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In this paper, a new method of constructing three-dimensional modulation formats with constant power is introduced. Constellations designed by the method have slightly larger minimum Euclidean distances (MEDs) than the conventional ones. No repetitive algorithm to maximize MED is used so that the new method has little computational complexity. Since signal points in the new formats are distributed regularly and symmetrically, an error control coding with systematic set-partition is applicable. We also present theoretical symbol error probability (SEP) of the new constellations in an additive white Gaussian noise environment, and demonstrate that the theoretical results are accurate. As the new modulation formats have almost the same or slightly lower SEPs than the conventional ones, they are appropriate for implementing a highly reliable optical communication system.
© 2011 Optical Society of America
1. Introduction
Designing a good multilevel modulation format with the constraint of achieving a given symbol error probability (SEP) in an additive white Gaussian noise (AWGN) environment is equivalent to constructing a constellation having the maximized minimum Euclidean distance (MED) for a certain power level [1]. Multidimensional modulation formats have been studied widely [2], [3]. In the field of optical communications, three-dimensional (3-D) constellations having a constant power are preferred to realize a digital modulation scheme such as polarization shift keying (POLSK) [4–8]. In the POLSK modulation, Stokes parameters which describe the state of polarization (SOP) are on the surface of Poincaré sphere.
An easy way to design a constellation with constant power is to use vertices of a regular polyhedron inscribed in a sphere [4]. Besides regular tetrahedron and hexahedron, no regular polyhedron has vertices of a power of 2. Thus, this method can design only 4-ary and 8-ary constellations. Betti et. al. [5], [6] presented the coordinates of optimum constant-powered multilevel 3-D constellations and an efficient structure for direct detection of polarization modulated signals. Here, a repetitive algorithm called downhill simplex method was used to compute the coordinates of 8-ary and 16-ary constellations with constant power. Both constellations have been optimum from the viewpoint of MED. However, the geometry of signal points is not regular. Neither a unique decision region for theoretical analysis nor an error control coding (ECC) with systematic set-partition such as trellis coding [7] is applicable. In addition, the downhill simplex method results in significant computational burdens. Due to the drawbacks, those constellations have hardly been employed in practical applications.
In this paper, we propose a simple method to build 3-D 8-ary and 16-ary modulation formats with constant power. The method expands multilevel phase shift keying (MPSK) to 3-D signal space to make the MED as large as possible. As a result, the presented constellations have slightly larger MEDs than the ones in [6]. We also investigate symbol error probability (SEP) of the 8-ary constellation and performance bounds for the 16-ary constellations. Computer simulation verifies that the theoretical SEPs are very accurate. Since the new modulation formats have regular and symmetric structures, an ECC with systematic set-partition algorithm is applicable to achieve further improved error performance. Hence, those modulation formats are appropriate for applications in high-quality optical communication systems.
2. Modulation formats with constant power
To construct practical modulation formats with constant power, we expand MPSK constellations into 3-D signal space. The design procedure consists of two steps as follows:
- Step 1: The symbols of an MPSK constellation are divided into disjointed sub-groups Gi. That is, any symbol Sk ∈ Gi must be Sk ∉ Gj for i ≠ j and 0 ≤ k ≤ M – 1, where M is the number of symbols. We assign four symbols to every sub-group to avoid computational complexity in finding the optimum number of symbols in each sub-group. Thus, i and j are 0 or 1 for 8-ary constellation and 0 ≤ i, j ≤ 3 for 16-ary constellation.
- Step 2: The same number of sub-groups is placed on the upper and lower hemispheres as shown in Fig. 1. Then compute the optimum vertical angle(s) which maximizes the MED.
Fig. 1 Design procedure (a) the new 3-D 8-ary constellation based on 8-ary PSK (b) the new 3-D 16-ary constellation based on 16-ary PSK.
When the vertical angle is increased, the intersymbol distance of a sub-group is decreased. With decrease of the angle, the Euclidean distance between two symbols in different sub-groups is decreased. Hence, the optimal vertical angle should be determined to balance the intersymbol distances of each sub-group and adjacent sub-groups. If we apply this principle to construct 8-ary constellation, the symbol pair (S0, S1) and (S0, S5) in Fig. 1(a) must have the same Euclidean distance. Thus, the optimum vertical angle φ satisfies
and is computed as φ = cos−1(4/(4 + 21/2))1/2 ≅ 30.736°. The 8-ary constellation designed by this method has very similar geometry to the one presented in Sloane’s work [9].In designing 16-ary constellation, there are four sub-groups as depicted in Fig. 1(b), and the x – y plane on which the symbols of a sub-group lie is rotated 45° to make the intersymbol distance between different sub-groups as large as possible. There are two vertical angles to be optimized. When applying the design principle, we have the following relationship concerning the intersymbol distances among (S0, S1), (S0, S5), and (S5, S8)
where 0 < φ2 < φ1 < 90°. Solving (2), we have φ1 ≅ 51.489° and φ2 ≅ 13.631°. It is, therefore, considered that the new 3-D constellations with constant power can be constructed simply by proper grouping of the symbols and computation of the optimum vertical angles.3. The theoretical symbol error probability
Gram-Schmidt orthogonalization procedure is useful to get orthogonal basis functions for a multidimensional signal set [2], [10]. We can apply it to produce transmitted signals s(t) = xϕx(t) + yϕy(t) + zϕz(t), where x, y, and z are the coordinates of 3-D symbols. ϕx(t), ϕy(t), and ϕz(t) are orthogonal basis functions. The received signals in an AWGN environment is r(t) = s(t) + n(t), where n(t) = nxϕx(t) + nyϕy(t) + nzϕz(t) is a Gaussian random process. nx, ny, and nz are independent and identically distributed Gaussian random variables with zero-mean and variance of σ2.
As the new constellations have regular geometries in the signal space, average SEP of a constellation is identical to that of a symbol. Decision region of a symbol is a 3-D space of which the boundaries bisect the space among adjacent symbols. Since recovered 3-D signal can be located either inside or outside a sphere, outer surface of the decision region extends to infinity as illustrated in Fig. 2. Probability of correct decision is the statistical volume of the decision region. The volume of 8-ary constellation is the same as an octant of the 3-D space [2], [11]. Thus, the SEP of a symbol in the 8-ary modulation format in terms of the MED dm is
where erfc(·) is the well-known complementary error function. N0 = 2σ2 is the single-sided noise spectral density.In the case of 16-ary constellation, as shown in Fig. 2(b), the boundaries for correct decision of a symbol are very close to a cone. With a little modification to Eq. (7) in [11], the SEP for a cone-shaped decision region can be represented as
where is signal-to-noise power ratio (SNR). Es is average symbol energy. ϕd is deviation angle from the central axis of a cone. Since ϕd varies with the sub-groups, we compute the deviation angles for SEP lower and upper bounds. When the decision region is a 16th of the entire volume, the angle for lower bound is ϕd,l = cos−1(1 – 2/M). The angle for upper bound ϕd,u can be computed for the smallest cone-shaped decision region.When M = 2 (a binary antipodal constellation), (4) is given as
This is the same as the average SEP of binary PSK [10]. This proves that (4) is reasonable and applicable to an arbitrary 3-D modulation format with constant power.4. Performance analysis
4.1. Structure of modulation format
The spherical coordinates of 16-ary constellations constructed by Betti’s method [6] and the new method are presented in Table 1, where Φ, −π ≤ Φ < π, and Ψ, −π/2 ≤ Ψ < π/2, are horizontal and vertical angles of a symbol, respectively.
Table 1. Spherical coordinates of the 16-ary constellations.
It is obvious that distribution of the symbols produced by Betti’s method is not regular. Thus, a coded modulation based on systematic set-partition algorithm [7] is not applicable. It also implies that we cannot implement a highly reliable optical communication system with the conventional modulation formats. As compared to the conventional scheme, symbols of the new 16-ary constellation are distributed uniformly on the surface of a sphere.
The MEDs of 8-ary and 16-ary constellations are compared in Table 2. The 8-ary cube is easy to design but has the least MED. Betti’s method known to have optimized MED at the expense of significant complexity has larger MED than the cube constellation. The proposed formats have slightly larger MEDs than Betti’s works. It can be concluded that both new constellations have better geometric features than the conventional ones.
Table 2. MEDs of the constellations
4.2. Simulation results
To analyze error performance of the proposed formats in an AWGN environment, computer simulation has been carried out. The system model is established with Gram-Schmidt orthogonalization procedure. The number of 3-D symbols for simulation is 1012. Considering an application to optical communication systems, we set Ps = 10−9 as a reference SEP for analysis.
SEPs of the proposed modulation formats are compared in Fig. 3. In the case of the 8-ary constellation, the SEP given in (3) is almost the same as the simulation result. Substituting the deviation angle ϕd,l ≅ 28.955° and ϕd,u ≅ 26.122° to (4), the lower and upper bounds on the SEP of the 16-ary constellation are computed. Both bounds have around 0.9 dB difference at the reference SEP. The simulation result is bounded by the performance bounds, and has around 0.6 dB higher and 0.3 dB lower Es/N0 at Ps = 10−9 than the lower and the upper bound, respectively. Hence, the theoretical SEPs are considered quite accurate.
Average symbol error rates (SERs) obtained by simulation are plotted in Fig. 4. As it can be expected, the cube has the worst error performance of the three 8-ary constellations. It can be observed that the proposed 8-ary constellation has almost the same SER as Betti’s method. In the case of 16-ary constellations, the proposed method has slightly lower SER than Betti’s method. Therefore, the simulation results correspond well with the analysis about MEDs.
5. Conclusions
A new method to construct good 3-D modulation formats with constant power is proposed. The method exploits MPSK constellations to design regular and symmetric constellations with little computational complexity. The 8-ary and 16-ary constellations constructed by the new method have slightly larger MEDs than the conventional ones. That makes the proposed modulation formats have almost the same or slightly improved error performance. We also studied SEPs of the proposed constellations in an AWGN environment, and demonstrate that the theoretical SEPs are accurate. Since a coded modulation employing a systematic set-partition technique is applicable, the new modulation formats are appropriate for implementation of a highly reliable optical communication system.
Acknowledgments
This work was supported partly by the Ministry of Knowledge Economy (MKE), Korea, under the ITRC Support Program supervised by the NIPA (NIPA-2011-C1090-1131-0007), and was supported in part by Basic Science Research Program through the NRF, Korea, funded by the Ministry of Education, Science and Technology (MEST) (No. 2011-0009443).
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