Continuous Phase Modulation with Faster-Than-Nyquist Signaling for Channels with 1-Bit Quantization and Oversampling at the Receiver Rodrigo R

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,..., xn] . Likewise, n [ T T ]T energy consumption of the analog-to-digital converter (ADC) sequences of vectors are written as y = y1 ,..., yn .A k [ ]T scales exponentially with the resolution in amplitude [3]. segment of a sequence is given by xk−L = xk−L,..., xk k [ T T ]T 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 ∞ 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 (t) = 2πh αk f (t − kTs) + ϕ0, (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 (·) is the phase response, ϕ 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 ∈ ±1, ±3,..., ±(M − 1) . 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 (·) 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 Telecomunicaçoes˜ Pontif´ıcia response is characterized by Universidade Catolica´ do Rio de Janeiro, Rio de Janeiro CEP 22453-900, ( { } 0, if τ ≤ 0, Brazil, (email: alencar, lukas.landau @cetuc.puc-rio.br). f (τ) = This work has been supported by the ELIOT ANR18-CE40-0030 and 1, if τ > T , FAPESP 2018/12579-7 project. 2 cpm 2

xk CPM RX Filter Decimation zk 1-bit ADC yk CPM xˆk Modulator G D Q(·) Demod.

Fig. 1: Discrete time description of the CPM system with 1-bit quantization and oversampling at the receiver

φ(t) ψ(t) the appearance of zero-crossings can be adjusted, as proposed 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 ejψk (s˜k ), 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 g(t) 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 (η + 1) 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 MD(η + 1) × MD(Lg + η + 1) 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 , 0MD(η+1)], where the time-dependency, a time invariant trellis is constructed by T Ts Ts g = [g(LgTs), g( MD (Lg MD − 1)),..., g( MD )]. 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 ψ(t) = φ(t)+2π∆ f t, effect. This receive filtering yields an increase of memory in where ∆ f = h(Mcpm −1)/2Ts. In this context, Fig. 2 shows the the system by Lg symbols, where (Lg − 1)Ts < 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 = (αk + 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 ψ(t) 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 M(η +1)× MD(η +1). 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 ψ(τ + kTs) = βk−Lcpm (2) Pcpm "r #! E ψk 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 (τ + lT ) k l cpm s (3) l=0 τ where the quantization operator Q(·) is applied element- + πh Mcpm − 1 + Lcpm − 1 + ϕ0, T wise. The quantization of zk is described by yk,m = s sgn(Re zk,m ) + jsgn(Im zk,m ), 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 ∈ k k−Lcpm {1 + j, 1 − j, −1 + j, −1 − j}. 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 CPMWAVEFORM 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. Nyquist signaling can yield a benefit for the design of zero- The corresponding tilted CPM phase ψ(τ+kTs) for one symbol crossings [11], which is key for channels with 1-bit quan- interval, i.e., 0 ≤ τ < Ts, is then discretized into MD tization at the receiver. In this section, a new subclass of samples, which composes the vector denoted by ψk (s˜k ) = CPM waveforms is introduced which provides relatively high, Ts Ts T [ψ( MD (kMD + 1)), ψ( MD (kMD + 2)), . . ., ψ(Ts(k + 1))] , signaling rate and high spectral efficiency at the same time. where M is the oversampling factor, and D is a higher resolu- Illustrated special configurations of the proposed waveform tion multiplier. The tilt of the phase can be established in the only require low-complexity at the transmitter and receiver. actual communication system by receiving at an intermediate In the sequel the proposed waveform is considered with the frequency (IF). With this, we can consider that different low- rectangular frequency pulse with duration Tcpm, but the exten- IF frequencies can be used, which motivates the definition of sion to frequency pulses like raised cosine and or Gaussian ψ (t) = ψ(t) + 2π nIF t. Choosing n > 0 is promising because pulses would be straight forward. IF Ts IF 3

0 jψ(t) ψ (t) ψ(t) Im e ψ(t) 3π 1 0 3π 2 2 π π π 1 2 jψ(t) π t t Re e 2 0 0 t (a) Ts (b) 3Ts (a) (b) Ts Fig. 3: 8-symbol CPFSK (a) and three-symbol-period FTN-CPM (b) tilted trellises Fig. 5: Tilted CPM constellation diagram (a) and trellis (b) of the proposed FTN-CPM with Tcpm = Ts, h = 1/4 and φ0 = π/4

adjustment of the modulation index h and the length of the frequency pulse Tcpm. In the sequel the FTN-CPM waveform conﬁgurations are detailed which are promising in the presence of 1-bit quantization at the receiver.

B. FTN-CPM for 1-bit quantization at the receiver A widely used waveform design criterion for channels with 1-bit quantization at the receiver is given by the maximization

0 0 of distance to the decision threshold [14]. By assuming that / / Fig. 4: Equi-bandwidth (Bc Bc) lines due to Carson’s criterion, Ts Ts = 3, Mcpm = 2 the receive filter g(t) only marginally changes the signal phase ψ(t) at the receiver, zero-crossings appear whenever the phase crosses integer multiples of π . Considering that sampling A. Proof-of-concept of Faster-than-Nyquist CPM 2 rate is equal to the FTN signaling rate, the illustrated FTN- In this section, it is shown that CPM signals can be con- CPM phase tree on the RHS of Fig. 3 is optimal in terms structed with significantly reduced symbol duration as com- of distance to decision threshold. The corresponding binary pared to standard CPFSK, which convey the same information FTN-CPM constellation diagram is shown in Fig. 5, which per time interval and occupy the same bandwidth. Because of implies that a zero-crossing conveys the transmit symbol 1 its mathematical tractability, Carson’s bandwidth criterion, as and 0 else. In order to achieve a spectral efficiency similar to used in [12], [13], is considered in this section. Following the the corresponding conventional CPFSK waveform, the length steps in [12], Carson’s bandwidth for CPM signals with i.u.d. of the frequency pulse Tcpm can be increased, cf. Fig. 4, where input and rectangular frequency pulse can be expressed as different cases are examined in the sequel. q 2 −1 −1 Bc = h (Mcpm − 1)(3TsTcpm) + Tcpm. (4) IV. CPM DEMODULATION As a reference, a standard CPFSK signal shall be considered, This section describes two demodulation methods. whose parameters are indicated with 0. The reference CPFSK 0 0 0 1 0 0 0 A. MAP Detection is fully described by Ts , Mcpm, h = 0 , Tcpm = Ts and Bc. Mcpm Now, it is aimed to construct a FTN CPM signal with a shorter The MAP decision metric for each symbol xk is given by the 0 ( | n) symbol duration T , such that the ratio T /T is an integer value, A Posteriori Probability (APP) P xk y . For the considered s s s ( | n) as shown in Fig. 3. The conveyed information per time interval system, approximated APPs Paux xk y can be computed 0 0 Ts /Ts via a BCJR algorithm [15] that is based on an auxiliary is equal for both signals by defining Mcpm = Mcpm. With this, channel law. Depending on the receive filter, noise samples are the relation between the bandwidth of the FTN-CPM signal correlated which then implies dependency on previous channel and the reference CPFSK signal can be expressed as k−1 n outputs, such that the channel law has the form P(yk |y , x ). −1 s 0 In this case the consideration of an auxiliary channel law W(·) 0 s 2T /T B T M2 − 1 T M s s − 1 c s © cpm s ª © 1 cpm ª is required [6], which can be described by 0 = h + ® 1 + 0 ® . B Ts 3Tcpm/Ts Tcpm Ts /Ts 3 ® k−1 n k−1 k c M W(y |y , x ) = P(y |y , β − − , x ) « ¬ cpm k k k−N k L N k−L−N+1 « ¬ k k (5) P(y |βk−L−N, x ) = k−N k−L−N+1 , (6) 0 k−1 k−1 T P(y |β , x ) For the case of predefined design parameter s and M , the k−N k−L−N k−L−N+1 Ts cpm relation in (5) is a function of modulation index h and relative where the dependency on N previous channel outputs is taken frequency pulse length Tcpm/Ts. Aiming for a high spectral into account. With this, an extended state representation is efficiency for the FTN-CPM signal a low relative bandwidth required which is denoted by 0 ( (5) is promising. An example case is illustrated for Ts /Ts = [ k ] βk−L−N+1,xk−L−N+2 , if L + N > 1, 3 and M = 2. As can be seen, the bandwidth increase sk = (7) cpm [β ], L N brought by the higher signaling rate can be compensated by k if + = 1, 4 where L + N is the total memory of the system. Based on the ψ(t) ψ(t) ψ(t) state notation in (7) the probabilities for the channel law can 3π k | k−1 | 2 be cast as P yk−N sk, sk−1 and P yk−N sk−1 . The auxiliary channel law probabilities (6) involve a multivariate Gaussian π integration in terms of π ¹ 2 k k k t t t P yk−N |sk, sk−1 = p(zk−N |sk, sk−1)dzk−N , (8) 3Ts 3Ts 3Ts k ∈ k (a) (b) (c) zk−N Yk−N zk Fig. 6: Simple receive strategy: decide for 0 (dashed line) and 1 (solid line); Different where k−N is a complex Gaussian random vector that FTN-CPM configurations are shown: (a) Tcpm = Ts, (b) Tcpm = 1.5Ts and (c) describes the input of the ADC, with a mean vector de- Tcpm = 2Ts hq E ψk i fined by µ = DG s e k−N −Lg , and covariance matrix x Ts 2 H T filter bank is hardware demanding and not compatible with R = σn DGG D , with D and G as introduced before with η = N. The integration interval is expressed in terms of the the considered 1-bit approach. Table I gathers the simulation k parameters for all considered CPM waveforms. 4-CPFSK [6] quantization region Yk−N that belongs to the channel output k serves a standard reference waveform, which provides reliable symbol yk−N . After rewriting (8) as a real valued multivariate Gaussian integration, how it is done in equation (21) in communication without additional coding when considering [6], the algorithm in [16] can be applied. Finally the BCJR 1-bit quantization. The same holds for 8-CPFSK [7] which n serves as reference waveform that does not require additional algorithm provides the probabilities Paux (sk−1, sk |y ) which are subsequently used for computing the symbol APPs via coding for M = 5 and optimized low-IF with nIF = 0.25. The P (x |yn) = Í P (s , s |yn). The multivariate proposed FTN-CPM is represented by the running example aux k sk−1,sk ⊇xk aux k−1 k 1 Gaussian integration∀ (8) becomes computationally expensive from Section III and IV specified by Mcpm = 2, h = 4 and rectangular frequency pulse with different durations Tcpm. Note when M, Mcpm and the channel memory are high, as detailed in [7]. Note that for uncorrelated noise samples, subsequent that for the considered FTN-CPM schemes the receive filter k−1 n is such that noise samples are uncorrelated and an auxiliary channel outputs are independent, such that P(yk |y , x ) = n channel law (specified by N) is not required. P(yk |x ), obviating the need for an auxiliary channel law. Waveform Simulation Parameters M = 4, L = 1, T = 0.5T , B. The Proposed Simple Demodulator 4-CPFSK [6] cpm cpm g s h = 1/4, nIF = 0, φ0 = π/4, N = 0 In order to relieve the computational load at the receiver, M = 8, L = 1, M = 5, T = 0.5T , 8-CPFSK [7] cpm cpm g s some versions of the proposed FTN-CPM scheme can be h = 1/8, nIF = 0.25, φ0 = π/8, N = 0 M = 2, M = 1, T = T , demodulated with an alternative simple strategy. The receive Proposed FTN-CPM cpm g s h = 1/4, nIF = 0, φ0 = π/4 strategy for the binary FTN-CPM case with h = 1 , and 4 TABLE I: Considered waveforms sufficiently small Tcpm, like Tcpm = 2Ts, only involves the evaluation of a change in real or imaginary part, depending In the sequel the adjustable power containment bandwidth on the previous sample yk−1, which can be cast as B90% is considered, where we refer to 90% power containment ( as default and use 95% as an alternative for some cases. 1 |{Re{y } − Re{y }|, if y ∈ {1 + j, −1 − j}, 2 k k−1 k−1 The considered SNR is defined by xˆk = 1 |{Im{yk } − Im{yk−1}|, if yk−1 ∈ {1 − j, −1 + j}. 2 lim 1 ∫ |x (t)|2 dt T→∞ T T Es −1 Fig. 6 illustrates the receiver decisions in a noise-free scenario SNR = = (TsB90%) , (10) N0 B90% N0 for Tcpm = Ts in Fig. 6(a), Tcpm = 1.5Ts in Fig. 6(b) and q E Tcpm = 2Ts in Fig. 6(c), where the phase distortion brought s jψ(t) where x(t) = T e is the complex low-IF representation by the receive filter is neglected for illustration purpose. Note s of the signal and the noise power density N0 corresponds to that for larger values for Tcpm the noise sensitivity increases. the variance of the noise samples in (3). A special case is given by Tcpm = 2Ts which results in the same number of equidistant constellation points as the A. Spectral Efficiency and Effective Oversampling Ratio corresponding 8-symbol CPFSK. The spectral efficiency w.r.t. B90% reads

V. NUMERICAL RESULTS Ibpcu spectral eff. = [bit/s/Hz], (11) In order to preserve the transmit waveform and its zero- B90%Ts crossings, a suboptimal short bandpass receive filter is consid- where Ibpcu is the achievable rate w.r.t. one symbol duration ered as follows Ts, which is computed by applying the methods developed in s t − T /2 [6]. The effective oversampling ratio w.r.t. B90% is given by 1 g j2π∆f (t−Tg /2) g(t) = rect · e , (9) 0 −1 Tg Tg OSR = M (B90%Ts) . (12) which is similar to the integrate and dump receiver considered Table II displays computed values for effective oversampling in [17]. Note that the common receiver based on a matched factor and maximum spectral efficiency for the waveforms 5

4 Proposed Simple Rx BCJR 10−1 3

10−3

2 BER Proposed FTN-CPM, M = 1, Tcpm = 2Ts 8-CPFSK, M = 5 [7] Proposed FTN-CPM, T = 2T Proposed FTN-CPM, M = 1, Tcpm = 1.6Ts cpm s Proposed FTN-CPM, T = 1.6T 8-CPFSK, M = 5, [7] cpm s 1 −5 Proposed FTN-CPM, Tcpm = Ts Proposed FTN-CPM, M = 1, Tcpm = Ts 10 4-CPFSK, M = 2 [6] Spectral Eff. [bit/s/Hz] 4-CPFSK, M = 4 [6] 4-CPFSK, M = 4 [6] 4-CPFSK, M = 2 [6] 0 0 5 10 15 20 25 30 −5 0 5 10 15 20 25 30 SNR [dB] SNR [dB] Fig. 8: BER performance of the considered CPM waveforms Fig. 7: Spectral Efﬁciency with respect to the 90% power containment bandwidth

binary FTN-CPM signal can be detected with an extremely considered in Table I. Moreover, Fig. 7 illustrates the spectrum simple detector with a performance close to MAP detection. efficiency w.r.t. B90% versus SNR. For this bandwidth criterion choosing Tcpm ≥ 1.6Ts can yield a higher spectral efficiency as the corresponding CPFSK waveform for medium and high REFERENCES SNR. When referring to the B95% it is required to choose [1] J. Anderson, T. Aulin, and C. E. Sundberg, Digital Phase Modulation. ≥ New York: Plenum Press, 1986. Tcpm 2Ts for approaching the spectral efficiency of the [2] C. E. Sundberg, “Continuous phase modulation,” IEEE Commun. Mag., corresponding CPFSK, cf. Table II. vol. 24, no. 4, pp. 25–38, April 1986. [3] R. Walden, “Analog-to-digital converter survey and analysis,” IEEE J. T log M log M Waveform cpm M 2 cpm 2 cpm OSR0 Sel. Areas Commun., vol. 17, no. 4, pp. 539 –550, Apr. 1999. Ts B90%Ts B95%Ts [4] S. 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