## 3 Performance Analysis of the PSI Technique  
### Analysing the Error Vector Magnitude (EVM)  
Error vector magnitude (EVM) is a commonly used figure of merit to quantify the performance of a transmitter or receiver. It quantifies to what extent constellation points deviate from their ideal locations. It should be noted that in many works, root square mean (RMS) of EVM is used [25; 26; 27]; however, here, in order to analyse the worst case, we consider the maximum of EVM in each symbol as:  
\[EVM_{max}=\left(\sqrt{\frac{1}{\frac{K}{\sum_{n=0}^{K-1}(I_{i}-I_{m})^{2}+(Q_{ i}-Q_{m})^{2}}{|P_{max}|}}}\times 100\right)_{max} \tag{6}\]  
where \(I_{i}\) and \(Q_{i}\) are the I and Q of an ideal signal without PSI modification, and \(I_{m}\) and \(Q_{m}\) are I and Q of a modified signal after being processed with PSI algorithm. Therefore, \((I_{i}-I_{m})^{2}+(Q_{i}-Q_{m})^{2}\) is the measured error between the modified I and Q, and the ideal I and Q. In order to provide a more comprehensive test of the technique, LTE signals were used for experimentation [28]. The \(EVM\) for signal processing on the transmitter should be minimised as much as possible. Here, over 100 randomly generated modulated signals with Offset QPSK (OQPSK) were tested and the \(EVM\) was measured as presented in Table 2. OFDM signals with various lengths of inverse fast Fourier transform (IFFT) (\(K=512,1024,2048,4096\), and 8192) were tested with different values of \(RP\), and \(SR\), and the \(EVM_{max}\) in each case was measured and is presented. It was observedthat the maximum EVM captured in Scenario 4 when \(K=512\), \(RP=10\), \(SR=0.5\), and \(NS=16\), and the \(EVM_{max}\) had the lowest value in Scenario 6, when \(K=2048\), \(RP=5\), and \(SR=0.5\). Table 2 presents six scenarios, as in Scenario 1, \(RP=5,SR=0.5,NS=16\); Scenario 2, \(RP=5,SR=0.5,NS=32\); Scenario 3, \(RP=10,SR=0.5,NS=16\); Scenario 4, \(RP=10,SR=0.5,NS=32\); Scenario 5, \(RP=5,SR=0.4,NS=16\); and Scenario 6, \(RP=10,SR=0.6,NS=16\). PAPR Reduction Performance; Complementary Cumulative Discrete Function (CCDF), and Time Domain Signal Power Peak Reduction  
In order to gauge the effectiveness of the proposed techniques, simulations were first carried out in Matlab [29]. When comparing signals with \(K=512\), 1024, 4096, and 8192 analytically, it was expected to experience higher PAPR as \(K\) was increased. This is due to a higher probability of the presence of high peaks in general. In order to measure this probability the complementary cumulative distribution function (CCDF) was used, which can be represented as:  
\[CCDF=\mathbb{P}(PAPR(S(t))\geq PAPR_{th}), \tag{7}\]  
where \(PAPR_{th}\) is a threshold PAPR, and \(\mathbb{P}\) denotes the probability. Again, the modulation scheme used in these tests is offset quadrature shift key (OQPSK), and \(K\) is varied from 512 to 8192. It should be noted that the LTE toolbox used did not exceed IFFT length of 2048 [26,30]; however, as carrier aggregated signals in LTE-A and 5G technologies had increased IFFT lengths [31, 32, 33], in order to validate that this scheme is applicable for LTE-A signals and 5G signals, the test was performed for larger IFFT lengths of \(K=4096\) and 8192. The comparison of the original signal and the signal with PSI applied in the time domain is shown in Figure 8a, and the CCDF plot is presented in Figure 8b, in which the common point to look at the CCDF for performance comparison is at \(10^{-4}\). This means that, for example, an original signal with length of \(K=1024\) is 0.01% likely to have PAPR of higher than 12 dB. It is shown that this value is reduced to about 7.2 dB by applying the PSI technique. It should be noted that any improvement in CCDF more than 4 dB aligns with IEEE standards [34], and can be considered a useful technique in practice. A full comparison is displayed in Table 3, which shows in case of \(K=8192\), the PAPR is reduced by 4.4 dB.