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Performance vs. Spectral Properties For Single-Sideband Continuous Phase Modulation

This study reports on a single side band spectrum directly generated by a continuous phase modulation (CPM) signal. This signal is analyzed in terms of modulation indices, pulse lengths, and pulse widths, all of which affect error probabilities, bandwidths, and SSB property. The error probability performance is based on an approximation of the minimum Euclidean distance. A numerical power spectral density calculation for this particular SSB modulation in terms of modulation index is presented. Reasonable tradeoffs in designing modulation schemes have been defined using multi-objective optimization to ensure sizable improvements in bit error rate (BER) and spectral efficiencies, without losing the property of being a SSB signal. Performance comparisons are made with known CPM schemes, e.g., Gaussian Minimum Shift Keying (GMSK) and Raised Cosine based CPM (RC).


I. INTRODUCTION
C ONTINUOUS phase modulation (CPM) [1] is particularly suitable for long-distance wireless communications due to spectral efficiency and high energy efficiency as a result of its constant signal envelope. For these attractive properties, many digital communication systems use the CPM family extensively, e.g., satellite communications, deep-space, optical fiber, telemetry, etc. Further to these advantages, the authors in [2] presented a CPM scheme, which has the original feature of directly generating a single-sideband (SSB) spectrum, which in turn, provides a highly compact frequency occupancy. By directly we mean that SSB property results from the original modulation, not from post-filtering [3]. This waveform is hereafter referred to as single-sideband Frequency Shift Keying (SSB-FSK). SSB-FSK uses a generic phase derivative pulse with Lorentzian shape and a 2π phase increment. The Lorentzian pulse belongs to specific shapes addressing fundamental quantum physics, particularly the ondemand injection of a single electron in a quantum conductor. Digital transmissions based on CPM are the first immediate application of classical levitonics [4]. In [3], we presented the principle of the modulation and its mathematical justification. We did not, however, provide any result about detection schemes or the interplay between error probability performance, bandwidth, and SSB property. Besides, CPM signals suffer from receiver complexity due to the memory property introduced by the phase continuity, which should also K be considered alongside the error probability, bandwidth, and SSB property. In this paper we suggest several tradeoffs under all these considerations to get the full potential of SSB-FSK signals.

A. Related Work On Single-Sideband CPM
This modulation scheme originates from the theoretical proposal of Levitov et al. [5], who demonstrated how to create a pure single electron without any perturbation in the quantum conductor. It is widely accepted that a quantum conductor is full of many free-moving electrons. When one electron is excited by an elementary pulse, others tend to move to fill the hole created, generating undesirable neutral plasmonic waves. To overcome this issue, Levitov et al. predicted that one could inject an electron without causing any disturbance if and only if the voltage pulse has a Lorentzian time shape. Using the Lorentzian pulse, we can generate a pure single electron state, called a "Leviton", a new quasi-particle with minimal excitation. The Levitov idea is given further experimental illustration by the authors in [6] and [7], who showed that using a Lorentzian pulse, the electron energy distribution becomes SSB-the left part of the energy distribution disappears. Based on this experimentation, the authors in [4] proposed a new CPM modulation signal, which directly provides a SSB signal. Then in [2] and [8], we gave the first system model of SSB-FSK as a CPM modulation signal. Additionally, we introduced properties, such as the symbol-bysymbol coherent demodulation scheme based on the orthogonality of SSB-FSK signals. Due to the memory generated by the CPM signals, we developed an average matched filter detector, making decisions according to a given observation window. We then analyzed the power spectral density (PSD) for the binary case, leading us to demonstrate that the PSD of the SSB-FSK signal is unilateral with respect to the carrier frequency (f c ), and almost all of the power is concentrated in the first period of the frequency band. Finally, in [3], we presented an analytical expression for the power spectral density (PSD) for binary and quaternary SSB-FSK.

B. Main Contributions
In this paper, we report a complete study of the SSB-FSK performance regarding error probability, spectral efficiency, and SSB property depending on the following parameters: • modulation index h: even if increasing h consistently increases the bandwidth (BW) occupancy, it does not carry a monotonic impact on the error probability performance or the SSB property preservation. • pulse length L: expanding L results in a complete response of the Lorentzian filter and will hence introduce a better BW occupancy and lower side lobes, although the larger L, the larger the receiver complexity. • modulation alphabet size M : increment in M produces a gain in error probability performance while degrading the BW occupancy and increasing complexity. • pulse width w: it is a new tuning parameter introduced specially for SSB-FSK, which plays an essential role in error probability and BW occupancy performances.
It is worth emphasizing the need for a functional interplay between all these parameters, which, respectively, lead to opposite effects on the error probability and spectral efficiency. Therefore, we need to define certain tradeoffs between the parameters as long as an optimal scheme considering all performance metrics is impossible.
Four key contributions of this paper are: 1) The study of the Maximum Likelihood Sequence Detector (MLSD) performance on the SSB-FSK signals, and the effects of all parameters (h, w, L, M ) on the error probability performance-mainly the effect of pulse width w. Additionally, we highlight some points related to receiver complexity (memory performed by the Viterbi algorithm [9]). We gave the error probability performance in terms of an approximation of the achievable minimum Euclidean distance.
2) The effect of (h, w, L, M ) on the power spectral density (PSD)-especially for h = 1, and compared it with the results obtained for the particular case of h = 1 (responsible of pure SSB signal) [2]. Here we exhibit a better spectrum efficiency, with h = 1 and L ≤ 12 (lower receiver complexity) for a variety of w, with a slight loss in SSB property. 3) An optimization method to select the best parameter combinations, to obtain the optimal performance of the SSB-FSK scheme; this optimization method is based on the Pareto optimum, which is a multi-objective optimization method. 4) Energy-bandwidth comparisons for SSB-FSK signals, where we were able to define a parameter configuration, with acceptable tradeoffs between energy consumption and BW occupancy for SSB-FSK that outperforms popular CPM schemes (e.g., GMSK, RC).
The rest of the paper is organized as follows. In Section II, we briefly introduce the signal model. In Section III, we evaluate the error probability performance of SSB-FSK signals using the union bound. In Section IV, we present a numerical method to compute the power spectral density, using the autocorrelation approach. In Section V, we present the Pareto optimum, a multi-objective optimization method, which is used alongside a brute force to obtain the best possible tradeoffs for SSB-FSK scheme. In Section VI, we illustrate several simulation results for error probability performance, power spectrum performance, and energy-bandwidth comparison, respectively. Finally, conclusions are drawn in Section VII.

II. SINGLE-SIDEBAND CPM SYSTEM MODEL
The SSB-FSK signal is defined as CPM [1] with the complex representation where E s is the energy per transmitted symbol and T s is the duration of the symbol. The information-carrying phase is defined as where α i is the transmitted symbol that takes values from the M-ary alphabet 0, 1, . . . , (M − 1). For instance, in order to preserve the SSB property, we can only use positive information symbols, responsible of the right side band spectrum. The negative values will introduce a lower side in the spectrum. Therefore, no antipodal coding is allowed [2,8]. Furthermore, h = 2h, where h is the modulation index used to ensure a 2π phase increment. The phase response φ 0 (t) is represented as where g(t) is a Lorentzian frequency pulse truncated to a symbol duration L > 1 (partial-response), defined as The variable w is the pulse width, a vital tuning parameter that significantly impacts the performance, mainly the spectral efficiency of the transmitted signal s(t, α) [2,8]. As the Lorentzian pulse is characterized by a slow decrease, a pulse truncation is needed to get acceptable values of pulse lengths (regarding especially receiver complexity). As a consequence of this truncation, µ is introduced as a correcting factor to keep a 2π phase increment to sustain the SSB property. The correcting factor µ is defined as the ratio between the total phase increment without any truncation and the one obtained after Lorentzian truncation: LTs/2 −LTs/2 The effect of w on the frequency pulse g(t) for L = 4 is presented in Fig.2, for different values of w = 0.3, 0.7, 1.3. As shown in Fig.2, it is obvious that increasing w rises the tails of frequency pulse g(t), which means an increase in inter-symbol interference (ISI).

III. SINGLE-SIDEBAND CPM ERROR PROBABILITY
In this section, we consider the performance of the proposed signals when transmitted over an AWGN (additive white Gaussian noise) channel, assuming there is no channel coding. To evaluate the performance of the signal, we used the same method shown in [1,Ch. 3]. We consider the union bound of the error probability, as the asymptotic performance for high values of E b /N 0 , where E b is the mean energy per information bit. The union bound is defined as in [1,Ch. 3,p. 55]  where d 2 min is the minimum squared Euclidean distance. The function Q(.) is the error Gaussian function defined by The normalized squared Euclidean distance (d 2 ) is the Euclidean difference between any two transmitted SSB-FSK data sequences α and α . The general form of the normalized squared Euclidean distance for any set of signals is given by where N T s is the observation symbol intervals. The normalized squared Euclidean distance d 2 (α, α ), can be represented as a function of a single difference data sequence γ, where γ = α − α . Therefore, (8) can be rewritten as (see appendix A for the derivation details)[1, ch.2] where φ(t, γ N ) is the phase difference trajectories. To calculate the minimum normalized squared Euclidean distance (d 2 min ) for N observation symbols, we need to find pairs of γ that minimize (9). It is clear that (9) has positive integral, which means that the higher the N we observe, the higher the Euclidean distance we obtain. Consequently, if we obtain an upper bound for N → ∞, we obtain an upper bound at any defined N . This suggests the search for the difference phase trajectories φ(t, γ) with the shortest duration time τ , where φ(t, γ) is equal to zero at all times t ≥ τ . Hence, an upper bound (d 2 B ) on the Euclidean distance d 2 (γ) can be obtained using only φ(t, γ) with the shortest length τ . To better clarify the idea, we use in Fig. 3 the difference phase tree for general modulation index h , as an example to illustrate the method we used to obtain the boundary. Since we are presenting different phase trees, the first symbol γ 0 cannot be zero, since α 0 and α 0 must not be the same. Using the difference phase tree, we can evaluate γ with the shortest length of τ . We search for the merger points between two different paths using the different phase trajectories (φ(t, γ)) that start at t = 0, and coincide with the x-axis at a certain time t = t k , and coincides all the time after that for all t > t k . In Fig. 3, we showed two of the merger points A and B, and two of the best γ combinations with the shortest τ (presented as marked lines). We can directly see that the difference phase tree with the best γ combinations (presented as marked lines) coincide at t equal to 4 and 5 with the x-axis respectively at the points A and B, and continue to be aligned with the x-axis from this points to the end. Usually, the time instant of the first merger point is obtained at t = (L + 1)T s , for non-weak modulation schemes [1, Ch. 3, p. 73]. The phase difference tree, with the best difference sequence γ is defined by given that The variable m is the index of the merger points; the higher the number of mergers m we select, the tighter upper bound d 2 B we obtain (in Fig. 3, we showed only two mergers points A and B for m = 2). As the exact numbers of merger m is not known [1, Ch. 3, p. 74], m is selected by trial and is increased until there is no change in the upper bound d 2 B , with the knowledge that, there is no case where mergers higher than L is needed [1, Ch. 3, p. 74].
The calculation of d 2 min is obtained using the sequential search algorithm proposed in [10]. The search algorithm require the upper bound d 2 B ; the main objective of the upper bound d 2 B is to accelerate the algorithm search. For instance, any pair of phase trajectories with the corresponding γ N for a specified h and w, having a distance value defined in (9) larger than d 2 B , will not be used again. The steps to obtain the minimum normalized squared Euclidean distance (d 2 min ) are summarized below: • Step 1: select the number of mergers m. • Step 2: use exhaustive search to calculate the set S γ of difference sequences γ according to (10) and (11). • Step 3: compute d 2 (γ) based on (9) for each difference sequence obtained in the set S γ . • Step 4: the upper bound of the minimum distance d 2 B is equal to the minimum bound of all normalized squared Euclidean distance obtained from step 3 (d 2 B = min γ∈Sγ d 2 (γ)).
• Step 5: calculate d 2 min using the algorithm presented in [10], using the d 2 B obtained from step 4.

IV. SINGLE-SIDEBAND CPM POWER SPECTRUM
The spectrum occupancy plays an important role in the overall performance of the transmitted signal. The radiofrequency spectrum is a limited and essential resource. Therefore, it is important to minimize the usage of this resource as much as possible. In this section, we apply a numerical method to compute the power spectral density and, consequently, to analyze the spectrum efficiency of the SSB-FSK signals for different parameters (h, L, w, M ). The method is based on [1, Ch. 4] [11] and [12,Ch. 3], which is a generalization for the solution developed by Greenstein for time-limited phase response CPM signals [13].

A. Non-Integer Modulation Index
The power spectrum calculation using this method is based on the evaluation of the autocorrelation of the Fourier transform function, with the fact that s(t, α) is cyclostationary. Therefore, we first introduced the autocorrelation function, where we assumed a baseband signal, given by The variable τ = τ + mT s is the time difference over the interval [0, (L + 1)T s ], with 0 ≤ τ < T s and m = 0, 1, 2, . . .. The variable P k is the priori probabilities of the data symbols k. The power spectrum is obtained by using the numerical Fourier transform of (12), given by for h values for which |C α | < 1. The summation in (13) converges, leading to Inserting (15) into (13), we obtain the PSD expression given by However, this PSD equation is only valid for |C α | < 1, i.e. for non-integer h. Therefore, in the Subsection IV-B, we apply certain modifications to obtain the PSD for integer values of h.

B. Integer Modulation Index
For |C α | = 1 (integer values of h), (15) is not valid anymore. However, with |C α | = 1 we can set Based on (17), the summation in (15) can be rewritten as [12, Ch. 3, p. 141] inserting (18) into (13) leads to the complete PSD expression for integer h, which covers discrete and continuous components of the spectrum. Moreover, based on (18) the discrete spectrum lines are located at frequencies V. MULTI-OBJECTIVE OPTIMIZATION SSB-FSK has several conflicting objectives that need to be optimized, where the major challenge is to find tradeoffs for designing a competitive SSB CPM system. We have thus divided this section into three parts: -Objective functions: in this part, we summarize the performance metrics considered in the optimization analysis. -Design space: A massive number of configurations has to be tested due to the variety of the parameters. Therefore, it is interesting to reduce this number of combinations by excluding situations that cannot be envisaged, i.e., by defining a feasible region of the optimization problem variables. -Optimization method: as we have several objectives to optimize, different methodologies are possible. One solution among many is to optimize one performance metric when considering others constrained to a certain threshold (drawn from well-known other CPM schemes). Using this methodology, we define plenty of optimization problems (as many as possible combinations between considered objective functions and fixed thresholds for constrained performance metrics). Therefore, the best course of action is to jointly optimize all performance metrics by defining several intermediate configurations leading to different tradeoffs depending on the order of priority given for these objective functions.

A. Objectives Functions
• Minimum Normalized Squared Euclidean Distance (d 2 min ): measures the energy consumption of the signals, based on the error probability bound given in (6). It is clear that an increase of d 2 min can be seen as a reduction in the mean energy per information bit E b for the same targeted probability error. • Bandwidth Occupancy (BW): estimates the BW occupancy of the signal. We divided the BW occupancy into two measurements; the first one computes 99% of the signal power inside BT b , where T b = T s / log 2 M . The second measurement computes 99.9% of the power inside BT b , where with the second measurement, we obtain a better idea about the out-of-band leakage. We noted the two measurements B 99 and B 999 , respectively. The objective of these measurements is to estimate the BW occupancy of BT b . We note that the bandwidth BT b is normalized to the data rate in terms of bits carried per second so that schemes with different modulation levels M can be compared. Moreover, B represents the total bandwidth occupancy and not half of it due to the unsymmetrical spectrum shape form of the SSB-FSK modulation. Due to the enormous computational complexity of all of these metrics, we have to establish an order of priority order among them. Therefore, we provide the highest priority degree to the signal energy consumption and BW occupancy, where we try to find certain tradeoffs between these two objective functions. Based on these tradeoffs, we subsequently try to reduce the SSB-LOSS. It is worth highlighting that we can also consider the complexity (based on the number of states of the Viterbi algorithm performing the MLSD) as a fourth objective function. However, we chose not to include it in this paper. It will be presented in another study, discussing optimal and sub-optimal receivers.

B. Space Design
The initial step for solving a multi-objective function is to define a feasible region for the problem. In this study, we are mainly interested in two objective functions: energy consumption and BW occupancy. Therefore, the feasible region for the constraints is discussed hereafter : , we used a pulse length L = 12. Therefore, we define the pulse length range as 2 ≤ L ≤ 12 with increment step size L s = 1. We will not exceed this maximum value of L = 12 to not explose the receiver complexity earning very little on the SSB property. Furthermore, we would not consider L = 1 (Full response), due to inefficient performance in power consumption and bandwidth occupancy BT b . 3) Modulation alphabet M : an increase in M , will create an increase in complexity [12, ch. 4, p. 248], and degradation in BW occupancy BT b for the same modulation index (h) [1]. Therefore we selected M to take only these discrete values {2, 4, 8}. 4) Pulse Width w: w is a new parameter introduced by the SSB-FSK. Based on [2], the only idea we have about w is that increasing w will impact the SSB-FSK PSD exponential decrease, which can be quantified to exp(−4πw/T b ) for the particular case h = 1, therefore, it is interesting to study the effect of the pulse width w more deeply on both energy computation and BW occupancy BT b . To fully cover the effect of the pulse width w, we need to define a numerical range that covers up the useful region (a reduced set into the feasible region). The pulse width w is only found in the frequency pulse given by (4). It is clear from this expression that no restrictions are found to delimit possible values of w (expect that it has to be a non-null positive real). Therefore, to cover the full feasible region, we should take into consideration all values of pulse width w, i.e., w ∈]0, ∞[. Consequently, we define w lim a new limit on the pulse width w, where for w lim ≤ w ≤ ∞ we obtain approximately the same frequency pulse g(t), hence, it will be possible to reduce the study to w ∈ ]0, w lim ], which is for us now the useful region. The calculation of w lim is straightforward using the 2-norm defined as The error is then calculated using where x is the frequency pulse g ∞ (t), for w = ∞ (in simulation we take w = 1000). Furthermore,x is the estimated frequency pulse g lim (t), for w = w lim . We calculate w lim for each L in the defined interval. The criteria to obtain w lim from (21) is to maintain an error ≤ 10 −1 . In Table I, we present w lim with respect to the pulse length L, for binary SSB-FSK. Finally, we define the pulse width range 0.1 ≤ w ≤ w lim . It is worth noting that w lim is increasing with respect to the pulse length L, therefore, more configurations have to be considered when increasing L as the defined range of w is getting wider. Moreover, the increment step size is w s = 0.1.

C. Optimization Method
In multi-objective problems, there is no single global optimal solution. Therefore, Pareto optimum is used to find all optimal points. A solution is said to be Pareto optimum (P opt ), if there is no other solution that can improve at least one of where F 1 (x) and F 2 (x) are respectively the 10log 10 (d 2 min /2) and the inverse of the bandwidth occupancy (1/BT b ). The constraints variables (h, L, M, w) are denoted here by x ∈ R 4 . The definition of Pareto optimum is given by (we present the weak Pareto optimum) : For more information about the Pareto optimum, please refer to [18,19]. We used a Brute Force method to obtain all the values of F 1 (x) and F 2 (x) for the space Z, where Z is constrained by the parameters h, L, M, w and there defined ranges. Based on this Brute Force, we apply the Pareto optimum optimization to obtain all the optimum solutions for Z. Remarks: • We calculate the minimum normalized squared Euclidean distance (d 2 min ) using the algorithm introduced in Section III, where we consider all the mergers for merger index m = 3 (the number of mergers m found by trials). Moreover, we also consider N max = 30, where N is the number of observation symbols. • We calculate B 99 and B 999 BW occupancy (BT b ) using the method presented in IV. Hence, we obtain different P opt for each bandwidth BW occupancy BT b measures.

Figs 4 and 5 present the
Pareto optimum for binary LSSB-FSK 1 using B 99 and B 999 , respectively. Moreover, the GMSK performance for BT = 0.3 is considered here as a reference point. Hence, for the rest of the paper, we consider only the pareto front values which have a bandwidth BW occupancy that is less than or equal to the GMSK bandwidth BW occupancy, and a minimum normalized squared Euclidean distance d 2 min greater than or equal to the GMSK distance d 2 min . These values are illustrated in the upper left quarter of the plane delimited by the axis passing by the GMSK reference point in Figs 4 and 5. We will refer to these values as P opt−max . In Fig. 4, P opt−max is achieved with 26 different configurations (parameters combinations). In Fig. 5, P opt−max is achieved with 29 different configurations. Note that each point of P opt−max corresponds to a particular configuration, and the number of configurations depends on the parameters step used in the search (L s,Ws , h s ).     (y-axis, 10log 10 (d 2 min /2)) and bandwidth occupancy metrics (x-axis); tables II and III give absolute optimum values, performing better than the GMSK, for each performance metric alone and the resulting effect on the other one. Furthermore, in Fig. 5, we can see that the normalized minimum Euclidean distance d 2 min performance has a small variation between Config. A and Config. C (the line is quite horizontal). Therefore, Config. C seems an interesting point (another good tradeoff), since it is offering almost the same normalized minimum Euclidean distance d 2 min while decreasing significantly the bandwidth BW occupancy. Config. C corresponds to the following parameters combination (L = 12, w = 0.7, h = 0.99); leading to d 2 min-C = 3.216, BW C = 1.043.

VI. SIMULATION RESULTS
In this section, we present the results obtained based on Section III and IV. Additionally, we discuss many of the critical points about the effect of the CPM parameters on SSB-FSK performance, particularly the pulse width w. Indeed, this section is essential to understand in detail, the choices we made in the optimization section (Section V). In this section, we also present the simulation of BER, to verify the effectiveness of d 2 min by illustrating the symbol error performance of SSB-FSK signals. Moreover, this section also presents the performance of the SSB-FSK scheme compared to other CPM schemes.

A. SSB-FSK CPM error probability Performance
The author in [1,Ch. 3] describes, in detail, the effect of the parameters h, M, L on d 2 min for a variety of CPM schemes; the SSB-FSK has a quite similar scheme compared to the ones presented in [1,Ch. 3] (unlimited time CPM schemes with a symmetrical form with respect to zero). However, the pulse width w is a specific parameter introduced particularly for SSB-FSK signals. Therefore, it is interesting to study its effects on the d 2 min . Fig. 6 depicts the variations of the upper bound d 2 B as a function of modulation index h for binary 5SSB-FSK, for pulse width w = 0.1, 0.3, 0.5, 0.7, 0.9, 1.1, 1.3. Consequently, we are able to quantify the double effect of both parameters h and w on the error probability performance of SSB-FSK. Based on Fig. 6, we obtain a better upper bound d 2 B using a low pulse width w, e.g., w = 0.1 for low modulation index region h, e.g., for h ≈ 0.5. Therefore, for small modulation index h, it is always better to use small pulse width w. In addition, we also show that for higher modulation index h, a higher pulse width w is preferred, e.g., for h ≈ 1.5 we obtain the best d 2 B for w = 1.3. This behavior is very similar to the effect of increasing the pulse length L, which is explained in details in [1, ch.3, p.75-76]. However, it is not always better to use higher pulse width w for higher modulation index h. Since we notice that the d 2 B with w = 0.9 is better than w = 1.3, for the same h ≈ 1.9. Therefore, we cannot conclude a general can be seen directly from Fig. 2, where it is obvious that the tails of the frequency pulse g(t) goes up with the increase of pulse width w, which means an increase in inter-symbol interference (ISI). Fortunately, when the ISI is properly compensated at the receiver side, this can be seen as an increase on the redundant information (increase in memory effect), and hence this introduces sizable gains in the upper bound d 2 B values. Fig. 7 illustrates the performance of minimum normalized squared Euclidean distance d 2 min as function of h for binary 5SSB-FSK with pulse width w = 1.3. This configuration (L = 5 and w = 1.3) is chosen related to the results of the Pareto optimum analysis given in the previous Section (it is Config. A given also in Table II row 1 relative to  the binary scheme). Moreover, the results are illustrated for different observation symbol intervals N = 5, 10, 15, 30. In addition, the GMSK is also shown as x mark, and the d 2 min−opt for binary B 99 SSB-FSK (Table. II row 1 or Config. A in Fig. 4) is also shown as reference point. Fig. 7 shows that the most significant d 2 min value slightly higher than 5 can be reached, for h = 1.61 with N = 30: this is almost the absolute optimum without considering any other metric, and it is denoted by d 2 (abs) min-opt (this is visible in Fig. 4, it is one of the extreme points at the upper right quarter of the plane). Compared to GMSK, we obtained a gain of ≈ 4.83 dB. Note that this configuration, leading to d 2 (abs) min-opt , offers very poor BW performance (this is also visible in Fig. 4). That is why in Table. II, we have chosen the modulation index h = 0.78 (Config. A), which is shown as a reference point in the figure. This result, which is different from d 2 (abs) min-opt , is obtained because we are considering the bandwidth BW occupancy alongside the minimum normalized squared Euclidean distance d 2 min . We can notice the direct decrease of modulation index h (from h = 1.61 to h = 0.78), where better bandwidth BW occupancy can be obtained for lower modulation index h. As a consequence, the energy consumption gain compared to GMSK decreases to 1.36 dB. Besides, we can observe that, for certain specific modulation index h, the d 2 min cannot reach the upper bound d 2 B , specially for modulation index h = 3/2. This effect appears for all CPM schemes, and it is called weak modulation indices [1,10].

B. SSB-FSK BER
The authors in [10], [20], showed that the minimum Euclidean distance is sufficient for illustrating the performance in terms of bit error probability for a large variety of CPM schemes. Since SSB-FSK is a new CPM scheme, in this section we present its bit error rate (BER) simulation in order to verify the effectiveness and the tightness of our derivations. We obtained the BER using the Viterbi algorithm, which is the optimal detector used to achieve MLSD for CPM schemes. Fig. 8 shows the BER for Config. A and Config. B with M = 2. For sake of comparison, the union bound of each configuration is also given as a dashed line. It is shown that BER curves are approaching the bounds with high SNR values, which is consistent with what we previously stated, i.e., the union bound depicts the performance of the error probability for high SNR regime. For low SNR, the bounds are quite loose. From Fig. 8, we can conclude that the minimum Euclidean distance d 2 min may be the right metric to adopt in order to depict the bit error probability performance.

C. SSB-FSK CPM Power Spectrum Performance
Similar to Section VI-A, the effects of the parameters are already detailed for a wide variety of pulses in [1,Ch. 4]. Therefore, we present only the effect of the parameter w on the PSD of the SSB-FSK signals. Moreover, in this section, we also present the effect of h on the SSB-LOSS.  Table II (Config. A in Fig. 5), used also in Figs 6 and 7 to depict the effect of the pulse width w on the d 2 min . However, in this part, we show the effect of pulse width w on the power spectral density PSD. Therefore, we added two more plots for different pulse widths w = 0.3, 0.7, to clarify the selection of pulse width w = 1.3 in our optimization solution. Based on the closeup figure made around h = 0.78 in Fig. 6, we can see that increasing w, in this particular region, is not decreasing d 2 B that much (consequently d 2 min ). Nevertheless, from Fig. 9, it is clear that increasing w accentuates the power exponential decay of the SSB-FSK modulated signal [3]. Moreover, increasing the pulse width w condenses the power spectral density, particularly in the frequency interval of width 1/T s , which is mostly clear when we compare the plots of pulse width w = 0.3 and the others. Based on these observations, it is clear that increasing the pulse width w will reduce the bandwidth BW occupancy, and improve the adjacent-channel interference. However, we cannot increase the pulse width w indefinitely. For instance, from Fig. 6, we can see that increasing the pulse width w reduces the d 2 min specially for low modulation index h. Accordingly, we can understand the optimization choice; which is a practical tradeoff aiming not only to optimize the d 2 min , but also the bandwidth BW occupancy. Fig. 10 represents the power spectral density of the binary 12SSB-FSK, for w = 0.8 and different h = 0.5, 0.8 and 1.04. In this part, we show the effect of the modulation index h on the SSB-LOSS. As in the previous part, we used the parameters obtained from the optimization section (V), i.e, the first row from Table III which corresponds to Config. A in Fig. 5 (result of the Pareto optimum for B 999 ). In this particular context, we notice that we obtain the lowest SSB-LOSS. It is evident that increasing the modulation index h increases the bandwidth BW occupancy. It is especially visible for the frequency region between 0 and 1/T s . On the other hand, the modulation index h has the inverse effect on the SSB-LOSS, where for h = 0.5 we obtain an SSB-LOSS of 2.06%, and by increasing the modulation index to h = 0.8, the SSB-LOSS decreases to 1.63%, and then for h = 1.04 it reached the lowest point with an SSB-LOSS of 0.366%. However, for different parameters, e.g., 6SSB-FSK and pulse width w = 0.37, increasing the modulation index will increase the SSB-LOSS. Therefore, it is not possible to deduce a general trend about the modulation index h on the SSB-LOSS. Otherwise, it is always true that taking an integer modulation index h will reduce the SSB-LOSS; this decrease is due to the spectral lines' appearance, which takes a part of the transmitted power. We can notice for h = 1.04, the appearance of these spikes (spectral lines). This observation is due to selecting a modulation index h around an integer (≈ 1). Even if these spikes seem undesirable because a part of the transmitted power is wasted, they could be of great interest for synchronization purposes [1, Ch. 9, Sec. 1] [21]. Table IV presents a performance comparison between 12SSB-FSK (h = 1.04 and w = 0.8), which is config. A from Table. III; and 12SSB-FSK (h = 1, w = 0.37), which is the study case from the original proposal in [2,8]. Based on Table IV, it is clear that config. A is outperforming that of the original proposal. More precisely, config. A offers a better spectrum occupancy, whereas maintaining a better energy consumption performance (≈ 2.47 dB ). Besides, we have a small improvement in SSB-LOSS performance using config. A (a decrease of around 0.1031%). Furthermore, the spikes are decreased compared to the configuration considered in the original proposal, because the modulation index h is chosen to be h = 1.04 (a slightly higher than an integer).

D. Single-Sideband CPM Energy-Bandwidth Comparison
In this section, we show comparison plots between SSB-FSK and other CPM schemes. We used SSB-FSK parameters based on the results obtained from the optimization study given in Section V. The plots are characterized by the SSB bandwidth occupancy in the x-axis, where it is noted as BW, and 10 log 10 (d 2 min /2) for the y-axis, which reflects the energy consumption of the signal. Moreover, we divided this section into two parts, one presents the occupied BW as 99% of the power inside BT b , and one presents 99.9% of the power in the BT b (Note the 99.9% is a larger measure compared to the 99%; therefore, 99.9% curves are located in the right side compared to 99% curves). Besides, the optimum normalized minimum squared Euclidean distance d 2 min−opt and the optimum bandwidth BW occupancy BW opt are also presented in the plots, denoted respectively, by Config. A and Config. B for B 99 . Likewise, they will be denoted by Config. A and Config. B for B 999 . A subscription is used to refer to the modulation level, e.g., Config. A 2 refers to Config. A for the binary case. The curves with better performance are located toward the upper left side. Furthermore, the legend for SSB-FSK is presented with the same notation used in Section V as P opt−max , followed by the modulation level M ( e.g., P opt−max−4 , for M = 4). The legends of the RC pulse, given for sake of comparison, are given starting with the pulse type followed by the modulation level M , and the pulse length L (e.g., RC-2-3 is the raised cosine pulse for modulation level M = 2 and pulse length L = 3).
1) 99% Bandwidth occupancy: Fig. 11 illustrates the performance for B 99 of SSB-FSK signals compared to LRC for different modulation levels M. Besides, the GMSK signal for BT = 0.3 is also shown as x mark in the plots. The SSB-FSK P opt−max for all modulation levels M outperform the GMSK, where we obtain an energy consumption gain of 1.35, 3.03, and 2.67 dB, for respectively Config. A 2 , Config. A 4 , and Config. A 8 for approximately the same bandwidth BW occupancy. Similarly, we obtain a bandwidth BW occupancy gain of 0.125, 0.26, and 0.233, respectively for Config. B 2 , Config. B 4 , and Config. B 8 for almost the same energy consumption. Moving to the binary RC, the P opt−max−2 outperforms the 2RC; however, we obtain the same performance compared to binary 6RC while slightly reducing the complexity of the system, where for Config. A 2 and Config. B 2 we use a pulse length L = 5 instead of L = 6 (for 6RC). On the other hand, the P opt−max−4 and P opt−max−8 outrun the binary RC for all cases. Likewise, for M = 4, using the P opt−max−4 , we obtain almost the same performance as quaternary 3RC; however, as before, we achieve a lower complexity using a pulse length L = 2 instead of L = 3. Similarly, for M = 8, we obtain a slightly better perfor- mance using P opt−max−8 compared to 8-ary 3RC. However, for all bandwidth occupancies, the P opt−max−4 outperforms the P opt−max−8 , meaning that increasing the modulation level M will not always increase the performance. Besides, we can remark the effect of increasing the modulation level M specially between 2 and 4, we obtain an energy consumption gain of 1.68 dB between Config. A 2 and Config. A 4 for the same bandwidth occupancy. Similarly, considering now the bandwidth occupancy, we obtain a gain of 0.135 between Config. B 2 and Config. B 2 for almost the same energy consumption.
2) 99.9% Bandwidth occupancy: This subsection conduct the same study as that given in the previous one with the only difference being Fig. 12, where the plots are given for 99.9% of bandwidth BW occupancy. Using B 999 , we can obtain a better idea about the out-of-band leakage. As in the previous part, we compare SSB-FSK with GMSK (shown as x mark in the plot), we achieve a gain of 2.8 dB, 2.3 dB, and 2.4 dB respectively for Config. A 2 , Config. A 4 , and Config. A 8 at nearly the same bandwidth BW occupancy. Likewise, we obtain a gain of 0.19, 0.227, and 0.226, respectively, for Con- fig. B 2 , Config. B 4 , and Config. B 8 at almost the same energy consumption. Config. C is reported for sake of comparison and it is clearly an interesting operating point as discussed in the the optimization study given in Section V. Besides, the P opt−max for different modulation level M outperform the binary RC for pulse length L = 3 and 6. However, the pulse length L used in P opt−max−2 is always greater than or equal to 6, specially for Config. A 2 where we have a pulse length L = 12. With L = 12, we have a disastrous impact on the receiver complexity. For modulation level M = 4, we note a different behavior compared to the previous part. In B 99 , we obtained the same performance using P opt−max−4 and quaternary 3RC. However, for B 999 , the quaternary 3RC outrun the P opt−max−4 and P opt−max−8 for all points. Unlike what we noticed for the quaternary case, we do not have a unique behavior for the binary case. For instance, the P opt−max−2 outperforms the P opt−max−4 for all BT b ≥ 1, and outperforms the P opt−max−8 for all BT b ≥ 1.01. Moreover, it also has a similar or better performance than the quaternary 3RC for points defined on the B 999 plot.
Overall, these plots reveal two concluding remarks for improving the performance: • strong effect of increasing the alphabet size of M , especially for M = 4 in B 99 case. • huge effect of increasing the pulse length L on the performance, particularly for binary 12SSB-FSK in the 99.9% BW occupancy.

VII. CONCLUSION
In this paper, we have investigated a CPM modulation that can directly generate a SSB spectrum. To exploit the characteristics and the advantages of this modulation scheme, we investigated the error probability based on the derivation of the minimum Euclidean distance as a function of the modulation index, pulse width, and pulse length. Furthermore, we explored the spectrum of SSB-FSK by quantifying its signal power bandwidth occupancy using a numerical method. We presented an optimization method to select the best parameter combinations to obtain the SSB-FSK scheme's maximum performance. Based on the parameters chosen from the optimization solution, we achieved a better spectrum occupancy and a gain in error probability performance. Our optimization method leads to practical tradeoffs through certain parameter combinations. These tradeoffs differ because they prioritize performance metrics differently. The choice depends on the target application and, consequently, the resources we want to allocate. Finally, we have applied an energy-bandwidth comparison, where we achieved a similar or better performance compared to certain powerful CPM modulations. in particular, for B 99 case, SSB-FSK scheme with modulation level M = 4 gives the best performance compared with other SSB-FSK modulation levels M as it was expected in [10]. For B 999 case, the binary 12SSB-FSK presents an excellent performance: the advantage of this particular scheme is twofold. From one side, this scheme is the only modulation with good performance (similar to RC) while having almost an integer modulation index h ≈ 1 (in the B999 case): with integer modulation indices, other pulses behave as weak modulation schemes [1]. From another side, using an approximately integer modulation indices generates the spectral lines (spikes), which can be efficiently used in synchronization in comparison with other modulations (e.g., RC) [21]. Therefore, combining these two aspects, we think that SSB-FSK has the potential to go beyond which has been achieved with well-known CPM schemes, when considering all performance metrics together.

APPENDIX A
In this appendix, we detail how (9) was obtained from (8) where E s = 2E is the energy per transmitted symbol and w 0 is the carrier frequency. To simplify the derivation, the carrier frequency w 0 is assumed much larger than 2π/T s . We suppose that the two signals s(t, α i ) and s(t, α j ) differ over an interval N . The Euclidean distance between the two signals over the interval N is defined as [12] N Ts Since both signals are constant envelope with energy E over an interval T s (we obtain E by integrating the square of (25) over an interval T s ), the second line of (26) can be expressed as Using the trigonometric function cos(a)cos(b) = 1 2 (cos(a − b)+cos(a+b)) and assuming large carrier frequency, the third line in (26) can be presented as From (27) and (28), we can give (26) in the form where ∆φ(t) denotes the phase difference between the two signals. Here E is the energy obtained for an interval and not for a data bit. Therefore, for fair comparison sake between many modulation schemes with different modulation levels M , E should be normalized to obtain the energy per data bit E b . The normalization of E is defined as Placing (29) and (30) in (8), we obtain (9).