## 3 Methods  
### Physical Layer of OFDM Communication System  
OFDM is a parallel transmission system that allows spectra of subchannel to overlap with subcarrier tones and coherent detection maximizing the available bandwidth. The physical layer (PHY) of the OFDM communication system is shown in Figure 1.  
In the OFDM system, serial high data-rate sub-streams are divided into low data-rate sub-streams and encoded onto orthogonal subcarriers. The spectral shape of the subcarriers is such that the discrete subchannels at the orthogonal subcarrier frequencies have a spectrum of 0 and there are no Inter Carrier Interference (ICI) among the subchannels. As seen in Figure 1, the encoded bit stream is mapped onto a complex-valued in-phase and quadrature (IQ) constellation plane and converted to IQ data according to the chosen modulation techniques in the modulation block. Deployed modulation techniques will be further discussed in Section 3.1.3. The serial data stream is then converted into parallel data stream and a frequency domain OFDM symbol, \(X(k)\), is generated by inserting training data and guard bands into the IQ data. The training data are the pilot symbols (known data-sequence) that are inserted in the data stream to carry out channel estimation [44]. For pilot insertion, there are multiple techniques based on the location and order of pilots in the transmission. We have used comb-type pilot insertion with linear interpolation as it provides improved results [44]. In comb-type pilot insertion, pilots are inserted into specific OFDM subcarriers that are continuously transmitted throughout the communication. The frequency domain OFDM symbol, \(X(k)\), is then transformed into the time-domain OFDM symbol, \(x_{t}(n)\), using an N-point inverse discrete Fourier transform (IDFT) followed by parallel-to-serial (P/S) transmission. IDFT transformation reduces the system complexity of parallel symbols and removes any pulse shift that occurred in the modulation process [7]. The IDFT equation is given below,  
\[x_{t}(n)=\frac{1}{N}\sum_{k=0}^{N-1}X(k)e^{\frac{2\pi nx}{N}}, \tag{1}\]  
Figure 1: Basic block diagram of OFDM physical layer with frequency–domain and time domain symbol representation. where \(j=\sqrt{-1},\ N\) is the DFT length, \(X(k)\) is the frequency-domain OFDM symbol for \(k=0,\ 1,\ \cdots,\ N-1\), and \(x_{t}(n)\) is the resultant time-domain OFDM symbol. In the next step, a guard interval known as a cyclic prefix (CP), is sliced from the end of \(x_{t}(n)\) symbol and prepended to \(x_{t}(n)\) to form the full time-domain OFDM symbol, \(x_{cp}(n)\), which extends the portion of the OFDM symbol in a cyclic manner. The CP mitigates the Inter-Symbol Interference (ISI) and ICI from multipath radio channels and neighboring OFDM symbols, and it is selected to be larger than the anticipated delay spread. OFDM symbol with CP length \(N_{g}\) is given below:  
\[x_{cp}(n)=\left\{ \begin{matrix}x_{t}(N+n),&-N_{g}\leq n\leq\cdots,N-1\\ x_{t}(n),&otherwise\end{matrix} \right.. \tag{2}\]  
The signal, \(x_{cp}(n)\), is then elevated to the radio frequency (RF) by the RF front end and transmitted through the air via wireless channel. During the transmission over the air, the signal experiences the effect of multiple paths and fading effect that creates duplicate signals with different time lags, phases, and amplitudes at the receiver. Applied wireless channel models will be further discussed in Section 3.1.1. The radio signal is then captured and re-converted to IQ samples by the receiver front end. The carrier synchronizer recovers the time-domain OFDM symbols at the receiver. the CP is then removed from \(y_{cp}(n)\) to yield the remaining IQ samples, \(y(n)\), which is given by:  
\[y(n)=\left\{ \begin{matrix}y_{cp}(n),&-N_{g}\leq n\leq\cdots,N-1\\ y_{cp}\big{(}n+N_{g}\big{)},&otherwise\end{matrix} \right.. \tag{3}\]  
The symbols are then converted to parallel stream and is transformed into the frequency-domain OFDM symbol, \(Y(k)\), using discrete Fourier transform (DFT).