IEEE COMMUNICATIONS LETTERS, VOL. 6, NO. 12, DECEMBER 2002
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Asymptotic Performance of Space-Frequency Codes Over Broadband Channels Jibing Wang, Ezio Biglieri, Fellow, IEEE, and Kung Yao, Fellow, IEEE
Abstract—In this letter we show that when the number of receive antennas is large, the Euclidean distance among codewords dominates the performance of space-frequency codes, the same result as for space-time codes. Therefore, in the presence of a large number of receive antennas, space-frequency codes can be optimized by using the Euclidean-distance criterion valid for additive white Gaussian noise channels. Simulation results show that this conclusion is also valid when the number of antennas is small. Index Terms—Euclidean distance, MIMO OFDM, space-frequency coding.
I. INTRODUCTION
A
N EFFECTIVE approach to increasing the data rate as well as the power efficiency over wireless channels consists of introducing temporal and spatial correlation into signals transmitted from different antennas (see, e.g., [1] and references therein). In [2] and [3] it was shown that, for coherent systems over quasi-static flat fading channels, the design of space-time (ST) codes for asymptotically high signal-to-noise ratio (SNR) scenarios should follow the rank-and-determinant criterion (see below). Later on, several researchers have pointed out that the Euclidean-distance criterion is more appropriate when the number of receiver antennas is large (see, e.g., [6], [7] and references therein). Most work assumes flat-fading channels [2]. However, in broadband wireless communications, frequency selective channels are often encountered. Concatenation of space-time coding with orthogonal frequency-division multiplexing (OFDM) has gained much interest recently (see, e.g., [4]). It has been shown that properly designed space-frequency (SF) codes, which amount to coding across space and frequency, can achieve both spatial diversity and frequency diversity [4], [5]. For asymptotically high SNRs, the design criteria for SF codes differ significantly from those for ST codes over flat fading channels [4]. However, when the number of receive antennas is large, the frame error rates of interest are achieved at very small SNRs, so that the design criteria of [4] may not be optimal. In this letter, by drawing an analogy between SF and ST codes, we show that when the number of receive antennas is large, the minimum Euclidean distance among code words Manuscript received July 24, 2002. The associate editor coordinating the review of this letter and approving it for publication was Dr. R. Blum. This work is supported in part by NASA/Dryden under Grant NCC2-374 and by the University of California CoRe grant sponsored by ST Microelectronics, Inc. J. Wang and K. Yao are with the Electrical Engineering Department, University of California, Los Angeles, CA 90089 USA (e-mail:
[email protected]). E. Biglieri is with the Dipartimento di Elettronica, Politecnico di Torino, Turin 101029, Italy. Digital Object Identifier 10.1109/LCOMM.2002.806463
dominates the performance of SF codes. Therefore, SF codes can be optimized by using the Euclidean-distance criterion valid for AWGN channels. Simulation results are given to show that the results valid for a number of receive antennas tending to infinity still provide correct indications when the number of antennas is small. The following notations are adopted throughout this letter: scalars are denoted in lower case, vectors are column vectors unless otherwise indicated and are denoted in lower case bold, denotes the conjugate while matrices are in upper case bold. and denote transpose and transpose of the matrix , and is the elementwise conjugate of , respectively. identity matrix. II. CHANNEL MODEL AND PEP Consider a multiple-input multiple-output (MIMO) fretransmit quency-selective multipath fading channel with receive antennas. Assume there are significant paths and in the channel. The input–output relation is given by [4] (1) denotes the where is the discrete-time index, transmitted signal vector and the received signal is an additive white Gaussian vector, respectively. is an complex random matrix modnoise vector. eling the th “tap” of the MIMO; fading channel impulse reare i.i.d. circusponse. We assume that the elements of larly-distributed complex Gaussian random variables with mean zero and variance , and furthermore that the elements of are uncorrelated with the elements of when . To inantennas from each sure that the total power received by varies, we make the transmit antenna remains constant as following normalization: (2) Note that (2) is only a normalization which is relevant only as the number of antennas grows to infinity. Otherwise, it can be easily introduced/removed by redefining the SNR. In MIMO OFDM systems, the data streams are first modulated by OFDM modulators. Let denote the block length. For the th transmit . antenna, the data symbol transmitted on the th tone is Define the frequency response of the channel as
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IEEE COMMUNICATIONS LETTERS, VOL. 6, NO. 12, DECEMBER 2002
We further pack the data symbols on the th tone together . It can be into a vector shown that the following input–output relation holds [4]: (4) is the reconwhere is structed data vector at the receiver for the th tone, the channel frequency response at the th tone given by , and is additive white Gaussian noise . By stacking the satisfying vectors according to
SNRs. In this scenario, from the bound in (9) it follows that dominates the performance. After some manipulations, we can is actually equal to the squared Euclidean distance show that between the codewords and (10) The above considerations lead us to conclude that, for a very large number of receive antennas, the sensible design criterion for SF codes is the maximization of the minimum Euclidean distance between any pair of codewords. III. ASYMPTOTIC PERFORMANCE
we obtain an input–output relation in matrix form: (5) diag where We assume that the channel-state information (CSI), that is, the realization of , is perfectly known at the receiver. For maximum-likelihood (ML) detection, the pairwise error probability is given by (6) where
We now analyze the performance of SF codes when the of receive antennas grows to infinity while the number of transmit antennas is finite, and when both number tends to a numbers grow to infinity but their ratio . We derive results similar to those in [6] using constant the following lemmas. is a random matrix with i.i.d. comLemma 1: For each , . plex Gaussian entries with mean zero and variance , the elements of are i.i.d. Proof: For all . According to our assumptions, the elements of are uncorrelated with the elements of , if . The lemma and the follows from the definition normalization (2). Lemma 2 [8]: If, As , a.s. and a.s. a.s. then ,
A. Asymptotic Pairwise Error Probability, Finite A Chernoff bound on PEP was derived in [4], and has the form
From Lemma 1 and the strong law of large numbers we have [6]
(7) where
a.s. as For each tone, as
(11)
almost surely we have (12)
(8) with and Following [7], we have
, , , denotes the Kronecker product.
and hence, using Lemma 2
(9) , is the product of the nonzero where , and , the “diversity advantage”, eigenvalues of Under the assumption of is given by asymptotically high SNRs, the design criteria advocated in [4] are based on the rank-and-determinant criterion [2], [3]. The rank criterion consists of maximizing the minimum rank for any codeword pair and , and the determinant criterion of for any codeword pair maximizing the minimum product and . On the other hand, for a large number of receive antennas, the frame error rates of interest are typically achieved at small
(13) Thus, we have shown that as the number of receive antennas grows to infinity, the pairwise error probability (PEP) of spacefrequency codes depends only on the Euclidean distances between pairs of the code words. B. Asymptotic PEP,
,
From [6], as are
, the matrices a.s. asymptotically
and free
WANG et al.: ASYMPTOTIC PERFORMANCE OF SF CODES OVER BROADBAND CHANNELS
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We compare the performance of the CHL code with a spacefrequency code for OFDM [4] employing the 4-state, rate-1/2 space-time code (TSC code) of [2]. The figure of merit is bit error probability, which we average over independent fading channel realizations, each realization consisting of one OFDM symbol. Fig. 1 shows the bit error probability versus SNR for the CHL code and the TSC code. We can see that CHL outperforms TSC scheme, the gap between the two increasing with the number of receive antennas. V. CONCLUSIONS In this letter, which extends results in [6] to the frequency-selective fading channel, we observe that with a large number of receive antennas the performance of space-frequency codes is determined by the Euclidean distance between pairs of code words. Therefore, in the presence of a large number of receive antennas, space-frequency codes can be optimized under the Euclidean-distance criterion valid for AWGN channels. Simulation results show that this conclusion is also valid when the number of antennas is small.
Fig. 1. Bit error probability versus SNR for CHL code and TSC code.
[9, p. 146 ff.]. Therefore, under the assumption that is finite
REFERENCES
(14) which yields the same result as for
finite.
IV. SIMULATION RESULTS We provide some simulation results that corroborate the theoretical analysis and show how this yields conclusions that are valid even for a small number of antennas. We assume there are 3 resolvable paths in the frequency-selective fading channel of fast with uniform power delay profile. The block length Fourier transform is assumed to be 128. The number of transmit is equal to 2, while the number of receive antennas antennas is chosen to be 2, 4, or 8. A binary, 4-state, rate-2/4 convolutional code (CHL code) optimized for the AWGN channel [10] is used as space-frequency codes in the simulation. The outputs of the convolutional codes are mapped onto QPSK constellations and then transmitted through different antennas and tones.
[1] E. Biglieri, G. Caire, and G. Taricco, “Recent results on coding for the multiple-antenna transmission systems,” in Proc. IEEE ISSSTA 2000, Sept. 2000, pp. 117–121. [2] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: Performance criterion and code construction,” IEEE Trans. Inform. Theory, vol. 44, pp. 744–765, Mar. 1998. [3] J.-C. Guey, M. P. Fitz, M. R. Bell, and W.-Y. Kuo, “Signal design for transmitter diversity wireless communication systems over Rayleigh fading channels,” in Proc. IEEE VTC ’96, pp. 136–140. [4] H. Bölcskei and A. J. Paulraj, “Space-frequency coded broadband OFDM systems,” in Proc IEEE WCNC 2000, Sept. 2000, pp. 1–6. [5] B. Lu and X. Wang, “Space-time code design in OFDM systems,” in Proc IEEE GLOBECOM 2000, Dec. 2000, pp. 1000–1004. [6] E. Biglieri, G. Taricco, and A. Tulino, “Performance of space-time codes for a large number of antennas,” IEEE Trans. Inform. Theory, vol. 48, pp. 1794–1803, July 2002. [7] D. Aktas, H. E. Gamal, and M. P. Fitz, “Toward optimal space-time coding,” IEEE J. Select. Areas Commun., submitted for publication. [8] P. Billingsley, Convergence of Probability Measures. New York: Wiley, 1999. [9] F. Hiai and D. Petz, The Semicircle Law, Free Random Variables and Entropy. Providence, RI: American Math. Soc., 2000. [10] J.-J. Chang, D.-J. Hwang, and M.-C. Lin, “Some extended results on the search for good convolutional codes,” IEEE Trans. Inform. Theory, vol. 43, pp. 1682–1697, Sept. 1997.
