1 Continuous Phase Modulation With Faster-than-Nyquist Signaling for Channels With 1-bit Quantization and Oversampling at the Receiver Rodrigo R. M. de Alencar, Student Member, IEEE, and Lukas T. N. Landau, Member, IEEE, Abstract—Continuous phase modulation (CPM) with 1-bit construction of zero-crossings. The proposed CPM waveform quantization at the receiver is promising in terms of energy conveys the same information per time interval as the common and spectral efficiency. In this study, CPM waveforms with CPFSK while its bandwidth can be the same and even lower. symbol durations significantly shorter than the inverse of the signal bandwidth are proposed, termed faster-than-Nyquist CPM. Referring to the high signaling rate, like it is typical for faster- This configuration provides a better steering of zero-crossings than-Nyquist signaling [9], the novel waveform is termed as compared to conventional CPM. Numerical results confirm a faster-than-Nyquist continuous phase modulation (FTN-CPM). superior performance in terms of BER in comparison with state- Numerical results confirm that the proposed waveform yields of-the-art methods, while having the same spectral efficiency and a significantly reduced bit error rate (BER) as compared to a lower oversampling factor. Moreover, the new waveform can be detected with low-complexity, which yields almost the same the existing methods [7], [6] with at least the same spectral performance as using the BCJR algorithm. efficiency. In addition, FTN-CPM can be detected with low- complexity and with a lower effective oversampling factor in Index Terms—1-bit quantization, oversampling, continuous phase modulation, faster-than-Nyquist signaling. comparison with the state-of-the-art methods. The sequel is organized as follows: Section II defines the I. INTRODUCTION system model, whereas Section III describes the proposed Continuous phase modulation (CPM) yields spectral ef- waveform. Section IV details the detection, which includes ficiency, smooth phase transitions and a constant envelope also the proposed low-complexity method. Section V discusses [1], [2], which allows for the use of energy efficient power numerical results, while Section VI shows the conclusions. n T amplifiers with low dynamic range. At the receiver side, the Sequences are denoted with x = »x1;:::; xn¼ . Likewise, n » T T ¼T energy consumption of the analog-to-digital converter (ADC) sequences of vectors are written as y = y1 ;:::; yn .A k » ¼T scales exponentially with the resolution in amplitude [3]. segment of a sequence is given by xk−L = xk−L;:::; xk k » T T ¼T Hence, in this study a low resolution ADC is considered, and yk−L = yk−L;:::; yk . where the ADC provides only sign information about the received signal. In order to compensate for the loss in terms II. SYSTEM MODEL of the achievable rate, oversampling with respect to (w.r.t.) the The considered system model is based on the discrete time signal bandwidth is considered. In this context, it is shown that system model described before in [6]. Later in Section IV oversampling yields a significant gain in terms of achievable the model is extended by different CPM demodulators for rate for the noiseless [4] and for the noisy channel [5]. processing the quantized received signal, as illustrated in As the information is implicitly conveyed in phase transi- Fig. 1. In the sequel the individual building blocks are detailed. tions of CPM signals, resolution in time is more promising 1) CPM Modulator: The information conveying phase term than resolution of amplitude. CPM signals with channels of the constant envelope CPM signal [1] reads 1 arXiv:1910.06465v1 [cs.IT] 15 Oct 2019 with 1-bit quantization and oversampling has been considered Õ before in [6], where the achievable rate is studied and max- φ ¹tº = 2πh αk f ¹t − kTsº + '0, (1) imized via optimization of sequences. Later, more practical k=0 approaches were proposed in [7], where the intermediate Kcpm where Ts denotes the symbol duration, h = is the frequency and the waveform is considered in a geometrical Pcpm modulation index, f ¹·º is the phase response, ' is a phase- analysis of the phase transitions. Moreover, in [8] it is pre- 0 offset and α represents the kth transmit symbol. For an even sented how to exploit the channel with 1-bit quantization and k modulation order M , such transmit symbols are taken from oversampling by using iterative detection with sophisticated cpm an alphabet described by α 2 ±1; ±3;:::; ±(M − 1º . In channel coding for CPM signals. k cpm order to obtain a finite number of phase states K and P In the present study, a CPM waveform is introduced with a cpm cpm must be relative prime positive integers. The phase response symbol duration that is only a fraction of the symbol duration function f ¹·º shapes the sequence of CPM symbols to the of an equivalent CPFSK, which is promising in terms of continuous phase signal with smooth transitions. The phase The authors are with Centro de Estudos em Telecomunicac¸oes˜ Pontif´ıcia response is characterized by Universidade Catolica´ do Rio de Janeiro, Rio de Janeiro CEP 22453-900, ( f g 0; if τ ≤ 0; Brazil, (email: alencar, lukas.landau @cetuc.puc-rio.br). f ¹τº = This work has been supported by the ELIOT ANR18-CE40-0030 and 1; if τ > T , FAPESP 2018/12579-7 project. 2 cpm 2 nk xk CPM RX Filter Decimation zk 1-bit ADC yk CPM xˆk Modulator G D Q¹·º Demod. Fig. 1: Discrete time description of the CPM system with 1-bit quantization and oversampling at the receiver the appearance of zero-crossings can be adjusted, as proposed φ¹tº ¹tº π in [7]. Nevertheless, in the considered examples nIF = 0 is 2 π chosen for simplicity. t Regarding the discrete system model in Fig. 1, the CPM t modulator takes the input sequence xn and generates the (a) (b) q E transmit signal s ej k ¹s˜k º, where E is the symbol energy, Ts s Fig. 2: CPM trellis (a) and its tilted version (b), Mcpm = 2, h = 1/2, φ0 = 0, i.e., it already takes into account the frequency offset. Lcpm = 1 and linear phase transition 2) Receive filter and 1-bit quantization: The receive filter g¹tº has an impulse response of length Tg. In the discrete where Tcpm defines the CPM memory in terms of Lcpm = model for expressing a subsequence of ¹η + 1º oversampling dTcpm/Tse transmit symbols. In general, the corresponding output symbols it is represented in a matrix form with G, as phase trellis of (1) is time variant, which means that the a MD¹η + 1º × MD¹Lg + η + 1º Toepliz matrix, as described in possible phase states are time-dependent. In order to avoid T T equation (17) in [6], whose first row is »g ; 0MD¹η+1º¼, where the time-dependency, a time invariant trellis is constructed by T Ts Ts g = »g¹LgTsº; g¹ MD ¹Lg MD − 1ºº;:::; g¹ MD º¼. A higher tilting the phase according to the decomposition approach in sampling grid in the waveform signal, in the noise generation [10]. The tilt corresponds to a frequency offset applied to the and in the filtering is adopted to adequately model the aliasing CPM signal, i.e., the phase term becomes ¹tº = φ¹tº+2π∆ f t, effect. This receive filtering yields an increase of memory in where ∆ f = h¹Mcpm −1)/2Ts. In this context, Fig. 2 shows the the system by Lg symbols, where ¹Lg − 1ºTs < Tg ≤ LgTs. tilted version of a MSK signal. Taking into account the tilted This motivates the definition of the overall memory in terms trellis, a different symbol notation xk = ¹αk + Mcpm − 1)/2 can of L = Lcpm + Lg. be considered, which then corresponds to the symbol alphabet The filtered samples are decimated to the vector zk k−η X = 0; 1;:::; Mcpm − 1 . The tilted CPM phase ¹tº within according to the oversampling factor M, by multiplication with one symbol interval with duration Ts, letting t = τ+kTs, can be the D-fold decimation matrix D, as described in equation (16) h k i fully described by the state definition s˜k = βk−L ; x cpm k−Lcpm+1 in [6], with dimensions M¹η +1º× MD¹η +1º. Then, the result k k in terms of zk−η is 1-bit quantized to the vector yk−η. These operations 2π can be represented by the following equations ¹τ + kT º = β (2) s k−Lcpm " #! Pcpm r k E k k s k−η−Lg k Lcpm−1 yk−η = Q zk−η = Q DG e + nk−η−L , Õ Ts g + 2πh 2x − − M + 1 f ¹τ + lT º k l cpm s (3) l=0 τ where the quantization operator Q¹·º is applied element- + πh Mcpm − 1 + Lcpm − 1 + '0, T wise. The quantization of zk is described by yk;m = s sgn¹Re zk;m º + jsgn¹Im zk;m º, where m denotes the over- where the absolute phase state βk−Lcpm can be reduced to sampling index which runs from 1 to M and yk;m 2 k k−Lcpm f1 + j; 1 − j; −1 + j; −1 − jg. The vector n contains Õ k−η−Lg β = ©K x ª mod P , k−Lcpm ­ cpm l® cpm complex zero-mean white Gaussian noise samples with vari- l=0 ance σ2 = N . « ¬ n 0 which is related to the 2π-wrapped accumulated phase contri- butions of the input symbols that are prior to the CPM memory. III. PROPOSED FASTER-THAN-NYQUIST CPM WAVEFORM In the sequel a discrete time description is considered which As known from linear modulation schemes, a faster-than- implies that the CPM phase is represented in a vector notation.