# Null Subcarrier Index Modulation in OFDM Systems for 6G and Beyond
## 2 Conventional OFDM Systems  
Orthogonal frequency division multiplexing (OFDM) is a multicarrier transmission technique that is widely used in wireless communication systems. In the Long-Term Evolution (LTE) systems, OFDM is the key waveform constructed in the in-parallel form of orthogonal subcarriers. The subcarriers' orthogonality is achieved by carefully choosing each carrier frequency in the spectrum. As a basic concept, the available frequency bandwidth is divided into a number of narrowband frequencies, and they are positioned orthogonal to one another along the spectrum [18]. In contrast to the traditional frequency division multiplexing (FDM) schemes, in the OFDM system, the subcarriers that use the allocated frequency bandwidth are allowed to overlap each other, and thus, more carriers are deployed in the transmission. The basic diagram of the OFDM system is shown in Figure 1. It consists of the transmitter, receiver, and wireless communication channel. The binary data are given to the constellation mapper, and then, they are converted to the complex data. The constellation mapper operates with a predefined modulation order as a power of two (2, 4, 8, etc.). The complex data at the output of the constellation mapper are used for the frequency domain representation. The inverse fast Fourier transform (IFFT) block is fed by the constellation outputs, and by processing the complex inputs, the orthogonal subcarriers in the time domain are obtained. Each subcarrier at a different center frequency is modulated by the IFFT to carry the complex data. The OFDM symbol is constructed by adding the Cyclic Prefix (CP), also known as the guard time interval, at the beginning of the signal. The CP insertion allows for more efficient communication against the intersymbol interference (ISI). The OFDM signal generated in the time domain can be expressed as:  
\[ \begin{split} x(\ell)&=IDFT\{X(k)\}\quad\ell=0,1, \ldots,N-1\\ &=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}X(k)e^{j\frac{2\pi k}{N}\ell} \end{split}  \tag{1}\]  
where IDFT indicates the inverse discrete Fourier transform, \(\frac{1}{\sqrt{N}}\) denotes the scaling factor, and the time domain OFDM symbol samples are indicated by \(x(\ell)\), where \(\ell=0,\cdots,N-1\), for each subcarrier. The subcarrier index \(k=0,\cdots,N-1\), represents the frequency domain samples. Figure 1: Conventional OFDM system. The OFDM guard interval can be inserted in a number of different ways: one of them is the time domain zero padding (ZP), which pads the signal with zero samples, while the other is the cyclic prefix addition, where the last \(N_{CP}\) samples of the signal are copied and then inserted into the front of the OFDM time signal after the IFFT process, which modulates the subcarriers with the complex data [19]. By the CP addition, the OFDM signal \(x(\ell)\) is then extended to the length of \(N+N_{cp}\) before the transmission. The extended version of the OFDM signal \(x_{cp}(\ell)\) is given by:  
\[x_{cp}(\ell)= \begin{cases}x(N+\ell),&\ell=-N_{cp},-N_{cp}+1,\ldots,-1\\ x(\ell),&\ell=0,1,2,\ldots,N-1\end{cases}  \tag{2}\]  
The advantage of adding a guard interval to the OFDM symbols is that, with a sufficiently long cyclic prefix, the periodicity of subcarriers remains alive, and thus, it allows more efficient communication against ISI [20]. In a multipath channel environment, the length of the OFDM cyclic prefix is chosen to be equal to or greater than the length of the maximum path delay [21]. At the receiver, the OFDM signal \(y(\ell)\) is received from the wireless channel, and then, the frequency domain representation of the signal is obtained, following the guard interval removal. The signal \(y(\ell)\) in Equation (3) is the convolution of the transmitted OFDM signal \(x(\ell)\) and the channel impulse response \(h(\ell)\). \[ \begin{split} y(\ell)&=x(\ell)*h(\ell)+w(\ell)\\ Y(k)&=X(k)H(k)+W(k)\end{split}  \tag{3}\]  
where \(w(\ell)\) denotes additive white Gaussian noise (AWGN) and \(W(k)\) is the frequency domain representation of the noise vector \(w(\ell)\).