Distributed Space-Time-Frequency Block Codes for Multiple-Access-Channel with Relaying Onur O˘ guz, Abdellatif Zaidi, J´ erˆ ome Louveaux and Luc Vandendorpe Communications and Remote Sensing Laboratory, Universit´ e catholique de Louvain 2, place du Levant, B-1348 Louvain-la-Neuve, Belgium, {oguz,zaidi,louveaux,vandendorpe}@tele.ucl.ac.be Abstract—In this paper, we investigate diversity gain in coding for a 2-user multiple-access-channel (MAC) with cooperating transmitters–the MAC with relaying. We propose a simple dis- tributed space-time-frequency block coding (D-STFBC) scheme and analyze the offered diversity gain. In particular, we show that full diversity (order 3) is possible if collaboration is well enough and rigorous signal processing is assumed both at the transmitters and the receiver. Bit-error-rate (BER) analysis and curves are provided for illustrative purposes. I. I NTRODUCTION The exploding demand for efficient high-quality and volume of digital wireless communications, along with high mobility, are driving recent developments in communication technolo- gies for broadband wireless systems. Mitigating signal fading and co-channel interference (which are the main impairments that transmission might be subject to in real contexts) are among the main challenging problems. Diversity techniques [1] (be it spacial or due to multiple-antenna, time or frequency) play a central role in this trend. For example, it is by now of common knowledge that space-time (ST) codes can offer dramatic performance gain in multiple-input multiple- output (MIMO) systems [2]- [4]. Also, to cope with inter- symbol-interference (ISI) caused by multipath propagation, orthogonal frequency-division multiplexing (OFDM) has been utilized in conjunction with ST, space-frequency (SF) or the more involved space-time-frequency (STF) block codes [5]- [8]. More recently, cooperative diversity schemes in which multiple terminals in a network cooperate to realize spatial diversity gain in a distributed manner have been introduced [9]- [11]; and various coding techniques have been proposed (and analyzed). In particular, efficient block coding techniques based on either ST, SF or STF, have been reported [11]- [15]. It is demonstrated that, for channels with multiple relays, cooperative diversity with appropriately designed codes can offer full spatial diversity gain [13]- [15]. This work builds upon the classical 2-user multiple-access channel (MAC) with relaying, which nicely models two co- operating transmitters communicating with a common access point or next hop in a wireless network. We examine the problem of creating and exploiting additional degree of di- versity gain by an appropriate design of transmitted signals (for each of the two users) in terms of space (through signalling), time (through two-fold repetition) and frequency This work is supported by the European project COOPCOM and the national project COSMOS. (through circular shift in allocated OFDM subchannels (most like [16])). As a convenient by-product of this (space-time- frequency) code design third-order diversity is achieved, by creating and combining additional time/frequency degrees of freedom while maintaining full spatial (i.e., cooperative) diver- sity well established (by means of the well known Alamouti scheme [3]). Time diversity gain is brought up by having each user sending his information at two successive time transmit periods. Cooperative diversity is allowed by having each user sending both his own information and also other user’s information (learned in the signalling period) at the second transmit period. The additional frequency diversity gain is brought up by allowing each information symbol to be carried over two (nearly) uncorrelated subchannels (i.e., carriers). We shall mention that it is the (here appropriate) combination of these three (type of) diversity mehods that achieves third-order diversity. We provide performance analysis of the proposed scheme through derivation of pair-wise error probability and illustrate the corresponding improvement through Monte-Carlo based simulations. The remaining of this paper is organized as follows. Section II describes our system model for the 2-user MAC with relaying under consideration. Section III characterizes the performance of the proposed protocol in terms of offered diversity gain. Section IV illustrates the results and provides some comparisons from a number of perspectives, and Section V draws some concluding remarks. Notation: All bold face letters indicate vectors (lower case) or matrices (upper case). For a vector x, x (i) denotes its i th element. For a matrix X, diag(X) designates the vector formed by its diagonal elements and X| n m designates the inner square matrix having m th to n th diagonal elements of X on its diagonal. Notations ¯ (·), (·) T and (·) H denote conjugate, transpose, and conjugate transpose operations, respectively. We use Q, H xy , and Λ xy = QH xy Q * to denote the N × N normalized FFT matrix, the N × N circulant channel matrix (relative to link X → Y ) and the corresponding diagonal subchannel gain matrix. II. SYSTEM MODEL Consider the three-terminal wireless network shown in Fig.1. Both terminals A and B send private information and also assist each other (through relaying) communicate with the destination terminal D. This forms a 2-user multiple-access channel (MAC) with relaying, which might correspond to 2