Full Diversity Block Diagonal Codes for Differential Space-Time-Frequency Coded OFDM Qian Ma, Cihan Tepedelenlio˘ glu Telecommunications Research Center Dept. of Electrical Engineering Arizona State University Tempe, AZ 85287-7206 USA {qian.ma, cihan}@asu.edu Zhiqiang Liu Dept. of Electrical and Computer Engineering The University of Iowa Iowa City, IA 52242 USA zhiqiang-liu@uiowa.edu Abstract— Focusing on Orthogonal Frequency Division Multi- plexing (OFDM) transmissions over frequency selective Rayleigh fading channels, we consider full diversity block diagonal codes for differential space-time-frequency (DSTF) coded OFDM with no channel state information (CSI). Resorting to subcarrier grouping, we convert the system into a set of DSTF systems, within which DSTF coding is considered. Through Pairwise Error Probability (PEP) analysis, the code design criteria are derived over correlated channels and shown to be not dependent on the channel correlations. In [7], it is proved that full diversity gain can be achieved by appropriately using the existing unitary diagonal Space-Time (ST) codes without CSI over the frequency selective channels. We show here that, at higher data rates there exist fuller non-diagonal matrices that have the necessary unitary and full diversity properties, and outperform the diagonal codes in [7]. Two design methods are proposed for block diagonal codes with full diversity which are based on existing designs for flat fading channels. I. I NTRODUCTION Utilizing multiple transmit antennas, space-time (ST) coding has evolved as one of the most promising transmit diversity techniques [1]. Multipath diversity is available when frequency selectivity is present, which is the typical situation for broad- band wireless channels. Because Orthogonal Frequency Divi- sion Multiplexing (OFDM) converts the frequency selective channel into a set of flat fading subchannels, it makes sense to combine ST coding with OFDM over frequency selective channels [2]. As either the number of antennas or the fade rate increases, it is desirable to develop techniques that do not require channel state information (CSI) at the receiver, for multiantenna sys- tems operating over frequency selective channels. A technique to cope with unknown channels is differential modulation. Hughes [5], and Hochwald and Sweldens [6] independently propose a differential unitary ST modulation scheme for frequency flat fading channels. In [10], all full spatial diversity groups of finite order are classified and as a result, many excellent-performing group constellations are found. The infi- nite groups with full spatial diversity are classified in [11]. Using a modified probabilistic information-theoretic design criterion, Cayley codes are presented in [12]. Reference [13] designs unitary constellations with good diversity products for both high and low SNR channels. Treating each subchannel as a transmit antenna, Liu and Giannakis [3] propose a differen- tial encoding scheme for single-antenna OFDM transmissions over frequency selective channels with maximum multipath diversity. Recently, B¨ olcskei and Borgmann have proved full space and multipath diversity can be achieved in the noncoher- ent case, where simultaneous coding over space and frequency is proposed [4]. Addressing maximum multipath and spatial diversity, similar work in differential ST coding over frequency selective channels has been done in [9]. In [7], we have proposed a novel differential scheme which performs joint coding over space, time and frequency and achieves maximum diversity gain by appropriately using the existing unitary diagonal ST codes without CSI over frequency selective channels. Taking into account the time domain processing and incorporating subcarrier grouping enable a much lower complexity in code design and decoding than noncoherent space-frequency (SF) coding in [4]. In this pa- per, we consider the unitary non-diagonal codes, which are block diagonal and have better performance compared to their diagonal counterparts in [7]. Based on existing designs for flat fading channels, two design methods for block diagonal codes with full diversity are proposed. Also we discuss the code design criteria over correlated channels while existing works in noncoherent ST communications only consider uncorrelated channel taps for simplicity. II. SYSTEM MODEL We consider a multiantenna wireless communication system with N t transmit antennas and N r receive antennas, where OFDM with N c subcarriers is employed at each antenna. The fading channel between the μth transmit antenna and the ν th receive antenna is assumed to be frequency selective. The channel at time τ between the μth transmit antenna and the ν th receive antenna is described by the discrete- time baseband equivalent impulse response vector h τ μν := [h τ μν (0), ..., h τ μν (L)] T ∈ C (L+1)×1 , with L representing the channel order. The transmitted sample matrix at subcarrier p is defined as X(p) with [X(p )] τμ = x μ τ (p), where x μ τ (p) is the data symbol transmitted on the pth subcarrier from the μth transmit antenna through the τ th OFDM symbol interval. We also define the GLOBECOM 2003 - 868 - 0-7803-7974-8/03/$17.00 © 2003 IEEE