## 3 Methods
### Physical Layer of OFDM Communication System
#### 3.1.3 M-QAM Modulation
\begin{table}
\begin{tabular}{c c} \hline \hline
**Communication Systems** & **M-QAM Utilized** \\ \hline IEEE 802.11n/g/ad/ay & 16, 64 \\ IEEE 802.16m (WiMAX 2) & 16, 64 \\ IEEE 802.11 ac/af/ah & 16, 64,256 \\ DVB-T2 (Digital Video & 16, 64,256 \\ Broadcasting-Terrestrial) & 16, 64,256 \\ IEEE 802.22b & 16, 64,256 \\ TS 36.331 (Release 14-LTE Advanced Pro) & 16, 64,256 \\ TS 36.331 (Release 15-5G support) & 16, 64,256, 1024 \\ IEEE 802.11 ax (Wi-fi 6) & 16, 64,256, 1024 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Applications utilizing M-QAM Modulation for modern communication standards. independently, where \(2d\) is the adjacent Euclidean distance between the two constellation points given by [56]  
\[d=\sqrt{\frac{3 log_{2}M\cdot E_{b}}{(M^{2}-1)}}, \tag{14}\]  
where \(E_{b}\) is the bit energy and \(M\) is the order of QAM. Bit Error Rate (BER) is a vital factor in evaluating the effectiveness of constellation designs in modulation schemes. Later sections will cover the analysis of the experimental results, but the theoretical bit error calculation of SQAM is given by [56]  
\[B_{e}=\frac{1}{log_{2}\sqrt{M}}\sum_{i=1}^{log_{2}\sqrt{M}}B_{e}(i), \tag{15}\]  
where \(B_{e}(i)\) is the bit error probability of the \(i^{th}\) bit. When designing constellation points with odd length of symbols (32-QAM, 128-QAM, 512-QAM) in RQAM, the constellation points are usually distributed to form a perfect rectangle, either with a horizontal shape parallel to the in-phase axis or with a vertical shape parallel to the imaginary axis, both having the same average energy; however, this is not desirable as it increases the peak and average powers. To overcome this, [57] proposes SQAM with the outer corner constellation points relocated for odd length of symbols forming a cross shape constellation design referred to as cross-constellation M-QAM (XQAM). XQAM reduces the peak and average energy and provides at least 1 dB gain over prefect rectangular shape constellation design [57; 59]. The generalized GP for higher order XQAM constellation design can be given as \(\left(1+\frac{1}{\sqrt{2M}}+\frac{1}{3M}\right)\)[59]. For the BER calculations, authors in [57] proposed an approximate expression which was reproduced by authors in [59] for XQAM as  
\[B_{e}(M)\cong G_{p}\frac{A_{n}}{log_{2}M}\frac{1}{2}erfc\left(\frac{d}{\sqrt{ N_{0}}}\right), \tag{16}\]  
where \(G_{p}\) is the GP, \(A_{n}\) is the average number of nearest neighbors of a constellation point [given by \(\left(4-\left(\frac{6}{\sqrt{2M}}\right)\right)\) for XQAM of M \(\geq 32\)], \(N_{0}\) is the noise density, \(2d\) is the adjacent Euclidean distance between the two constellation points and \(erfc(\cdot)\) is the error function defined as:  
\[erfc(x)=\frac{2}{\sqrt{\pi}}\int_{x}^{\infty}e^{-t^{2}}dt. \tag{17}\]  
The decision region for SQAM is simple because the IQ component has distinct vertical and horizontal decision regions. However, for XQAM constellations, only horizontal and vertical decision regions are not sufficient as their corner constellation points are not available. In XQAM, the end columns symbols are relocated to new cross-type positions in the constellation, forming three types of symbols, edge symbols, corner symbols, and interior symbols [57; 59], as shown in Figure 3. The inner symbols form a closed square decision region, while the edge symbols form semi-infinite rectangles. The decision region for the corner symbols forms a 45-degree angle along the horizontal and vertical axes. Due to this irregular pattern, some of the noisy symbols are not transformed into the mapped decision region, resulting in higher bits in error for XQAM constellation design [5; 55]. Higher order M-QAM utilizes a greater number of bits per symbol and can significantly increase the data rate.