## 3 4-SSB System Design
### The Concept of Applying 4-SSB into OFDM  
Wireless communications have become a hot research topic these days due to their huge demands in terms of throughput, quality of service (QoS) and coverage. Various techniques have been developed to improve the performance of the applicable traditional orthogonal frequency division multiplexing (OFDM). As the OFDM scheme is applied by recent wireless systems [21], this section then is dedicated to giving a brief concept of the OFDM scheme.  
The transmission in OFDM based system is divided into several subsequent bits carried by different overlapped sub-channels. The sub-channel's bandwidth is designed to make sure each sub-channel has less than the coherent bandwidth of the channel. Consequently, each subchannel will occur in flat fading so that ISI can be minimized.  
The process starts by transmitting a bit sequence of information. After being mapped into Gray QAM, the two symbols of M-QAM are combined by 4-SSB modulation process from the integrated profile spectrum. Then 4-SSB M-QAM scheme is divided into N parallel streams. Each symbol stream is modulated to a set of \(N\) subscribers. An inverse discrete Fourier transform (IDFT) block is employed to transform the sub-stream into time domain signal. We denote the \(l\)th symbol on the \(k\)th subcarrier as the time signal \(X_{\text{1,4 SSB}}[k]\), where \(l\in[0,\infty)\) and \(k\in[0,N-1]\):  
\[x_{\text{1,4 SSB}}(t) = \Re\left\{\frac{1}{T_{sym}}\sum_{l=0}^{\infty}\left[\sum_{k=0}^{ N-1}X_{\text{1,4 SSB}}[k]e^{j2\pi f_{k}(t-IT_{sym})}\right]\right\}. \tag{21}\]  
where \(\Re\) is real part, \(T_{sym}\) is the time duration of a single OFDM symbol equaled to \(\text{NT}_{S}\). where \(T_{S}\) is the duration of the original symbol \(X_{\text{1,4 SSB}}[k]\) in OFDM. \(f_{k}\) is the frequency of subcarrier k.  
After applying the 4-SSB modulation scheme on two symbols, we can obtain the \(x_{\text{4-SSB}}\) signal as shown in Equation (21). In this way, we can transmit the obtained 4-SSB symbols by using OFDM.  
The IDFT produces the discrete time-based OFDM signal at \(t=IT_{sym}+nT_{s}\) which can be identified as:  
\[x_{\text{1,4 SSB}}[n]=\sum_{k=0}^{N-1}X_{\text{1,4 SSB}}[k]e^{\frac{j2\pi n}{N}} \quad for\ n=\ 0,1,2,\ldots,N-1. \tag{22}\]  
In the receiver side, the receiving continuous time OFDM symbol can be expressed by the following equation:  
\[y_{\text{1,4 SSB}}(t)=\sum_{k=0}^{N-1}X_{\text{1,4 SSB}}\ [k]e^{j2\pi f_{k}(t-IT_{sym})},\ IT_{sym}<t< IT_{sym}+nT_{s} \tag{23}\]  
\(X_{\text{1,4 SSB}}\) can be demodulated by using orthogonal sub-carrier for low complexity where we neglect channel and White Gaussian noise from the expression:  
\[\begin{array}{l}Y_{\text{1,4 SSB}}[k]=\frac{1}{T_{sym}}\int_{-\infty}^{\infty }y_{\text{1,4 SSB}}(t)e^{  
Then, the sampled signal is received by the OFDM symbol \(y_{\text{1,4 SSB}}(l)\) at \(t=IT_{sym}+nT_{s}\) with discrete time \(\left\{y_{\text{1,4 SSB}}(n)\right\}_{n=0}^{N-1}\). Using fast Fourier transform (FFT) for the discrete time equation, we have:  
\[\begin{split} Y_{\text{1,4 SSB}}[k]=\sum\limits_{i=0}^{N-1}y_{ \text{1,4 SSB}}(n)e^{\frac{-2n\text{ln}}{N}}=\sum\limits_{k=0}^{N-1}\left\{ \frac{1}{N}\sum\limits_{i=0}^{N-1}X_{\text{1,4 SSB}}\left[i\right]e^{\frac{2n \text{ln}}{N}}\right\}e^{\frac{-2n\text{ln}}{N}}\\ =\sum\limits_{k=0}^{N-1}\sum\limits_{i=0}^{N-1}X_{\text{1,4 SSB}} \left[i\right]e^{\frac{2n(i-k)n}{N}}\end{split} \tag{25}\]