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We introduce the multi-dimensional permutation-modulation format in coherent optical communication systems and analyze its performance, focusing on the power efficiency and the spectral efficiency. In the case of four-dimensional (4D) modulation, the polarization-switched quadrature phase-shift keying (PS-QPSK) modulation format and the polarization quadrature-amplitude modulation (POL-QAM) format can be classified into the permutation modulation format. Other than these well-known modulation formats, we find novel modulation formats trading-off between the power efficiency and the spectral efficiency. With the increase in the dimension, the spectral efficiency can more closely approach the channel capacity predicted from the Shannon’s theory. We verify these theoretical characteristics through computer simulations of the symbol-error rate (SER) and bit-error rate (BER) performances. For example, the newly-found eight-dimensional (8D) permutation-modulation format can improve the spectral efficiency up to 2.75 bit/s/Hz/pol/channel, while the power penalty against QPSK is about 1 dB at BER=10−3.
© 2015 Optical Society of America
1. Introduction
1.1. General background
The recent development of digital coherent optical communication technologies enables us to modulate an optical electric field in a four-dimensional (4D) manner using two degrees of freedom of the complex amplitude of the electric field, that is, in-phase and quadrature (IQ) components, and two degrees of the freedom of the state of polarization (SOP) [1]. Then, the transmitted optical electric field E is written as the following 4D vector:
where suffixes x and y represent x- and y-polarization components, respectively, and I and Q denote in-phase and quadrature components, respectively. In addition, when we use k orthogonal wavelength channels, the dimension of the signal is extended to n = 4k. Similarly, using orthogonal channels composed of k time slots and the block coding, we can also achieve the signal dimension of n = 4k. With such n-dimensional vectors, we can design the constellation diagram of the signal in the n-dimensional vector space. The well-designed constellation in the n-dimensional vector space can improve the power efficiency and/or the spectral efficiency toward the Shannon limit.A symbol of the conventional dual-polarization (DP) quadrature phase-shift keying (QPSK) format is transmitted by one of the sixteen vectors given as
On the other hand, several novel 4D modulation formats, whose spectral/power efficiencies are better than those of DP-QPSK, have been demonstrated so far.
The concept of the densest non-overlapping arrangement of hyperspheres in the multidimensional vector space has been applied to find novel multi-dimensional modulation formats. The polarization-switched QPSK (PS-QPSK) format has the highest power efficiency in all of the 4D modulation formats. The following eight vectors are transmitted per symbol in this format [2, 3]:
The polarization quadrature-amplitude-modulation (POL-QAM) format is another example [4]. This format consists of the union of 4D vectors of DP-QPSK and those of PS-QPSK [3], where twenty five hyperspheres are aligned most densely in the 4D space and one at the origin is eliminated.
Another approach to find novel 4D modulation formats is set-partitioning of constellations of 4D-QAM (SP-4D-QAM). The 4D-MQAM format (i.e., the DP-MQAM format) has M2 constellation points in the 4D space, from which thinned-out constellation points of SP-4D-QAM are generated through the set-partitioning process [5]. The SP-4D-QAM format increases the minimum Euclidean distance between constellation points, sacrificing the spectral efficiency. For example, 128-ary SP-QAM (128-SP-QAM) is generated from first-step set-partitioning of the 4D-16QAM constellation having 256 signal points in the 4D space, and 32-SP-QAM is generated by successive set-partitioning of the 128-SP-QAM constellation. These SP-4D-MQAM formats have good trade-off between the power efficiency and the spectral efficiency. It is possible to increase the dimension of SP-MQAM higher than four, although such modulation formats have not been investigated.
The other approach to design multi-dimensional modulation formats has been proposed on the basis of the linear block error-correction code [6]. Each element of n-dimensional vectors is modulated in a bipolar binary manner (i.e., +1 and −1) according to binary logic levels of n-dimensional linear block codes. For example, it is shown that the 24-dimensional modulation format using the extended Golay code can achieve 3-dB coding gain at the bit-error rate (BER) of 10−3 against the DP-QPSK format.
1.2. Purpose of this paper
To find novel multi-dimensional modulation formats other than those above, this paper investigates multi-dimensional permutation modulation, where one of either 1, −1, or 0 is allotted to each element of vectors. A permutation modulation format is expressed as (n,m), where n and m are integers satisfying n ≥ m. In such a format, m elements of the n-dimensional vector (n ≥ m ≥ 1) are either +1 or −1, and remaining (n − m) elements are 0. This modulation format was first proposed in [7] and applied to designs of M-ary modulation systems [8] and optical communication systems [9]. Varying m for a given n, we analyze the performance of the spectral efficiency and the power efficiency using the geometry of n-dimensional constellations.
For 4D modulation (n = 4), we find out a novel modulation format (4,1) ∪ (4,3), combining two permutation modulation formats (4,1) and (4,3). Its power efficiency is maintained almost the same as that of DP-QPSK, while its spectral efficiency is as high as 2.66 bit/s/Hz/pol/channel. As the number of dimensions increases, the spectral efficiency can more closely approach the channel capacity predicted from the Shannon’s theory [10]. For example, the spectral efficiency of (8,3) ∪ (8,5) is 2.78 bit/s/Hz/pol/channel and its power efficiency is better than that of DP-QPSK by 0.83 dB. These results are also confirmed by computer simulations of the symbol-error rate (SER) and BER.
We also discuss the adaptive-equalizer configuration suitable for demodulation of 4D permutation modulation formats. Such an adaptive equalizer is driven by the decision-directed least-mean-square (DD-LMS) algorithm [11] and is similar to that used in the conventional digital coherent receiver [1, 12] except for the 4D decision circuit. In the case of 8D modulation, the channel separation process in the time domain or the frequency domain increases computational complexity significantly. On the other hand, the transmitter configuration for 4D permutation modulation formats is the same as that for the DP-QPSK modulation format; however, symbol mapping for 8D formats is much more complex.
1.3. Organization of this paper
The organization of this paper is as follows: in Sec. 2, we describe the definition of permutation modulation formats. In Sec. 3, we discuss the relation among the spectral efficiency, the power efficiency, the ratio of energy per bit to noise power spectral density Eb/N0, and the channel capacity. Section 4 analyzes characteristics of multi-dimensional permutation modulation formats in terms of the power efficiency and the spectral efficiency. Simulation results on SER and BER performances of the proposed formats are shown in Sec. 5. Section 6 deals with implementation issues, which include configurations of the transmitter and the receiver. Finally, we conclude this paper in Sec. 7.
2. Permutation modulation formats
We express the permutation-modulation format as (n, m), which means that m elements of an n-dimensional vector are allotted to either +1 or −1 and the other (n m) elements are set to 0. All of the vectors belonging to the format (m,n) are generated as follows: we take all of the possible permutations of m elements in the n-dimensional vector and assign either +1 or −1 to each element, while the other elements are set to 0. Therefore, the total number of the symbols of (n,m) is given as
where nCm = n!/(m!(n − m)!).In the extreme case where n = m, all of the vector elements are independently modulated in a bipolar binary manner. In the 4D case (n = 4), this modulation format is nothing but DP-QPSK (see Eq. (2)). Another extreme where m = 1 is called the biorthogonal modulation format, where the power efficiency is highest in n-dimensional modulation formats, while the spectral efficiency is reduced significantly. Especially in the 4D case, the format (4,1) is equivalent to PS-QPSK (see Eq. (3)). In the region between these two extreme cases (n > m > 1), permutation modulation can create novel modulation formats trading-off between the spectral efficiency and the power efficiency within the limit determined from the Shannon’s theorem, as discussed in Sec. 4.
3. Relation among spectral efficiency, power efficiency, Eb/N0, and channel capacity
The power efficiency of a modulation format is given as [2]
where is the minimum squared Euclidean distance (MSED) between constellation points and Eb is the average energy per bit. The energy of the signal is defined as the squared Euclidean distance measured from the origin. The power efficiency defined by Eq. (5) is unity when the modulation format is QPSK, and hence 1/γ of a given format represents the power penalty against QPSK. Note that all of QPSK, DP-QPSK, binary PSK (BPSK), and DP-BPSK have γ = 1. In addition, it should be stressed that the power efficiency given by Eq. (5) is an asymptotic value, which is valid when SER is sufficiently low. When SER is sufficiently low, symbol errors most likely occur between two closest symbols, that is, SER is determined from dmin. In such an asymptotic case, it is effective to estimate the power penalty against QPSK from 1/γ.On the other hand, the spectral efficiency per polarization and channel [bit/s/Hz/pol/channel] is given as
where M is the number of symbols and D is the dimension of the modulation format. Since we need to transmit IQ components of the light simultaneously, the minimum dimension is two. When we employ frequency slots to increase the dimension, we assume that there is neither frequency gap between them nor inter-symbol interference (ISI) as in the orthogonal frequency-division multiplexing (OFDM) system. In such a case, any degradation of SE never occurs from Eq. (6).The power efficiency γ and the spectral efficiency SE are determined simply from the geometry of the constellation diagram of a given modulation format. Thus, the performance of a modulation format is indicated by a point plotted on the SE-versus-(1/γ) chart [2].
Meanwhile, the channel capacity C is given as [10]
where W is the signal bandwidth, S the average signal power, and N the average noise power. Equation (7) can be rearranged as where N0 is the power spectral density of noise. The parameter Eb/N0 means the ratio of energy per bit to noise power spectral density, and C/(W (D/2)) is the Shannon limit of the spectral efficiency per polarization and channel. Substituting SE for C/(W (D/2)) in Eq. (8), we haveThus, we can calculate the Shannon limit of SE as a function of Eb/N0 using Eq. (9) [13].
On the other hand, from Eq. (5), we can obtain the relation between 1/γ and Eb/N0 as
When Eb/N0 is sufficiently large and SER is low enough, SER of any modulation format is approximately determined from the distance between closest two constellation points and written in the additive white-Gaussian-noise (AWGN) channel as [13]
where erfc(*) represents the complimentary error function. Equation (11) yieldsAssuming SER= 10−5, for example, where Eq. (11) is valid and the reduction of mutual information due to symbol errors is negligible, we have from Eq. (12). Using this value and Eq. (10), we can relate 1/γ of a given modulation format with Eb/N0 at SER ≃ 10−5. Although the choice of the SER value is arbitrary, Eb/N0 obtained from Eq. (10) is not so critically dependent on it; when SER= 10−6, for example, Eb/N0 obtained from Eq. (10) is only 0.94-dB larger than that at SER= 10−5. Thus, the performance of a modulation format is also indicated by a point plotted on the SE-versus-Eb/N0 chart. In Sec. 4, using the SE-versus-Eb/N0 chart, we show how closely permutation modulation formats can approach the Shannon limit.
4. Theoretical analyses of multi-dimensional permutation modulation
4.1. SE and γ of permutation modulation formats
Using Eqs. (4) and (6), the spectral efficiency of the permutation modulation format (n,m) can be calculated as
The average energy per symbol of the format (n,m) is always equal to m, because m elements of the vectors of this set have the absolute value of 1 and the energy of each vector is given as its squared magnitude. In addition, MSED of (n,m) is 2 when m ≠ n and 4 when m = n. From the average energy per symbol, the number of symbols, and MSED, the power efficiency of (n,m) can be calculated when 1 ≤ m ≤ n − 1 as
When m = n, we have γ = 1.
4.2. 4D permutation modulation formats
4D permutation modulation formats are given by (4,m) (1 ≤ m ≤ 4). All of the vectors belonging to each set are obtained by permutations of ±1 in the vector shown in the lower row of Table 1.
Table 1. Four sets of 4D permutation modulation formats. All of the vectors in each set are obtained by permutation of ±1. The superscript T means to transpose the vector.
The performance of these formats can be obtained from the results given in 4.1 and is summarized in Table 2. In Fig. 1, we plot SE (vertical axis) and 1/γ (lower horizontal axis) for these 4D permutation modulation formats by blue dots. In addition, the upper horizontal axis represents Eb/N0, which is transformed from 1/γ using Eq. (10) at SER≃10−5. The Shannon limit of SE is given by Eq. (9) and plotted by the solid curve as a function of Eb/N0 in this figure. SE and 1/γ of QPSK are also shown for comparison by the black dot.
Table 2. Performances of 4D permutation modulation formats.
Fig. 1 SE as a function of 1/γ of 4D (blue dots) and 8D (red dots) permutation modulations. The upper horizontal axis shows Eb/N0 corresponding to 1/γ when SER ≃ 10−5. Those of (4,1) ∪ (4,3) and ((8,3) ∪ (8,5) are shown by blue and red diamonds, respectively. For comparison, those of QPSK are shown by the black dot. We also plot the Shannon limit of SE by the solid curve as a function of Eb/N0.
The format (4,1) is equivalent to PS-QPSK, which is the most power efficient in all of the 4D modulation formats. The format (4,2) is equivalent to POL-QAM as described in [3]. On the other hand, (4,3) has not been studied yet. Its spectral efficiency is higher than that of DP-QPSK ((4,4)) by 0.5 bit/s/Hz/pol/channel; however, the power efficiency is slightly worse than that of DP-QPSK.
Taking the union of two sets, we may improve the modulation performance because the number of symbols is increased. However, during such a combination process, we need to take care to prevent from decreasing MSED of the total set. We find that (4,1) and (4,3) can be combined without the decrease in MSED. Hereafter, we refer to this format as (4,1) ∪ (4,3). In (4,1), the distance between [1,0,0,0]T and [0,1,0,0]T is minimum, for example. On the other hand, in (4,3), the distance between [1,1,1,0]T and [1,1,0,1]T is minimum, for example. Then, MSED is 2 in both cases. When we take the union, the distance between [1,0,0,0]T (originally in the set of (4,1)) and [1,1,1,0]T (originally in the set of (4,3)) is minimum in the set of (4,1) ∪ (4,3), and MSED is maintained at 2.
The characteristics of (4,1) ∪ (4,3) are shown in the right-hand side of Table 2 and plotted by the blue diamond in Fig. 1. We find that the spectral efficiency of (4,1)∪(4,3) is the highest in all of the 4D permutation modulation formats, almost keeping the power efficiency of DP-QPSK.
4.3. 8D permutation modulations
Next, we discuss permutation modulation whose dimension is higher than four. As an example, we consider eight-dimensional (8D) permutation modulation formats, where two additional dimensions are given either from orthogonal wavelength channels or from time slots. Symbols of each set (8,k) (k = 1, 2,⋯, 8) are generated in the same way as 4D modulation formats. Table 3 and Fig. 1 show the characteristics of these modulation formats.
Table 3. Performances of 8D permutation modulation formats.
As plotted by red dots in Fig. 1, some of 8D permutation modulation formats can approach the Shannon limit more closely than 4D formats. Especially, (8,3), (8,4), and (8,5) provide better performances than DP-QPSK both in terms of the power efficiency and the spectral efficiency.
Similarly to (4,1) ∪ (4,3), MSED of (8,3) ∪ (8,5) is the same as those of (8,3) and (8,5). Therefore, as shown in Table 3 and by a red diamond in Fig. 1, its spectral efficiency is highest of all the 8D permutation modulations and the power efficiency is slightly better than that of DP-QPSK.
5. Simulation results
Modulation formats (4,1) ∪ (4,3) and (8,3) ∪ (8,5) have good trade-off relations between the power efficiency and the spectral efficiency; thus, we conduct computer simulations on SER performances as a function of Eb/N0 to verify their characteristics in the AWGN channel. In the simulation, we randomly select one of the symbols at the transmitter and add white Gaussian noise to the symbol. At the receiver, we calculate Euclidean distances between the received multi-dimensional vector and all of the possible symbols and decide the symbol which gives the minimum distance. This process is called the maximum-likelihood estimation (MLE).
Figure 2 shows SERs calculated as a function of Eb/N0. SER characteristics of (4,1) ∪ (4,3) and (8,3)∪(8,5) are shown by the blue curve and the red curve, respectively. The SER performance of QPSK is also shown by the black curve for comparison.
Fig. 2 Symbol-error rates of (4,1) ∪ (4,3) (blue curve), (8,3) ∪ (8,5) (red curve), and QPSK (black curve) as a function of Eb/N0.
The power penalty or the sensitivity improvement against QPSK of each modulation format approaches their asymptotic power efficiency given in Sec. 3 as the SER decreases. However, at SER=10−3, (4,1) ∪ (4,3) has a power penalty of about 1 dB. Such a performance is almost the same as that of (8,3)∪(8,5). As the SER decreases, the performance of (8,3) ∪ (8,5) becomes better than that of (4,1) ∪ (4,3) because the asymptotic power efficiency of (8,3) ∪ (8,5) is larger than that of (4,1) ∪ (4,3). On the other hand, spectral-efficiency improvements against QPSK are 0.66 bit/s/Hz/pol/channel for (4,1)∪(4,3) and 0.78 bit/s/Hz/pol/channel for (8,3)∪(8,5), respectively, as shown in Sec. 4.
However, it should be noted that the number of symbols of each format is not a power of two. Therefore, to map binary data on the signal constellation, we must discard some symbols so as to make the number of the symbols a power of two. In the case of (4,1) ∪ (4,3), we must discard eight symbols from 40 symbols and refer to this modulation format as (4,1) ∪(4,3)′. Its spectral efficiency reduces from 2.66 bit/s/Hz/pol/channel to 2.5 bit/s/Hz/pol/channel. All of the discarded symbols should be chosen from (4,3) in order to reduce the average energy per symbol; then, its asymptotic power efficiency becomes 0 dB. Even if any eight symbols are discarded from (4,3), MSED of the set and hence the power efficiency are unchanged. On the other hand, if we discard 192 symbols from (8,3)∪(8,5), the number of symbols becomes 2048. We refer to this modulation format as (8,3)∪(8,5)′. All of the discarded symbols should be chosen from (8,5) to reduce the average energy per symbol, and we can discard any 192 symbols from (8,5). Therefore, (8,3) ∪ (8,5)′ consists of 448 symbols of (8,3) and 1600 symbol of (8,5). The spectral efficiency and the asymptotic power efficiency are 2.75 bit/s/Hz/pol/channel and 0.81 dB, respectively.
Note that it is impossible to apply the Gray mapping to these modulation formats; instead, we employ the natural binary mapping, which results in small degradation of the BER performance. Blue, red, and black curves in Fig. 3 respectively show BERs of (4,1) ∪ (4,3)′, (8,3)∪(8,5)′, and QPSK calculated as a function of Eb/N0. At BER=10−3, (4,1) ∪ (4,3)′ and (8,3) ∪ (8,5)′ have almost the same power penalty of about 1 dB against QPSK, while the spectral efficiency of (8,3) ∪ (8,5)′ can be improved up to 2.75 bit/s/Hz/pol/channel.
Fig. 3 Bit-error rates of (4,1) ∪ (4,3)′ (blue curve), (8,3) ∪ (8,5)′ (red curve), and QPSK (black curve) as a function of Eb/N0.
6. Configurations of the transmitter and the receiver
Figure 4 shows the transmitter setup for 4D permutation modulation, which is the same as that for DP-QPSK modulation. Bit-to-symbol mapping is carried out through digital signal processing (DSP) in the bit-to-symbol mapper. The x-polarized beam and the y-polarized beam are prepared from a single laser diode (LD) by using a half-wave plate (λ/2) and a polarization beam splitter (PBS). Each polarization component is modulated with an optical IQ modulator (IQMx,y). We can generate the 4D-modulated signal, combining the two polarization components with a polarization beam combiner (PBC). In the case of the 8D permutation modulation, the transmitter configuration is the same, but computational complexity is increased because the symbol mapping should be done over two time slots or two frequency slots.
Fig. 4 Configuration of the transmitter for 4D permutation modulation. LD: laser diode, IQMx,y: optical IQ modulator, λ/2: half-wave plate, PBS: polarization beam splitter, PBC: polarization beam combiner.
Figure 5 shows the configuration of the receiver for 4D permutation modulation. Four finite-impulse response (FIR) filters (wxx, wxy, wyx, wyy) are placed in the butterfly configuration [12] and equalize received complex amplitudes of two polarization components Ein,x and Ein,y based on the DD-LMS algorithm [11]. Using real and imaginary parts of equalized complex amplitudes of x- and y-polarization components, we constitute the 4D vector and perform MLE-based 4D decision. Note that computational complexity for MLE-based 4D decision increases exponentially with the increase in the number of symbols included in a permutation modulation format. Then, the error signal for the complex amplitude of each polarization component is sent to the equalizer. In the case of the 8D permutation modulation, the necessity for channel separation in the time domain or the frequency domain makes computational cost for DSP much higher.
Fig. 5 Configuration of the receiver for 4D permutation modulation. wxx, wxy, wyx, and wyy denote FIR filters in the butterfly configuration, whose tap coefficients are updated with the DD-LMS algorithm.
7. Conclusion
We have introduced multi-dimensional permutation modulation in coherent optical communication systems and analyzed modulation characteristics of 4D and 8D formats in terms of the power efficiency and the spectral efficiency. We propose two novel modulation formats (4,1) ∪ (4,3) and (8,3) ∪ (8,5), combining two permutation modulation formats.
These formats have good trade-off between the power efficiency and the spectral efficiency. We verify their characteristics through computer simulations of SER performances. In addition, (4,1)∪(4,3) and (8,3)∪(8,5) are modified to (4,1)∪(4,3)′ and (8,3)∪(8,5)′, respectively, so as to make the number of symbols a power of two. Simulation results on the BER performance show that (8,3) ∪ (8,5)′ has about 1-dB power penalty against QPSK at BER=10−3, while the spectral-efficiency improvement is 0.75 bit/s/Hz/pol/channel.
We have also discussed configurations for the transmitter and the receiver for 4D/8D permutation modulation formats. Computational complexity for 8D permutation modulation formats is much higher than that for 4D modulation formats, although their modulation characteristics are better than those of 4D modulation formats.
Acknowledgments
This work was supported in part by Grant-in-Aid for Scientific Research (A) ( 25249038), the Ministry of Education, Culture, Sports, Science and Technology in Japan.
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