## 2 Method
### System Description
The transmitted PAM signals are as follows:  
\[S(t)=\left[ \begin{array}{c}s_{1}\\ s_{2}\end{array} \right]=\sqrt{\frac{E_{s}}{N_{T}\cdot N_{slot}}}\left[ \begin{array}{cc}S_{1}&-S_{2}\\ S_{2}&S_{1}\end{array} \right]\left[ \begin{array}{c}p(t)\\ p(t-T_{d})\end{array} \right]=\sqrt{\frac{E_{s}}{4}}\left[ \begin{array}{c}S_ {1}p(t)-S_{2}p(t-T_{d})\\ S_{2}p(t)+S_{1}p(t-T_{d})\end{array} \right], \tag{1}\]  
\[\mbox{where}\int\limits_{2T_{d},}{s_{1}}^{2}(t)dt=\int\limits_{2T_{d}}{s_{2}}^ {2}(t)dt=\frac{E_{s}}{2},\]  
where \(S_{j}\) is the transmitted signal of transmitter \(j\), \(T_{d}\) is the transmitting signal duration, and the value of \(T_{d}\) is usually much larger than the pulse duration. \(E_{s}\) is the average transmitted energy per symbol. In a BPAM, each symbol takes only two values \(S=\{+1,-1\}\) and is assumed to be independent and identically distributed (i.i.d.). The second derivative of the Gaussian waveform is used here as the transmit waveform \(p(t)\) at the nanosecond scale. The second derivative of Gaussian waveform \(p(t)\) can be described by the following expression:  
\[p(t)=\left(1-\frac{t^{2}}{2\sigma^{2}}\right)\exp\left(-\frac{t^{2}}{2\sigma^{ 2}}\right) \tag{2}\]  
where \(\sigma\) is the standard deviation of the Gaussian wave  
The channel state information (CSI) is available at the receiver, which is the channel impulse response between a receiving antenna and a transmitting antenna. A ray-tracing channel model was developed to calculate the channel state information for the UWB-MIMO system. The ray-tracing technique can deal with radio wave propagation in any environment. A ray-tracing approach based on frequency domain was applied to the deterministic simulation for the indoor channel. A pincushion of rays was sent out from the transmitter and traced over a given environment until either it had reached the receiver or it had lost enough power, and its contribution to the received signal was negligible. During transmission, the ray was transmitted, reflected, or diffracted before reaching the receiver. The receiver collected a narrowband signal around the given frequency range and converted the signal to a complex equivalent baseband signal. The goal of channel modeling was to determine the impulse response for any given transmitter-receiver location in the system. \[h_{ij}(t)=\sum\limits_{L=1}^{N_{L}}\beta_{ij,L}\delta(t-\tau_{ij,L}), \tag{3}\]  
where \(h_{ij}\) is the channel impulse response between a receiving antenna \(i\) and a transmitting antenna \(j\), \(L\) is the path index, \(\beta_{ij,L}\) and \(\tau_{ij,L}\) are the amplitude and the time delay of \(L\)-th path between receiving antenna \(i\) and transmitting antenna \(j\). \(N_{L}\) is the number of paths. The impulse response with input transmitted signal \(S(t)\) and received signal \(Y(t)\) is described by convolution. The corresponding received signals are as follows:  
\[Y(t)=\left[ \begin{array}{c}y_{1}\\ y_{2}\end{array} \right]=\left[ \begin{array}{c}h_{11}*S_{1}+h_{12}*S_{2}\\ h_{21}*S_{1}+h_{22}*S_{2}\end{array} \right], \tag{4}\]  
where * denotes the convolution operator. After receiving the signals through the channel, they are processed with a correlator in receivers with rake finger \(N_{F}\). With rake receivers, the ability to improve the received signal can be increased in multi-path fading channels. The different signal paths propagated by different paths are combined by rake receivers [31; 32; 33]. These programs produce time diversity. Combining different signal components increases the signal-to-noise ratio (SNR) and also improves the link performance. The rake receiver is designed to compensate the UWB channel's multipath. It is assumed that the receiver noise \(n(t)\) at the corresponding time instant, including the effect of correlation, is Gaussian white noise with zero mean and variance \(N_{0}/2\).