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Design and Performance of Space–Time Codes for Spatially Correlated MIMO Channels Bruno Clerckx, Claude Oestges, Luc Vandendorpe, Fellow, IEEE, Danielle Vanhoenacker-Janvier, and Arogyaswami J. Paulraj, Fellow, IEEE
Abstract—Space–time code (STC) designs classically rely on the assumption of independent and identically distributed (i.i.d.) Rayleigh channels. However, poor scattering conditions may have detrimental effects on the performance of STCs. In this letter, we derive code-design criteria leading to robust STCs in a large variety of slow-fading propagation conditions. No channel knowledge is assumed at the transmitter. Codes satisfying these criteria are shown to perform much better on real-world channels than codes designed only for i.i.d. channels. As examples, the robustness of various spatial multiplexing schemes, linear dispersion codes, and space–time trellis codes is discussed based on those criteria.
assumed at the transmitter. The proposed design criterion is more general than those proposed in [2] and [3]: it does not depend on the channel gain distribution and can be applied to both rank-deficient and full-rank STCs. Moreover, codes designed based on the proposed criterion are shown to outperform previously developed codes in a number of propagation scenarios.
Index Terms—Correlated channels, multiple-input multipleoutput (MIMO), space–time coding, spatial multiplexing (SM).
A. System Model
I. INTRODUCTION
A
COMMON assumption in the design of space–time coding for multiple-input multiple-output (MIMO) systems is to consider the fading coefficients between the pairs of transmit–receive antennas as independent and identically Rayleigh distributed (i.i.d.). This is, however, an idealistic situation. In practice, correlation between fading coefficients exists, and highly influences the capacity as well as the performance of space–time codes (STCs) [1]. While receive correlations affect all STCs equally, transmit correlations induce interactions within the transmitted codewords. This interaction is particularly detrimental for rank-deficient codes, such as spatial multiplexing (SM) [1]. In [2], a phase randomization was introduced in order to achieve good performance of full-rank STCs in line-of-sight (LOS) scenarios. In [3], based on a virtual channel representation, precoding matrices are optimized in order to guarantee the robustness of SM in correlated channels. In this letter, we formalize the interactions between the channel and the code, and derive a new design criterion for codes evolving in spatially correlated slow-fading channels. No channel knowledge is
Paper approved by H. El-Gamal, the Editor for Space–Time Coding and Spread Spectrum of the IEEE Communications Society. Manuscript received May 15, 2004; revised February 8, 2006. The work of B. Clerckx and C. Oestges was supported by the Belgian Fund FRIA and the Belgian National Science Foundation, respectively. B. Clerckx was with the Microwave Laboratory, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium. He is now with Samsung Advanced Institute of Technology, Gyunggi-do 449-712, South Korea (e-mail:
[email protected]). C. Oestges and D. Vanhoenacker-Janvier are with the Microwave Laboratory, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium (e-mail:
[email protected];
[email protected]). L. Vandendorpe is with the Communications and Remote Sensing Laboratory, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium (e-mail:
[email protected]). A. J. Paulraj is with the Information Systems Laboratory, Stanford University, Stanford, CA 94305 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TCOMM.2006.885063
II. DESIGN CRITERIA
transmit and reWe consider a MIMO system with ceive antennas, communicating through a frequency flat-fading channel. Over symbol durations, a codeword of size is transmitted through transmit antennas. At the th time instant, transmitted and received signals are related by the following relationship: (1) is the received signal vector, is the where channel matrix, is a zero-mean complex additive white Gaussian noise (AWGN) vector with1 , is the identity matrix of dimension . is an energy normalization factor. The channel is considered as constant over symbol durations. We assume that the instantaneous channel realizations are unknown at the transmitter and perfectly known at the receive side. Maximum-likelihood (ML) decoding is performed so that the receiver computes the estimates of the transmitted codeword according to , where the minimization is performed over all possible codeword vectors . For a given channel realization , the pairwise error probability (PEP) can be written as (2) where
is the Gaussian
-function.
B. “Sum-Wise” Catastrophic Codes and Robust Codes evolves in the Let us first derive how presence of poor scattering conditions at the transmit side. can always be decomposed into The channel matrix
E
1In this letter, stands for expectation, for transposition, for elementwise for conjugate transpose; vec( ) is the operator that forms a conjugation, vector from successive columns of a matrix , : is the absolute value, : is is the trace of matrix , and is the Kronecker the Frobenius norm, Tr product.
0090-6778/$25.00 © 2007 IEEE
fAg
A A jj
A
kk
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the summation of multipath contributions , each contribution generated by a specific scatterer or a cluster of scatterers. Using the narrowband array assumption, we can , where write is the transmit array response in the DOD corresponding to the scatterer . For a horizontal uniform linear array, , with it is given by , being the interelement spacing, the wavelength, and the direction of departure (DOD). In a balanced antenna array, the magnitudes of all elements of are equal and normalized to 1. Definition 1: A MIMO channel is said to be degenerate in the DOD if all scatterers surrounding the transmitter are located along the same direction . Since robust codes should perform well whatever the channel conditions, they should be robust even with very small angle spreads. The worst situation occurs when the channel becomes for degenerate in any direction . In this case, all , and we get
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We show in the simulations that sampling the spatial domain is apparently not sufficient to guarantee good performance. Equation (3) also leads to the next results.2 Proposition 1: In the presence of degenerate channels in a should be DOD , a SIMO transmission based on a code preferred3 to a MIMO transmission if , where is the minimum squared Euclidean distance4 . of Corollary 1: In the presence of a degenerate channel in the DOD , a “sum-wise” catastrophic STC in that direction is always outperformed by a SIMO transmission, whatever its rate. C. Relationships With Transmit Diversity Under Rayleigh Fading Let us show how the previous criterion is related to achieving transmit diversity in Rayleigh fading channels. The PEP averaged over a slow Rayleigh fading channel with spatial correlacan be evaluated as tion matrix
(3) Equation (3) expresses that in poor scattering conditions at the transmit side, the MIMO channel degenerates into a single-input transmitted multiple-output (SIMO) channel, where the . Note that codeword is given by is nothing else than the Euclidean distance between the codewords and . Since an STC designed for i.i.d. channels is only concerned with and , its interaction is not taken into account. So, there is no guarantee with that an STC designed for i.i.d. channels will perform satisfactorily on correlated channels. Definition 2: For a code characterized by the set of codewords , we define
(5) with and the rank of the transmit angle spread
, denoting its nonzero eigenvalues. As reduces to a single DOD , , with the receive correlation matrix. eigenvalues of become null, and the PEP becomes a function of the DOD through remnant nonzero eigenvalues the
(4) (6) We then derive the new notions of “sum-wise” catastrophic and robust codes. Definition 3: A “sum-wise” catastrophic STC in the DOD satisfies . , a robust STC satisfies Definition 4: For some . Focusing on the minimization of the maximum PEP (i.e., optimization of the worst case), the following criterion is proposed based on (3) in order to design codes that perform well in both rich and poor scattering conditions (large and small angle spread at the transmitter). Let us denote by the space of codes that perform well on i.i.d. slow Rayleigh fading channels. with Code-Design Criterion 1: Choose a robust code over all DODs. large Code-design criterion 1 requires the knowledge of the signal constellation, the interelement distance, and the geometry of the transmit antenna array. Contrary to [3], this criterion is not limited to SM and Rayleigh fading. Moreover, it is a continuous function of the angle of departure , which is not the case in [3], where the spatial domain is sampled into virtual directions.
Since the code cannot benefit from transmit diversity on highly correlated channels, its performance is only determined by its coding gain. Therefore, the design of codes robust in poor scattering conditions comes to find codes that maximize the coding gain. This is exactly what design criterion 1 does. Remark 1: In practice, degenerate channels do not exist. However, a lot of channels behave as degenerate at realistic signal-to-noise ratios (SNRs). It is sufficient that the correlation be sufficiently high and/or the SNR sufficiently low. In that , the smaller case, (6) can still be written. The larger the range of SNRs where the code will appear as close to “sum-wise” catastrophic. From (6), we have the following result. Proposition 2: A non-“sum-wise” catastrophic code achieves a transmit diversity order 1 on any MIMO channel at high 2The proofs of several propositions, corollaries, or lemmas are not given in this text, but can be found in [4]. 3To be more rigorous, the distance spectrum should be taken into account. 4With
the same total transmit power.
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SNR.5 A “sum-wise” catastrophic code achieves a null diversity order on degenerate channels for some DODs. D. Full-Rank and Rank-Deficient Codes Let us define smallest eigenvalue of with all pairs of codewords Lemma 1: The minimum of DODs is lower bounded by
as the evaluated over
(8)
. over all
(7) The lower bound is achieved if and only if (iff), for a DOD , lies in the space spanned by the . eigenvectors of corresponding to Lemma 1 leads to the following proposition and criterion. Proposition 3: A full-diversity code is never “sum-wise” catastrophic. A rank-deficient code is “sum-wise” catastrophic in lies in the null space of . the DOD iff with large Code-Design Criterion 2: Choose a code . Unlike design criterion 1, criterion 2 allows designing robust STCs independently of the interelement spacing and the shape of the transmit antenna array. However, as shown in the examples, it is based on a loose lower bound, and can only be applied to full-rank codes. Moreover, it is generally difficult to find . Note codes that have large minimum trace [5] and large that a result similar to design criterion 2 was also derived in [2] in the case of LOS channels. E. Robust and “Sum-Wise” Catastrophic Linear Codes A linear dispersion code (LDC) [13] can be expressed in its , where general form as are complex basis matrices of size , with being the codeword duration relative to the symbol duration, stands for the complex information symbol taken from phase-shift keying (PSK) or quadrature amplitude modulation (QAM) constellations, is the number of complex symbols transmitted over a codeword, and and stand for the real and imaginary parts. SM is a well-known particular case of an LDC, in which independent symbols are transmitted per symbol period through antennas. This code is particularly sensitive to channel correlation, as explained by the next proposition. Proposition 4: The PSK/QAM-based classical SM is “sumwise” catastrophic for a balanced transmit antenna array. The following proposition deals with the ultimate perforunitary basis mance of an LDC based on wide matrices, such that . Proposition 5: For a DOD , a PSK/QAM-based LDC consisting of unitary basis matrices achieves the largest equal to iff the matrices satisfy the conditions with , and where is the minimum squared Euclidean distance of the constellation used. 5i.e.,
k C 0 E) a ()k (E =4 ) 1.
such that (
Corollary 2: In order to satisfy Proposition 5, it is sufficient be pairwise skew-Herthat the unitary basis matrices with . Orthogonal codes [6] mitian are the most robust linear codes based on unitary matrices. Defining the following quantity:
conditions of Proposition 5 can be re-expressed as , which leads to the following design criterion for linear codes based on unitary basis matrices. conCode-Design Criterion 3: Choose the linear code sisting of unitary basis matrices , such that
(9) Unlike design criteria 1 and 2, it does not require the knowledge of the signal constellation used. It does, however, require knowing the shape of the transmit array and the interelement spacing. Corollary 2 leads to the following design criterion. conCode-Design Criterion 4: Choose the linear code , such that sisting of unitary basis matrices (10) This criterion does not require any information about the array or the constellations. Orthogonal codes are the most robust linear codes whatever and the signal constellation the structure of vectors used. This was intuitively predictable. Indeed, the structure of , orthogonal codes is such that with a scalar, indicating that the code excites all directions equally if the transmit antenna array is balanced. In this case, and is independent of . However, the skew-Hermitianity condition of Corollary 2 is sufficient. Depending on the knowledge of the interelement spacing, the signal constellation, and/or the shape of the transmit antenna array, we approach the robustness of orthogonal codes by applying design criteria 1, 2, 3, or 4. It is trivial that the deeper the knowledge we have about the transmit array and/or the constellation used, the more robust the code will be. So, if possible, code-design criterion 1 should always be considered first. From Proposition 5, we also get the following results. Corollary 3: On degenerate channels, an LDC consisting of unitary basis matrices and based on a constellation with a min, is always outperformed imum squared Euclidean distance by an uncoded SIMO transmission based on a constellation with . a minimum squared Euclidean distance larger than This shows that most LDCs will always be outperformed by a SIMO transmission on degenerate channels. Hopefully, propagation conditions vary a lot, so highly correlated channels should be only encountered during a small percentage of time.
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Fig. 1. G rad.
of several SM schemes as a function of the angle of departure
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Fig. 2. BER of several SM schemes in i.i.d. and correlated (in the DOD = 0 and = 0:63) channels with n = 2 and n = 2.
III. EXAMPLES OF CODE DESIGN Since applying a unitary precoder in front of an STC does not modify its performance on i.i.d. channels but affects its performance on correlated channels, a simple way to derive good codes consists of optimizing a unitary precoder in order to max(code-design criterion 1). In this imize the section, we focus on a horizontal uniform linear array with two transmit antennas and quaternary (Q)PSK modulation. We assume an interelement spacing at the transmit side larger than 0.5, so that minimization in design criterion 1 can be done for varying over . The correlated channel is generated following the model developed in [7]. We consider a 2 GHz uplink with broadside configuration at the base station. The mobile station is evolving at 1000 m apart from the base station, in a Rayleigh environment with directional scattering conditions. The mobile station antennas are two half-wavelength dipoles oriented side . At the base station, an intereleby side and separated by is assumed. ment distance of A. Linear Codes of various SM (“classical In Fig. 1, the SM,” SM with “new precoder,” SM with “unitary diagonal precoder” [3], SM with “unitary general precoder” [3]) have been displayed as a function of the angle of departure . The classical SM and the diagonal unitary precoder have the can be quite low, even drawback that for some DODs, the zero for some of them. Those codes are “sum-wise” catastrophic in those directions. This should lead to bad error performance if the dominant component or if all scatterers are oriented in this direction. The general unitary precoder is not “sum-wise” catastrophic, and the new precoder gives the best performance from the point of view of design criterion 1. It should thus lead to good performance, whatever the propagation conditions. are conIn Fig. 2, performance predicted based on the firmed through bit-error rate (BER) simulations. Two array orientations are considered, where all scatterers are mainly located
Fig. 3. G rad.
of several linear codes as a function of the angle of departure
around direction and around direction , corresponding to directions where the classical SM and diagonal unitary precoder are “sum-wise” catastrophic, respectively. Clearly, the diversity achieved by “sum-wise” catastrophic codes can be very poor. Performance match exactly the predic: the larger the of a code in a given tions based on direction, the better its performance in terms of error rate. The SM code with new precoder has the same diversity on correlated channels and i.i.d. channels whatever the DOD, and there is no direction where it has weak performance. It also outperforms codes designed based on the virtual channel model [3]. to full-rank linear In Fig. 3, we apply the concept of the ” [10]. Thanks to their codes “Yao” [8], “Dayal” [9], and “ nonvanishing determinant properties, “Yao” and “Dayal” codes can be shown to be approximately universal [11]. This is a high , we can, moreover, predict that SNR vision. Based on the the “Dayal” code will largely outperform the other codes in correlated channels.
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minimum trace (much larger than “TSC” minimum trace). This motivates the design of STCs based on two criteria: one for uncorrelated channels and one for correlated channels. It is worth noting that the “CYV” code achieves the lower bound (7) for some DODs. While this lower bound is not afis highly affected. fected by the use of a precoder, the This explains why even a code with a small lower bound (7) can perform well in correlated channels. Other design examples can be found [4]. IV. CONCLUSIONS
Fig. 4. G and frame-error rate of several 4-state STTCs in i.i.d. and correlated (in the DOD = =4) channels with n = 2 and n = 4.
B. Space-Time Trellis Codes (STTCs) In i.i.d. channels, assuming a large number of receive antennas in the design of STCs leads to the trace criterion [5]. Codes satisfying this criterion have been shown to outperform previously developed codes based on the rank and determinant criterion [12]. Hence, codes performing well, in terms of the trace criterion, have been chosen and a unitary precoder has been optimized in the sense of design criterion 1. is evaluated for three QPSK 4-state codes: In Fig. 4, the “TSC,” proposed in [12], “CYV,” proposed in [5], and a new proposed code consisting of the “CYV” code combined with an optimized unitary precoder. The “TSC” code presents a good in all DODs. On the other hand, the “CYV” code has can be quite low. The the drawback that for some DODs, proposed code combining the “CYV” and the unitary precoder , compared with the “CYV” code. presents a much better Its performance on correlated channels should then be better. By combining a code optimized following the trace criterion and a unitary precoder designed following design criterion 1, the code should present good performance on i.i.d. channels and also be robust against fading correlations. This is confirmed in Fig. 4 by simulations of the frame-error rate in the presence of i.i.d. and correlated Rayleigh channels for four receive antennas. Each frame consists of 130 symbols out of each transmit antenna. The scatterers are mainly situated in the direction corre. Those direcsponding to an angle of departure equal to tions are such that the simulated codes present their worse performance. We see that the “CYV” code, while performing well on i.i.d. channels, highly suffers from correlations. On the other hand, the degradations undergone by the “TSC” code are much more limited, but its performance on i.i.d. is not that good. The precoded 4-state “CYV” code offers the best performance on i.i.d. and correlated channels. It results from the combination of (that is quite similar to “TSC” ) and a large a large
STC designs commonly assume i.i.d. Rayleigh channels. Real-world channels are, however, correlated, and code performance is affected by the nonideality of those propagation conditions. Design criteria providing robust codes in the presence of correlated channels have been derived. No channel knowledge is assumed at the transmitter. Codes satisfying these criteria have been shown to perform much better on correlated channels than codes designed only for i.i.d. channels. The robustness of various SM schemes, LDCs, and STTCs is predicted in order to illustrate the usefulness of the proposed design criteria. REFERENCES [1] H. Boelcskei, M. Borgmann, and A. J. Paulraj, “Impact of the propagation environment on the performance of space-frequency coded MIMO-OFDM,” IEEE J. Sel. Areas Commun., vol. 21, no. 3, pp. 427–439, Apr. 2003. [2] H. El Gamal, “On the robustness of space-time coding,” IEEE Trans. Signal Process., vol. 50, no. 10, pp. 2417–2428, Oct. 2002. [3] Z. Hong, K. Liu, R. W. Heath, and A. Sayeed, “Spatial multiplexing in correlated fading via the virtual channel representation,” IEEE J. Sel. Areas Commun., vol. 21, no. 5, pp. 856–866, Jun. 2003. [4] B. Clerckx, “Space-time signaling for real-world MIMO channels,” Ph.D. dissertation, Université catholique de Louvain, Louvain-la-Neuve, Belgium, Sep. 2005. [5] Z. Chen, J. Yuan, and B. Vucetic, “Improved space-time trellis coded modulation scheme on slow Rayleigh fading channels,” Electron. Lett., vol. 37, no. 7, pp. 440–441, Mar. 2001. [6] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1456–1467, Jul. 1999. [7] C. Oestges and A. J. Paulraj, “A physical scattering model for MIMO macrocellular wireless channels,” IEEE J. Sel. Areas Commun., vol. 21, no. 5, pp. 721–729, Jun. 2003. [8] H. Yao and G. W. Wornell, “Structured space-time block codes with optimal diversity-multiplexing tradeoff and minimum delay,” in Proc. IEEE Globecom, San Francisco, CA, Dec. 2003, vol. 4, pp. 1941–1945. [9] P. Dayal and M. K. Varanasi, “An optimal two transmit antenna spacetime code and its stacked extensions,” in Proc. Asilomar Conf. Signals, Syst., Comput., Monterey, CA, Nov. 2003, pp. 987–991. [10] 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, no. 3, pp. 753–760, Mar. 2002. [11] S. Tavildar and P. Viswanath, “Approximately universal codes over slow fading channels,” IEEE Trans. Inf. Theory, vol. 52, no. 7, pp. 3233–3258, Jul. 2006. [12] 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, no. 2, pp. 744–765, Mar. 1998. [13] B. Hassibi and B. M. Hochwald, “High-rate codes that are linear in space and time,” IEEE Trans. Inf. Theory, vol. 48, no. 7, pp. 1804–1824, Jul. 2002.
