IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 9, SEPTEMBER 2008 1469 High-Rate Diversity Across Time and Frequency Using Linear Dispersion Jinsong Wu, Member, IEEE, and Steven D. Blostein, Senior Member, IEEE Abstract—To improve performance of orthogonal frequency division multiplexing (OFDM) for fading channels, this paper proposes increasing frequency and time diversity using linear dispersion codes (LDC-OFDM). Methods of LDC-OFDM pro- cessing are proposed for both zero-padding (ZP) and cyclic-prefix (CP) type guard intervals. A two-step-estimation (TSE) decoding strategy is proposed that decouples symbol estimation from LDC decoding. This paper analyzes the upper bound diversity order of LDC-CP-OFDM, which is equal to the full diversity order available in the channels. A criterion for full frequency-time diversity design is derived, a rate-one code is provided and performance is examined through simulations. This paper also investigates LDC-CP-OFDM and LDC-ZP-OFDM performance under imperfect channel estimation and low complexity receiver structures, respectively. In addition, TSE is shown to have performance close to that of full complexity one-step estimation (OSE). Index Terms—Linear dispersion codes, OFDM, diversity, COFDM, equalization, signal estimation, MMSE, SINR. I. I NTRODUCTION I N recent years, multicarrier communications systems, es- pecially those employing orthogonal frequency division multiplexing (OFDM) [1], have received increasing attention for high-data-rate communications in frequency selective fad- ing environments [2]. In practical OFDM system design, it is important to notice that uncoded OFDM cannot provide the same order of diversity as uncoded single-carrier systems in severe frequency-selective fading environments, since the frequency responses of channel space branches differ from one another. One technique to mitigate the above problem is to combine interleaving and forward error correction across all subchannels at the price of reduced bandwidth efficiency, i.e., coded OFDM (COFDM) [3]–[5]. Coding rate is a critical issue related to bandwidth efficiency for high-data-rate transmission. In conventional COFDM, the coding rate is typically less than one, and achieving appro- priate trade-offs between coding rate and error probability is critical to the system design. As a recent alternative to error control coding, linear constellation precoding has been combined with OFDM to maximize achievable frequency diversity and coding gain [6]. However, LCP-OFDM is not Paper approved by C. Schlegel, the Editor for Coding Theory and Tech- niques of the IEEE Communications Society. Manuscript received May 13, 2006; revised June 22, 2007. This work was supported in part by Samsung and in part by NSERC Discovery Grant 41731. This work has been presented in part at the IEEE Int. Conf. on Commun. 2004. J. Wu and S. D. Blostein are with the Department of Electrical and Com- puter Engineering, Queen’s University, Kingston, Ontario, Canada, K7L3N6 (e-mail: wujs@ieee.org, steven.blostein@queensu.ca). Digital Object Identifier 10.1109/TCOMM.2008.060298. able to exploit time diversity over different OFDM blocks in the channels. Recently, Hassibi and Hochwald proposed a high-rate space-time coding framework, known as linear dispersion codes (LDC) [7], which can support arbitrary configurations of transmit and receive antennas. These LDC are designed to optimize the mutual information between the transmitted and received signals. This paper proposes and analyzes a new high- rate LDC approach to jointly achieve both frequency diversity and time diversity (LDC-OFDM). The paper is organized as follows: LDC is defined in Section II. In Section III, the construction of an LDC-OFDM block is proposed. The proposed TSE based LDC-OFDM system is discussed in Section IV, and the proposed receiver structure is illustrated. An analytical discussion of diversity properties of LDC-OFDM is given in Section V, and then a rate-one full diversity LDC-OFDM design is provided. Performance analysis and comparison is presented in Section VI. The following notation is used:(·) † denotes matrix pseu- doinverse, (·) T matrix transpose, (·) H matrix transpose con- jugate, I K denotes identity matrix of size K × K, 0 m×n denotes zero matrix of size m × n, A ⊗ B denotes Kronecker (tensor) product of matrices A and B, C m×n denotes a complex matrix with dimensions m × n, and F M denotes the discrete Fourier transform (DFT) matrix, representing the M -point fast Fourier transform (FFT) with entries, [F M ] a,b =  1/ √ M  exp (−j 2π(a − 1)(b − 1)/M ) . II. LINEAR DISPERSION CODING AND ITS MATRIX FORM A. Definition of Linear dispersion codes Assume the data sequence has been modulated using complex-valued symbols chosen from an arbitrary, e.g. r-PSK or r-QAM, constellation. A linear dispersion code (LDC), S LDC , was first defined for multi-input, multi-output (MIMO) systems with M transmit antennas, N receive antennas, T channel uses and Q source constellation symbols as [7] S LDC = Q  q=1 α q A q + jβ q B q (1) where the LDC matrix is S LDC ∈ C T ×M , A q ∈ C T ×M , B q ∈ C T ×M ,q =1, ..., Q are called dispersion ma- trices, which transform data symbols into a space-time matrix. The constellation symbols are defined by s q = α q + jβ q ,q = 1, ..., Q. This paper applies LDC to multicarrier systems, and the data symbol coding rate of LDC in such systems is defined as R sym LDC = Q MT . 0090-6778/08$25.00 c  2008 IEEE Authorized licensed use limited to: Queens University. Downloaded on May 2, 2009 at 19:44 from IEEE Xplore. Restrictions apply.