Simplified Receivers for Generic Binary Single Side Band CPM Using PAM Decomposition

This paper investigated the pulse amplitude modulation (PAM) decomposition of a class of continuous phase modulation (CPM) signal, which has the property to be a single-side band. We used the PAM decomposition as a convenient solution to provide a large reduced complexity trellis detection to approach the theoretical optimal performance. Moreover, we developed an algorithm to obtain the necessary PAM pulses to approach the optimal performance bound using suboptimal receivers. The algorithm is generic; it can provide the results for any parameter combinations. The proposed demodulation system exhibits excellent performance with minimal complexity with respect to the maximum likelihood sequence detection (MLSD) optimal receiver.


I. INTRODUCTION
Continuous phase modulation (CPM) [1] schemes are good candidates for long-distance wireless communication due to their high energy efficiency arising from their constant signal envelopes. For this attractive property, the CPM schemes have been widely used in digital communication systems, e.g., in satellite and deep-space communications, optical fiber, telemetry, etc. In the present paper, we consider a CPM scheme having the original feature of directly generating a single side-band (SSB) spectrum providing a very compact frequency occupation [2]. By directly, we mean that SSB property results from the original modulation, not from post-filtering [3]. The new CPM is hereafter called Single Side-Band Frequency Shift Keying (SSB-FSK), which uses a generic phase derivative pulse with a Lorentzian shape and a 2π phase increment. The original idea behind the Lorentzian pulse and the SSB property was first investigated in quantum physics [4] and [5] with the demonstration of a new quasi-particle called ''Leviton''.
Digital transmissions based on CPM constitute the first immediate applications of classical levitonics [6]. In [3], we gave the principle of the modulation and its mathematical justification. In [7], we presented a complete study of the The associate editor coordinating the review of this manuscript and approving it for publication was Yuan Gao . SSB-FSK performance regarding error probability, spectral efficiency, SSB property, and complexity. We investigated the error probability based on the derivation of the minimum squared Euclidean distance. We explored the spectrum of the SSB-FSK by quantifying its signal power bandwidth occupancy using a numerical method. We estimated the complexity from the number of states required by the Viterbi algorithm (VA) to implement the MLSD optimal receiver. Finally, we used these metrics in two different optimization methods to illustrate the full potential of this waveform. Based on this work, we learned that the SSB-FSK has the advantage of being highly tunable and can go beyond what has been achieved with well-known CPM schemes (e.g., RC and GMSK). This work allowed us to determine some interesting configurations, some with acceptable receiver complexity and others requiring more work to design sub-optimal receivers to reduce their high complexity. Therefore, this paper aims to reduce the receiver complexity of the SSB-FSK signals using a suboptimal receiver without affecting the performance. The SSB-FSK signals studied in this work is retrieved from the results obtained in [7] for the specific binary case (modulation level M = 2).
A variety of methods have been suggested to decrease the receiver complexity [1], [8]. In this paper, we are particularly interested in using the pulse amplitude modulation (PAM) decomposition of CPM signals.

A. RELATED WORKS ON CPM PAM DECOMPOSITION
The first connection between linear modulations and CPM was addressed by Laurent [9]. Laurent showed that any binary non-integer single-h CPM could be presented by a superposition of PAM pulses. He also showed that using a reduced number of PAM pulses or even only keeping the main pulse can often approximate very well the CPM signal. Mengali and Morelli followed [10] by extending the PAM decomposition of CPM to M-ary signaling. Then, Huang and Li provided the PAM decomposition for the particular case of CPM schemes with integer modulation index [11]. Soon after, Perrins and Rice [12], and Wylie-Green [13] proposed a generalization of this decomposition to include multi-h CPM schemes. Moreover, Perrins and Rice also developed a PAM decomposition for ternary CPM [14], which was revisited and simplified by Othman et al. [15].
Following Laurent's PAM decomposition, Kaleh [16] derived a suboptimal MLSD receiver in AWGN using the PAM decomposition. He also showed that with a considerable decrease in the number of matched filters, the suboptimal PAM-based could reach the optimal MLSD receiver. Moreover, Kaleh provided a boundary for the performance of the suboptimal PAM-based receiver. However, Perrins and Rice [17] showed a lack of accuracy in Kaleh bound. Therefore, they derived an exact performance bound for the suboptimal PAM-based receiver in AWGN channel based on the pairwise error probability. This performance bound has been used to design reduced complexity PAM-based receivers for some CPM schemes [18].

B. MAIN CONTRIBUTIONS
In this paper, a reduced-complexity receiver based on the PAM decomposition is designed and analyzed for binary nonantipodal SSB-FSK. The three key contributions of this paper are: 1) The derivation of a PAM representation for binary nonantipodal SSB-FSK, based on Laurent [9] and Huang and Li [11] PAM decomposition. This representation is not restricted to a specific waveform but could be used for any binary non-antipodal CPM scheme. 2) The design of a simplified receiver for binary SSB-FSK based on Kaleh's [16] suboptimal receiver. 3) The proposal of an algorithm computing the number of PAM pulses required by Kaleh's [16] suboptimal receiver to attain a specific performance bound. This algorithm is based on the performance bound derived by Perrins and Rice [17] and the PAM mean-square approximation [10]. The rest of the paper is organized as follows: we introduced the signal model in Section II. We presented the equivalent PAM representation of SSB-FSK signals for integer and non-integer modulation indices in Section III. Then, in Section IV, we described the MLSD optimal receiver alongside a low-complexity receiver. In Section V, we developed a simple algorithm to decide on the number of PAM pulses required to attain a specific performance bound. The algorithm is generic; it can provide the results for any parameter combinations. In Section VI, we applied the PAM decomposition alongside the algorithm developed on the binary SSB-FSK results retrieved from [7]. In Section VII, we showed that the simplified PAM-based receiver could reach a similar bit error rate (BER) performance compared to the optimal receiver with a considerable decrease in complexity. Finally, conclusions are drawn in Section VIII.

II. SSB-FSK SIGNAL MODEL
The complex envelope of the SSB-FSK signal is defined as where E s is the signal energy per symbol, T s is the symbol interval,h = 2h, where h is the modulation index used to ensure a 2π phase increment, and α = {α i } denotes the information binary sequence. The information symbols α i are assumed to be independent and identically distributed.
It is important to note that no antipodal coding is performed in order to preserve the SSB property, i.e. α i ∈ {0, 1}. Furthermore, φ 0 (t) is a Levitonic phase-shift function [3] and VOLUME 9, 2021 is given by: where g(t) is a truncated Lorentzian pulse of duration LT s > 1 (partial-response) symbol durations, defined as The parameter w is the pulse width, and µ is a correcting factor introduced to keep a 2π phase increment after frequency pulse truncation. The correcting factor µ is defined as the ratio between the total phase increment without any truncation and the one obtained after Lorentzian truncation: In other words, the derivative of the total phase φ(t, α) is then a sum of overlapping Lorentzians 2µw 2 (t−kT s ) 2 +w 2 , centered on kT s weighted by the symbols α i and truncated to the length LT s . For the non-truncated Lorentzian pulse (L = ∞), µ = 1 (no correction is required).

III. PAM REPRESENTATION OF SSB-FSK SIGNAL
Despite its originality coming essentially from the Lorentzian pulse, the SSB-FSK signal is nothing else than a CPM signal without any antipodal coding at the symbol level. Therefore, we proposed to apply the same approach given in [9], [11] for the PAM decomposition of CPM signals with integer/noninteger modulation indices. To do so, we had to rewrite the SSB-FSK signal in a way that we could retrieve the same initial assumptions considered in the derivations presented in [9] and [11]. Hence, the signal s(t, α) given in (1) is strictly equivalent to According to (6), the baseband SSB-FSK signal s(t, α) could be viewed as a product of two independent signals: s 1 (t,α), depending on the antipodal coded information symbolsα, and s 2 (t) a deterministic signal that does not carry any information.

A. PAM DECOMPOSITION OF SSB-FSK WITH NON-INTEGER MODULATION INDEX
For non-integer modulation indices, we followed the derivations given by Laurent in [9], where s 1 (t,α) can be reformulated as where Q = 2 L−1 is the number of pulses required for an exact representation of the signal. The PAM pulse c k (t) is defined as: The function u(t) is given by Note that the parameters β k,i can only take the values 0 or 1, and are obtained from the equality Additionally, D k is the k-th pulse duration, defined as Finally, the mapping between the pseudo-symbols and the information data α i follows the expression

B. PAM DECOMPOSITION OF SSB-FSK WITH INTEGER MODULATION INDEX
For integer modulation indices, we follow the derivations given by Huang and Li [11], stating that the signal s 1 (t,α) can be represented as where J = cos (hπ). The PAM pulses h 0 (t) and h k (t) are defined as The parameters β k,i are determined by the equality Additionally, L k is the k-th pulse duration, defined as Finally, the mapping follows the expression We will always refer to these two methods with the same notations, v k , ρ k , and k , for the PAM pulse, the pseudo symbols, and the pulse duration, respectively.
From what we presented in this section, we could deduce that the PAM decomposition is only applied on the s 1 (t,α) signal since it contains the complete transmitted information. Thus, the first step on the receiver side is to extract s 1 (t,α) from s(t, α) by discarding s 2 (t), which presents no difficulty since s 2 (t) is a deterministic periodic signal with no information. Besides, to assure that we have no degradation in BER performance after discarding s 2 (t), in Appendix. A, we showed that s 1 (t,α) and s(t, α) have the same error probability performance, using the union bound and the minimum squared Euclidean distance (d 2 min ).

IV. RECEIVER FOR SSB-FSK MODULATION
In this section, we introduced the detection methods used on the received signal. We based our study on the development proposed in [1, ch.7] and [18]. Assuming an additive white Gaussian noise (AWGN) channel, the complex baseband representation of the received signal is where n(t) is a complex baseband AWGN with zero mean and power spectral density N 0 .

A. OPTIMUM MAXIMUM LIKELIHOOD DETECTION
Due to the assumption of AWGN noise, the optimal receiver should be able to find a sequenceα, which maximizes the log likelihood function given by: Since s(t) has a constant envelope, minimizing (19) is equivalent to maximizing the correlation function [1] Due to the trellis-like structure of the CPM modulation, the computation of the correlation function (20) can be done using the Viterbi Algorithm (VA). To highlight the connection between the VA and the correlation function (20), we need to express the phase trajectories φ(t, α) as where the right side θ(t, α n ) presents the correlative state vectors (α n−1 , α n−2 , . . . , α n−L+1 ). The number of all possible correlative state vectors is M L−1 . The left side θ n is the phase state, where the number of phase states could takes only p unique values modulo-2π. The number of phase states p is obtained from the modified modulation indexh = 2m/p, whereh takes only positive rational values. Based on (21), it is now possible to present the SSB-FSK signal in the form of a trellis with a finite number of states, where each branch of the trellis is presented with the starting state (S n ), ending state (E n ), and the (L + 1)-tuple (σ n ) defined as S n = (θ n−L , α n−1 , α n−2 , . . . , α n−L+1 ), σ n = (θ n−L , α n , α n−1 , . . . , α n−L+1 ), The total number of states in the trellis is given by From the arrangement of (21) and the definition of the starting states, ending states, and the (L + 1)-tuple (S n , E n , σ n ) (22), we computed (20) recursively with where e −jθ n−L is the cumulative phase, and Z n (·) is the sampled matched filter output, given by: The number of matched filters required for the VA is M L , obtained from the different possible sequences presented in the (L + 1)-tuple vector σ n , and the number of branches pM L , when considering all possible p phase states.
The performance of the error probability for the SSB-FSK scheme in AWGN noise is quantified by the Union Bound [1, ch.3]

defined in Appendix. A (38).
A detailed study of the error probability performance for SSB-FSK signals is given in [7].

B. REDUCED-COMPLEXITY RECEIVER FOR SSB-FSK
First, we have to recall that the transmitted signal s(t, α) is composed of two parts: one information-dependent component s 1 (t,α), and one information-independent component s 2 (t). However, so as to use the PAM decomposition, we need to process only s 1 (t,α). Therefore, after extracting s 2 (t) from the received signal r(t), the input of the proposed detector is expressed as whereñ(t) is an equivalent noise resulting from extracting the signal s 2 (t), which remains Gaussian. Using the PAM decomposition presented in Section III, we obtained a Viterbi-like receiver based on the PAM decomposition by inserting s 1 (t,α) (7)/(13) (non-integer/integer modulation index h) into (19) to get the recursion [16]: The maximum number of PAM pulses N is given by: The number of states for the PAM-based detector is [18] where From (27) and (31), we simplified the receiver complexity (number of states and matched filters) by considering a fewer number of pulses [16]. Based on the selection of the number of pulses, the performance and the complexity of the PAM-based receiver may vary. Therefore, it is essential to define thresholds between the number of PAM pulses (complexity) and the performance. In the next section, we will present an algorithm inspired by the methods used in [18] to obtain these thresholds.

V. DECISION ON THE NUMBER OF PAM PULSES
This section discusses how to select the number of PAM pulses for a targeted error probability P e . In Fig. 1, we presented a flowchart for the associated algorithm used to select the number of PAM pulses. The algorithm depends on two important factors: the PAM pulses obtained from the Mean-Square approximation and the performance bounds of the MLSD and the PAM-based receivers.

A. PAM MEAN-SQUARE APPROXIMATION
Laurent showed in [9] that the signal power is unevenly distributed between the PAM pulses, and for most cases, almost all the power is concentrated in the first few pulses. We will refer to these pulses as the main pulses. Laurent obtained a good approximation of the original CPM signals using only the main pulses (N ) instead of (N = Q−1) pulses. Following Laurent's derivation, Mengali and Morelli [10] showed that a better approximation of the original CPM signals using an optimum PAM pulse p k (t) instead of v k (t). This optimum pulse p k (t) is given by minimizing the meansquare error (MSE) with respect to the original signal. The optimum pulse p k (t) is given by: where w k,i (m) are weights whose calculations are detailed in [10]. Furthermore, in [10], it was shown that for a modulation index h = 1 2 , using p k (t) or v k (t) has the same performance in approximating the original signal, which means that v k (t) is already optimal in the MSE sense.
In Table 2, we present the MSE between the original signal and the approximated one using v k (t) (σ ) and p k (t) (σ ), for three CPM schemes with a different number of main pulses (N ). We can observe from the last two columns that we have similar results between (σ ) and (σ ). Therefore, we deduced that also for integer modulation index h = 1, p k (t) and v k (t) have similar performance in approximating the original signal. As a consequence, we can find in Fig. 1 a decision box that selects the PAM pulse p k (t) or v k (t) based on the modulation index value h.

B. PERFORMANCE BOUND OF THE PAM-BASED RECEIVER
The performance of the PAM-based receiver can be computed using the pairwise error probability union bound given in [17]. The union bound for binary single h schemes is presented in the form where γ min (different from zero) is the difference sequence (α i − α j ) corresponding to the modified minimum distance (d ), and R is the span of observation symbols. The function W (γ min ) is the Hamming weight of the difference vector γ min . The modified minimum distance d is given by [19]: where withs(t) the approximated signal using N PAM pulses.

REMARK
The modified Euclidean distance (d ) is only an upper bound of the normalized minimum squared Euclidean distance (d 2 min ) for the PAM-based receiver. Moreover, it is FIGURE 1. Flowchart of the algorithm used to select the number of PAM pulses required for a specific SNR difference α dB between the optimal MLSD receiver performance bound and the PAM-based receiver performance bound for P e = 10 −5 .
also known from the analysis of the union bound for the optimal receiver that the minimum normalized Euclidean distance d 2 min will never reach the upper bound d 2 B for several modulation indices h (weak modulation index h [1]). These weak modulation indices h have the same effect on the PAM-based receiver. Due to the high computational efforts on obtaining the exact minimum squared Euclidean distance (d 2 min ) for PAM-based receiver [1, ch.8], we will always use the upper bound (d ) to presents the performance of PAM-based receiver. To ensure that this upper bound (d ) is optimal, we first computed the minimum squared Euclidean distance (d 2 min ) and the upper bound (d 2 B ) for the optimal receiver. We calculated the difference between the two obtained minimum squared Euclidean distances. If the difference is slight (≤ 0.1), we assumed that the normalized minimum squared Euclidean distance (d 2 min ) could reach the upper bound (d 2 B ), which means that the upper bound (d 2 B ) can present the minimum squared Euclidean distance (d 2 min ) for the optimal receiver (tight upper bound). Consequently, d can present the minimum squared Euclidean distance (d 2 min ) for the PAM-based receiver. Therefore, we can find in Fig. 1 a decision box, which computes the difference between the normalized minimum squared Euclidean distance (d 2 min ) and the upper bound (d 2 B ), to tests if it exists an upper bound d that reaches the error probability P e .

VI. CASE STUDIES FOR SSB-FSK
We present in this section some case studies for several SSB-FSK schemes, which are showed in Table 3. The parameters selection for these schemes was conducted in [7] using the Pareto optimum multi-objective optimization. In Fig. 2, we showed how the performance bound for the PAM-based receiver approaches the optimal MLSD performance bound using the algorithm ( Fig. 1) with (α < 0.5) for the configuration 6SSB-FSK from Table 3. The plot for N = 0 is not presented, wherefrom (13) v 0 (t) (h 0 (t)) does not carry TABLE 3. The minimum squared Euclidean distance, the 99.9% bandwidth occupancy, and the complexity (number of states and matched filters) of LSSB-FSK studied cases for an optimum MLSD receiver. VOLUME 9, 2021 any information, which means a flat performance bound. Therefore, v 0 (t) is not considered in N as one of the required pulses for the PAM-based receiver. From Fig. 2, we noted that increasing the number of PAM pulses N sometimes induces a wide leap toward the optimal bound (especially noted between N = 3 and N = 4). Nevertheless, this is not a general trend: after N = 4 increasing N has a slight increase towards the optimal bound. Then, it is possible to define different thresholds between the required number of PAM pulses (complexity N ) and the performance. The performance is computed in terms of SNR required to reach a specific error probability.   Similarly, in Fig. 4 we showed the performance bound for the PAM-based receiver approaches the optimal MLSD performance bound using the algorithm (Fig. 1) with (α < 0.5)  Table 3, to obtain the required number of PAM pulses N . bound.
for the configuration 2SSB-FSK from Table 3. We observe that using only one pulse N = 1 leads to the optimal bound. This observation can be explained from Fig. 5, where most of the signal energy is contained in v 0 (t).

A. LINEAR DETECTOR
Based on the observation of Fig. 4, we conclude that we only need one pulse to reach the optimal MLSD bound. For this reason, it is interesting to decrease the complexity at the receiver side by applying a linear detector on the received signal instead of the PAM Kaleh receiver.
Following Kaleh's approach in [16], the linear detector is composed of a matched filter v * 0 (−t) followed by a Wiener filter, which plays the role of an equalizer that reduces the effect of intersymbol interference (ISI) from v 0 (t) and v 1 (t).
In Fig. 6, we presented the coefficients of this filter, which are computed according to [16] for the 2SSB-FSK.  Fig. 7 shows a BER comparison between the optimal receiver and Kaleh's receiver for 6SSB-FSK. The 8 states receiver is near optimal, as the BER degradation does not exceed 0.1 dB at BER =10 −5 . On the other hand, we obtain a vast decrease in complexity since we went from 32 to 8 for the number of states and 64 to 8 for the number of matched filters. We have also plotted the performance bounds showing a fair reflection of the BER performance, which confirms the results obtained from the algorithm (Fig. 1). Fig. 8 presents a BER comparison between the optimal MLSD receiver and the linear receiver for the configuration 2SSB-FSK. The linear filter with Wiener filter reaches a BER = 10 −5 at a SNR slightly lower than the optimal MLSD  bound with a loss equal ≈ 0.2 dB. Without the Wiener filter, the degradation increases to ≈ 0.4 dB. Table 4 presents the simulation results summary of the BER performance and the receivers complexity for the binary 6SSB-FSK and 2SSB-FSK. From these results, we can conclude the importance of the PAM decomposition and the algorithm (Fig. 1) in reducing the detection complexity for the SSB-FSK signal.

VIII. CONCLUSION
In this paper, we studied a suboptimal receiver for SSB-FSK CPM schemes using PAM decomposition. First, we obtained the linear PAM decomposition of the nonantipodal integer/non-integer modulation index h SSB-FSK. Using the linear PAM decomposition, we proposed a suboptimal receiver. Due to the PAM decomposition's flexibility, we developed an algorithm with many performance capabilities and with different complexity reductions. One of the most important contributions of this paper is to reduce the complexity at the receiver side using Khaleh and linear receivers based on the PAM decomposition, with an acceptable slight decrease in performance.

APPENDIX A
In this appendix, we show that s 1 (t,α) and s(t,α) have the same error probability performance using the union bound and the minimum squared Euclidean distance. First, let us define the union bound as presented in [1,Ch. 3,p. 55] for any CPM signal where d 2 min is the minimum squared Euclidean distance. The function Q(.) is the Gaussian error function defined by To prove that the two signals s 1 (t,α) and s(t,α) have the same error probability, we need to prove that they have the same minimum squared Euclidean distance.
Based on [1,Ch. 3], the general form of the normalized squared Euclidean distance for any CPM signals is given as The variable NT s is the observation symbol interval, γ is the single difference data sequence, where γ = α − α , and φ(t, γ N ) is the phase difference trajectories.
Based on (40), it is proved that s 1 (t,α) and s(t,α) have the same d 2 min if they have the same phase difference trajectories φ(t, γ N ). The phase difference trajectories for s(t,α) is where the difference sequence γ can take the values γ = {0, ±1} for the binary case (M = 2). Similarly, the phase difference trajectories for s 1 (t,α) is and the difference sequenceγ for binary case (M = 2) can take the valuesγ = {0, ±2}.
As we proved that the two modulations had the same phase difference trajectories, we showed that they had the same minimum normalized Euclidean distance d 2 min , and consequently, the same error probability.