## 3 Methods
### Physical Layer of OFDM Communication System
#### 3.1.3 M-QAM Modulation  
Most practical OFDM communication systems exhibit passband frequency responses, and to achieve high throughput with bandwidth and power efficiency, digital passband modulation is frequently employed in these systems. Digital passband modulation schemes can be categorized according to the variation in amplitude, phase, or frequency of the transmitted signal with respect to the message signal, such as ASK, PSK, FSK, and QAM. Among them, M-QAM is an advanced modulation technique that varies both amplitude and phase of the transmitted signal simultaneously, resulting in greater bandwidth and power efficiency which can be extended to advanced systems, such as MIMO [5; 6]. Additionally, high data rate can be achieved when utilizing higher order M-QAM; however,using higher order M-QAM for higher data rates comes with a cost of higher BER, cell-to-cell interference, smaller coverage area, and hardware complexity [5; 6; 54]. Despite these challenges, M-QAM is extensively used in various IEEE standards and wireless communication systems, such as 5G digital video broadcast communications, satellite communications, Wi-Fi, WiMAX, VDSL, and more. A few applications with different order of M-QAM for modern communication standards are given in Table 1[55]. The M-QAM signal waveforms are composed of two independently amplitude-modulated carriers that are in quadrature to each other as shown below [56]  
\[s(t)=I(t)\cos(2\pi f_{c}t)-Q(t)\sin(2\pi f_{c}t),\;0\leq t\leq T, \tag{12}\]  
where \(T\) time duration of symbol, \(f_{c}\) is the carrier frequency, and \(I(t)\) and \(Q(t)\) are the respective IQ components of the signal. Wireless communication systems aim to achieve high data-rate transmission with efficient utilization of limited bandwidth for the best quality of service (QoS), and high spectral and power efficiency. This can be achieved by reducing the average transmit power of the constellation design in the modulation schemes [5; 6; 54]. Most of the M-QAM constellation designs proposed decades ago (from early 1960s) are still being used in commercial communication systems. M-QAM has the unique capability to encode information in both amplitude and phase of the transmitted signal, allowing higher spectral efficiency. This enables more bits to be encoded per symbol for a given average energy in the constellation design [5; 6]. Rectangular M-QAM (RQAM) constellation design is widely used due to its improved error performance and use of a simple maximum likelihood detection method, which reduces system complexity [55; 57]. Square M-QAM (SQAM) is a type of RQAM constellation that is optimized for even-length symbols (16-QAM, 64-QAM, 256-QAM, and 1024-QAM) [58]. The distribution of constellation points (symbols) in SQAM forms a perfect square lattice with equal rows and columns and maximizes the minimum Euclidean distance between points for a given average symbol power. In SQAM, each constellation points shares boundaries with a maximum of four neighboring points and differs by only one bit, therefore it has perfect gray coding, with a Gray Code penalty (GP) value 1 [57]. The GP was introduced in [57] as the average difference in bits between adjacent symbols in a constellation  
\[G_{P}=\frac{1}{M}\sum_{i=1}^{M}G_{P}^{S_{i}}, \tag{13}\]  
where \(M=2^{k}\), \(k\) is the length of symbol and \(G_{P}^{S_{i}}\) is the GP for the \(i^{th}\) data symbol, \(S_{i}\). In SQAM, the \(k\) bits of the serial data stream are represented on a two-dimensional constellation design using gray coding. The \(I(t)\) and \(Q(t)\) distributed over the set \(\{\pm d,\pm 3d,\ldots,\pm(M-1)d\}\)