On 64-APSK Constellation Design Optimization

On 64-APSK Constellation Design Optimization Konstantinos P. Liolis

Nader S. Alagha

Space Hellas S.A. R&D and Applications Division Athens, Greece e-mail: [email protected]

European Space Agency (ESA) Research and Technology Centre (ESTEC) Noordwijk, The Netherlands e-mail: [email protected]

Abstract — This paper investigates the design of power and spectrally efficient coded modulations based on amplitude phase shift keying (APSK) with application to broadband satellite communications. Emphasis is put on 64APSK constellations. The APSK modulation has merits for digital transmission over nonlinear satellite channels due to its power and spectral efficiency combined with its inherent robustness against nonlinear distortion. This scheme has been adopted in the DVBS2 Standard for satellite digital video broadcasting. Assuming an ideal rectangular transmission pulse, for which no nonlinear inter-symbol interference is present and perfect precompensation of the nonlinearity takes place, we optimize the 64APSK constellation design by employing an optimization criterion based on the mutual information. This method generates an optimum constellation for each operating SNR point, that is, for each spectral efficiency. Two separate cases of interest are particularly examined: (i) the equiprobable case, where all constellation points are equiprobable and (ii) the nonequiprobable case, where the constellation points on each ring are assumed to be equiprobable but the a priory symbol probability associated per ring is assumed different for each ring. Following the mutual information-based optimization approach in each case, detailed simulation results are obtained for the optimal 64APSK constellation settings as well as the achievable shaping gain. Keywords - additive white Gaussian noise (AWGN); amplitude phase shift keying (64-APSK) modulation; broadband satellite communications; mutual information criterion; shaping gain.

I.

INTRODUCTION

Amplitude Phase Shift Keying (APSK) represents an attractive modulation scheme for digital transmission over nonlinear satellite channels due to its power and spectral efficiency combined with its inherent robustness against nonlinear distortion. Thus, it has been recently adopted in the state-of-the-art standard for digital video broadcasting via satellite, DVB-S2 [1]. Specifically, DVB-S2 has selected the 16APSK and 32APSK modes, which are mainly targeted at professional applications, due to the higher requirements in terms of available Signal to Noise Ratio (SNR), but they can also be used for broadcasting when higher spectral efficiency is needed. Detailed optimization for up to 32APSK constellation design has been performed for linear and nonlinear satellite channels in [2] and [3], respectively. In addition to the minimum Euclidean distance-based optimization criterion [4], a new optimization criterion based on the mutual information maximization was introduced in [2],[3], which generates an

optimum constellation for each spectral efficiency. Recently, there has been a commercial interest to go well beyond the DVB-S2 standardized modulation schemes and use up to 64APSK constellations especially in professional applications, such as Digital Satellite News Gathering (DSNG), where even higher spectral efficiency is needed and, consequently, higher requirements in terms of available SNR apply. On the 64APSK constellation, design optimization results are reported in [4] as part of the ESA project MHOMS but only taking into account the minimum Euclidean distance-based optimization criterion. Moreover, note that these results on 64APSK constellation design optimization have been recently adopted in the CCSDS Orange Book for space telemetry applications [5]. The scope of this paper is to determine through detailed computer simulations the optimal 64APSK constellation design based on the new optimization criterion, that is, the maximization of the mutual information that can be carried through an Additive White Gaussian Noise (AWGN) channel. Assuming perfect pre-compensation of the nonlinearity [2], we optimize the 64APSK constellation design for two separate cases of interest: (i) equiprobable constellation points; and (ii) non-equiprobable constellation points. In the equiprobable case, based on the maximum mutual information-based optimization criterion, we determine the optimal relative radius and phase shift of each ring with respect to the inner ring of a 64APSK constellation. Our analysis focuses on 4+12+20+28-APSK and 4+12+16+32-APSK signal constellations. Moreover, and in line with modulation and coding selection of practical systems such as in [5], we examine particularly operating SNR points with relatively high spectral efficiency, corresponding to coding rates r between 0.8 and 0.9. The optimization results obtained through the mutual information maximization criterion are compared against the ones obtained through the minimum Euclidean distance maximization criterion [4],[5] and the relative differences over the SNR range of interest are reported. In the case of non-equiprobable constellation points, 64APSK constellation shaping is examined in order to achieve the so called “shaping gain” [6],[8]. To this end, assuming equally-probable constellation points per each ring, a new 64APSK constellation design optimization problem is formulated and numerically solved in order to calculate a priori probabilities of constellation points on each ring and also to calculate the corresponding shaping gain.

The rest of the paper is organized as follows. Section II describes the system model assumed under the ideal case of a rectangular transmission pulse. Section III briefly describes APSK signal sets, presents the maximum mutual information optimization criterion, and solves the formulated optimization problem for the equiprobable and non-equiprobable cases. Section IV provides some useful numerical results for each case. Conclusions are finally drawn in Section V. II. SYSTEM MODEL The baseband equivalent of the transmitted signal at time t, sT(t), is given by L −1

sT ( t ) = P ∑ x ( k ) pT ( t − kTs )

APSK constellation. The impact of nonlinear ISI as well as other realistic demodulation effects such as timing and phase recovery has been considered in [3] for 16- and 32-APSK constellations. III. 64APSK CONSTELLATION DESIGN OPTIMIZATION The optimization criterion used for the 64APSK constellation design is the maximization of the mutual information that can be carried through an AWGN channel assuming an ideal transmission pulse with zero excess bandwidth, for which no nonlinear impairments and no ISI are present.

(1)

k =0

where P is the signal power, x(k) is the kth transmitted symbol, drawn from a complex-valued 64APSK signal constellation X64-APSK with |X64-APSK |=M=64, pT(.) is the transmission filter impulse response, and Ts is the symbol duration corresponding to one channel use. Without loss of generality, we consider transmission of L-symbol frames. The spectral efficiency n is defined as the number of information bits conveyed at every channel use. The signal sT(t) passes through a high-power amplifier (HPA) operated close to saturation. In this region, the HPA shows nonlinear characteristics that induce phase and amplitude distortions to the transmitted signal. The amplifier is modeled by a memoryless nonlinearity, with an output signal sA(t) at time t given by

(

)

s A ( t ) = F sT ( t ) e

(

)

j ⎡⎢φ ( sT ( t ) ) +Φ sT ( t ) ⎤⎥ ⎣ ⎦

Figure 1a: 4+12+20+28-APSK constellation

(2)

where we have implicitly defined F(A) and Φ(A) as the AM/AM and AM/PM characteristics of the HPA amplifier for a signal with instantaneous signal amplitude A, respectively. The signal amplitude is the instantaneous complex envelope, so that the baseband signal is decomposed as sT(t)=| sT(t)|exp[jφ(sT(t)]. In this paper, we assume an (ideal) signal modulating a train of rectangular pulses. These pulses do not create intersymbol interference (ISI) when passed through an amplifier operated in the nonlinear region. Under these conditions, the channel reduces to an AWGN, where the modulation symbols are distorted following (2). Let xA denote the distorted symbol corresponding to x=|x|exp[jφ(x)] ∈ X64-APSK, that is, xA = F ( x ) e

j ⎡⎣φ ( x ) +Φ ( x ) ⎤⎦

. After matched filtering and sampling at time kTs, the discrete-time received signal at time k, y(k) is then given by y ( k ) = Es x A ( k ) + n ( k ) , k = 0,1,..., L − 1

(3)

with Es the symbol energy, given by Es=PTs, xA(k) is the symbol at the kth time instant, as defined above, and n ( k ) ∼ N ( 0, N 0 ) is the corresponding noise sample. This simplified model suffices to describe the nonlinearity up to the nonlinear ISI effect, and allows us to easily design

Figure 1b: 4+12+16+32-APSK constellation

The 64APSK signal constellation points are complex numbers, drawn from a set X64-APSK given by Χ 64 − APSK ⎧r e j ( ( 2π n1 )i +θ1 ) , ⎪1 ⎪⎪r2 e j (( 2π n2 )i +θ2 ) , =⎨ j ( ( 2π n3 ) i +θ3 ) , ⎪r3 e ⎪ j (( 2π n4 )i +θ4 ) , ⎪⎩r4 e

i = 0,..., n1 − 1 i = 0,..., n2 − 1 i = 0,..., n3 − 1 i = 0,..., n4 − 1

( Ring ( Ring ( Ring ( Ring

= 1) = 2)

(4)

= 3) = 4)

where n , r and θ are defined as the number of points, the radius and the relative phase shift for the th ring. X64-APSK is further assumed normalized in energy, that is, 2 Ε ⎡ Χ 64 − APSK ⎤ = 1 , which implies that the radii r are ⎣ ⎦

normalized such that ∑ 4=1 n r 2 = 1 . Since the focus here is on 4+12+20+28-APSK and 4+12+16+32-APSK constellations, the values of interest are {n1 , n2 , n3 , n4 } ={4,12,20,28} and {4,12,16,32}, respectively. In Figure 1, the two multi-ring 64APSK constellations of interest are depicted. Furthermore, in order to reduce the dimensionality of the optimization problem, instead of optimizing the phase shifts θ and the ring radii r in absolute terms, the objective here is their optimization in relative terms. That is, we are looking for the optimum values of the phase shift of the th ring with respect to the inner ring, φ = θ − θ1 , and of the relative radius of the th ring with respect to the inner ring, ρ = r r1 , which result in the maximum mutual information that can be carried through an AWGN channel. In particular, φ1 =0 and ρ1 =1. Thus, we are interested in finding a 64APSK constellation, defined by the parameters ρ=(ρ1=1,ρ2,ρ3,ρ4) and φ=(φ1=0,φ2,φ3,φ4) such that a given cost function f(X64-APSK) reaches a maximum. The cost function used here is the mutual information of the AWGN channel [2],[3]. Interestingly, following this approach, for a given SNR, or equivalently, for a given spectral efficiency n, an optimum X64-APSK constellation can be obtained. Specifically, two separate cases are examined, that is, that of equiprobable constellation points and that of nonequiprobable constellation points. In the former case, all X64-APSK constellation points are equiprobable with probability 1/64. In the latter case, the X64-APSK constellation points on each ring are assumed to be equiprobable but the a priory symbol probability associated per ring, P , =1,2,3,4, is assumed different for each ring so that ∑ 4=1 n P = 1 . A. Equiprobable constellation points Assuming equiprobable symbols, the mutual information for a given signal set X64-APSK provides the maximum transmission rate (in bits/channel use) at which error-free transmission is possible with such signal set, and is given by [7, Eq.(5)] f eq ( X 64 − APSK ) = 6−

(

63 2 ⎡ E 1 63 2 ⎪⎧ ∑ Ew ⎨log 2 ∑ exp ⎢ − s x k + w − xi − w 64 k = 0 ⎪⎩ i=0 ⎣ N0

)⎤⎥⎦ ⎪⎫⎬⎪⎭

(5)

Thus, the optimization problem to be solved is formulated as * Ceq = max f eq ( X 64 − APSK )

ρ ,φ

(6)

In (5) we have used expectation over the normally distributed noise variable w which is complex with variance N0. Ceq* in (6) is numerically computed using the Gauss-Hermite quadrature rules as was done in [2],[3] for 16- and 32-APSK case.

B. Non-equiprobable constellation points In the case of non-equiprobable constellation points, 64APSK constellation shaping is examined in order to achieve some “shaping gain” and, thus, to decrease the relatively large penalty with respect to the Shannon limit experienced with equiprobable constellation points (as shown in Section IV-A). The main idea behind constellation shaping is that signals with large norm are used less frequently than signals with small norm, thus improving the overall gain by adding shaping gain to the original coding gain [6],[8]. Theoretically, when constellation points are selected according to a continuous Gaussian distribution at every dimension, the maximum achievable shaping gain in the limit for infinite transmission rates is 1.53dB [6],[8]. Practically, a smaller gain can be achieved in finite constellations as is the case here. To this end, considering that the constellation points on each ring are equiprobable but the a priori symbol probability associated per ring, P , is different so that ∑ 4=1 n P = 1 , and fixing the values of the relative radius and phase shift of the each ring with respect to the inner ring to the optimized ones found in the equiprobable case (see Section IV-A), a new constellation design optimization problem is formulated, which allows the calculation of the shaping gain. The cost function in this case is given by [7, Eq.(3),(5)] f non − eq ( X 64 − APSK ) 3

15

k =0

k =4

= − P1⋅ ∑ Q − P 2 ⋅ ∑ Q − P3 ⋅

35

63

k =16

k = 36

(7)

∑ Q − P4 ⋅ ∑ Q

where ⎧ ⎫ 3 ⎡ 2 ⎛ Es 2 ⎞ ⎤ x k + w − xi − w ⎟ ⎥ ⎪ ⎪ ⎢ P1⋅ ∑ exp ⎜ − i =0 ⎪ ⎢ ⎝ N0 ⎠ ⎥⎪ ⎪ ⎢ ⎥⎪ 15 ⎪ ⎢ + P 2 ⋅ exp ⎛ − Es x k + w − x i 2 − w 2 ⎞ ⎥ ⎪ ∑ ⎜ ⎟ ⎥⎪ ⎪ ⎢ N0 i=4 ⎪ ⎝ ⎠ ⎥ ⎬⎪ (8) Q = Ew ⎨log 2 ⎢ 35 ⎢ ⎥⎪ 2 ⎛ ⎞ Es 2 ⎪ x k + w − xi − w ⎟ ⎥ ⎪ ⎢ + P3 ⋅ ∑ exp ⎜ − ⎪ i =16 ⎝ N0 ⎠ ⎥⎪ ⎢ ⎪ ⎢ ⎥⎪ ⎪ 63 ⎢ + P 4 ⋅ ∑ exp ⎛⎜ − Es x k + w − x i 2 − w 2 ⎞⎟ ⎥ ⎪ ⎪ ⎢ i = 36 ⎪⎩ ⎝ N0 ⎠ ⎥⎦ ⎭⎪ ⎣ Thus, the optimization problem to be solved is formulated as

(

)

( ( (

* Cnon − eq = max f non − eq ( X 64 − APSK ) P1 , P2 , P3

) ) )

(9)

where, in this case, the parameters to be calculated are the a =1,2,3 (P4 is calculated from priory probabilities, P , 4 ∑ =1 n P = 1 ) since the values of ρ,φ are now fixed. Note that,

since the probabilities per ring P , =1,2,3,4 are now varying, the signal set X64-APSK needs to be normalized in energy accordingly so that unit power is maintained [8].

IV.

NUMERICAL RESULTS

A. Equiprobable constellation points Following the optimization approach described in Section III-A, the optimum 64APSK constellation settings and the corresponding upper bound on the spectral efficiency n are determined for each operating SNR point. The upper bound on spectral efficiency n is used to derive the coding rate r for an encoded system with ideal performance. This is to establish a baseline for practical systems. Since the coding rates of interest are from r = 0.8 to 0.9, we particularly look at moderate to high SNR values corresponding to these coding rates. Moreover, the optimization results obtained through the mutual information maximization criterion as described in Section III-A [2],[3] are compared against the ones obtained through the minimum Euclidean distance maximization criterion [4],[5] and the relative differences over the whole SNR range of interest are reported hereinafter. A1) Case of 4+12+20+28-APSK Following the optimization approach described above for the equiprobable 4+12+20+28-APSK signal constellation set, we obtained optimal constellation settings for a range of operating SNR. Figure 2 illustrates the resulting modulation constrained capacity (the upper bound of the spectral efficiency) along with the Shannon capacity bound versus the operating SNR. In this case, the SNR range corresponds to coding rates r from 0.7 to 0.9. As shown in this figure, the penalty with respect to the Shannon limit ranges between 0.5dB and 1.5dB within the SNR (or coding rate r or spectral efficiency n) range of interest.

noticeable dependence on the relative phase shift of the th ring with respect to the inner ring, φ = θ − θ1 ( = 2, 3, 4 ) and so, it can be further assumed that φ2 = φ3 = φ4 = 0 .

TABLE I: OPTIMIZED PARAMETERS FOR 4+12+20+28-APSK CONSTELLATION DESIGN (ASSUMING FIXED RELATIVE PHASE SHIFTS φ2 = φ3 = φ4 = 0 ) EQUIPROBABLE CASE

Similar observation regarding the insensitivity to the constellation rotation was reported in [2],[3] for 16- and 32APSK constellations which was further verified here for the 64-APSK case, as well. The simulation results verifying the absence of dependence on the relative phase shifts φ in the case of 4+12+20+28-APSK constellation are reported in Table II. In this simulation activity, the values of ρ opt ( = 2,3, 4 ) have been fixed to the ones reported in Table I for each given spectral efficiency n, and the optimization has been performed only over the φ = θ − θ1 ( = 2, 3, 4 ) . As can be observed, even though the resulting optimized φ values are generally different than 0 (as assumed in Table I), the resulting capacity achieved at each SNR point is exactly the same as in the case where all φ are set equal to 0. That is, there is no noticeable dependence on the φ with respect to the maximum mutual information, which further reduces the dimensionality of the optimization problem. TABLE II: OPTIMIZATION OF 4+12+20+28-APSK CONSTELLATION DESIGN OVER RELATIVE PHASE SHIFTS φ SHOWING NO DEPENDENCE ON φ EQUIPROBABLE CASE

Figure 2: Capacity of 4+12+20+28-APSK optimized constellation (equiprobable case)

Table I provides the optimized parameters taken into account in Figure 2 for various coding rates, giving an optimal 4+12+20+28-APSK constellation for each given spectral efficiency n. The coding rates reported refer to the ones specified in the CCSDS Orange Book [4],[5] which correspond to the ACM formats 25, 26, 27. As can be seen, only the optimized values of the relative radius of the th ring with respect to the inner ring, ρ opt ( = 2,3, 4 ) are of rather importance and, therefore, reported. This is because there is no

Finally, Figure 3 depicts the capacity penalty due to the suboptimal 4+12+20+28-APSK signal constellation setting, if the minimum Euclidean distance maximization results (ρ2=2.73, ρ3=4.52 and ρ4=6.31 [4],[5]) are instead taken into account. As pointed out in [2],[3], these results refer to the high SNR asymptotic case and so, are not optimized over the whole SNR range for each given spectral efficiency n. The latter statement is illustrated in Figure 3 where the optimal results in terms of mutual information maximization approach converge to the ones optimized based on minimum Euclidean distance maximization approach as high SNR values. It can also be observed that there is a small gain (of about 0.25dB at SNR corresponding to r=0.7 which gets even less as the SNR, or r or

n increases) in using the optimized constellation for every n, rather than the calculated one with the minimum Euclidean distance (or high SNR) criterion. Moreover, it can be seen that the relatively large penalty with respect to the Shannon limit within the SNR range of interest still remains in this case.

performance criteria such as the signal peak-to-average power ratio, synchronization issues, impact on non-linearity, etc [3].

Figure 4: Capacity of 4+12+16+32-APSK constellation based on suboptimal parameters obtained for high SNR (equiprobable case) Figure 3: Capacity of 4+12+20+28-APSK constellation obtained through mutual information maximization and minimum Euclidean distance maximization approaches (equiprobable case)

A2) Case of 4+12+16+32-APSK Based on the findings in the case of 4+12+20+28-APSK above, there is no significant capacity gain in using the optimized constellation for each spectral efficiency n rather than the calculated one with the minimum Euclidean distance (or high SNR) criterion. Therefore, for the 4+12+16+32-APSK case, we have simply tested the previous optimization approach at high SNR and considered the (slightly) suboptimal results obtained over the whole SNR range for each coding rate r or spectral efficiency n. Following this approach, Table III provides the suboptimal parameters obtained for 4+12+16+32-APSK constellation design at high SNR. Assuming these values (as well as the relative phase shifts φ all equal to 0) fixed over the whole SNR range of interest, the resulting modulation constrained capacity achieved along with the Shannon capacity bound is plotted versus the operating SNR in Figure 4. The SNR range depicted corresponds to coding rates r from 0.7 to 0.9. As can be seen, the penalty with respect to the Shannon limit ranges between 0.75dB and 1.5dB within the SNR range of interest. TABLE III: SUBOPTIMAL PARAMETERS OBTAINED FOR 4+12+16+32-APSK CONSTELLATION DESIGN AT H IGH SNR - EQUIPROBABLE CASE

Finally, based on Figures 3 and 4, there is no significant difference between the capacity bounds for 4+12+20+28APSK and 4+12+16+32-APSK constellations in the equiprobable case. Therefore, the preference of one constellation versus the other has to be based on other

B. Non-Equiprobable constellation points B1) Case of 4+12+20+28-APSK Following the optimization approach described above for the non-equiprobable 4+12+20+28-APSK signal constellation set, we obtained the optimal a priori probabilities P , =1,2,3,4 for the given optimized constellation settings previously found in the equiprobable case. The values of ρ opt ( = 2,3, 4 ) have been fixed to the ones reported in Table I and optimization over P for three specific operating SNR points has been performed. These optimization results are reported in Table IV and illustrated in Figure 5, as well. For the sake of comparison, the Shannon capacity bound and the capacity results for the equiprobable case have been plotted in Figure 5, as well. As can be seen, the constellation shaping decreases the penalty with respect to the Shannon capacity bound. Namely, for spectral efficiency n (or equivalently coding rate r or operating SNR) equals to 5 bits/channel use, the penalty with respect to the Shannon limit decreases from 1dB (equiprobable case) to 0.5dB (non-equiprobable case), whereas for n=5.5 bits/channel use, it decreases respectively from 1.5dB to 1.1dB.. Therefore, the shaping gain for 5 and 5.5 bits/channel use is 0.5dB and 0.4dB, respectively. Further information on how this shaping gain can be achieved is provided in [6],[8]. TABLE IV: OPTIMIZATION OF 4+12+20+28-APSK CONSTELLATION DESIGN OVER A PRIORI PROBABILITIES, P1,P2,P3,P4 NON-EQUIPROBABLE CASE

capacity bound and the capacity results for the equiprobable case have been plotted in Figure 6, as well. As can also be seen here, the constellation shaping decreases the penalty with respect to the Shannon capacity bound. The shaping gain in this case is 0.5dB for spectral efficiency n=5 bits/channel use, whereas for n=5.5 bits/channel use, it is 0.3dB. Further information on how this shaping gain can be achieved is provided in [6],[8]. V.

Figure 5: Capacity of 4+12+20+28-APSK optimized constellation (non-equiprobable case) and shaping gain

TABLE V: OPTIMIZATION OF 4+12+16+32-APSK CONSTELLATION DESIGN OVER A PRIORI PROBABILITIES, P1,P2,P3,P4 NON-EQUIPROBABLE CASE

CONCLUSIONS

This paper presented the results of detailed computer simulations devised in order to determine an optimal design of 64APSK signal constellation by maximizing the mutual information that can be carried through an AWGN channel under a transmitted power constraint. Following this optimization approach, the optimal constellation settings are determined as a function of the operating SNR corresponding to an upper bound on spectral efficiency (the channel capacity). Two separate cases were specifically examined: (i) the equiprobable case, where all constellation points are equiprobable and (ii) the non-equiprobable case, where the constellation points on each ring are assumed to be equiprobable but the a priory symbol probability associated per ring is assumed different for each ring. In the equiprobable case, detailed simulation results indicated, from the maximum capacity point of view, that there is no noticeable dependence on the relative phase shifts between the rings. The optimum normalized radii of the constellation rings were calculated as a function of the operating SNR or, equivalently, of the spectral efficiency. Moreover, it was shown that there is a slight capacity penalty (less than 0.25dB in all cases simulated) if the optimal settings at high SNR are adopted for a wider SNR range corresponding to the coding rates of interest.

Figure 6: Capacity of 4+12+16+32-APSK optimized constellation (non-equiprobable case) and shaping gain

B2) Case of 4+12+16+32-APSK Similar to the non-equiprobable case 4+12+20+28-APSK reported above, for the non-equiprobable 4+12+16+32-APSK signal constellation set, we obtained the optimal a priori probabilities P , =1,2,3,4 for the given optimized constellation settings previously found in the equiprobable case. The values of ρ opt ( = 2,3, 4 ) have been fixed to the ones reported in Table III and optimization over P for three specific operating SNR points has been performed. These optimization results are reported in Table V and illustrated in Figure 6, as well. For the sake of comparison, the Shannon

In the non-equiprobable case, the a priory probabilities associated with each ring were calculated through detailed computer simulations assuming that the radii and phase shifts of each ring are the same as those optimized in the equiprobable case. In this way, the achieved shaping gain was calculated and the relatively large penalty with respect to the Shannon limit experienced before with equiprobable constellation points was shown to decrease due to the constellation shaping. The achieved shaping gain ranges from 0.3dB to 0.5dB for the simulated cases referring to relatively high coding rates r or, equivalently, to relatively high operating SNR. REFERENCES [1]

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