## 2 Efficient Modulation  
The increasing demand for frequency resources is becoming a tough problem when allocation and reallocation of frequency bandwidth are periodically repeated. Higher level modulations are used in solving the problem. However, these solutions are all at the expense of increased energy. Furthermore, the order grows in powers of two, while the constellation is becoming dense and difficult to divide. As a result, only binary modulation or binary keying make sense to easily and fairly measure the bandwidth efficiency.  
Efficiency modulation was first proposed by [PERSON], who holds several patents on the technique. After his cooperation with Photron Science Company, these patents were registered as ultra spectral modulation (USM), which has pretty high bandwidth efficiency. In some other literatures, from a signal bandwidth rather than power bandwidth point of view, such a system can be referred to as a carrier-synchronized ultra-wide band system (CS-UWB) [22], which can produce a high narrow carrier spectrum when the practical duty-cycle is low.  
From a unified expression perspective, all of these techniques are actually a special EBPSK system that is defined as follows:  
\[\begin{split} g_{0}(t)&=A\sin 2\pi\!\!\!/_{c}t,\ \ 0 \leq t<T\\ g_{1}(t)&=\begin{cases}B\sin\bigl{(}2\pi\!\!\!/_{c}t +\theta\bigr{)},\ \ 0\leq t<\tau,\,0\leq\theta\leq\pi\\ A\sin\bigl{(}2\pi\!\!\!/_{c}t\bigr{)},\ \ \ \ \ \ \tau\leq t<T\end{cases}\end{split} \tag{1}\]  
where \(g_{0}\) and \(g_{1}\) are the modulation waveforms corresponding to bit \"0\" and bit \"1\", respectively; \(T=N/\!\!\!/_{c}\) is the bit duration, \(\tau=K/\!\!\!/_{c}\) is the phase modulation duration, \(f_{c}\) is the carrier frequency, and \(\theta\) is the modulating angle. If we set \(\tau=T\) and \(\theta=\pi\), then Equation (1) becomes the classical binary phase shift keying (BPSK) modulation. Moreover, T controls the bit rate, (_i.e._, the bit rate with N = 5 is 4 times than with N = 20), so if we want to increase the bit rate, we can use a short bit duration N. As an example, Figure 1(a) below is the waveform of EBPSK modulation and Figure 1(b) is the waveform of SIF output.  
According to the FCC's bandwidth definition, there should be a total 99% signal power hold in the band. Such a power reservation criterion is practically equivalent to the \(-20\) dB attenuation bandwidth, indicating that spectral attenuation from the peak power to the cutoff frequency point is no less than 20 dB. The \(-40\) dB attenuation bandwidth of EBPSK modulation is only several Hertz [11]. In order to full understand the characteristics of EBPSK modulation, its power spectra should be analyzed. According to reference [20], the power spectrum density (PSD) of EBPSK can be written as follows:  
\[S(f) = \frac{1}{8T\pi^{2}\left(\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
factor 1/T. In this paper we focus on the high bit rate of the system, and so the short bit duration T (or N) should be used. Moreover, the linear spectra containing no information can be removed with some optimization and the phase shift remains in the waveform with amplitude changed but no infection [23].