# Co-Efficient Vector Based Differential Distributed Quasi-Orthogonal Space Time Frequency Coding
## 1 Introduction
However, in contrast, for practical scenarios where the relay nodes are compelled to differentially decode and re-transmit information signals received from the source node, the co-efficient vector design is better suited. The co-efficient vector design reflects its utilisation specifically for applications like Adhoc and sensor networks that require inherent data correlation at intermediate nodes. The integration of differential schemes with DSTFC in broadband networks is even more challenging because of the presence of multiple paths in frequency-selective fading channels and multiple broadcast phases between the source and destination. Furthermore, all the aforementioned works employ orthogonal codes and are thus only capable of providing full code-rate and full diversity for a maximum of two-relay cooperative networks. In order to cater for high data rate in cooperative networks, our proposed scheme employs rotated constellation quasi-orthogonal codes, which achieves full code-rate for any number of relay nodes. In addition, we show how the quasi-orthogonal codes are constructed from a proper choice of constellation sets. Moreover, the full mapping scheme and differential recipe for utilizing co-efficient vectors in cooperative networks are presented. Motivated by all the aforementioned, we propose co-efficient vector differential distributed quasi-orthogonal space-time-frequency coding (DQSTFC) for cooperative networks. This study thus aims to make the following contributions:  
* Due to the involvement of multiple broadcast phases in cooperative networks, our proposal of designing the STF codes at the source node aims to simplify the operation of relay nodes. Here, the codewords are carefully distributed in the temporal and frequency dimensions to relax the assumption that the channel is quasi-static in the time and frequency domains for long symbol periods;* We provide a systematic construction of full rate quasi-orthogonal codes that can exploit full STF diversity and non-coherent signal detection. The full differential procedure using our proposed co-efficient vector design is also provided in this work. Through pair-wise error probability analysis, we derive the necessary conditions for our code to achieve full STF diversity while the coding gain is maximized as much as possible;
* The simulation carried out also provides an analytical study of the performance of our co-efficient vector differential DQSTFC scheme in frequency-selective fading and time-selective fading environments. We also compare and distinguish between the unitary matrices and co-efficient vector designs in terms of Bit Error Rate (BER) performance thus highlighting the robustness of our proposed non-coherent scheme against highly selective fading environments. In addition, we generalize the co-efficient vector design to cooperative networks with four, six, and eight relay nodes. The remaining part of this paper is organised as follows: Section 2 presents the cooperative system and channel models and showing how the STF codes are designed at the source node and the relay nodes. Furthermore, Section 3 covers the differential encoding and decoding procedure using our proposed co-efficient vector design while Section 4 presents the pair-wise error probability analysis. Finally, Section 5 presents some simulation results and Section 6 contains the conclusion.