Adaptive constellation diagram reducing the impact of phase distortions

Abstract

Disclosed herein is a method of encoding and/or decoding data for optical data transmission along a transmission link, as well as corresponding transmitters and receivers. The data is encoded based on an adaptive constellation diagram in a 2-D plane, said constellation diagram including a first and a second pair of symbols, wherein the symbols of the first pair of symbols are located at opposite sides of the origin of the 2-D plane at a first distance di from each other, and wherein the symbols of the second pair of symbols are located at opposite sides of the origin of the 2-D plane at a second distance d2 from each other. The method comprises a step of adapting the constellation diagram by varying the ratio of the first and second distances d.sub.1, d.sub.2 such as to minimize or nearly minimize a bit error rate in the transmitted data.

Claims

1. A method of processing data for optical data transmission along a transmission link, the method comprising the steps of: encoding the data based on an adaptive constellation diagram in a 2-D plane, the adaptive constellation diagram including at least a first pair of symbols and a second pair of symbols, wherein respective symbols of the first pair of symbols are located at opposite sides of an origin of the 2-D plane at a first distance d.sub.1 from each other, wherein respective symbols of the second pair of symbols are located at opposite sides of the origin of the 2-D plane at a second distance d.sub.2 from each other, wherein a line in the adaptive constellation diagram connecting the first pair of symbols is perpendicular to a line in the adaptive constellation diagram connecting the second pair of symbols, adapting the adaptive constellation diagram for data transmission by varying a first ratio of the first distance d.sub.1 and the second distance d.sub.2 to minimize or nearly minimize a bit error rate in the transmitted data, wherein the first distance d.sub.1 does not equal the second distance d.sub.2 in the first ratio once the adaptive constellation diagram for data transmission has been adapted according to the adapting step to minimize or nearly minimize the bit error rate in the transmitted data, wherein the first ratio is varied in the adapting step to a degree that is determined based on information corresponding to: a second ratio of the amount of a phase noise and an amount of an additive noise to which an optical data transmission along the transmission link is subject, the phase noise characterized by a random rotation of at least one constellation point location with respect to the origin of the 2-D plane, and the additive noise characterized by at least one random isotropic shift of at least one constellation point in the 2-D plane.

2. The method of claim 1, wherein the step of adapting the adaptive constellation diagram for data transmission by varying the first ratio comprises scaling a base distance d.sub.0 with corresponding scaling factors α, β according to
d.sub.1=d.sub.0.Math.α, and
d.sub.2=d.sub.0.Math.β, so as to keep constant a time-averaged power of the optical data transmission, in particular by providing for α.sup.2+β.sup.2=2.

3. The method of claim 1, wherein the optical data transmission comprises transmission, in a same direction, of both data signals and a pump wavelength.

4. The method of claim 1, wherein at least one of the amount of the phase noise, the amount of the additive noise, the information corresponding to the second ratio, is estimated based on a statistical analysis of at least one of sample points of data signals received at a receiving side of the transmission link, sample points of data signals of a reverse data transmission received at a transmitting side of the transmission link.

5. The method of claim 4, wherein the information corresponding to the second ratio corresponds to an asymmetry in a statistical distribution of sample points in a direction parallel to a line in the adaptive constellation diagram interconnecting two adjacent non-adjusted symbols in the 2-D plane.

6. The method of claim 4, wherein at least one of the amount of the additive noise, the amount of the phase noise, is estimated based on a standard deviation of a distribution of sample points in the 2-D plane.

7. The method of claim 4, wherein the amount of the additive noise is estimated based on a distribution of respective distances of sample points from the origin of the 2-D plane.

8. The method of claim 7, wherein the amount of the phase noise is estimated based on a distribution of angular offsets of constellation points assigned to a same symbol, while accounting for a contribution of additive noise to the distribution of angular offsets, and wherein a standard deviation of the phase noise σ.sub.phase is estimated as ${\sigma }_{phase}=\sqrt{{\sigma }_{\mathrm{angle}}^2-a{\cos \left(1-\frac{1}{2}{\left(\frac{{\sigma }_{\mathrm{radius}}}{R_{\mathrm{symbol}}}\right)}^2\right)}^2},\mathrm{or}{\sigma }_{phase}=\sqrt{{\sigma }_{\mathrm{angle}}^2-{\left(\frac{{\sigma }_{\mathrm{radius}}}{R_{\mathrm{symbol}}}\right)}^2},\mathrm{or}{\sigma }_{phase}=\sqrt{{\sigma }_{\mathrm{angle}}^2-a{\tan \left(\frac{{\sigma }_{\mathrm{radius}}}{R_{\mathrm{symbol}}}\right)}^2},$ wherein R.sub.symbol denotes the respective distance of a given one of the sample points from the origin of the 2-D plane, and a tan represents an inverse tangent function and a cos represents an inverse cosine function.

9. The method of claim 4, wherein the amount of the phase noise is estimated based on a distribution of angular offsets of constellation points assigned to a same symbol, while accounting for a contribution of additive noise to the distribution of angular offsets.

10. The method of claim 1, wherein the 2-D plane is a complex plane, and each constellation point Z.sub.symbol of each of the first pair of symbols and the second pair of symbols can be expressed as one of
Z.sub.symbol=α.Math.(X.sub.0+i.Math.Y.sub.0),with X.sub.0,Y.sub.0∈{−1,1},
Z.sub.symbol=β(X.sub.0+i.Math.Y.sub.0),with X.sub.0,Y.sub.0∈{−1,1}, up to a common normalization factor and a common arbitrary phase factor, wherein the additive noise is defined as Z.sub.add as follows:
Z.sub.add=n.sub.x+i.Math.n.sub.y, wherein n.sub.x and n.sub.y are random variables distributed according to a same statistical distribution, wherein the phase noise is represented by a factor
exp{i.Math.φ}, wherein φ is a random variable having a corresponding statistical distribution, wherein a given sample point Z originating from a given constellation point Z.sub.symbol subjected to the additive noise and the phase noise is a random variable defined as
Z=Z.sub.symbol.Math.exp{i.Math.φ}+Z.sub.add.

11. A transmitter for transmitting an optical data signal along a transmission link, comprising an encoding unit configured for encoding data to be transmitted by the optical data signal, the encoding being based on an adaptive constellation diagram in a 2-D plane, the adaptive constellation diagram including at least a first pair of symbols and a second pair of symbols, wherein respective symbols of the first pair of symbols are located at opposite sides of an origin of the 2-D plane at a first distance d.sub.1 from each other, wherein respective symbols of the second pair of symbols are located at opposite sides of the origin of the 2-D plane at a second distance d.sub.2 from each other, wherein a line in the adaptive constellation diagram connecting the first pair of symbols is perpendicular to a line in the adaptive constellation diagram connecting the second pair of symbols, wherein the encoding unit is further configured for adapting the adaptive constellation diagram for data transmission by varying a first ratio of the first distance d.sub.1 and the second distance d.sub.2 to minimize or nearly minimize a bit error rate in the transmitted data, wherein the first distance d.sub.1 does not equal the second distance d.sub.2 in the first ratio of the adaptive constellation diagram for data transmission once adapted by the encoding unit to minimize or nearly minimize the bit error rate in the transmitted data, wherein the varying of the first ratio is to a degree that is determined based on information corresponding to a second ratio of the amount of a phase noise and an amount of an additive noise to which an optical data transmission along the transmission link is subject, the phase noise characterized by a random rotation of at least one constellation point location with respect to the origin of the 2-D plane, and the additive noise characterized by at least one random isotropic shift of at least one constellation point in the 2-D plane.

12. The transmitter of claim 11, wherein the varying of the first ratio comprises scaling a base distance d.sub.0 with corresponding scaling factors α, β according to
d.sub.1=d.sub.0.Math.α, and
d.sub.2=d.sub.0.Math.β, so as to keep constant a time-averaged power of the optical data transmission, in particular by providing for α.sup.2+β.sup.2=2.

13. A system comprising: a transmitter configured to transmit an optical data signal along a transmission link, and a receiver configured to receive the optical data signal transmitted along the transmission link, wherein the transmitter comprises an encoding unit configured for encoding data to be transmitted by the optical data signal, the encoding being based on an adaptive constellation diagram in a 2-D plane, the adaptive constellation diagram including at least a first pair of symbols and a second pair of symbols, wherein respective symbols of the first pair of symbols are located at opposite sides of an origin of the 2-D plane at a first distance d.sub.1 from each other, wherein respective symbols of the second pair of symbols are located at opposite sides of the origin of the 2-D plane at a second distance d.sub.2 from each other, wherein a line in the adaptive constellation diagram connecting the first pair of symbols is perpendicular to a line in the adaptive constellation diagram connecting the second pair of symbols, wherein the encoding unit is further configured for adapting the adaptive constellation diagram for data transmission by varying a first ratio of the first distance d.sub.1 and the second distance d.sub.2 to minimize or nearly minimize a bit error rate in the transmitted data, wherein the first distance d.sub.1 does not equal the second distance d.sub.2 in the first ratio of the adaptive constellation diagram for data transmission once adapted by the encoding unit to minimize or nearly minimize the bit error rate in the transmitted data, wherein the varying of the first ratio is to a degree that is determined based on information corresponding to a second ratio of the amount of a phase noise and an amount of an additive noise to which an optical data transmission along the transmission link is subject, the phase noise characterized by a random rotation of at least one constellation point location with respect to the origin of the 2-D plane, and the additive noise characterized by at least one random isotropic shift of at least one constellation point in the 2-D plane, wherein the receiver comprises a decoding unit configured for decoding the received optical data signal based on the adaptive constellation diagram.

Description

SHORT DESCRIPTION OF THE FIGURES

(1) FIG. 1 shows a distribution probability of data samples resulting from constellation points being subjected to additive noise and phase noise according to a default constellation diagram.

(2) FIG. 2 is a diagram illustrating how the contribution of the additive noise to the distribution of angular offsets can be estimated.

(3) FIG. 3 shows the distribution of data samples resulting from constellation points being subjected to additive noise and phase noise according to an adjusted constellation diagram.

(4) FIG. 4 shows similar constellation diagrams as FIGS. 1 and 3 as well as the corresponding decision regions and the distribution of radii.

(5) FIG. 5 shows an achievable BER as a function of scaling factor α for different values of the ratio of standard deviations.

(6) FIG. 6 shows the optimum scaling factor α as a function of the ratio of standard deviations of phase noise and additive noise.

(7) FIG. 7 shows histograms of the probability density of the radii and of the angular offsets of sample points included in a data signal transmitted via a transmission link.

(8) FIG. 8 shows the estimated ratio of the standard deviations of phase noise and additive noise as a function of BER.

(9) FIG. 9 shows two parallel transmission links connecting locations A and B, in which the optimum scaling factor α is transmitted from location B to location A via a supervisory channel.

(10) FIG. 10 shows the same two parallel transmission links as FIG. 9, but without communication of the value a via the supervisory channel.

(11) FIG. 11 shows in the left column probability distributions of sample points for varying degrees of phase noise, in the middle column the real parts of the sample points and in the right: the imaginary part of the sample points. In the diagrams of the second and third columns, a dashed curve is added which represents a mirrored version of the solid curve, with a vertical mirror axis located at the maximum of the respective distribution.

(12) FIG. 12 shows an adjustable constellation diagram including 16 symbols.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

(13) For the purposes of promoting an understanding of the principles of the invention, reference will now be made to a preferred embodiment illustrated in the drawings, and specific language will be used to describe the same. It will nevertheless be understood that no limitation of the scope of the invention is thereby intended, such alterations and further modifications in the illustrated method and apparatus and such further applications of the principles of the invention as illustrated therein being contemplated as would normally occur now or in the future to one skilled in the art to which the invention relates.

(14) FIG. 1 shows an ordinary normalized QPSK constellation diagram in the complex plane. The constellation diagram comprises a first pair of symbols located at (−1, −i) and (1, i); and a second pair of symbols (−1, i) and (1, −i). Accordingly, each symbol can be represented by a complex number Z.sub.symbol=X.sub.0+i.Math.Y.sub.0, with X.sub.0, Y.sub.0 ∈{−1, 1}.

(15) Upon transmission along a transmission link, the symbols included in the data signals will be subjected to additive noise and phase noise. As already explained, the locations of the symbols emitted by a transmitter in the constellation diagram are called constellation points in the following, whereas the term sample points is used for the locations of the data samples detected by the receiver. Additive noise is characterized by random isotropic shifts of constellation points in the 2-D plane with regard to the true symbol locations. Phase noise is characterized by a random rotation of the constellation point locations with respect to the origin of the complex plane with regard to the true symbol location.

(16) In mathematical terms, additive noise Z.sub.add may be defined as Z.sub.add=n.sub.x+i.Math.n.sub.y, wherein n.sub.x and n.sub.y are random variables distributed according to a same statistical distribution, which in the following explanations will be assumed to be a Gaussian distribution. The phase noise may be represented by a factor exp{i.Math.φ}, wherein φ is a random variable having a corresponding statistical distribution, which in the following is likewise assumed to be a Gaussian distribution. Accordingly, the constellation points Z.sub.symbol subjected to additive noise and phase noise, or in other words, the sample points actually received after the transmission along the transmission link, can be statistically described by a random variable defined as Z=Z.sub.symbol exp{i.Math.φ}+Z.sub.add.

(17) FIG. 1 shows the statistical distribution of the random variable Z, or, in other words, of the received sample points. In absence of phase noise, the width of the distribution is represented by the circles to around each of the true symbol locations Z.sub.symbol, where the radius of the circle to corresponds to the standard deviation of the Gaussian additive noise. The standard deviation of the additive noise is also referred to as σ.sub.radius, as it corresponds to the standard deviation of the distribution of radii of the sample points in the constellation diagram, wherein the “radius” is the distance of a sample point from the origin of the complex plane (not to be confused with the radius of circle 10). The distance between two constellation points of the same pair is d.sub.0 and corresponds to 2.Math.√{square root over (2)} in the normalized constellation diagram of FIG. 1. The radius of the circle 10 corresponding to σ.sub.radius implies that under the additive noise only, 39.35% of all sample points will lie within the circle 10. However, due to the phase noise, the distribution of the sample points is further spread out in circumferential direction as indicated from the elongate shapes 12 in FIG. 1. This aspect is illustrated by the contour lines connecting locations with constant probability density.

(18) The angle σ.sub.angle shown in FIG. 1 corresponds to the standard deviation of the distribution of angular offsets of sample points with regard to the true angular location of the symbol. Note that σ.sub.angle is not the same as the standard deviation of the phase noise, σ.sub.phase, because the additive Gaussian noise likewise leads to an angular spread. As illustrated on the left of FIG. 2, the angular offset 13 can be decomposed into two contributions, namely variations of the phase due to phase noise 14 and contributions from additive noise 15. As illustrated on the right, the standard deviation σ.sub.phase of the phase noise corresponds to good approximation to the length of the arc 14 indicated in FIG. 2, divided by the radius R.sub.symbol, which in the normalized diagram of FIG. 1 corresponds to √{square root over (2)}, bearing in mind that all angles are expressed in radians. In view of the geometry shown in FIG. 2, and under the assumption that the phase contributions from additive noise and phase noise can be considered as uncorrelated random variables whose variances sum up to the overall variance, the following approximate expressions for the standard deviation σ.sub.phase of the phase noise may be considered:

(19) $\begin{array}{cc}{\sigma }_{phase}=\sqrt{{\sigma }_{\mathrm{angle}}^2-a{\cos \left(1-\frac{1}{2}{\left(\frac{{\sigma }_{\mathrm{radius}}}{R_{\mathrm{symbol}}}\right)}^2\right)}^2}& \left(1\right)\end{array}$
where “a cos” represents the inverse cosine function, or

(20) $\begin{array}{cc}{\sigma }_{phase}=\sqrt{{\sigma }_{\mathrm{angle}}^2-{\left(\frac{{\sigma }_{\mathrm{radius}}}{R_{\mathrm{symbol}}}\right)}^2}.& \left(2\right)\end{array}$

(21) In particular in the presence of small phase variations, an improved estimate at the expense of increased mathematical complexity can be obtained with the equation

(22) $\begin{array}{cc}{\sigma }_{phase}=\sqrt{{\sigma }_{\mathrm{angle}}^2-a{\tan \left(\frac{{\sigma }_{\mathrm{radius}}}{R_{\mathrm{symbol}}}\right)}^2}& \left(3\right)\end{array}$
where “a tan” stands for the inverse tangent function.

(23) FIG. 3 shows a modified constellation diagram as employed in the encoding/decoding method of the invention. As is seen in FIG. 3, a first distance d.sub.1 between the first pair of symbols originally located at (−1, −i) and (1, i) is decreased with regard to d.sub.0 by a scaling factor α<1, while the second distance d.sub.2 between the second pair of symbols originally located at (−1, i) and (1, −i) is increased by a scaling factor β>1. Herein, the scaling factors α and β are chosen such as to obey the condition α.sup.2+β.sup.2=2, such as to keep the time-averaged power of the signal unaffected by the distance variations. The centrically symmetrical placement of symbols belonging to the same pair of symbols to the origin is not affected by this variation.

(24) With reference to FIG. 4, it is demonstrated how this modification of the constellation diagram leads to a decrease of the bit error rate. On the left side of FIG. 4, a spread out distribution of sample points in an ordinary QPSK constellation diagram subjected to additive noise and phase noise is shown. The alternating light and shaded regions indicate decision regions, meaning that whenever a sample point falls into this region, it is assigned to the symbol located in the region. In case of FIG. 4, the decision regions are simply the quadrants of the coordinate system. Further shown on the left part of FIG. 4 is the probability distribution of the radii of the sample points.

(25) The right part of FIG. 4 shows a spread out distribution of sample points using the modified constellation diagram of the invention. Due to the modification of the constellation diagram, the configurations of the decision regions likewise change. The boundaries of the decision regions are lines where the probability that a sample point belongs to one of the adjusted symbols is the same. Serving as an example, a circle 16 is marking in each case a location on the border between two decision regions. As can be seen by comparison of the left and right parts of FIG. 4, due to the modification of the constellation diagram, the probability of a sample point lying on the border is significantly decreased, which allows for less errors in the assignment of sample points to the symbols, or, in other words, a reduced BER.

(26) However, it is seen that whether and to which extent such a modification of the constellation diagram leads to an improvement depends critically on the nature of the noise encountered. FIG. 5 shows the BER versus the scaling factor α for different ratios of standard deviations σ.sub.phase/σ.sub.radius. In each case, the total noise is chosen such that without modification, i.e. at α=1, the BER is 10.sup.−2. As can be seen from FIG. 5, in case of no or very little phase noise, shifting of the first pair of constellation points closer together only leads to a continuous performance degradation. However, a significant improvement can be achieved with increasing amounts of phase noise.

(27) FIG. 6 shows the relationship between the ratio of standard deviations σ.sub.phase/σ.sub.radius and the shift of the constellation points (the scaling factor α) that leads to optimum transmission performance. It is seen from FIG. 6 that in a wide range of parameter values, the shift providing optimal performance decreases with growing ratio σ.sub.phase/σ.sub.radius. This plot clearly demonstrates that the applied shift of the constellation points should indeed be adapted to the ratio of the two noise contributions, and further precisely defines how it should be chosen.

(28) As explained above, the amount of phase noise encountered upon transmission along the transmission link, and more precisely the ratio of the amount of phase noise with regard to the amount of additive noise, can be estimated based on a statistical analysis of the deviation of the sample points of received data signals from the true symbol location. FIG. 7 shows on the left a histogram of the radii of sample points, from which it can be seen that the radii obey a Gaussian-like distribution centered at √{square root over (2)}, where the radius of a sample point is again its distance from the origin. The fact that the radii are statistically distributed rather than all precisely located at a value of √{square root over (2)} is due to the additive noise explained above.

(29) On the right of FIG. 7, a histogram of the angular offset of the sample points in the received signal from the phase of the true symbol is shown, which again amounts to a Gaussian distribution. As explained above, the width of the angular distribution or phase distribution is only partly due to phase noise, but is also due to additive noise.

(30) FIG. 8 shows the estimated value of the ratio of the standard deviations σ.sub.phase/σ.sub.radius versus bit error rate obtained by using equation (2), for different ratios of actual standard deviations that were used in simulating the noise. The actual ratio of standard deviations is constant for all data points on a curve. As is seen from FIG. 7, the estimated values are in good agreement with the actual ratio for BERs lower or equal to 10.sup.−2. Accordingly, as long as the overall noise is within reasonable limits allowing for a BER of 10.sup.−2 or below, the ratio of standard deviations σ.sub.phase/σ.sub.radius can be estimated with good precision from the statistical analysis of the received symbols.

(31) FIG. 9 shows two parallel transmission links 20 connecting locations A and B. Each transmission link 20 comprises optical fibers 22 and optical amplifiers 24, such as erbium-doped fiber amplifiers (EDFA). Further included in each link 20 is a pump source 26 for counterdirectional Raman pumping, and a pump source 28 for codirectional Raman pumping. At the transmitting side of each transmission link 20, a multiplexer 30 is provided, and at the receiving side of each transmission link 20, a demultiplexer 32 is provided. At each of locations A and B, a number of transponders 34 is provided, of which only two are shown at each location in FIG. 9. Each transponder comprises a transmitter and a receiver. The number of transponders 34 corresponds to the number of wavelength division multiplex channels (WDM) transmitted over the transmission links 20, but in FIG. 9, only two of such transponders 34 are shown at each end location A, B.

(32) Transponders are typically realized on cards that can be placed in a subrack of the communication equipment. Each card provides a data stream on one wavelength and receives another data stream on the same wavelength. However, transponders providing the functionality of several transmitters and receivers on a single card are known. Within this description, the term “transponder” is used for referring to the functionality of a transponder rather than designating a specific physical entity. In optical communications, sometimes a distinction is made between transceivers and transponders. Despite the fact that different definitions are used among experts, the addressed differences are not of relevance for the invention. Therefore, the term “transponder” as used in the following comprises transponders and transceivers.

(33) Each transponder 34 is adapted for coherent receiving of data signals, and for generating optical data signals using the coding scheme with the adaptive constellation diagram according to the present invention. In the situation depicted in FIG. 9, the rightmost transponder 34 at location B receives a data signal over the lower transmission link 20 and carries out a statistical analysis of the deviations of sample points in the received data signal as compared to the true symbol locations, and from this analysis determines the ratio of standard deviations σ.sub.phase/σ.sub.radius in the manner described above. Based on this ratio, an optimum scaling factor α can be determined from a suitable lookup table representing the relationship for the example shown in FIG. 6. This optimum scaling factor α is then transmitted to location A by means of a supervisory channel indicated by reference sign 36, and is further communicated to the rightmost transponder 34 at location A and used for the coding by this transponder, or by all of the transponders at location A, assuming that the ratio of phase noise and additive noise is identical for different wavelengths. The scaling factor α is likewise stored at the transponders 34 at location B, because it is of course also used for adapting the constellation diagram employed in decoding the data signals received at location B. In the same way, the optimum value of the scaling factor α may be determined from data signals received at the transponders 34 at location A via the upper transmission link 20 of FIG. 9, and this scaling factor α is communicated to the transponders at location B via a further supervisory channel (not shown). Of course, scaling parameter values might also be transmitted via the overhead sections of the data streams. However, this might require scanning the allowed parameters range for a suitable setting when installing a system for the first time.

(34) Alternatively, statistical data or the ratio of standard deviations can be communicated instead of the optimum scaling factor to the transmitter at location A for determining the optimum scaling factors. However, this solution requires to transmit the determined optimum scaling factor to the receiver at location B by a complementary supervisory channel transmitting data in opposite direction, e.g. from location A to location B.

(35) In the embodiment shown in FIG. 9, at each location A and B a scaling factor α is hence determined based on the statistical analysis of the data signals received, and this factor α is forwarded to the respective other location to be used in encoding data to be transmitted. However, since the two transmission links 20 are very similar, in practice it will often turn out that the ratio of phase noise versus additive noise will be very similar as well, such that the two respective values of a will be the same or almost the same. Accordingly, instead of forwarding the determined value of a to the other side for the purposes of encoding with an adapted constellation diagram, it is also possible that each side uses the locally determined value of a for its own encoding purposes. This is schematically shown in FIG. 10, which shows the same two transmission links 20 as FIG. 9, but which dispenses with the communication of the value of a via the supervisory channel. Instead, the value of a determined in the analysis in the leftmost transponder 34 at location A is also used for adaption of the constellation diagram for the purposes of encoding signals to be transmitted from location A to location B.

(36) Instead of directly determining the ratio of the amount of phase noise with regard to the amount of additive noise, it is likewise possible to base the adjustment of the constellation diagram on further information related to this ratio. An example of this is shown with reference to FIG. 11. In the left column of FIG. 11, the probability distribution of sample points is shown, where the ratio of phase noise to additive noise increases from top to bottom. The middle column shows with the solid line the distribution of the real parts (Re(Z)) of the sample points, and the right column shows the distribution of the imaginary parts (Im(Z)) of the sample points. In the diagrams of the second and third columns, a dashed curve has been added which represents a mirrored version of the solid curve, with a vertical mirror axis located at the maximum of the respective distribution. Deviations of the dashed and solid curves hence show an asymmetry in the distribution. As is seen from FIG. 11, this asymmetry clearly increases with an increasing ratio of phase noise. Accordingly, the asymmetry, or any parameter reflecting the asymmetry can be regarded as “information related to the ratio of the amount of phase noise with regard to the amount of additive noise”, and the adjustment of the constellation diagram may be based on this information as well.

(37) While in the previous description reference has only been made to constellation diagrams including two pairs of symbols, this shall not rule out that the constellation diagram includes more than these two pairs of symbols. FIG. 12 shows an example of an adjustable constellation diagram including eight pairs of symbols, which in the default condition, i.e. without variation of the distances or radii, are located on three different radii. In this constellation diagram, the two innermost pairs of symbols resemble the two pairs of symbols referred to in the summary of the invention. In FIG. 12, the default positions of the symbols are represented by the light symbols, while the shifted positions are indicated by the shaded symbols. As is seen in FIG. 12, the outer six pairs of symbols can be shifted in a similar manner as the innermost two pairs of symbols, with similar effects on an improved bit error rate.

(38) Although a preferred exemplary embodiment is shown and specified in detail in the drawings and the preceding specification, these should be viewed as purely exemplary and not as limiting the invention. It is noted in this regard that only the preferred exemplary embodiment is shown and specified, and all variations and modifications should be protected that presently or in the future lie within the scope of protection of the invention as defined in the claims.

REFERENCE SIGNS

(39) 10 circle indicating standard deviation of sample distribution due to Gaussian additive noise 12 elongate shape indicating sample distribution due to Gaussian additive noise and Gaussian phase noise 13 angular offset 14 arc representing standard deviation of Gaussian phase noise 15 arc representing contribution to angular sample distribution due to additive Gaussian noise 16 circle marking location on the border between two decision regions 20 transmission link 22 optical fiber 24 optical amplifier 26 pump source for counterdirectional Raman pumping 28 pump source for codirectional Raman pumping 30 multiplexer 32 demultiplexer 34 transponder 36 supervisory channel