# Co-Efficient Vector Based Differential Distributed Quasi-Orthogonal Space Time Frequency Coding
## 1 Introduction  
The fundamental idea behind space-time-frequency (STF) coding in cooperative networks is to provide a scheme through which spatial, temporal, and frequency coding is exploited. This has led to the design of distributed space-time-frequency coding (DSTFC) schemes. In DSTFC schemes, elements of the codeword matrices are simultaneously transmitted via relay nodes across multiple orthogonal frequency-division multiplexing (OFDM) time slots and frequency sub-carriers. The principal objective is to design a coding scheme that mitigates the performance degradation experienced by distributed space time coding-orthogonal frequency division multiplexing (DSTC-OFDM) schemes and distributed space-frequency coding (DSFC) schemes in severe time-selective and frequency-selective fading channels, respectively. Furthermore, in the DSTC-OFDM schemes of [1, 2, 3] and the DSFC schemes of [4, 5, 6, 7, 8], for example, the channel gain on adjacent OFDM [9, 10] time slots and frequency sub-carriers, respectively, is assumed to remain quasi-static to facilitate signal recovery at the destination. This assumption is impractical when signals with long symbol duration are transmitted or in cases involving large numbers of cooperating nodes. However, in the scheme proposed by this study, it is shown that DSTFC can be designed to mitigate this problem by simultaneously transmitting across OFDM time-slots andfrequency sub-carriers. Therefore, this ensures that the requirement for constant channel gain in the temporal and frequency dimensions is more relaxed. In order to implement cooperation in multi-hop networks generally, the most popular relaying protocols currently being used are the Decode-and-Forward (DF) [11; 12] and Amplify-and-Forward (AF) [13; 14] protocols. In the bid to cater for highly mobile environments where the nodes are unable to acquire channel state information (CSI), research efforts have been focused on the implementation of differential schemes for data transmission in non-coherent cooperative networks [15]. For example, a randomized differential scheme was proposed for a partially coherent cooperative network in [11] where the destination has partial CSI as opposed to full CSI. In other words, the source node is assumed to have CSI for the source-relay link while that of the relay-destination link cannot be acquired. The research in [16] provides the differential strategy for non-constant modulus constellations whereby the differential procedure is dependent on both amplitude and phase transitions. Furthermore, in their work, they focused on the reduction of the computational complexity of the differential procedure. Similarly, the studies carried out in [15; 17; 18] consider differential orthogonal coding schemes where neither the relays nor the destination are able to acquire CSI. Fundamentally, when differential strategies are incorporated with coding schemes, a principal aspect is how to generate the differentially encoded symbols and, subsequently, the transmission matrix (at the source and relay nodes, respectively) in such a manner that would ensure their recovery at the destination without the need for CSI. Differential strategies can be implemented in cooperative networks through the utilisation of differential unitary matrix design as adopted by the authors in all the aforementioned works [17; 19; 20; 21; 22]. In contrast to existing works in the literature [23], differential strategies can also be implemented in cooperative networks through the use of a differential co-efficient vector design, which is being proposed in this study. It is evident from [17; 24; 25] that differential distributed space time frequency coding (DSTBC) schemes adopting the unitary matrix design are amenable to the AF protocol. This is because the relay nodes are generally required to forego CSI acquisition such that differential decoding is only performed at the destination.