Constellation modulation &#x2013; an approach to increase spectral efficiency

Soumya Sunder Dash; Frederic Pythoud; David Hillerkuss; Benedikt Baeuerle; Arne Josten; Pascal Leuchtmann; Juerg Leuthold

doi:10.1364/OE.25.016310

1. Introduction

Increasing the spectral efficiency (SE) of data communication is a key challenge in both wired and wireless communications. As state-of-the art transmission systems are improving and getting closer to the Shannon limit, any SE improvement matters [1–4].

Advanced modulation formats like quadrature amplitude modulation (QAM) in combination with advanced digital signal processing (DSP) have made it possible to get closer to the Shannon limit [1, 5–7]. In addition, spectral shaping, Nyquist sinc-pulse shaping (also termed as Nyquist-WDM) and orthogonal frequency division multiplexing (OFDM) have helped to reduce the guard bands between carriers and led to another significant improvement [8–12]. Recently, widely accepted techniques such as Trellis coding [13], polar codes [14], multiple-input and multiple-output (MIMO) [15], spatial division multiplexing [16], time-space coding [17] or probabilistic coding [18] have also received quite some attention in optical communications. To further improve the granularity of the spectral efficiency and maximize the usage of the available bandwidth various scheme such as time-domain hybrid [19], bandwidth variable transceiver [20], hybrid subcarrier modulation [21] and rate-adaptive coded modulation [22] has been proposed and compared against each other [23]. However, there is still a measurable gap to the limit [2, 24] especially for low signal-to-noise ratio (SNR) values with typical bit-error ratio (BER) of 2×10⁻². These low SNR transmissions are interesting since, using forward-error-correction (FEC) with limited overhead can reduce significantly the BER [25–27]. Recently, constellation switching has been proposed as an option to increase spectral efficiency of binary phase shift keying (BPSK) and quadrature phase shift keying (QPSK) in wireless communications [28–30]. Yet, there is room to encode additional information.

In this paper, we introduce constellation modulation (CM) in a general form as a means to increase the spectral efficiency. For this, we encode information both in the transmitted symbol and in the choice of constellations from a set of single polarizations (2D) and dual polarization (4D) constellations. We demonstrate by simulations that constellation modulation is a means to increase channel capacity significantly. For instance, we show how the capacity increases by more than 33% when adding constellation modulation on top of a 2D-BPSK channel operating in low SNR links. This SE increase comes with a minor penalty of less than 0.2 dB on the required SNR. And while the concept of CM is more applicable to almost any modulation format, the spectral efficiency gain is largest for systems operating with lowest SNRs and constellations with few symbols only. Lastly, we introduce a novel 4D-CM-BPSK modulation format that promises a most power efficient operation when operated in an additive white Gaussian noise (AWGN) system with a record low required SNR of as little as 0.2 dB for a SE of 0.7 bit/s/Hz at BER of 2×10⁻². This corresponds to a SNR benefit of nearly 3 dB over BPSK at a similar BER.

2. Concept of constellation modulation

Usually information is encoded as symbols within a constellation. Constellation modulation provides an additional degree of freedom for modulation. With constellation modulation, the choice of a particular constellation from a set of possible constellations is used to encode additional information. In a practical implementation, one may encode blocks of L symbols with one constellation and then switch to a new constellation. The constellations then may change from block to block, see Fig. 1. The term meta-symbol describes such a block of symbols with identical constellation.

Fig. 1 Schematic representation of the symbol stream being transmitted in constellation modulation: the stream in constituted of meta-symbols. Each meta-symbol consists of a sequence of L consecutive symbols encoded with the same constellation.

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For CM, a set of constellations has to be constructed. Ideally, all constellations in the set should exhibit similar performances in terms of BER to achieve maximum capacity. Here we illustrate this with a simple example where the constellation set is obtained by rotation of an 8 QAM constellation (Fig. 2). The set can also be generated by mixing different geometrical transformations like scaling, shifting, even using completely different constellations or partitioning of a larger constellation to obtain the constellation set. Assuming a set of K constellations, a number of $k = l o g_{2} (K)$ bits can be transmitted for each choice of constellation. It should be noted, that symbols from different constellations sets might occupy the same location in the complex plane. What matters only is that the average hamming distance from one constellation to another is still sufficient.

Fig. 2 Example of a constellation set generated by rotation of the 8 QAM constellation.

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The challenge now lies in detecting the transmitted constellation correctly. Especially if the SNR is low, transmitted symbols of different constellations might be indistinguishable at the receiver. To solve this, multiple symbols are transmitted with the same constellation c_i and the transmitted constellation can be determined through averaging. In our example, the transmitted symbol stream consists of a meta-symbol comprising of a sequence of L symbols being encoded with the same constellation (see Fig. 1).

3. Implementation of constellation modulation

To implement constellation modulation, we implemented an encoder and decoder. The encoder is shown in Fig. 3. Here, the bit stream is split in two streams, the main symbol bit stream and the additional constellation modulation bit stream. The constellation modulation bit stream is encoded through the selection of the constellation that is subsequently used to transmit the symbol bit stream. These symbols with information encoded in constellation and symbol will subsequently be called meta-symbols.

Fig. 3 Schematic representation of the constellation modulation encoder. The input bit stream is segmented into blocks of k + (L × m) bits to be encoded onto a meta-symbol. In a first step, the first k bits are encoded by selecting one out of K constellations. Next the L × m bits are encoded by mapping the bits onto a consecutive sequence of L symbols each encoding m bits. Finally, the stream of meta-symbols comprising of bits encoded into constellations and symbols is transmitted.

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The decoder then detects the transmitted constellation and symbols as shown in Fig. 4. In a first step, we perform de-mapping of each meta-symbol by de-mapping its symbols onto all possible constellations c_i within the set of K constellations. This de-mapping can be performed in parallel as shown by Fig. 4. The process leads to a possible symbol bit stream with a meta-symbol demapping error $E_{i}$ for each of possible constellations c_i. The constellation and therefore the symbol bit stream with the smallest meta-symbol de-mapping error are then selected. The constellation demapping error $E_{i}$ for constellation c_i is defined as the quadratic sum of the magnitude of all symbol errors of a meta-symbol with respect to the constellation c_i:

E_{i} = \sqrt{\sum_{t = 1}^{L} {| s_{t} - s_{t}^{i} |}^{2}}

where

{| s_{t} - s_{t}^{i} |}^{2}

is the Euclidean distance in the complex plane between the received phasor

s_{t}

of the t^th symbol of a meta-symbol and the ideal phasor

s_{t}^{i}

of the associated constellation point in the constellation c_i_. The constellation with the minimum error now represents the data of the constellation modulation bit stream and the demodulated bits from this constellation provide the symbol bit stream.

Fig. 4 Schematic representation of the decoder for constellation demodulation. The meta-symbol input stream is mapped onto all possible constellations within the set of K constellations. For each of the K possible constellations, the total square error of the received with respect to the ideal symbols of the constellations are determined. The constellation with the smallest total error is then selected as the constellation for demodulation of the meta-symbol. The demodulated bits of constellation modulation bit stream and symbol bit streams are then merged to form the output bit stream.

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For the decoder to work properly we have assumed that carrier-, timing- and phase estimation are known from a previous demapping of a block and that values will be continuously adapted (decision directed algorithm). We further have assumed that the carrier linewidths are sufficiently narrow such that deviations within a block of length L are sufficiently low.

The complexity of implementation of the encoder for the proposed scheme is minimal since it doesn’t require any additional hardware and can be implemented in terms of software or firmware upgrade. For the decoder, the implementation penalty is low when compared to Viterbi decoder. At least two different decoder implementations are straight forward:

• Hardware implementation of K constellation decoder.
• Software implementation of K parallel decoders without any change to the hardware. The software can be realized with a sequential decoding of symbols for the set of K constellations.

4. Achievable spectral efficiency increase by constellation modulation

Constellation modulation has an interesting property: The bandwidth of the transmitted signal generally is not modified. This is guaranteed as constellation modulation affects neither the signal pulse-shape nor the symbol rate. However, while the bandwidth stays the same, the raw bitrate increases. This raw bitrate increase is of $l o g_{2} (K)$ bits per meta-symbol. The increase has to be compared to the number of bits transmitted in the primary data stream by the sequence of L symbols of the meta-symbol encoded using constellation c_i. The number of bits per meta-symbol in the primary stream is $L \times l o g_{2} (M_{i})$ , with $M_{i}$ being the number of symbols in the constellations c_i. Assuming that each constellation occurs with the same probability we get

Bitrate increase = \frac{l o g_{2} (K)}{L \times l o g_{2} {(M_{i})}_{i}}

with

{〈 \dots 〉}_{i}

meaning the arithmetic mean for all i values. E.g., for a constellation set consisting of K = 4 constellations: 8 QAM, 8 QAM rotated by 22.5°, 8 QAM rotated by 45°, and 8 QAM rotated by 67.5° and assuming a meta-symbol length of L = 8, the bitrate increase would be

8.3 %

.

Now, another interesting question is regarding the bit-error ratio (BER) of the new modulation scheme. This question is difficult to treat in a general way. However, the following simplified approach is proposed based on considerations of the minimum distance between symbols. Let us assume that the transmission is performed using only constellations c_i, with a given SNR. We assume in this case that the BER obtained is BER_i. Let us further assume that the overall BER ( $B E R_{0}$ ) is similar for all constellations:

B E R_{0} ≅ B E R_{1} ≅ B E R_{2} ≅ \dots ≅ B E R_{K}

Each of the BER_i depend mostly on the ratio of the minimum distance

d_{i}^{m i n, s}

between any two different symbols within the constellation c_i and the amount of noise σ [31]. For the sake of simplicity, we now implicitly assume that this minimum distance is the same for any constellation, i.e.

d_{1}^{m i n, s} ≅ d_{2}^{m i n, s} ≅ \dots ≅ d^{m i n, s}

We now go a step further and we define the minimum distance

d_{i, j}^{m i n, c}

between different constellations of the constellation set. (For the sake of simplicity, we thus neglect the special case where symbols from different constellations occupy the same location in the complex plane. If the probability of symbols from different constellations occupying the same location increases, then the required meta-symbol length L would increase.) This minimum distance is defined as the minimum pairwise distance between the symbols

s_{s}^{i}

of constellation i and the symbols

s_{t}^{j}

from constellation

j

. We now consider the minimum distance

D_{i, j}^{m i n}

between two different meta-symbols, the first one being a sequence of L symbols of constellation c_i, the second one being a sequence of L symbols of constellation c_j (with

i \neq j

). Assuming orthogonality of the individual symbols within the meta-symbol, the minimum distance between meta-symbols

D_{i, j}^{m i n}

now becomes

\begin{matrix} D_{i, j}^{m i n, c} = \sqrt{L} \times d_{i, j}^{m i n, c} & for i \neq j \end{matrix}

The factor

\sqrt{L}

originates from the quadratic addition of L equal distances

d_{i, j}^{m i n, c}

and from the time-orthogonality of the symbols within the meta-symbol.

We now take the following assumption: in order to achieve reliable detection, we need to detect the transmitted constellation correctly before we can demodulate the individual symbols within the meta-symbol. We are therefore interested in working with meta-symbols for which $D_{i, j}^{m i n, c} > d_{i}^{m i n, s}$ , where the resulting meta-symbol error probability is lower than the error probability for the individual symbols. Considering the fact that Eq. (5) depends on the symbol length L, it therefore means that for any set of constellations with $d_{i, j}^{m i n, c} > 0$ , it is always possible to find a length L of a meta-symbol that meets the condition

L > {(\frac{d_{i}^{m i n}}{d_{i, j}^{m i n}})}^{2}

If this condition is met, the constellation detection is reliable and we can expect to reach a similar overall BER as compared to the transmission without constellation modulation. We therefore get a

B E R

close to the

B E R_{0}

without constellation modulation. For extremely long meta-symbol lengths L, the amount of data carried by the constellation modulation and the error probability for the meta-symbol detection would become negligible. The increase in spectral efficiency would then become negligible as well, see Eq. (2). On the other hand, for extremely short meta-symbol lengths L (i.e. shorter than what is needed according to Eq. (6)), the resulting BER would be larger than the

B E R_{i}

of an individual constellation. Wrong constellation detection would lead to a burst of errors due to a false interpretation of all L symbols.

Therefore, the optimum meta-symbol length L has to be selected such, that the increase in spectral efficiency is maximum and the overall BER is similar to $B E R_{0}$

B E R \approx B E R_{0}

In summary, for any set of constellations, it is always possible to find a meta-symbol length L that provides a similar BER with the same SNR condition as without any constellation modulation. The proper value of L is hereafter found using simulations. Simulations can then lead to values of L different from the ones suggested in Eq. (6), since the above theoretical investigations assume the worst case by focusing on

d_{i, j}^{m i n, c}

and do not take into account statistical effects. The benefit of the constellation modulation is a net increase of the bitrate as described by Eq. (2), without any major degradation of the BER and minor impact on the required SNR.

5. Assessment of constellation modulation for different modulation formats

The potential of constellation modulation has been assessed by simulations using the Mathematica calculation program. Transmission streams have been generated following the encoder depicted in Fig. 3. Gaussian random noise has been added to each symbol individually. Finally, the symbols have been decoded with a decoder such as shown in Fig. 4. All simulations have been performed with complex valued symbols (in-phase and quadrature plane) in the frequency domain. Transmission effects, timing-, carrier- and phase estimation are not part of this investigation but most of the standard estimation schemes can be applied with little penalty and only small modifications [29]. We assume that hard-decision forward-error-correction (HD-FEC) codes that can deal with BER of 3.8 × 10⁻³ [27] or 1 × 10⁻³ [5] can be applied. In recent years, soft-decision forward-error-correction (SD-FEC) codes have been proposed in optical communication to move closer to Shannon limit. Such codes can deal with BERs as low as 2 × 10⁻² with an overhead of 15-25% [26, 27]. To judge the potential of the new coding we demonstrate the benefits at pre-FEC BERs of 1 × 10⁻³ and 2 × 10⁻².

Single polarization signals

First, we perform simulations for a single polarization communication link. We start with an initial constellation such as BPSK or QAM. Here, additional constellations for the constellation sets are then obtained by e.g. rotating the initial constellation by an angle of $π / (2 K)$ . Figure 5 shows a simple example for the case of a 16 QAM constellation in the presence of little noise. Even though this choice of constellation in this example might not be ideal, we will show that a spectral efficiency increase can be achieved with a small BER penalty only.

Fig. 5 Representation of the transmitted symbols for (a) a regular 16 QAM signal, as well as for (b) a constellation modulated signal with a set of K = 2 constellations. The second constellation (in red) is obtained from the constellation (a) by a rotation of $π / 4$ (right picture).

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The signal without constellation modulation (K = 1) serves as a reference. The SNR required for a BER of BER₀ = 10⁻³ is then determined for this reference. In the following, we kept the SNR at the value of the reference signal and added constellation modulation. For each constellation set size K, the meta-symbol length L was increased from L = 2 until the total BER dropped to an error below 1.2 × BER₀. This factor of 1.2 has been chosen in order to avoid infinite convergence to ${BER}_{0}$ , since the BER of a constellation modulated signal normally should be larger than the reference ${BER}_{0}$ . As a result, we must accept a certain SNR penalty to take the increase in BER into account. The penalty in constellation modulation comes simply from the fact that by changing the constellation (e.g. rotated constellation), the reliability of the transmission of individual symbols does not increase. On the other hand, if the constellation is identified correctly, it should not degrade either. In the few cases where a wrong identification of the constellation is done, most symbols within the meta-symbol will be identified incorrectly. Our simulations have shown that this leads to BER degradation below 2 × 10⁻³ rather than compared to a 1 × 10⁻³ BER with conventional modulation technique. This BER degradation was then converted into an SNR penalty in order to report a fair comparison. Numerically, we found this BER to correspond to an SNR penalty of about 0.1 dB. Similarly, SNR penalty for BER degradation when compared at 2 × 10⁻² is about 0.2 dB. This SNR penalty is absolutely in agreement with literature values which is calculated from the characteristic performance of the modulation format without FEC [31]. We therefore conservatively assume an SNR penalty with respect to the initial constellations of 0.2 dB for all CM formats at a BER₀ = 1 × 10⁻³ and a SNR penalty of 0.1 dB for all CM formats at a BER₀ = 2 × 10⁻².

All the simulations were performed with a symbol data stream of 10⁶ bits, this provides a typical confidence interval of > 99% for BER of 10⁻³ [32]. To further strengthen the confidence level, we performed simulation for CM-QPSK with data stream of 10⁸ bits and indeed no difference were found compared to a simulation with 10⁶ bits. In order to determine the total BER, we assumed Gray coding for constellation modulation, where one detection error for constellation modulation translates into only one bit error in the constellation stream. A constellation error will of course lead to a burst error in the symbol stream. The bit error ratio is then the sum of the number of bit errors in the symbol stream and bit errors in the constellation stream, divided by the total number of transmitted bits.

BER = \frac{bit {errors}_{symbol stream} + meta-symbol {errors}_{constellation stream}}{N^{ο} {symbols}_{symbol stream} \times \log_{2} (M) + N^{ο} meta-symbol s_{constellation stream} \times \log_{2} (K)}

with

N^{ο}

as “number of”. With these assumptions, the spectral efficiency increase shown in Fig. 6 and Table 1 have been derived.

Fig. 6 Spectral efficiency increase when working with sets of K rotated constellations for BPSK, 4 QAM, 8 QAM, 16 QAM, 32 QAM, and 64 QAM.

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Table 1. Summary of the performance of constellation modulation for transmission on a single polarization (2D) to reach total BER less than 1.2 × 10⁻³.

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Figure 6 shows that there is an optimum constellation set size K and Table 1 additionally shows that there is as well an optimum meta-symbol length L to reach a maximum spectral efficiency for a particular constellation. CM brings in two aspects: By increasing the constellation set size K the information content and the SE increases. Conversely, a higher set size K requires a longer L which comes at the cost of decreasing the SE. For modulation formats with a small number of symbols this optimum seems to be with very low K. For higher order modulation formats the optimum lies at the higher choice of constellation set K and longer meta-symbol length L. For BPSK, the spectral efficiency may increase by up to 33.3%, at the price of a minimal BER increase to 1.2 × BER₀ for K = 2 (a rotation by 90° in the complex plane) and meta-symbol length of L = 3. For 4 QAM and 8 QAM, spectral efficiency increases by up to 8.3% can be achieved. 16, 32, and 64 QAM still exhibit spectral efficiency increases of up to 5.9%, 4.7%, and 4.5%, respectively. These numbers also make it clear that constellation modulation offers the highest benefits for lower order modulation formats. It is therefore especially beneficial to increase the SE in situations with low-SNR or long reach systems. Table 2 summarizes similar simulations performed for a BER₀ = 2 × 10⁻².

Table 2. Summary of the performance of constellation modulation for transmission on a single polarization (2D) to reach a total BER of less than 2.2 × 10⁻².

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Dual-polarization signals

As we expect highest benefits for lower order modulation formats, we extended our investigations of BPSK towards dual-polarization or so-called 4D constellation modulation schemes. In this investigation, we discuss three different variations.

First, we discuss the constellation-modulated polarization-multiplexed BPSK (CM-PM-BPSK) modulation format, see Fig. 7. Here, the two polarizations carry a BPSK signal each and constitute a first constellation with M = 4 symbols. The constellation set then consists of 2 constellations c₁ and c₂, where c₂ is rotated by π/2 in the complex plane with respect to c₁.

Fig. 7 Polarization multiplexed BPSK (CM-PM-BPSK) encoded in two different sets of constellations. Each of the constellations c₁ and c₂ of the constellation set consists of two binary phase shift keying modulated signals multiplexed in two polarizations. To generate the second constellation c₂, the initial constellation c₁ is rotated by 90° in the complex plane.

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Second, we examine constellation-modulated polarization-switched BPSK (CM-PS-BPSK) modulation, see Fig. 8. Here too, every constellation encodes up to M = 4 symbols, see Fig. 8. The second constellation c₂ is obtained by rotating the symbols in the complex plane of the polarization-switched BPSK signal of c₁ by π/2.

Fig. 8 Constellation modulation by relying on polarization switched BPSK (CM-PS-BPSK). Each of the constellations c₁ and c₂ of the constellation sets consists of a polarization switched binary phase shift keying modulated signal. To generate the second constellation c₂, the initial constellation c₁ is rotated by 90° in the complex plane.

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Finally, we introduce a four-dimensional constellation-modulation BPSK (4D-CM-BPSK) format, see Fig. 9. The 4D-CM-BPSK modulation format comprises of the regular BPSK symbols with M = 2 elements s₁ = (1, 0) and s₂ = (−1, 0), where (.,.) denotes the electric fields in Jones vector nomenclature. The constellation set is now obtained by rotating and/or switching the polarization of the BPSK signal. As a result, every constellation encodes up to 2 additional bits by constellation modulation with a set comprising of K = 4 elements.

Fig. 9 Four-dimensional constellation modulation BPSK (4D-CM-BPSK). Each of the constellations c₁, c₂, c₃ and c₄ of the constellation set consists of a binary phase shift keying modulated signal. To generate the second constellation c₂, the initial constellation c₁ is rotated by 90° in the complex plane. The constellations c₃ and c₄ are then generated by switching the polarization of the constellations c₁ and c₂, respectively.

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We summarize the details of the chosen modulation schemes in Table 3.

Table 3. Descriptions of selected dual polarization constellation sets and their derivation from known modulation constellations.

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To judge the spectral-efficiency and data rate gains for a given SNR we followed the same procedure as for the single polarization formats. We obtained the required SNR for the initial constellation and subsequently determined the required meta-symbol length L for a BER below 1.2 × BER₀ (where BER₀ = 10⁻³ and allowed SNR penalty is 0.2 dB) and 1.1 × BER₀ (where BER₀ = 2 × 10⁻² and allowed SNR penalty is 0.1 dB). The results are shown in Table 4 and Table 5 respectively.

Table 4. Summary of the performance of constellation modulation in our 4D examples in order to reach a total BER less than 1.2 × 10⁻³.

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Table 5. Summary of the performance of constellation modulation in our 4D examples in order to reach a total BER less than 2.2 × 10⁻².

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As depicted in Table 4, for both, CM-PM-BPSK and CM-PS-BPSK, a spectral efficiency and data rate increase of up to 25% was achieved compared to initial modulation format of PM-BPSK and PS-BPSK respectively to reach a BER of 10⁻³. In Table 5 we perform the same simulations with the same modulation formats but for a BER of 2 × 10⁻². As expected the meta-symbol length L needs to be increased for reliable detection of the constellation at a higher BER. That is why the meta-symbol length L increases when going from Table 4 to Table 5. From Table 5 one also learns that the SE increase for CM-PM-BPSK and CM-PS-BPSK become 25% and 16% at a BER of 2 × 10⁻². The PS-QPSK modulation format offers a data rate increase of up to 7% and 4.7% with constellation modulation for BER of 10⁻³ and 2 × 10⁻², respectively. For 4D-CM-BPSK, the spectral efficiency per polarization is effectively decreased by 25% and 30% to reach a BER of 10⁻³ and 2 × 10⁻² as compared to initial modulation format of PM-BPSK because the signal now encodes the information onto two polarizations instead of one. However, as the signal is now spread onto two polarizations, the required SNR is decreased by 2.8 dB as well. When compared to a single polarization BPSK signal, the data rate increases by 50% and 40% to reach a BER of 10⁻³ and 2 × 10⁻² respectively.

Finally, the spectral efficiency of both the initial and the constellation modulated formats to reach a BER of 10⁻³ are plotted against the SNR in Fig. 10. The blue dots correspond to the SNR versus SE in one polarization of the original modulation formats at a BER of 10⁻³. The red dots then show the SNR versus SE for the constellation modulated formats. An improvement of the spectral efficiency of up to 33% in case of BPSK, or 25% in case of PS-BPSK for a BER of 10⁻³ can be seen. For Fig. 10, we include the estimated SNR penalty of 0.2 dB for constellation modulation formats to account for the 20% increase in BER (BER = 1.2 × 10⁻³) at the selected SNR. Similar to Fig. 10 in Fig. 11, we plot the spectral efficiency against SNR with and without constellation modulation for various modulation formats to reach a pre-FEC BER of 2.2 × 10⁻². Here, we include the estimated SNR penalty of 0.1 dB for the constellation modulation formats to account for the 10% increase in BER (BER = 2.2 × 10⁻²) at the selected SNR. This SNR penalty is the price for constellation mapping with a low block number length L. Moreover, we take into account a modest SD-FEC overhead of 20% [27] for the transmission to compute the plotted SE for the corresponding SNR. From both Fig. 10 and Fig. 11 we can see that constellation modulation helps in increasing the SE for any given modulation formats against a negligible penalty for required SNR.

Fig. 10 Spectral efficiency improvements through constellation modulation (red circle) for single polarization, (red square) for dual polarization 4D formats) compared to the original modulation (blue circle) for single polarization, (blue square) for dual polarization 4D formats)). These values are valid for a BER of 1 × 10⁻³ and are compared to the Shannon-Hartley limit (solid curve).

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Fig. 11 Spectral efficiency improvements through constellation modulation (red circle) for single polarization, (red square) for dual polarization 4D formats) compared to the original modulation (blue circle) for single polarization, (blue square) for dual polarization 4D formats)). These values are for a pre-FEC BER of 2 × 10⁻² together with a SD-FEC overhead of 20%. The Shannon-Hartley limit (solid black curve) is plotted for reference.

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The BER as a function of the SNR is plotted in Fig. 12 for BPSK (blue circle) and CM-BPSK with constellation set size K of 2 and meta-symbol length L of 6 (red circle), PS-BPSK (blue diamond)) and CM-PS-BPSK with constellation set size K of 2 and meta-symbol length L of 6 (red triangle) and PS-QPSK (blue diamond)), CM-PS-QPSK with a K of 2 and a meta-symbol length L of 6 (red diamond) and finally the novel 4D-CM-BPSK for constellation set size K of 4 and meta-symbol length L of 4 (red square). We can see that the overall BER curve of constellation modulated BPSK, PS-BPSK and PS-QPSK follows the BER curve of the respective standard modulation i.e., without constellation modulation. Thus, constellation modulation leads to a spectral efficiency increase with hardly any penalty. Since the constellation set and the length of the meta-symbol is optimized for a BER of BER₀ = 2 × 10⁻², we can see a penalty in constellation modulation as we go to even higher BERs. Conversely, we can see a gain in the SNR as we go to a lower BER regime. Finally, the novel 4D-CM-BPSK offers a SNR margin of ~3 dB against BPSK at a BER of BER₀ = 2 × 10⁻² with a SE of 0.7 bit/s/Hz. This is achieved owing to the fact that the minimum Euclidean distance in 4D-CM-BPSK is similar to BPSK (ref to Eq. (5)) but the signal energy is spread over both the polarization. Higher dimensional modulation format e.g., 8D codes [33] have been shown to increase the noise tolerance in the optical communication link. Such modulation formats can be stacked together with the proposed constellation modulation scheme to further increase the SE of the system.

Fig. 12 Simulation results showing the bit-error ratio (in an AWGN Channel) as a function of SNR for the BPSK (blue circle) and CM-BPSK with constellation set size K of 2 and meta-symbol length L of 6 (red circle), PS-BPSK (blue trinagle) and CM-PS-BPSK with constellation set size K of 2 and meta-symbol length L of 6 (red triangle) and PS-QPSK (blue diamond), CM-PS-QPSK K of 2 and meta-symbol length L of 6 (red diamond) and finally the novel 4D-CM-BPSK for constellation set size K of 4 and meta-symbol length L of 4 (red square).

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6. Experiment

Finally, to support our theoretical prediction and simulation, that constellation modulation increases the spectral efficiency of the communication system with a negligible penalty, we performed an experiment. In our experiment, we study the performance of the constellation modulation with QPSK with additive Gaussian noise for different meta-symbol lengths L = 6; 8; 10; 12; 14. The experimental setup is depicted in Fig. 13. At the transmitter, an external cavity laser with linewidth < 100 kHz provides the optical carrier to the IQ-modulator. The transmitted signal is generated in offline DSP as depicted in the Ref [29] with a 2¹⁵-1 PRBS and root raised cosine transmit filter with a roll-off factor (β) of 0.05. An arbitrary waveform generator (Agilent AWG – M8195A) operate at 60 GSa/s to generate 28 GBd signal. The channel under test is operated at 1550 nm. The channel is noise loaded with an amplified spontaneous emission (ASE) source. The optical signal to noise ratio (OSNR) is measured by the optical spectrum analyzer (OSA). The OSNR and the relative ASE noise loading were adjusted with a variable optical attenuator (VOA). The signal is subsequently fed into a polarization controller. The polarization controller tracks the polarization and guarantees the optimum power in the required single polarization. Then the signal is received with a coherent optical receiver. The coherent receiver used a separate ECL (linewidth < 100 kHz) as local oscillator. The real-time oscilloscopes digitized the signal with a sampling rate of 80 GSa/s and an electrical bandwidth of 33 GHz. All sampled signals are subsequently processed offline. Each experimental shot was recorded for > 10⁷ symbols. In order to guarantee a fair comparison between the signal with and without constellation modulation the same set-up was used for both the signals. The offline digital signal processing (DSP) is described in Refs [29, 30]. The overhead for pilot symbols is < 1%.

Fig. 13 Experimental setup: Transmitter: An external cavity laser provides the optical carrier at 1550 nm to the IQ-modulator that encodes the 28 GBd signal generated by an arbitrary waveform generator operating at 56GSa/s. Inset (a) depicts the constellation of the QPSK signal with constellation modulation constellation points with (blue square) represents the standard QPSK constellation and the constellation points with (green square)) represents the π/4 rotated QPSK constellation (Ref [29].). Noise loading: The signal is then noise-loaded with an amplified spontaneous emission source (ASE). The OSNR and the relative ASE noise loading were adjusted with a variable optical attenuator (VOA). Receiver: The OSNR is measured by the optical spectrum analyzer (OSA). The signal is fed into a polarization controller to track the polarization and guarantee optimum power in the single polarization. A separate ECL is used as a local oscillator. The signal is then received with a coherent receiver. The real-time oscilloscope stores the signal for the offline processing (orange boxes). The received spectrum is the same, inset (b) depicts the spectrum of 28 GHz signal (‒ (blue dash)) with constellation modulation (meta-symbol length L of 8) and (‒ (red dash)) without constellation modulation. The processed constellation of the constellation modulated QPSK signal with meta-symbol length L = 10 after the constellation decision and correction.

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The Fig. 13(b) depicts the spectra of the 28 GHz optical signal measured with the real-time oscilloscope. (‒ (red dash)) is the spectrum for signal without constellation modulation (QPSK) and (‒ (blue dash)) is the spectrum for signal with constellation modulation for meta-symbol length L of 8. We can clearly see that there is no increase in the bandwidth due to constellation modulation.

The experimental results along with the theoretical limits are shown in Fig. 14. The measurement of the bit-error ratio (BER) for different SNR is plotted in Fig. 14(a). The curve (‒) displays the theoretical BER as a function of SNR for a QPSK signal and (red square, red dash) curves displays the measured QPSK BER as a function of SNR. An experimental implementation penalty of ~1.1 dB is reported here. The calculated BER for different meta-symbol length L with constellation modulation show a behaviour similar to the one obtained by the theoretical study. For a fair comparison, we compare the experimentally measured QPSK without constellation modulation against the QSPK signal with constellation modulation. For meta-symbol lengths, $L \geq 8$ there is hardly any penalty between QPSK modulation (red square, red dash) and the constellation modulation (blue square). Figure 14(b) shows how increasing the meta-symbol length L reduces the penalty for constellation modulation in comparison with the QPSK BER (red dash). As illustrated earlier in both theory and simulation in section 4 and 5 respectively, the experimental result follows the similar trend wherein the increase in meta-symbol length of the constellation modulation decreases the BER penalty. Figure 14(c) shows the SNR penalty for constellation modulation with comparison to the QPSK to reach the BER of 10⁻³. For a meta-symbol length of L = 6 and 8 we have an SNR penalty ~0.35 dB and 0.15 dB respectively. This corresponds to 8.33% and 6.25% increase in the data rate. Figure 14(d) shows how constellation modulation leads to a SE gain while the SNR penalty remains small. SE gains of 8.33% and 6.25% are found for meta-symbol lengths of L = 6 and 8 respectively. The SE of QPSK and 8-PSK is used to plot a reference limit (current achievable SE limit) threshold to evaluate the benefit of the constellation modulation format. The reference curve (‒ ‒) plotted in the Fig. 14(d) is shifted to the right, taking into account the experimental implementation penalty.

Fig. 14 Experimental results for meta-symbol lengths L of 6 (green square), 8 (blue square), 10 (turquoise square), 12 (pink square), and 14 (orange square) in comparison to QPSK (red square) and the theoretical limits (‒). (a) BER measurements for different signal to noise ratios for all meta-symbol lengths L. Theoretical limit (‒) and experimental results (red square, red dash) for QPSK serve as a reference, (b) BER for different L at the SNR required for a QPSK signal to reach a BER of 10⁻³. (c) Required SNR for different meta-symbol lengths L in comparison to QPSK (red ‒) to reach the BER of 10⁻³. (d) Spectral efficiency (SE) at a BER = 10⁻³ for all tested formats. A spectral efficiency increase of 6.25% is found for a constellation modulation format with meta-symbol length L = 8 (blue square).

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The analytical results discussed in the previous section agree well with the experiments and simulations. We find constellation modulation has the potential to increase the spectral efficiency of a communication system at the price of a small penalty. A penalty of $\leq 0.1 dB$ for a meta-symbol length of L = 8 is likely due to the combined laser phase noise in the system i.e. 200 kHz. Moreover, an improved DSP algorithm can offer better performance.

7. Conclusion

In this paper, we have introduced constellation modulation in a general form to increase the spectral efficiency of transmission systems. By encoding information in the choice of a constellation, the spectral efficiency can be increased in low-SNR regimes, e.g. up to 33% for BPSK. Constellation modulation is of particular interest in situations with lowest SNRs. A newly introduced 4D-CM-BPSK for instances allows transmission with a spectral efficiency of 0.7 bit/s/Hz and polarization in systems with a low SNR of 0.3 dB to reach a BER = 2.2 × 10⁻². This is especially interesting since it is quite difficult to increase the spectral efficiency of a modulation format for low SNR. Interesting techniques, like Trellis coding, increase the robustness of coding schemes by encoding sequence of symbols rather than symbols. With this they are able to provide up to 6 dB gain against standard modulation formats. However, this gain is typically valid for low BER in the order 10⁻⁹. At lower SNR where the BER is about 2 × 10⁻², the Trellis coding often fails to provide gain whereas the suggested scheme offers an advantage. For high SNR scenarios, the meta-symbol length L decreases thereof increasing the SE, however in these scenarios coding algorithm like set-partitioning (Trellis coding) perform better than the proposed scheme. For constellation modulation, the sets of constellations may be obtained by applying geometrical transformations like rotation, translation, scaling, or even more abstract transformations to an initial constellation. Constellation modulation can be used on top of existing modulation, higher dimensional modulation, hybrid modulation formats or multiplexing schemes in order to increase the spectral efficiency without major penalties on BER or SNR. It has also the interesting advantage that in the most cases, it can re-use existing transmitter and receiver technology, and can be implemented in terms of software or firmware upgrade. It has also the advantage that for high bitrate optical communication, it can be implemented quite easily in hardware by parallelizing the receivers.

Funding

European Metrology Research Program (EMRP) (IND 51 MORSE, Metrology for Optical and RF Communication Systems); European Research Council (ERC) (670478)

Acknowledgments

The authors would like to thank the reviewer for the constructive inputs

References and links

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	Without Constellation Modulation		With Constellation Modulation
Constellation	SNR [dB]	SE [bits/s/Hz/pol]	Constellation set size K	Length of Meta-symbol L	SNR [dB]	SE [bits/s/Hz/pol]	Bitrate increase
BPSK	6.8	1	2	3	7	1.33	33.30%
			3	5	7	1.32	31.70%
			4	8	7	1.25	25.00%
			5	11	7	1.21	21.10%
4 QAM	9.8	2	2	6	10	2.17	8.30%
			3	10	10	2.16	7.90%
			4	16	10	2.13	6.30%
			5	22	10	2.11	5.30%
8 QAM	13.7	3	2	8	13.9	3.13	4.20%
			3	8	13.9	3.20	6.60%
			4	8	13.9	3.25	8.30%
			5	10	13.9	3.23	7.70%
16 QAM	16.5	4	2	10	16.7	4.10	2.50%
			4	10	16.7	4.20	5.00%
			6	11	16.7	4.23	5.90%
			8	14	16.7	4.21	5.40%
32 QAM	19.7	5	2	10	19.9	5.10	2.00%
			6	14	19.9	5.18	3.70%
			8	14	19.9	5.21	4.30%
			10	14	19.9	5.24	4.70%
			12	16	19.9	5.22	4.50%
64 QAM	22.5	6	2	9	22.7	6.11	1.90%
			4	12	22.7	6.17	2.80%
			8	12	22.7	6.25	4.20%
			12	14	22.7	6.26	4.30%
			14	14	22.7	6.27	4.50%
			16	16	22.7	6.25	4.20%

	Without Constellation Modulation		With Constellation Modulation
Constellation	SNR [dB]	SE [bits/s/Hz/pol]	Const. set size K	Meta-symbol length L	SNR [dB]	SE [bits/s/Hz/pol]	Bitrate increase
BPSK	3.24	1	2	5	3.34	1.20	20%
4 QAM	6.25	2	2	13	6.35	2.08	3.8%
8 QAM	10.21	3	6	18	10.31	3.14	4.8%
16 QAM	12.68	4	5	22	12.78	4.11	2.6%
32 QAM	16.21	5	12	32	16.31	5.11	2.2%
64 QAM	18.42	6	14	34	18.52	6.11	1.9%

Modulation	Initial const.	Phasors s_i of the symbols of the initial constellation in both polarizations (pol. x, pol. y)	Constellation set for constellation modulation with constellations c_i and the respective phasors	Description of the principle for constellation set generation.
CM-PM-BPSK	PM-BPSK	s₁: (1,1) s₂: (1,-1) s₃: (−1,1) s₄: (−1,-1)	$\begin{matrix} c_{1} : & {s_{1}, s_{2}, s_{3}, s_{4}} \\ c_{2} : & {\begin{matrix} (j, j), \\ (j, - j), \\ (- j, j), \\ (- j, - j) \end{matrix}} \end{matrix}$	BSPK multiplexed on 2 polarizations, both polarizations are rotated by 90° in the complex plane to obtain the second constellation in the set, see Fig. 7
CM-PS-BSPK	PS-BPSK	s₁: (1,0) s₂: (−1,0) s₃: (0,1) s₄: (0,-1)	$\begin{matrix} c_{1} : & {s_{1}, s_{2}, s_{3}, s_{4}} \\ c_{2} : & {\begin{matrix} (j, 0), \\ (- j, 0), \\ (0, j), \\ (0, - j) \end{matrix}} \end{matrix}$	Polarization-switched BPSK that is rotated by π/2 in the complex plane to obtain the constellation set, see Fig. 8
CM-PS-QPSK	PS-QPSK	s₁: ( $\begin{array}{l} 1 / \sqrt{2} + j / \sqrt{2} \end{array}$ ,0) s₂: ( $- 1 / \sqrt{2} + j / \sqrt{2}$ ,0) s₃: ( $- 1 / \sqrt{2} - j / \sqrt{2}$ ,0) s₄: ( $1 / \sqrt{2} - j / \sqrt{2}$ ,0) s₅: (0, $1 / \sqrt{2} + j / \sqrt{2}$ ) s₆: (0, $- 1 / \sqrt{2} + j / \sqrt{2}$ ) s₇: (0, $- 1 / \sqrt{2} - j / \sqrt{2}$ ) s₈: (0, $1 / \sqrt{2} - j / \sqrt{2}$ )	$\begin{matrix} c_{1} : & {\begin{matrix} s_{1}, s_{2}, s_{3}, s_{4}, \\ s_{5}, s_{6}, s_{7}, s_{8} \end{matrix}} \\ c_{2} : & {\begin{matrix} (1, 0), \\ (j, 0), \\ (- 1, 0), \\ (- j, 0) \\ \begin{matrix} (0, 1), \\ (0, j), \\ (0, - 1), \\ (0, - j) \end{matrix} \end{matrix}} \end{matrix}$	Polarization-switched QPSK [5] that is rotated by π/4 in the complex plane to obtain the constellation set (similar to CM-PS-BPSK)
4D-CM-BPSK	BPSK	s₁: (1,0) s₂: (−1,0)	$\begin{matrix} \begin{matrix} c_{1} : \\ c_{2} : \end{matrix} & \begin{matrix} {s_{1}, s_{2}} \\ {(j, 0), (- j, 0)} \end{matrix} \\ \begin{matrix} c_{3} : \\ c_{4} : \end{matrix} & \begin{matrix} {(0, 1), (0, - 1)} \\ {(0, j), (0, - j)} \end{matrix} \end{matrix}$	BPSK only in one polarization at a time, the constellation set is obtained by rotating the BPSK constellation by 90° in the complex plane and/or switching the other polarization of the signal, see Fig. 9

	Without Constellation Modulation		With Constellation Modulation
Const.	SNR [dB]	SE [bits/s/Hz/pol]	Constellation Modulation Format	Const. set size K	Meta-symbol length L	SNR [dB]	SE [bits/s/Hz/pol]	Bitrate increase
PM-BPSK	6.8	1	CM-PM-BPSK	2	2	7	1.25	25%
PS-BPSK	7.1	1	CM-PS-BPSK	2	2	7.3	1.25	25%
PS-QPSK	7.6	1.5	CM-PS-QPSK	2	5	7.8	1.6	7%
Single-pol BPSK	6.8	0.5	4D-CM-BPSK	4	4	4	0.75	50%

	Without Constellation Modulation		With Constellation Modulation
Const.	SNR [dB]	SE [bits/s/Hz/pol]	Constellation Modulation Format	Const. set size K	Meta-symbol Length L	SNR [dB]	SE [bits/s/Hz/pol]	Bitrate increase
PM-BPSK	3.2	1	CM-PM-BPSK	2	3	3.3	1.16	16%
PS-BPSK	3.9	1	CM-PS-BPSK	2	2	4	1.25	25%
PS-QPSK	4.7	1.5	CM-PS-QPSK	2	7	4.8	1.57	4.7%
Single-pol BPSK	3.2	0.5	4D-CM-BPSK	4	5	0.3	0.7	40%

Constellation modulation – an approach to increase spectral efficiency

Abstract

1. Introduction

2. Concept of constellation modulation

3. Implementation of constellation modulation

4. Achievable spectral efficiency increase by constellation modulation

5. Assessment of constellation modulation for different modulation formats

Single polarization signals

Dual-polarization signals

6. Experiment

7. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (14)

Tables (5)

Equations (8)

Optics Express