## 5 BER Analysis  
Pulse shape modulation is more useful for M-ary communication where the individual orthogonal pulses are added together and transmitted as a composite pulse. The typical M-ary PSM symbol for the \(k\)th user for the \(i\)th symbol can be expressed as [56]:  
\[S_{i}^{(k)}(t)=\sqrt{E_{tx}^{(k)}}\sum_{j=0}W_{i}(t-jT_{f}-c_{j}^{(k)}T_{c}) \tag{4}\]  
where \(W_{i}(t)\) is the orthogonal pulse, \(T_{f}\) is the pulse repetition interval, \(c_{j}^{(k)}\) is the pseudo random code for the \(k\)th user in the \(j\)th pulse repetition interval, \(T_{c}\) is the chip duration, and \(E_{tx}^{(k)}\) is the transmitted power of the \(k\)th user. Figure 4: Proposed receiver architecture for the UWB system using PSM. Figure 3: Proposed transmitter architecture for the ultra-wide band (UWB) system using pulse shape modulation (PSM). The multipath channel model proposed by the IEEE 802.15.3a can be expressed as the following discrete time impulse response [70]:  
\[h^{(k)}(t)=\sum_{l=1}^{Lp}\alpha_{t}^{(k)}\delta(t-\tau_{l}^{k}) \tag{5}\]  
where, \(\tau_{l}^{k}\) is the delay of the \(k\)th user, and \(\alpha_{t}^{(k)}\) is the \(l\)th path gain of the \(k\)th user assuming there are \(Lp\) number of total paths for all the users. If there are \(Nu\) number of users and each user experiences a different channel model, then the received signal can be expressed as :  
\[r(t)=\sum_{k=1}^{Nu}\sum_{l=1}^{Lp}\alpha_{l}^{(k)}S^{(k)}(t-\tau_{l}^{k})+n(t) \tag{6}\]  
where \(\tau_{l}^{k}\) is the time delay of the \(l\)th path for the \(k\)th user, \(\alpha_{l}^{(k)}\) is the \(l\)th path gain of the \(k\)th user assuming there are \(Lp\) number of total paths for all the users, and \(n(t)\) is the AWGN. Besides the AWGN, there could be sources of error from inter-symbol interference (ISI) due to inter-pulse interference (IPI), multiple-pulse interference (MPI), and multiple-access interference (MAI). The probability of error in the lth correlator in the presence of ISI and MAI can be expressed as:  
\[P_{l}=Q\left(\sqrt{\frac{(S_{q})^{2}}{2(\sigma_{IPI}^{2}+\sigma_{MPI}^{2}+ \sigma_{MAI}^{2}+\sigma_{N}^{2})}}\right) \tag{7}\]  
where \(Q\) is the error function, \(S_{q}\) is the signal with \(q\)th order pulse, \(\sigma_{IPI}^{2}\) is the variance of IPI, \(\sigma_{MPI}^{2}\) is the variance of MPI, \(\sigma_{MAI}^{2}\) is the variance of MAI, and \(\sigma_{N}^{2}\) is the variance of white noise component. Each decision is independent; the average probability of bit error can be evaluated as:  
\[P_{avg}=\frac{1}{N}\sum_{l=0}^{N-1}P_{l} \tag{8}\]  
where \(N\) is the total number of correlators in the \(N\) bit symbol transmission. The correct decision of the \(l\)th correlator is \((1-P_{l})\). The decisions are independent, so the decision for a symbol can be defined as:  
\[P_{avg}=\prod_{l=0}^{N-1}(1-P_{l}) \tag{9}\]  
where, \((1-P_{c})\) gives the error probability of the \(N\) bit symbol. Bit error rate (BER) is an important parameter while analyzing the performance of any communication system. It is a dimensionless quantity and is the ratio of bits with errors to the total number of bits transmitted over a given time. For the BER performance analysis, the complete system was developed in MATLAB and the analysis was carried out. Random message bits (\(10^{6}\)) were generated and the number of erroneous bits at the receiver end was counted. The received signals at the receiver end were demodulated using the correlator block. The threshold value for the correlation was set to 21%. This threshold value was chosen from the analysis of the correlation between the composite and the individual pulses. The BER plot obtained while transmitting random information bits transmission (\(10^{6}\)) is shown in Figure 5.  
From the BER plots it can be clearly observed that the bit error rate at the receiver end is \(10^{-4}\) when the \(Eb/N_{0}\) is 13.8 dB. The reason for the bit being erroneous bit is that the threshold is hard set to 0.21.