# OFDM Network Optimization Using a QPSK Based on a Wind-Driven Genetic Algorithm
## 3 GAWDO for OFDM Network Optimization
### Quadrature Phase-Shift Keying Enhancement  
QPSK is a digital modulation scheme that transmits data by varying the phase of the carrier signal. The carrier wave variations have to be with great precision [6]. QPSK can also be referred to as 4-QAM (quadrature amplitude modulation). The root concepts of QPSK and 4-QAM are different, but the resulting modulated waves are the same [19; 20]. QPSK uses four points on a constellation diagram. It can encode 2 bits per symbol. QPSK systems remain susceptible to noise and high bit errors because they use binary data [18]. For QPSK systems to give optimum results, linearity needs to be maintained in the system. However, this is impossible to achieve due to the presence of components that present transient and non-linear behavior, e.g., high-power amplifiers [21]. QPSK also encounters phase ambiguity and carrier offset drift. This will automatically introduce ISI and ICI to the transmitted signal [8]. Considering that all channels already consist of noise, these interferences further increase the signal distortion and hinder demodulation of the transmitted signal with minimum bit errors. However, QPSK is highly preferred for OFDM  
Figure 1: GAWDO structure. systems because it ensures spectral efficiency, good noise immunity, and a low probability of error as compared to other modulation techniques [32]. These advantages that it presents are very important because communication channels can only use a limited amount of bandwidth issued by the Federal Communications Commission [33]. In OFDM systems, the transmitted QPSK is defined by the following benchmark functions [4,8,19] in Equations (1)-(4):  
\[S_{i}(t)=\sqrt{\frac{2E}{T}}\Big{[}2\pi f_{c}t+(2i-1)\frac{\pi}{4}\Big{]} \tag{1}\]  
for \(0\leq t<T\) & \(f_{c}=\frac{n_{c}}{T_{b}}\)  
whereby \(f_{c}\) is the carrier frequency, \(n_{c}\) is the number of carriers, and \(T\) is the time period, the pair of quadrature carriers is defined using the following benchmarks:  
\[\phi_{1}(t)=\sqrt{\frac{2}{T}}\ \cos(2\pi f_{c}t\ ) \tag{2}\]  
\[\phi_{2}(t)=\sqrt{\frac{2}{T}}\ \sin(2\pi f_{c}t\ ) \tag{3}\]  
for \(0\leq t<T\) & \(f_{c}=\frac{n_{c}}{T}\)  
The bit error probability of QPSK over an additive white Gaussian noise (AWGN) channel is given by the following benchmark functions:  
\[P_{b}=\frac{1}{2}erfc\left(\sqrt{\frac{E_{b}}{N_{0}}}\ \ \right) \tag{4}\]  
whereby \(\frac{E_{b}}{N_{0}}\) is the energy per bit to noise spectral density. In this research, the GAWDO was used for deducing and assigning the phase components of the QPSK. The phase assignments can be values within the range of 0 to \(2\pi\). In this research, a quartic function was used to determine the values that would be considered by the GAWDO. The quartic function was reflected along the X and Y axis so that it would ensure the same domain of values but in four different constellations as shown in Figure 2. During transmission, the conventional QPSK uses four fixed points on a constellation diagram which greatly differs from the proposed methodology which uses varying but optimum points. The graph shows that the GAWDO values would mostly be in the range of 0 to 4 and would not necessarily extend from 0 up to the maximum which is \(2\pi\). Values beyond 4 would cause phase ambiguity within the system. The quartic function is given in Equation (5). \[f(x)=x^{4}-15x^{2}+20 \tag{5}\]  
The radian value '\(\Phi\)' obtained by the GAWDO would be incorporated into the QPSK phase '\(\theta\)' from Equations (1)-(3) as shown below in Equations (6) and (7):  
\[\theta=\ \left[\frac{2\times\pi\times\overline{\alpha}}{\lambda}\ \right]+\phi \tag{6}\]  
\[\beta=e^{i\ (\theta+\theta_{int})} \tag{7}\]  
This is where \(\overline{\alpha}\) is the message signal and \(\lambda\) is the alphabet size. \(\lambda\) is an integer power of 2. The message signal \(\overline{\alpha}\) would consist of integers between 0 and \(\lambda\ -1\). For two or more dimensional signals, each column would be considered a channel. \(\theta_{int}\) is the initial phase.