## 4 Error Performance Analysis  
There are three types of errors that can occur in a DPIM pulse train, these being erasure, false detection and wrong slot error. Erasure occurs when a bit is erased due to attenuation and the received signal falls below the threshold. False detection is the case when a 1 is wrongly detected in place of a zero due to the signal exceeding the threshold due to noise. A wrong slot is when a pulse occupies the wrong time slot due to dispersion. In DPIM, the effect of an error spills over into the next symbol, unlike PPM, where the effect of an error is restricted to the specific symbol. An erasure causes two symbols to merge into one, while a false detection causes a symbol to split into two [32]. The error performance of DPIM is similar to differential PPM (DPPM), the difference being that the wrong slot error is restricted, only to the next symbol, and does not go beyond that used in DPPM [33].  
The expression for the probability of error in a DPIM symbol is given by Equation (2) as [32]  
\[Pe_{DPIM}=\frac{1}{4}\left(2 erfc\Bigg{(}(1-\infty)\sqrt{\frac{CNR}{2}}\Bigg{)} +(2^{M}-1)\;erfc\left(\propto\sqrt{\frac{CNR}{2}}\right)+erfc\Bigg{(}\left( \propto-e^{-(\frac{T_{b}}{T_{b}})^{2}}\right)\sqrt{\frac{CNR}{2}}\Bigg{)}\right) \tag{2}\]  
where \(\propto\) is the threshold, \(M\) is the modulation order, while \(T\)s and \(T\)i represent the slot duration and received pulse width, respectively. \(CNR\) in Equation (2) stands for carrier-to-noise ratio and may be given by,  
\[\frac{C}{N}=\frac{E_{b}}{N_{o}}\frac{F_{b}}{B} \tag{3}\]  
where \(F_{b}\) is the bit rate, \(C\) is the total carrier power, \(N\) is the total noise power and \(B\) is the bandwidth.  
The three terms in Equation (2) represent the erasure, false detection and wrong slot error. Alternately, Equation (2) may be rewritten as Equation (4) [33]  
\[P_{es}=P_{e}+\frac{P_{w}}{2}+\frac{L_{avg}-1}{2}\;P_{f} \tag{4}\]  
where \(P_{e}\) is the probability of erasure, \(P_{w}\) is the probability of wrong slot detection, \(P_{f}\) is the probability of false alarm and \(L_{avg}\) is the average symbol length. Unlike in the case of conventional DPIM, the end of a DS-PIM symbol is also marked with a 1 (Table 3), making  
Figure 6: Locations of ON subcarriers for 2-bit index data word (**a**) 00 (**b**) 01 (**c**) 10 (**d**) 11.  
the wrong slot error less likely. This effect is negligible, especially at higher values of CNR. The error rate of _mDPIM_ may be obtained by rewriting Equation (4) as Equation (5).  
\[P_{es\ mDPIM}=P_{e}+\frac{L_{avg}-1}{2}\ P_{f} \tag{5}\]  
where this results in a symbol error rate (SER) that is significantly lower than that of conventional DPIM and close to that of PPM.  
Equation (6) gives the probability of two adjoint events.  
\[\mathrm{P}\Big{(}\mathrm{A}\bigcup\mathrm{B}\Big{)}=\ \mathrm{P}(\mathrm{A})+\ \mathrm{P}( \mathrm{B})-\mathrm{P}(\mathrm{A}\cap\mathrm{B}) \tag{6}\]  
where P(A) and P(B) represent the probability of the occurrence of event A and B, respectively. \(\mathrm{P}(\mathrm{A}\cap\mathrm{B})\) in Equation (6) is the probability of both A and B occurring simultaneously. The probability of error in a SIM OFDM symbol is the adjoint probability of error in the index symbol and the probability of error in modulation symbol. This can be shown modifying Equation (6) to obtain Equation (7).  
\[Pe_{Total}=(Pe_{Modulation}+Pe_{index})-((Pe_{Modulation})(Pe_{index})) \tag{7}\]  
The probability of error in a DS-PIM-SIM OFDM symbol can be given by Equation (8)  
\[Ps_{mDPIM-SIM\ OFDM}=(Pe_{QAM}+Pe_{mDPIM})-((Pe_{QAM})(Pe_{mDPIM})) \tag{8}\]  
where \(Pe_{QAM}\) is the symbol error probability in a \(QAM\) symbol, while \(Pe_{mDPIM}\) is the probability of error in a DS-PIM symbol.  
Probability of symbol error in a QAM symbol has been discussed in detail, and an expression has been derived in [34].