise received by an MMSE equalizer. This model becomes an accurate representation when , the number of transmitted symbols, is large. Under this model, threshold is equivalent to detection of a single 4-QAM detection of symbol in additive Gaussian noise. , of the error From (5), we see that the covariance matrix, is given by [40] vector (9)

Since an MMSE equalizer is employed, the interference caused by the other transmitted symbols will not be totally eliminated and the error in the th symbol of the equalized signal vector will contain residual interferences. We note that since both the

where the subscript of the error covariance matrix stands for symbol error. Now, since the transmitted signal is of 4-QAM which consists of one bit in each of its real and imaginary parts

3150


of the symbol, to evaluate the bit error probability, we must then separate the transmitted and the detected 4-QAM symbols into their respective real and imaginary parts and examine the error and in both parts. Writing which are vectors with and denoting respectively the real and imaginary parts of a complex quantity, and employing standard transformation for complex matrices [41], the covariance matrix of the bit error is determined by

(10) where the subscript are all of dimension

stands for bit error, and , and and are respectively defined as

accurate expression for the BER when is sufficiently large. which, in This bit error probability is a function of turn, is a function of (and thus of ). In the next section, we seek the design of the space-time coding matrices such that this bit error probability in (15) can be minimized subject to a constraint of transmission power as expressed in (2). IV. DESIGN OF MINIMUM BER LINEAR SPACE-TIME BLOCK CODES WITH MMSE RECEIVER Our goal in this section is to characterize all the optimal linear space-time block codes that minimize the average asymptotic BER given by (15) which, as noted, is a function of the design variable matrix . Therefore, the optimization problem can be formally stated as (16)

(11) standing for conjugate. We also note that with . The th diagonal element of , denoted by , is the MSE of the th bit of . We note that the real and imaginary parts of (the zero-mean Gaussian interference and noise) in the model of (8) are independent and have equal variance. Thus, detection of the real and imaginary parts of the th 4-QAM symbol will yield identical BER expression in both parts. Following standard procedures of calculation [35], the BER in the detection of the th symbol of in (5) is given by (12) is the signal to interference plus noise ratio (SINR) where associated with the th bit of the equalized signal vector and is the expectation taken over all the random channel matrices . Now, for the system of (4) in which an MMSE equalizer is employed, the SINR in the th bit can be expressed in terms of the MSE [40] such that

Equation (16) is solved in two stages such that a lower bound on the average asymptotic BER is first minimized, and this is followed by showing how this minimized lower bound can be achieved. To do that, we first note that function is convex in the interval . is positive in This can be verified by showing that the range of the values of . Now, the matrix inside the brackets of (10) is Hermitian symmetric positive semidefinite (PSD) , implying that such that1 . Therefore, the diagonal elements [42]. Now, Jensen’s inequality [43] states that for any convex func, and for , where is the domain of tion , we have , with are equal. Since the function inside equality holding iff all the braces in (15) is convex, applying Jensen’s inequality, we obtain

(13) Substituting (13) into (12), the average probability of error for th bit can be written as

(17) where the equality holds iff the following condition is satisfied (18)

(14) Thus, the bit error probability averaged over all the can be approximated by

bits in

(15) Equation (15) yields an expression for the asymptotic average bit error probability of the MIMO system that transmits a 4-QAM signal and is fitted with an MMSE receiver. This is an

Equation (17) represents a lower bound for the average asymptotic BER of (15). Let us now solve the optimization problem of (16) by first minimizing this lower bound and then finding a solution within the remaining free variables to meet the condition in (18). Our main result so obtained is stated formally in the following theorem. The detailed derivations are shown in the Appendix. 1For two matrices PSD matrix.

P and Q, the notation P Q denotes that (P 0 Q) is a


Theorem 1: For a MIMO system with a full rate linear STBC, under the Gaussian approximation of interference at the receiver, the average BER for an MMSE detector has a lower bound given by

3151

the MMSE detector is closely approximated by [36], [37], [44]

(19)

where the equality holds iff the individual coding matrices satisfy the following two conditions: must be trace-orthogonal, i.e., for i) , where is the Kronecker delta, or ; equivalently, ii) must be unitary up to a constant factor, i.e., . Theorem 1 gives us the lower bound of the asymptotic average bit error probability for a 4-QAM modulated space-time-coded MIMO system equipped with an MMSE receiver. We note that this lower bound is independent of the design parameters and therefore is the absolute minimum value that can be achieved from a design point of view. The theorem also indicates to us the structures of the STBC that will yield this best performance. The following remarks will provide further insight to this optimal solution and the necessary code structure. A) From the proof of Theorem 1 (see the Appendix), we observe that it is the trace-orthogonal structure of the individual code matrices that causes the lower bound of the asymptotic average BER to be reduced to its minimum value, whereas it is the unitary structure of the individual code matrices that enables this minimized lower bound to be reached. From a physical point of view, we can see that for the asymptotic average BER to reach this minimized lower bound, the equalized MSEs for each symbol must be all equal, which in turn, necessitates the even distribution of the signal power at the input of the receiver in each dimension and the even distribution of the noise power at the output of the MMSE equalizer in each dimension as well. It is for this reason that we call a code having this combined structure a power-distributed linear unitary trace orthogonal (PLUTO) code. It should be noted that the uniform distributions of both (the average) signal and noise power are implemented without the channel state information at the transmitter. Thus, we have implicitly shown that for arbitrarily channels, the PLUTO code family is the one among all unitary matrices that minimizes the average BER of the MMSE detector without any channel information at the transmitter. B) While our derivation has been performed by considering the 4-QAM modulation, the same technique can be extended to show that this combined unitary trace-orthogonal code structure is also optimum in minimizing the average asymptotic BER for a general square QAM signaling with bits per symbol. Here, the BER for

where and . Assuming uniform bit, and . loading, we can drop the subscripts of Under high SNRs, the function is convex [39], [40] and hence Jensen’s inequality can be applied. Following a route similar to that of the derivation of Theorem 1, we obtain

(20)

where equality holds iff the component matrices of satisfy the unitary trace-orthogonal requirement. C) We can also extend the principle to other modes of modulation. We note that [40] at high SNRs, the performance the MMSE detector is dominated by the largest diagonal (worst case). entry of the error covariance matrix Therefore, we can regard the minimization of the maximum diagonal entry of as an effective means to minimize the average symbol error probability of the MMSE detector at high SNR. This leads to the following optimization problem: Since

(21)

(22) minimizing the worst case of the average MSE of the space-time-coded subchannel is reduced to first minimizing the overall average MSE and then finding a code that renders the MSE of each subchannel equal for any channel realization. Using parallel arguments and mathematical procedures similar to those for the derivation of Theorem 1, we can show the PLUTO structure is op-

3152


timum in minimizing the maximum value of the average MSE of the received symbols [28]. D) For the design of a linear STBC that minimizes the worst case average pair-wise error probability for the ML detector [29], the concept of the unitary trace-orthogonal structure can also be of great help. Here, the unitary traceorthogonality provides a guideline for the structure of a general good code and, by confining our consideration to the necessary conditions satisfied by such a good space-time code, the parameter space for design can be dramatically reduced [29]. E) It should be mentioned here that the property (or equivalently, is trace-orthogonal) has been observed by several authors to be optimum in ergodic channel capacity for Gaussian channels. Specifically, being square (full-rate), it is stated in [17] that for is capacity optimal for linear a unitary structure of dispersion code (LDC) design. On the other hand, by assuming the unitary structure, trace-orthogonality was obtained as a sufficient condition for capacity optimality in [30], together with a sufficient condition such that for minimizing the union bound of the error probability. Using mathematical techniques was similar to those in proving Theorem 1, also established in [45] as a necessary and sufficient condition for maximizing the mutual information. It is further noted that by assuming no knowledge of the channel at all and by employing a game theoretic-approach, the condition of equal transmission power allocation to the signal vector has been shown to be optimum for channel capacity in [46]. As pointed out in Section I, it is very interesting to observe that the space-time codes in [21], [22] which were designed to achieve full rate and full diversity indeed comply with the unitary trace-orthogonal requirement. V. OPTIMUM CODE GENERATION AND SIMULATION RESULTS We now present an algorithm for the generation of codes possessing the unitary trace-orthogonal properties described in the previous section. For the generation of the coding matrices we transmitter antennas. Thus, we need coding have to be generated. The original idea matrices of this generation method was stated in [15]. Here, we present a systematic description of the method extended to the generation ) coding matrices. of rectangular (i.e., , such 1) We first re-align all the matrices , that they are renamed where is the number of transmitter antennas and is the number of time slots. row permutation matrix such that 2) Now, we form a

where is the 3) We then form the with

identity matrix. normalized DFT matrix being its th column such that .

4) Finally, we generate the coding matrices according to the following equation:

where is the matrix obtained by putting the eleinto an diagonal matrix, ments of the vector zero matrix. Thus, totally and is an unitary coding matrices will be generated. In the following, we present five examples of computer simulations showing the error performance comparisons of our designed codes with other codes for Rayleigh flat-fading MIMO channels. Example 1: This example compares the error performance of V-BLAST [5] with that of our optimum code when 4-QAM signals are used for transmission in both schemes. V-BLAST employs transmitter antennas and transmits one symbol per antenna per time slot and is therefore full rate. The V-BLAST receiver utilizes a zero-forcing equalizer, and then, based on an order established by the postequalization SNR, detect the symbols one by one. A symbol is immediately cancelled after detection. This symbol detection-cancellation based on the ordering according to SNR yields superior BER performance compared to that of a similar scheme without the ordering. We note that while the V-BLAST scheme satisfies the trace-orthogonal structure, it transmits uncoded symbols and thus its coding matrix does not satisfy the individual unitary property. Hence, it is expected that the V-BLAST code will have inferior performance to PLUTO with an MMSE receiver. In addition, V-BLAST employs a comparatively more sophisticated receiver than the linear MMSE receiver. We now examine the BER performance of V-BLAST fitted with a zero-forcing equalizer and a symbol-ordering/cancellation detector, that of PLUTO equipped with the MMSE equalizer followed by a threshold detector, and that of an uncoded transmission system (such as that employed in the V-BLAST scheme) fitted with the same MMSE detector as PLUTO. These three schemes are transmitter antennas applied to a MIMO system having receiver antennas. Therefore, the transmission data and 16 bits pcu. The average BER performance of the rate is trials at each systems are then evaluated by averaging over of the various SNR. Simulation results are shown in Fig. 2. We can see that PLUTO outperforms not only the uncoded scheme, but also V-BLAST (in spite of having a more sophisticated zero-forcing equalizer and a symbol-ordering/cancellation detector). Example 2: We now present an example comparing the BER performance of PLUTO with a code that, like V-BLAST, only satisfies one of the requirements of unitary trace-orthogonality. This code, (designated by HP-A here), is obtained from Example A of [17], and, unlike V-BLAST, satisfies the individual unitary structure but does not satisfy the trace-orthogonality. We simulate the error performance under the same channel condiand and tion as that in [17] where 4-QAM signals are transmitted. The symbol transmission data symbol pcu. Since PLUTO is designed rate is , to make a fair comparison, we have to adjust for


Fig. 2. Average BER performance comparison of V-BLAST schemes with our design.

3153

Fig. 4. BER performance of our code versus OSTBC and HP-B.

good BER performance. However, the good error performance of OSTBC is achieved at the expense of transmission data rate. To ensure a fair comparison of these two schemes we fix the transmission data (bit) rate of both coding systems to be the same and compare their BER performance. In this example, PLUTO is chosen to transmit 4-QAM signals and the data rate 6 bits pcu. On the other hand, it is known that is the highest symbol rate of OSTBC [8], [9], [11] in this case is 3/4 symbols pcu. Here, we choose a minimum delay nonlinear OSTBC [11]

(23)

Fig. 3. Comparison of average BER performance of our design and HP-A, in which M = 4; N = 1; T = 4 and L = 4 and 4-QAM signals are transmitted. Both codes employ linear MMSE detection.

the PLUTO structure to suit the transmission condition by reducing the unfolded coding matrix to a tall matrix . This can be facilitated by employing the code generation algorithm provided at the beginning of this section to -matrix first, and then arbitrarily choose any obtain a four column vectors of to form . It is easily verified that such satisfies the unitary trace-orthogonal structures as well as the transmission conditions. Simulation results are shown in Fig. 3. Since the HP-A code only satisfies one of the two optimal structures, its performance is therefore expected to be worse than that of PLUTO. Example 3: In this example, we consider a MIMO system with three transmitter antennas and three receiver antennas; i.e., . We compare the error performance of PLUTO with that of OSTBC [6]–[11], which is known to have very

For this OSTBC to achieve the same transmission data rate as PLUTO, we must use a much larger, say, a 256-QAM constellation, so that the data rate is 6 bits pcu. Fig. 4 shows the comparison of the average BERs of the schemes by computer simulations, in which the BERs are evaluated averrandom channel realizations. For further comparaging over ison of performance under the same transmission environment, we have also included the performance of the HP-B code obtained by numerical optimization in Example B of [17] in which and thus has the same data rate as PLUTO in this example. Close examination of the HP-B code reveals that it complies with the requirements of unitary trace-orthogonality and, as confirmed here, has the same performance as PLUTO. From Fig. 4, it can be observed that for a very large range of SNRs, both PLUTO and HP-B are far superior to OSTBC. Only when the SNR is very high (approximate 24 dB and beyond), the performance of OSTBC becomes better. The reason for this is that at lower SNR, the denser signal constellation that is used by OSTBC to compensate for its poor data rate renders the BER much higher than that of PLUTO or HP-B. While at very high dB) and beyond, the diversity gain of OSTBC beSNR ( comes dominant and results in a better BER. Further comparisons [47] show that the range of SNR for which PLUTO is su-

3154


M T N

Fig. 5. Comparison of BER performance among our design, RBV, and MG = = = 4. codes for

perior in performance to OSTBC will increase dramatically if increases. Example 4: In this example, we examine and compare the average BER performances of PLUTO and the following two well-known STBC codes. • The RBV code [48]—This is so far, the best known code for ML detection. Examination of the RBV code shows that the trace-orthogonal structure is satisfied. However, each individual code matrix is orthogonal but not orthonormal. Hence, the RBV code does not satisfy the equal power of radiation in all directions. • MG code [22]—This code is designed for full rate and full diversity. It can be verified [47] that the MG code satisfies the two conditions and is therefore a member of unitary trace-orthogonal family. i) We first consider a MIMO system with and compare the performance of the RBV code and the MG code with that of PLUTO. Here, the transmitted signals for all the three STBC are 4-QAM and are detected by using a linear MMSE equalizer followed by a threshold detector. Fig. 5 shows the performance of the three different codes. Since the MG code has structures that comply with the optimal requirements for MMSE detection, we expect it to have the same performance as PLUTO. Indeed, it can be observed that the average BER performances of the two codes are almost indistinguishable in the figure. On the other hand, the RBV code satisfies most but not all of the optimal requirements. As a result, even though its performance is very close to that of PLUTO, it is inferior at higher SNRs. ii) We now compare the performance of PLUTO with that of the RBV code in an environment in which and . Again, for both codes, the signals are transmitted using 4-QAM. In addition, we also include the performance comparison when the signals are transmitted using 16-QAM. Comparing Figs. 5 and 6, we can observe that the performance of the two coding schemes have been improved dramatically with the increase of the number of receiver antennas. This improvement is due to the use

M T

N

Fig. 6. Comparison of BER performance between our design and RBV code = = 4 and = 6. for

of multiple receiver antennas which increases the receive array gain of the MIMO system. Fig. 6 also verifies that PLUTO maintains its superior performance to the RBV code when other modulation schemes are employed. Example 5: In this last example, we compare the performance of PLUTO to some LD codes. In particular, we choose to examine the performance of the LD codes designed by using (36) and (39) in [15]. We designate these two codes HH-1 and HH-2 respectively. In general, the block signal matrix for LD codes can be written as where and are the constituent coding matrices. The concept of the unitary trace-orthogonal structure can be extended readily to LD codes [49] by first vectorizing and forming ; and then constructing the unfolded coding matrix such that

The structures of LD codes corresponding to the unitary traceorthogonality of the linear STBC in this case are given by [49] i) for (“LD unitary”); (“LD trace-orthonormal”). ii) Careful examination on the codes HH-1 and HH-2 reveals that both codes are “LD trace-orthogonal” but not “LD trace-orthonormal.” Neither of the two codes satisfies the “LD unitary” condition. Computer simulation results shown in Fig. 7 confirm that PLUTO has superior BER performance to the HH1 and HH2 codes in [15] when linear MMSE detection is employed. It is further observed that at high SNR, the coded channel HH-2 tends to be singular and hence the marked deterioration in performance. VI. CONCLUSION In this paper, we have presented a new approach in the optimal design of STBC for a MIMO system in a flat-fading environment. Our emphasis here is that a linear MMSE equalizer and


3155

. To facilitate the development of the proof, we first introduce the following two lemmas on the trace of a matrix [42]. we have Lemma 1: For any square matrix . Lemma 2: For any nonsingular Hermitian symmetric PSD matrix

we have

where equality holds iff , i.e., iff is block diagonal. The starting point of our proof is the lower bound given by the right side of (17). We note that this bound is achieved iff (18) is satisfied. We first minimize this lower bound by reducing it to an expression completely independent of the design parameters: in (17) is given by (10). Now, the error covariance matrix Applying Lemma 1 to yields Fig. 7. Comparison of average BER performance of our design and the codes in [15], (36) and (39), in which M = 3; N = 1. 4-QAM signals are transmitted. All the codes employ linear MMSE detection.

(24) a threshold detector are to be employed at the receiver for simplicity of implementation. The MIMO system considered here employs 4-QAM in the transmission of the signals. Utilizing the convexity properties of the bit error probability of the system and its lower bounds, we arrived at an optimum code structure that minimizes the average asymptotic bit error probability. This optimum structure requires the coding matrices to be both individual unitary and trace-orthogonal. This structure has also been implicitly shown to be optimal among all the trace-orthogonal STBC when no channel information is available at the transmitter. Even though such structures have been tacitly assumed before by some other researchers, the derivation of the structures here is from a rigorous consideration of the convexity of the bit error probability and its lower bounds, and the conditions of optimality obtained are both necessary and sufficient. Such an approach also provides us with the insight of knowing the effect of each of the required optimal structures. We have also indicated that by choosing MSE of the received symbols as the objective function, the presented mathematical techniques can be extended to other modulation schemes of signal constellations. Further, under the assumption of high SNR, the proposed unitary trace-orthogonal structure can be proved to achieve the minimum asymptotic average BER for square QAM signals following similar mathematical derivations presented in this paper. Finally, we have provided a simple algorithm of generating such optimum codes, as well as simulation results showing that such a code is indeed optimum in performance under the conditions of operation. APPENDIX

is block diagonal with each block where matrix being . Furthermore, letting and applying Lemma 2 to (24), we have (25) where , are of . Equality in (25) holds iff

matrices on the diagonal is block diagonal, i.e.,

(26) On the other hand, since is convex with respect to positive definite matrix [43], applying Jensen’s inequality to this convex function and employing Lemma 1 lead to

(27) where matrix is defined by are equal, i.e., in (27) holds iff all

for

. Equality

(28)

Combining (25) with (27) results in

(29)

A. Proof of Theorem 1 The key in proving Theorem 1 is to derive an achievable minimized lower bound that is independent of the design variable

with equality holding iff (26) is satisfied. Since function is monotonically increasing with , putting (29)

3156


in (17) establishes a new lower bound for the average asymptotic such that BER

(30) where equality holds iff the conditions in (26) and (18) are met simultaneously. The lower bound in (30), though achievable, still depends on the design variable . Therefore, in the following our goal is to minimize this lower bound to one that is independent on . To do that, we perform an eigenvalue decomposition on the positive , where is the unitary definite matrix , i.e., eigenvector matrix and is the (positive) diagonal eigenvalue matrix. Since the elements of the channel matrix is i.i.d. and Gaussian, the stochastic properties will not change when postmultiplied by a unitary matrix [50]. Combining this property with (30), we have

(31) Furthermore, we note that there are ways of permuting the columns of the channel matrix by post-multiplying each time without with a permutation (unitary) matrix changing the expected value of the function. As a result, we have (32), shown at the bottom of the page.

Employing Jensen’s inequalities on the convex function on the right side of (32) and then, again on the convex funcinside the summation sign yield where, for the tion second inequality, we have used the fact that averaging the sum of all permuted eigenvalue matrices together with the power constraints results in an identity matrix. In addition, we note that both equalities in (33) and (34), shown at the bottom of the page, hold iff , and hence, , is an identity matrix, i.e., (35) Finally, combining (30) and (32) with (34), we obtain

(36) We observe that the lower bound in (36) is independent of the design matrix and is therefore the minimum value that can be achieved by a designer. The original lower bound of the BER given in (17) will equal this minimum value iff the three conditions in (26), (28) and (35) are satisfied simultaneously, or equivalently, iff is unitary. Since the th column of is being unitary is equivalent to . Thus, trace-oris a necessary and sufficient condition for the thogonality of lower bound of the BER to reach its minimum value given by (36). From (18), the necessary and sufficient condition for the BER to reach its lower bound is given by

(37)

(32)

(33) (34)


which should hold for any channel realization. Now, for unitary, we have

being

Also, the right-hand side of (37) is . Thus, the condition in (37) is reduced to

(38) for any channel realization .

. This can be true iff

REFERENCES [1] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun., vol. 6, pp. 311–335, Mar. 1998. [2] ˙I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Euro. Trans. Telecommun., vol. 10, pp. 585–595, Nov./Dec. 1999. [3] V. Tarokh, N. Seshadri, and A. R. Calderbank, “space-time codes for high data rate wireless communication: Performance criterion and code construction,” IEEE Trans. Inf. Theory, vol. 44, pp. 744–765, Mar. 1998. [4] G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J., pp. 41–59, Autumn 1996. [5] G. D. Golden, G. J. Foschini, R. A. Valenzuela, and P. W. Wolniansky, “Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture,” Electron. Lett., vol. 35, pp. 14–16, Jan. 1999. [6] A. V. Geramita and J. Seberry, “Orthogonal design, quadratic forms and hadamard matrices,” in Lecture Notes in Pure and Applied Mathematics. New York: Marcel Dekker, 1979. [7] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1451–1458, Oct. 1998. [8] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block code from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, pp. 1456–1467, Jul. 1999. [9] O. Tirkkonen and A. Hottinen, “Square-matrix embeddable space-time codes for complex signal constellations,” IEEE Trans. Inf. Theory, vol. 48, pp. 384–395, Feb. 2002. [10] E. G. Larsson and P. Stoica, Space-time Block Coding for Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2003. [11] X.-B. Liang, “Orthogonal designs with maximal rates,” IEEE Trans. Inf. Theory, vol. 49, pp. 2468–2503, Oct. 2003. [12] W. Su and X.-G. Xia, “Two generalized complex orthogonal spacetime block codes of rates 7/11 and 3/5 for 5 and 6 transmit antennas,” IEEE Trans. Inf. Theory, vol. 49, pp. 313–316, Jan. 2003. [13] X.-B. Liang and X.-G. Xia, “On the nonexistence of rate-one generalized complex orthogonal designs,” IEEE Trans. Inf. Theory, vol. 49, pp. 2984–2988, Nov. 2003. [14] S. Sandhu and A. Paulraj, “Space-time block coding: A capacity perspective,” IEEE Commun. Lett., vol. 4, no. 12, pp. 384–386, Dec. 2000. [15] B. Hassibi and B. M. Hochwald, “High-rate codes that are linear in space and time,” IEEE Trans. Inf. Theory, vol. 48, pp. 1804–1824, July 2002. [16] S. Sandhu and A. Paulraj, “Union bound on error probability of linear space-time block codes,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Salt Lake City, UT, May 2001, vol. 4, pp. 2473–2476.

3157

[17] R. W. Heath and A. J. Paulraj, “Linear dispersion codes for MIMO systems based on frame theory,” IEEE Trans. Signal Process., vol. 50, no. 10, pp. 2429–2441, Oct. 2002. [18] O. Tirkkonen and A. Hottinen, “Improved MIMO performance with nonorthogonal space-time block codes,” in Proc. IEEE Global Telecommunications Conf. (IEEE GLOBECOM), Texas, USA, Nov. 2001, pp. 1122–1126. [19] M. O. Damen, A. Tewfik, and J. C. Belfiore, “A construction of a spacetime code based on number theory,” IEEE Trans. Inf. Theory, vol. 48, pp. 753–760, Mar. 2002. [20] B. A. Sethuraman, B. S. Rajan, and V. Shashidhar, “Full-diversity, high-rate space-time block codes from division algebras,” IEEE Trans. Inf. Theory, vol. 49, pp. 2596–2616, Oct. 2003. [21] H. E. Gamal and M. O. Damen, “Universal space-time coding,” IEEE Trans. Inf. Theory, vol. 49, pp. 1097–1119, May 2003. [22] X. Ma and G. B. Giannakis, “Full-diversity full rate complex-field space-time coding,” IEEE Trans. Signal Process., vol. 51, no. 11, pp. 2917–2930, Nov. 2003. [23] O. Damen, A. Chkeif, and J.-C. Belfiore, “Lattice code decoder for space-time codes,” IEEE Commun. Lett., vol. 4, pp. 161–163, May 2000. [24] X. Wang and H. V. Poor, “Iterative (Turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, pp. 1046–1061, July 1999. [25] U. Erez and R. Zamir, “Achieving (1=2) log(1+SNR) on the AWGN channel with lattice encoding and decoding,” IEEE Trans. Inf. Theory, vol. 50, pp. 2293–2314, Oct. 2004. [26] G. D. Forney, “On the role of MMSE estimation in approaching the information-theoretic limits of linear Gaussian channels: Shannon meets Wiener,” in Proc. 2003 Allerton Conf., Oct. 2003, pp. 430–439. [27] D. Guo, S. Shamai, and S. Verdú, “Mutual information and minimum mean-square error in Gaussian channels,” IEEE Trans. Inf. Theory, vol. 51, pp. 1261–1282, Apr. 2005. [28] J. Liu, J.-K. Zhang, and K. M. Wong, “Design of optimal orthogonal linear codes in MIMO systems for MMSE receivers,” presented at the IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Montreal, ON, Canada, May 2004. [29] J.-K. Zhang, J. Liu, and K. M. Wong, “Trace-orthonormal full diversity triangular cyclotomic space-time minimizing worst case pair-wise error probability,” presented at the Int. Conf. Sensor Array Multichannel Signal Processing, Sitges, Barcelona, Spain, Jul. 2004. [30] S. Sandhu and A. Paulraj, “Unified design of linear space-time block codes,” Proc. IEEE Global Telecommunications Confe. (IEEE GLOBECOM), pp. 1073–1077, Nov. 2001. [31] O. Tirkkonen, “Maximal symbolwise diversity in nonorthogonal spacetime block codes,” in Proc. Int. Symp. Inf. Theory (ISIT), Jun. 2001, p. 197. [32] A. Hottinen, J. Vesma, O. Tirkkonen, and N. Nefedov, “High bit rates for 3G and beyond using MIMO channels,” in Proc. 13th IEEE Int. Symp. Personal, Indoor, Mobile Radio Communications, Sept. 2002, pp. 854–856. [33] R. H. Gohary and T. N. Davidson, “Design of linear dispersion codes: Some asymptotic guidelines and their implementation,” IEEE Trans. Wireless Commun., pp. 2892–2906, Nov. 2005. [34] A. Scaglione, G. B. Giannakis, and S. Barbarossa, “Redundant filterbank precoders and equalizers—Part I: Unification and optimal designs,” IEEE Trans. Signal Process., vol. 47, no. 7, pp. 1988–2006, Jul. 1999. [35] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2001. [36] H. V. Poor and S. Verdú, “Probability of error in MMSE multiuser detection,” IEEE Trans. Inf. Theory, vol. 43, pp. 858–871, May 1997. [37] J. Zhang, E. K. P. Chong, and D. N. C. Tse, “Output MAI distributions of linear MMSE multiuser receivers in DS-CDMA systems,” IEEE Trans. Inf. Theory, vol. 47, pp. 1128–1144, Mar. 2001. [38] S. Verdu, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [39] S. S. Chan, T. N. Davidson, and K. M. Wong, “Asymptotically minimum BER linear block precoders for MMSE equalization,” Proc. Inst. Elect. Eng. Commun., vol. 151, pp. 297–304, Aug. 2004. [40] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Joint Tx–Rx beamforming design for multicarrier MIMO channels: A unified framework for convex optimization,” IEEE Trans. Signal Process., vol. 51, no. 9, pp. 2381–2401, Sep. 2003.

3158


[41] R. A. Monzingo and T. W. Miller, Adaptive Arrays. New York: Wiley, 1980. [42] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1994. [43] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2001. [44] K. Cho and D. Yoon, “On the general BER expression of one and two dimensional amplitude modulations,” IEEE Trans. Commun., vol. 50, no. 7, pp. 1074–1080, Jul. 2002. [45] J. Liu, J.-K. Zhang, and K. M. Wong, “Optimal linear code design in improving BER for MIMO systems with MMSE receivers,” presented at the Int. Conf. Signal Processing (ICSP), Beijing, China, Sep. 2004. [46] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Uniform power allocation in MIMO channels: A game-theoretic approach,” IEEE Trans. Inf. Theory, vol. 49, pp. 1707–1727, Jul. 2003. [47] J. Liu, “Simulation studies of space-time block codes for MIMO systems,” Dept. ECE, McMaster Univ., Hamilton, ON, Canada, Res. Rep., Aug. 2004. [48] G. Rekaya, J.-C. Belfiore, and E. Viterbo, “Algebraic 3 3, 4 4 and 6 6 space-time codes with nonvanishing determinants,” presented at the Int. Symp. Information Theory Its Applications (ISITA), Parma, Italy, Oct. 2004. [49] J.-K. Zhang, J. Liu, and K. M. Wong, Trace-Orthogonal Full Diversity Cyclotomic Space-Time Codes. New York: Wiley, 2005, ch. 5. [50] R. J. Muirhead, Aspects of Multivariate Statistical Theory. New York: Wiley, 1982.

2

2

2

Jing Liu (S’02–M’02) received the B.S. degree in electronic science from Nan Kai University, Tianjin, China, in 1997 and the M.A.Sc. degree from McMaster University, Hamilton, ON, Canada, in 2004, where she conducted research in wireless communications. She is currently working towards the Ph.D. degree at the same university, continuing her research in wireless communications. She was previously employed in the Network Centre of Nan Kai University. Her research interests are in signal detection and estimation and the application of signal processing in communication systems. Ms. Liu received the Outstanding Thesis Award from McMaster University in 2004.

Jian-Kang Zhang (M’04) received the B.S. degree in mathematics and information science from Shaanxi Normal University, Xi’an, China, the M.S. degree in mathematics from Northwest University, Xi’an, China, and the Ph.D. degree in electrical engineering from Xidian University, Xi’an, China. He is now with the Department of Electrical and Computer Engineering at McMaster University, Hamilton, ON, Canada. His research interests include multirate filte banks, wavelet and multiwavelet transforms and their applications, number theory transform, and fast algorithm. His current research focuses on multiscale wavelet transform, random matrices, precoding, and space-time coding for block transmissions and multicarrier and wideband wireless communication systems.

Kon Max Wong (SM’81–F’02) was born in Macau. He received the B.Sc. (Eng.), DIC, Ph.D., and D.Sc. (Eng.) degrees, all in electrical engineering, from the University of London, U.K., in 1969, 1972, 1974, and 1995, respectively. In 1969, he started working at the Transmission Division of Plessey Telecommunications Research Ltd., U.K. In October 1970, he was on leave from Plessey pursuing postgraduate studies and research at Imperial College of Science & Technology, London, U.K. In 1972, he rejoined Plessey as a Research Engineer and worked on digital signal processing and signal transmission. In 1976, he joined the Department of Electrical Engineering at the Technical University of Nova Scotia, Canada, and in 1981, moved to McMaster University, Hamilton, ON, Canada, where he has been a Professor since 1985 and served as Chairman of the Department of Electrical and Computer Engineering from 1986 to 1987 and from 1988 to 1994. He was on leave as Visiting Professor at the Department of Electronic Engineering of the Chinese University of Hong Kong from 1997 to 1999. At present, he holds the NSERC-Mitel Professorship of Signal Processing and is the Director of the Communication Technology Research Centre at McMaster University. He is also serving another five-year term as Chair of the Department. His research interest is in signal processing and communication theory, and he has published over 200 papers in the area. Prof. Wong was the recipient of the IEE Overseas Premium for the Best Paper in 1989 and is a Fellow the Institution of Electrical Engineers, a Fellow of the Royal Statistical Society, and the Institute of Physics. He was an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING from 1996 to 1998 and served as Chair of the Sensor Array and Multi-Channel Signal Processing Technical Committee of the Signal Processing Society from 2002 to 2004. He was the recipient of a medal presented by the International Biographical Centre, Cambridge, U.K., for his “outstanding contributions to the research and education in signal processing” in May 2000 and was honored with the inclusion of his biography in the two books: Outstanding People of the 20th Century and 2000 Outstanding Intellectuals of the 20th Century published by IBC to celebrate the arrival of the new millennium.

On the Design of Minimum BER Linear Space-Time Block Codes for ...

On the Design of Minimum BER Linear Space-Time Block Codes for ...

Suggest Documents