## 3 4-SSB System Design
### Four Single-Sideband Demodulation  
This section describes the 4-SSB QAM demodulation as defined in [11; 19]. The brief process steps can be expressed as follows:  
\[u-\hat{v}=s_{\text{4 SSB,I}}+\hat{s}_{\text{4 SSB,Q}}, \tag{15}\]  
\[v+\hat{u}=-s_{\text{4 SSB,Q}}+\hat{s}_{\text{4 SSB,I}}, \tag{16}\]  
\[p+\hat{r}=s_{\text{4 SSB,I}}-\hat{s}_{\text{4 SSB,Q}}, \tag{17}\]  
\[r-\hat{p}=-s_{\text{4 SSB,Q}}-s_{\text{4 SSB,I}}, \tag{18}\]  
where \(\hat{s}_{\text{4 SSB,I}}\) is Hilbert transform of In phase 4-SSB and the \(\hat{s}_{\text{4 SSB,Q}}\) is a Hilbert transform of quadrature modulation phase 4-SSB [20].  
These equations show the partial de-combination of two transmitted complex signals. The Hilbert transform tap must be equal in transmitter and receiver to ensure the quality of the transmission process. Also, it is impossible to recover two complex signals without the use of 4-SSB M-QAM demodulation. The two complex signals can be represented by the upper sideband and lower sideband before feeding into the shadow equalizer as follows:  
\[d_{\text{1,LSB}}=\frac{1}{2}[(u-\hat{v}+j(\hat{u}+\hat{v})], \tag{19}\]  
\[d_{\text{2,USB}}=\frac{1}{2}[(p+\hat{r}+j(r-\hat{p})], \tag{20}\]  
where \(d_{\text{1,LSB}}\) denotes the lower sideband of 4-SSB and \(d_{\text{2,USB}}\) denotes the upper sideband of 4-SSB. Typically, by defining a spectral profile of 4-SSB signal in the similar way as we did in our previous work [19]. From this, we can identify the two symbols of QAM as shown in Equations (13) and (14). After the modulation of 4-SSB, the spectra are joined as one which includes the two symbols of \(d_{1}\) and \(d_{2}\). This spectrum is defined as LSB which includes \(d_{1}\) in the real and imaginary part as shown in Equation (19). In the same way, we can deduce the Equation (20).  
The Proposed OFDM Shadow Equalizer-based Scheme for Multiple-Input Multiple-Output (MIMO) System Using 4-SSB M-QAM  
In this section, firstly we introduce an overview of the basic concept of the OFDM system. Then, we present the proposed 4-SSB shadow equalizer-based scheme in MIMO. Additionally, we note that the proposed 4-SSB-based modulation scheme can be applied for Internet of Things (IoT) networks utilizing multiple sensors as the distributed nodes which can collect, process, and transmit data from sampled physical signals to the connected IEEE 802.11 Wi-Fi Access Points. These distributed sensor nodes then realize a new modulated transmitter design over OFDM/MIMO at the physical layer to enable wireless sensor networks (WSNs). Typically, the sensors can be coordinated as the modulated transmitters and can be removed or added to the distributed environment as long as the constructed sub-carriers do not overlap. Also, this kind of modulated IoT sensor-enabled communication scheme can enable the modulated IoT energy saving as energy-efficient transmission schemes is a requirement for the efficient data transmission in IoT networks. Typically, the recent IEEE 802.11 ah (Wi-Fi HaLow) is the latest low power wide area network (LPWAN) technique which extends the range of Wi-Fi network for the realization of IoT-based networks. IEEE 802.11 ah standard supports both OFDM and MIMO modulations with high data-rate and low energy consumption at the same time. This promising approach using multiple interconnected sensors in a communication medium will be detailed in our upcoming research.