# Energy and Spectrally Efficient Modulation Scheme for IoT Applications
## 2 The OSP-MSK Modulation System Model  
Figure 1 depicts the MIMO orthogonal shaping pulses minimum shift keying (OSP-MSK) modulation system model. As shown in the figure, the input binary data are firstly encoded by using a convolutional encoder accompanied with an interleaver to make the transmitted signal of the OSP-MSK more robust against the prospective disruptions (i.e., noise, interference, multipath propagation, etc.). The convolutionally encoded data are then partitioned into two distinct sets, and each dataset is divided into in-phase components (i.e., even bits) and quadrature-phase components (i.e., odd bits). These components are then modulated using constant envelope MSK modulation, which results in a signal with a unitary peak to average power ratio (PAPR). Therefore, an energy efficient nonlinear PA is utilized to amplify each modulated dataset. Hence, a superior enhancement in the power consumption is achieved by using the OSP-MSK modulation [26]. As such, the constant envelope signal of each dataset is given by:  
\[S_{1}(t)=A_{1}S_{1I}(t)cos\left(\frac{\pi t}{2T}\right)cos(2\pi f_{c}t)+A_{1}S_ {1Q}(t)sin\left(\frac{\pi t}{2T}\right)sin(2\pi f_{c}t), \tag{1}\]  
which can be rewritten as follows:  
\[S_{1}(t)=\pm\sqrt{\frac{2E_{b}}{T}}cos\left(\frac{\pi t}{2T}\right)cos(2\pi f _{c}t)\pm\sqrt{\frac{2E_{b}}{T}}sin\left(\frac{\pi t}{2T}\right)sin(2\pi f_{c} t), \tag{2}\]  
where \(A_{1}\) is the gain of the first amplifier, \(E_{b}\) is the energy per bit, \(T\) is the bit period, and \(f_{c}\) is the carrier frequency. \(S_{iI}(t)\) and \(S_{iQ}(t)\) with \(i=1,\ 2\) are the in-phase and the quadrature-phase components of each MSK signal, respectively. In the same way, \(S_{2}(t)\) is given by:  
\[S_{2}(t)=A_{2}S_{2I}(t)cos\left(\frac{\pi(t-T)}{2T}\right)cos(2\pi f_{c}t)+A_ {2}S_{2Q}(t)sin\left(\frac{\pi(t-T)}{2T}\right)sin(2\pi f_{c}t), \tag{3}\]  
Figure 1: The orthogonal shaping pulses minimum shift keying (OSP-MSK) modulation system model. \[S_{2}(t)=\pm\sqrt{\frac{2E_{b}}{T}}cos\bigg{(}\frac{\pi(t-T)}{2T}\bigg{)}cos(2\pi f _{c}t)\pm\sqrt{\frac{2E_{b}}{T}}sin\bigg{(}\frac{\pi(t-T)}{2T}\bigg{)}sin(2\pi f _{c}t), \tag{4}\]  
where \(A_{2}\) denotes the gain of the second amplifier. Therefore, the transmitted signal of the proposed OSP-MSK is expressed as follows:  
\[S(t)=S_{1}(t)+S_{2}(t). \tag{5}\]  
Confusion may be arisen here that both \(S_{1}(t)\) and \(S_{2}(t)\) are two constant envelope signals (i.e., unitary PAPR signals), but their sum (i.e., \(S(t)\)) is not a constant envelope signal. The OSP-MSK transmitted signal \(S(t)\) is actually not a constant envelope signal, but it is a low PAPR signal (see Appendix C), because either \(S_{1}(t)\) or \(S_{2}(t)\) is amplified separately before it is combined with the other signal. This is in contrast to the MIMO-OFDM system, where all of the subcarriers are combined (summed) before amplifying [27]. Hence, the low PAPR property of the OSP-MSK transmitted signal enables the one-bit ADC to be beneficially used with the OSP-MSK at the RX side. On its path, the OSP-MSK transmitted signal undergoes a fading channel, and it is corrupted by AWGN. At the RX side, a coherent demodulation is applied to the received signal, as shown in Figure 1. To extract the MSK in-phase components \(S_{1I}\) and \(S_{2I}\), the OSP-MSK received signal \(S(t)\) given by Equation (5) is firstly multiplied by \(cos(2\pi f_{c}t)\). Then, a low pass filter (LPF) is applied to the resulting signal in order to detect the summation of the two in-phase components \(S_{I}\) as follows:  
\[S_{I}=\frac{1}{2}S_{1I}cos\bigg{(}\frac{\pi t}{2T}\bigg{)}+\frac{1}{2}S_{2I}cos \bigg{(}\frac{\pi(t-T)}{2T}\bigg{)},\text{ i.e.,} \tag{6}\]  
\[S_{I}=\pm\sqrt{\frac{E_{b}}{2T}}cos\bigg{(}\frac{\pi t}{2T}\bigg{)}\pm\sqrt{ \frac{E_{b}}{2T}}cos\bigg{(}\frac{\pi(t-T)}{2T}\bigg{)}. \tag{7}\]  
Finally, the MSK in-phase components are separated by exploiting the orthogonality between the shaping pulses (i.e., \(cos\big{(}\frac{\pi t}{2T}\big{)}\) and \(cos\Big{(}\frac{\pi(t-T)}{2T}\Big{)}\)).