## 1 Introduction  
Increasing demand for wireless communication in various areas of human life has brought about an exponential increase in the number of wireless services. There will be a continuous increase in the demand for wireless spectrum in the foreseeable future with the introduction of internet multimedia applications such as online video, multimedia networks, and distributed gaming. This exponential increase has resulted in spectrum scarcity as the electromagnetic spectrum has become too crowded to incorporate all the upcoming wireless services. There is more of an increase in demand for spectrum than for development in technology, which aims at increasing the spectrum efficiency [1,2]. Meanwhile, due to growing demands for wireless multimedia services and the rollout of advanced radio transmission technologies wireless networks are consuming increasing amounts of energy and contribute a growing fraction to the CO\({}_{2}\) emissions of the information and communications technology industry. There is a need on environmental grounds to reduce the energy requirements of wireless communications [3], thus the energy consumption and energy efficiency issue is another research focus of the wireless communications research community. Recently, these problems have attracted a lot of attention of researchers and many new ideas have been proposed to mitigate the problem of spectrum scarcity and energy efficiency [4, 5, 6, 7, 8]. In order to satisfy the higher and higher demand for communication systems, a new technique called efficient modulation that can achieve high data rates and high spectrum efficiency is receiving the attention of many researchers [9,10]. In this research, another modulation technique referred to as extended binary phase shift keying (EBPSK) which has better bandwidth efficiency, and higher data rates was proposed by [PERSON] _et al_. [11]. The difference between the waveforms of EBPSK modulation corresponding to \"0\" and \"1\" is very tiny, if we use classical theory (such as by using the matched filter or correlation detection) to detect the signals, it has even higher demand on input SNR. Fortunately, a special impacting filter (SIF) which can produce high impact at the phase jumping point and great improvement in SNR, was applied at the receiver [12]. Therefore, a simple amplitude detector could perform the detection of EBPSK signals [13]. However, due to the characteristics of EBPSK modulation, if we increase the bit rate by using the short bit duration, the SIF output signals will interfere with the neighboring symbols which is the phenomenon of intersymbol interference (ISI) [14]. Furthermore, most transmission systems have band limitations imposed by either the natural band-width of the transmission medium or by regulatory conditions. If a narrowband band-pass filter (NBPF) is added at the transmitting end of the communications system to achieve a bandwidth-efficient transmission, the ISI is also hard to avoid, so it is difficult to detect the received signals by threshold decision (TD), because the signal amplitude of SIF output is no longer high [15]. Meanwhile, nonlinear detectors are specifically designed to get the optimum performance of blind multi-user detectors [16], and nonlinear channel equalization [17] and provide accurate posterior probability estimates for LDPC decoding [18]. All results have shown that the nonlinear detection technique performs like an optimum receiver. One of the goals of this paper is the analysis of a nonlinear detector based on support vector machine (SVM), together with the SIF. Although, the SVM approaches are less principled than Gaussian processes classification (GPC), it is impractical when the number of training samples of the GPC is not very low [19], so in this paper, we focus on the SVM technique for EBPSK signal detection and posterior probability estimates (PPEs). Some preliminary works on detecting the signal of SIF output have been presented in [13, 20, 21].