## 2 Method  
### System Description  
UWB combined with MIMO is a viable way to achieve high data rates in wireless communication. The authors focused on the application of RODs, because they are key to achieving low BER and cost reduction in MIMO-UWB systems. A plurality of complex orthogonal designs for narrowband is available in the literature [20; 24]. Figure 1 shows the basic idea of a binary pulse amplitude modulation (BPAM) for a UWB system consisting of \(N_{T}\) transmitting and \(N_{R}\) receiving antennas with \(N_{F}\) fake fingers. Each information symbol is being conveyed through monocycle pulses with a period of \(T_{d}\) and with different polarities. A frame is associated with a set of slots. Multi-antenna systems provide a diversity order without any feedback signal from the receiver to the transmitter. Diversity techniques are used to mitigate degradation in the error performance of the radio channel without using additional radiated power or spectral bandwidth [29]. The transmit symbols are applied to the ROD [20; 21], where each symbol with a slot time \(T_{S}\) seconds is pulse-shaped by the second order of Gaussian waveform in the transmitters. The received signal \(Y(t)\) passes through a matching filter whose impulse response is \(h_{ij}(t)\) and the sampled signal in the first frame is divided into two branches. These signals are fed to the rake receiver and combined with the MRC algorithm, producing signals \(r_{1}(t)\) and \(r_{2}(t)\) respectively. Finally, the transmitted symbols \(S_{1}(t)\) and \(S_{2}(t)\) can be independently decoded from the output \(r(t)\). The same procedure is repeated for the next frame. The signals \(\hat{S}(t)\) are summed up to collect all the diversities provided by multiple receive antennas, multi-time slot, and multipath. The figure shows that the inherent diversity of UWB signals in the temporal domain is collected by the rake receiver, while the diversity in the spatial domain is harvested by the use of the space-time code (STC). It is beneficial to employ an ROD code for collecting more diversities. An ROD of size \(n\) is an \(n\) by \(n\) orthogonal square matrix with entries in the indeterminate \(\pm S_{i}\) the rows of the matrix represent antennas, whereas its columns represent transmission time slots [24; 30]. The ROD of size 2 is as shown below:  
\[\left[ \begin{array}{cc}S_{1}&-S_{2}\\ S_{2}&S_{1}\end{array} \right].\]  
Figure 1: System block diagram. \(S_{1}\) and \(S_{2}\) are two information symbols transmitted simultaneously from two transmitted antennas (\(N_{T}=2\)). Then, the authors attempted to detect the information symbols \(S_{1}\) and \(S_{2}\) based on the consecutive time slot 1 and time slot 2 (\(N_{slot}=2\)). Finally, they combined all the spatial diversities provided by the receiving antennas (\(N_{R}=2\)) and the signal components that propagated through the channel by different paths. The ROD codes were developed, and the decoding algorithm was given. Based on the Hurwitz-Radon theorem, the full rate can be achieved by using RODs of size 8, that is, with eight transmitted antennas [20; 24]. The ROD of size 8 is as shown below:  
\[\left[ \begin{array}{cccccccc}S_{1}&-S_{2}&-S_{3}&-S_{4}&-S_{5}&-S_{6}&-S_{7}& -S_{8}\\ S_{2}&S_{1}&-S_{4}&S_{3}&-S_{6}&S_{5}&S_{8}&-S_{7}\\ S_{3}&S_{4}&S_{1}&-S_{2}&-S_{7}&S_{8}&S_{5}&S_{6}\\ S_{4}&-S_{3}&S_{2}&S_{1}&-S_{8}&-S_{7}&-S_{6}&S_{5}\\ S_{5}&S_{6}&S_{7}&S_{8}&S_{1}&-S_{2}&-S_{3}&-S_{4}\\ S_{6}&-S_{5}&S_{8}&-S_{7}&S_{2}&S_{1}&S_{4}&-S_{3}\\ S_{7}&-S_{8}&-S_{5}&S_{6}&S_{3}&S_{4}&S_{1}&S_{2}\\ S_{8}&S_{7}&-S_{6}&-S_{5}&S_{4}&-S_{3}&-S_{2}&S_{1}\end{array} \right]\]  
For this paper, an impulse radio with a BPAM was applied. Time-modulated UWB communication is based on the discontinuous emission of a very short waveform as a train of pulses. Each pulse has ultra-wide spectral requirements in the frequency domain. Because the pulse propagates well in the radio channel, it does not require additional carrier modulation. This radio concept is referred to as impulse radio.