Patent Description:
The capacity of an Additive White Gaussian Noise (AWGN) channel can be modeled by the Shannon-Hartley theorem: <MAT> where.

The ratio S/N is the signal to noise ratio.

Shannon's capacity limit is approached when a stream of symbols of a constellation, having undergone Probabilistic Amplitude Shaping, are transmitted to a modulator for modulation onto a carrier. In Probabilistic Amplitude Shaping, a frequency of occurrence of symbols in the stream is dependent upon a probability-amplitude distribution for symbols of the constellation. The probability-amplitude distribution is or is close to a Gaussian distribution to maximize capacity. The shape of the probability-amplitude distribution causes low-amplitude symbols to occur more frequently on average in the stream than high-amplitude symbols. At high signal to noise ratios, a significant gain can be achieved by shaping of the probability-amplitude distribution compared to equally probable amplitudes.

<CIT> discloses a transmitter that uses at least first and second fixed constellations to generate an outgoing data frame for each constellation, with the lower overall transmit energy data frame selected for transmission. It discloses an example transmitter being implemented with relatively low complexity by employing constellation mappers and demappers, that operate using relatively small look-up tables. It relates to frequency distributions that can be approximated by a Gaussian function.

Technical paper "<NPL>) provides a theoretical analysis of the effect of constellation shaping in the transmitter on multimodulus blind equalization in a receiver (abstract). It teaches that convergence failure occurs as the kurtosis of the constellation approaches that of a Gaussian source.

Neither document discloses the use of 64QAM, and as a consequence, neither document discloses a kurtosis target associated with 64QAM.

The invention is defined and limited by independent claim <NUM> which provides an apparatus for performing modulation onto a carrier following a probability amplitude distribution. Further embodiments are provided by the dependent claims.

illustrates an example of a communication system <NUM>. In this example, the system <NUM> is an optical communication system. In some, but not necessarily all, examples, it is a high speed coherent optical communication system. In this context high speed means an information rate of ><NUM> Gb/s or, in some circumstances, ><NUM> Tb/s.

A transmitter <NUM> is configured to transmit a modulated carrier signal <NUM> through an optical channel <NUM> to a receiver <NUM>. A stream of symbols has been modulated onto a carrier signal by a modulator <NUM> to form the modulated carrier signal <NUM>.

In this example, quadrature amplitude modulation is used to modulate the carrier signal. In quadrature amplitude modulation (QAM), an ordered group of N binary bits is converted to a unique combination of phase shift and amplitude. This unique combination can be represented as a point (X, Y) in an Argand diagram. A constellation diagram is an Argand diagram that represents the location, that is the constellation point (X, Y) of each modulation symbol. The set of M symbols for a given modulation is a modulation alphabet, where M=<NUM>N. A particular alphabet M is used for M-QAM modulation and this defines a constellation of M distinct points each of which represents a symbol. The modulator <NUM> modulates a stream of symbols of the constellation onto the carrier signal.

The transmitter <NUM> comprises a probabilistic amplitude shaping (PAS) symbol source circuit <NUM>. This circuit receives a sequence of bits <NUM> and produces a sequence of modulation symbols <NUM>. The produced symbols <NUM> have probabilistic amplitude shaping. The symbols produced are multi-level symbols which have multiple different amplitudes. The PAS symbol source circuit <NUM> performs probabilistic selection of the modulation symbols from a modulation symbol alphabet. Constellation points with higher amplitude are selected less frequently than constellation points with lower amplitude. A frequency of occurrence of symbols in the stream of symbols <NUM> produced by the PAS symbol source circuit <NUM> is dependent upon a probability-amplitude distribution for symbols of the constellation. The probability-amplitude distribution is or is close to a Gaussian distribution to maximize capacity. The shape of the probability-amplitude distribution causes low-amplitude symbols to occur more frequently on average in the stream <NUM> than high-amplitude symbols. At high signal to noise ratios, a significant gain can be achieved by this shaping of the probability-amplitude distribution compared to equally probable amplitudes.

The stream of symbols <NUM> produced by the PAS symbol source circuit <NUM> is provided to a constellation re-mapper circuit <NUM>. The constellation re-mapper circuit <NUM> is configured to receive a stream of symbols <NUM> of a constellation that has an associated first probability-amplitude distribution and produces an output stream of symbols <NUM> for the same constellation that has an associated second probability-amplitude distribution that is different to the first probability-amplitude distribution. The constellation re-mapper circuit <NUM> changes the statistics of the probability-amplitude distribution associated with the symbol stream. The constellation re-mapper circuit <NUM> is configured to lower a kurtosis of the probability-amplitude distribution that determines the frequency of occurrence of symbols in the stream of symbols.

In this example, but not necessarily all examples, the stream of symbols <NUM> from the constellation re-mapper circuit <NUM> is channel coded using a forward error correction (FEC) encoder <NUM>. The channel-encoded symbols are then provided to the modulator circuit <NUM> and are then modulated onto a carrier to produce the modulated carrier signal <NUM> transmitted through the optical channel <NUM>.

The optical channel <NUM> can, in some examples, be an optical fiber that travels over a short distance or a long distance. For example it may be used to span a distance of a few meters in a data center or thousands of kilometers in a transatlantic submarine cable.

The receiver <NUM> is configured to reverse the encoding stages of the transmitter and to recover an estimate <NUM> of the original bits <NUM> that were encoded by the transmitter <NUM>.

The modulated carrier signal <NUM> is received at the receiver <NUM> and is demodulated by demodulator <NUM> to recover the encoded modulation symbols. The encoded modulation symbols are then decoded by a channel decoder, for example a FEC decoder <NUM>, to produce estimates of the transmitted modulation symbols (after re-mapping). The inverse re-mapper circuit <NUM> reverses the mapping applied in the constellation re-mapping circuit <NUM> and the PAS symbol receiver circuit <NUM> performs the inverse process performed by the PAS symbol source circuit <NUM>.

The inverse constellation re-mapper circuit <NUM> is configured, in response to receipt of a stream of symbols of a constellation, to map the stream of symbols to a different stream of symbols of the constellation where the different stream of symbols can have a higher kurtosis than the stream of symbols.

The channel encoder <NUM> is systematic in that information bits are left unchanged by the encoding. The FEC encoder <NUM> adds parity bits only. The FEC encoding does not therefore impact on the kurtosis of the probability-amplitude distribution associated with the transmitted symbols. This is true because the parity bits are equiprobable (equal probability for <NUM> and <NUM>) and they become sign (or quadrant selection) bits for the constellation.

It will therefore be appreciated from the foregoing that <FIG> illustrates an apparatus <NUM> comprising: a probabilistic amplitude shaping symbol source circuit <NUM> configured, in response to receipt of input data <NUM>, to produce a stream of symbols of a constellation for modulation wherein a frequency of occurrence of symbols in the first stream of symbols <NUM> is dependent upon a probability-amplitude distribution for symbols of the constellation, wherein the probability-amplitude distribution is a probability-amplitude distribution that causes low-amplitude symbols to occur more frequently on average in the stream than high-amplitude symbols; and a constellation re-mapper circuit configured, in response to receipt of the first stream of symbols <NUM>, to map the first stream of symbols <NUM> to a second stream of symbols <NUM> of the constellation for modulation and to produce the second stream of symbols <NUM>, wherein the second stream of symbols <NUM> has a lower kurtosis than the first stream of symbols <NUM>.

The system <NUM> is an optical communication system comprising: a transmitter <NUM> configured to transmit a modulated carrier signal <NUM> onto which a stream of symbols of the constellation has been modulated and a receiver comprising: demodulation circuitry <NUM> configured, in response to receipt of the modulated carrier signal <NUM>, to demodulate the modulated carrier signal to obtain a stream <NUM> of symbols of the constellation and use the probability-amplitude distribution for symbols of the constellation to obtain an estimate <NUM> of the input data <NUM>. The receiver <NUM> comprises circuitry <NUM> configured to perform blind equalization to produce a stream of symbols <NUM>, a FEC decoder <NUM> to produce estimates of the transmitted modulation symbols <NUM>, an inverse constellation re-mapper circuit <NUM> configured, in response to receipt of the stream of symbols <NUM> of the constellation, to map the stream of symbols <NUM> to a different stream of symbols <NUM> of the constellation, where the different stream of symbols <NUM> can have a higher kurtosis than the original stream <NUM> of symbols. The receiver <NUM> also comprises circuitry <NUM> for recovering from the stream of symbols <NUM> estimated data bits <NUM>, which estimate the input data bits <NUM>.

The channel demodulator <NUM> performs blind equalization. In blind equalization a transmitted signal is inferred (equalized) from a received signal. In this example, but not necessarily all examples, the demodulator circuit <NUM> performs blind adaptive signal equalization using a least mean squares (LMS) filter. The LMS filter compensates for time-varying linear distortions in the optical channel <NUM> and equalizes the channel frequency response. It performs stochastic gradient descent, for example using the constant modulus algorithm (CMA). The constant modulus algorithm uses a second order cost function that includes a cross-correlation of the input signal and the output signal. The inventors have realized that this cross-correlation can be very small if the input signal is derived from symbols that have a probabilistic amplitude shaped constellation. This is particularly problematic when the frequency of occurrence of symbols in the stream is dependent upon a probability amplitude distribution of symbols of the constellation that is or which approximates to a Gaussian distribution.

The PAS signal source circuit <NUM> can for example use the Gaussian or quasi-Gaussian probability-amplitude distribution for producing the stream of symbols <NUM>. The constellation re-mapper circuit <NUM> reduces the kurtosis of the probability-amplitude distribution and therefore improves the performance of blind equalization in the channel demodulator <NUM>.

Kurtosis is a measure of the "tailedness" of a probability distribution. The standard measure of kurtosis is the fourth standardized moment. The fourth standardized moment is a normalized version of the fourth moment in which the fourth moment has been divided by a fourth power of the standard deviation which renders the standardized scaling variant. A central moment is a moment of a probability distribution of a variable about the variable's mean. It is the expected value of a specified power of the deviation of the variable from the mean. The fourth central moment uses an specified power of <NUM>. The kurtosis can therefore be defined as µ<NUM>/σ<NUM>, where µ<NUM> is the fourth central moment and σ is the standard deviation.

The probabilistic amplitude shaping symbol source circuit <NUM> is configured, in response to receipt of the input data <NUM>, to produce the first stream of symbols <NUM> of the constellation for modulation wherein a frequency of occurrence of symbols in the first stream is dependent upon a probability-amplitude distribution P for symbols of the constellation, wherein the probability-amplitude distribution P is a probability-amplitude distribution that causes low-amplitude symbols to occur more frequently on average in the stream than high-amplitude symbols.

The PAS symbol source circuit <NUM> can be configured to operate in a number of different ways to produce a stream of symbols <NUM> of the constellation, where a frequency of occurrence of symbols in the stream <NUM> is dependent upon a probability-amplitude distribution for symbols of the constellation and a shape of the probability-amplitude distribution causes low-amplitude symbols to occur more frequently on average in the stream than high-amplitude symbols.

For example, the PAS symbol source circuit <NUM> can for example use a shell mapping algorithm or comprise a multi-level distribution matcher or a combination of binary distribution matchers.

There are several algorithms for distribution matching and inverse distribution matching. The algorithm for distribution matching is uniquely reversable, which means that it retrieves the original information sequence from the received one.

For a M-QAM modulation, e.g. <NUM>-QAM, the modulation has an alphabet of size M and a symbol codeword size of N symbol bits where M=<NUM>^N, N=log<NUM>(M). The codeword is a sequence of N binary symbol bits.

In some examples, for example as illustrated in <FIG>, an amplitude distribution matcher <NUM> is used to separately define each bit of a symbol that encodes an amplitude of the symbol. Each symbol bit permutation that defines a different amplitude of a symbol has a different probability. An amplitude distribution matcher <NUM> can, for example, be used to produce a Probabilistically Amplitude Shaped (PAS) symbol with a target Gaussian probability density function.

A sequence of N data bits produces a symbol of size N from an alphabet of size M with the desired probability distribution. The PAS symbol source circuit <NUM> performs the data bit <NUM> to symbol <NUM> mapping.

In some but not necessarily all examples, the PAS symbol source circuit <NUM> is configured to separate a symbol of N symbol bits into a pair of symbol bits defining a quadrant of the Argand diagram and a sequence of N-<NUM> symbol bits defining a position (X,Y) in the Argand diagram. X is a real component of the symbol and Y is an imaginary component of the symbol.

The N-<NUM> symbols bits comprises N/<NUM> -<NUM> symbol bits defining a multi-level amplitude X and N/<NUM> -<NUM> symbol bits defining a multi-level amplitude Y. The multi-level amplitude X is an amplitude shift key in the x-direction defined by a code for X of N/<NUM>-<NUM> bits. The multi-level amplitude Y is an amplitude shift key in the y-direction, which is orthogonal to the x-direction defined by a code for Y of N/<NUM>-<NUM> bits.

An amplitude distribution matcher <NUM> separately controls a probability distribution Px for the combination of N/<NUM> -<NUM> symbol bits defining the multi-level amplitude X, creating a shaped combination of N/<NUM> -<NUM> symbol bits defining the multi-level amplitude X. This converts the code for X to a shaped code for X.

The amplitude distribution matcher <NUM> separately controls a probability distribution Py for the combination of N/<NUM> -<NUM> symbol bits defining the multi-level amplitude Y, creating a shaped combination of N/<NUM> -<NUM> symbol bits defining the multi-level amplitude Y. This converts the code for Y to a shaped code for Y.

The amplitude distribution matcher <NUM> illustrated in <FIG> uses N/<NUM>-<NUM> binary distribution matchers <NUM>. Each binary distribution matcher <NUM> controls a probability for one of the N/<NUM> -<NUM> symbol bits defining the multi-level amplitude (A e.g. X or Y).

For example, a first bit A(<NUM>) produced by a first binary distribution matcher <NUM> will be <NUM> with probability p1 and <NUM> with probability (<NUM>-p1) and a second bit A(<NUM>) produced by a second binary distribution matcher, produces <NUM> with probability p2, <NUM> with probability (<NUM>-p2).

The first bit A(<NUM>) and second bit A(<NUM>) define a code A(<NUM>) A(<NUM>) for the multi-level amplitude A that has been probability amplitude shaped in one dimension.

It will be appreciated that the above procedure is performed for A=X and A=Y, to produce a shaped code for multi-level amplitude X that has been probability amplitude shaped in a first dimension (x) and a shaped code for multi-level amplitude Y that has been probability amplitude shaped in a second dimension (y).

In the original symbol (of N symbol bits) the code for X of N/<NUM>-<NUM> bits is replaced by the shaped code for X of N/<NUM>-<NUM> symbol bits and the code for Y of N/<NUM>-<NUM> bits is replaced by the shaped code for Y of N/<NUM>-<NUM> symbol bits. This produces a shaped symbol of N symbol bits. The <NUM> quadrant bits are thus combined (unchanged) with the shaped code for X of N/<NUM>-<NUM> symbol bits and shaped code for Y of N/<NUM>-<NUM> symbol bits to produce a probabilistic amplitude shaped symbol of N symbol bits.

It will therefore be appreciated that the probabilistic amplitude shaping symbol circuit <NUM> can be configured to separate orthogonal amplitude values (X,Y) from a modulation symbol, to perform probabilistic amplitude shaping independently on each orthogonal amplitude value (X,Y) to create orthogonal probabilistic amplitude shaped amplitude values, and to recombine the orthogonal probabilistic amplitude shaped amplitude values to create a probabilistic amplitude shaped modulation symbol.

The probability-amplitude distribution Px has a shape that causes low-X symbols to occur more frequently on average in a stream than high-X symbols. It may be or may approximate to a Gaussian distribution.

The probability-amplitude distribution Py has a shape that causes low-Y symbols to occur more frequently on average in a stream than high-Y symbols. It may be or may approximate to a Gaussian distribution.

The overall probability-amplitude distribution has a shape that causes low-amplitude symbols to occur more frequently on average in a stream than high-amplitude symbols. It may be or may approximate to a Gaussian distribution.

The PAS symbol source circuit <NUM> can be implemented using hardware, for example as an application specific circuit.

The PAS symbol source circuit <NUM> can be implemented without using a running algorithm but instead using a look-up table.

The constellation re-mapper circuit <NUM> is configured, in response to receipt of a first stream of symbols <NUM>, to map the first stream of symbols <NUM> to the second stream of symbols <NUM> of the constellation for modulation. The second stream of symbols <NUM> has a lower kurtosis than the first stream of symbols <NUM>.

For example, the first stream of symbols <NUM> has an associated probability amplitude distribution that is quasi-Gaussian whereas the second stream of symbols <NUM> has modified symbol level statistics with lower kurtosis.

In the present example, but not necessarily all examples, the constellation re-mapper circuit <NUM> is an Δ-dimensional re-mapper, configured, in response to receipt of the first stream of symbols <NUM> of a constellation, to map each non-overlapping contiguous group of Δ symbols of the first stream of symbols <NUM> to a non-overlapping contiguous group of Δ symbols of the second stream of symbols <NUM> of the constellation, wherein Δ is greater than or equal to <NUM>.

The constellation re-mapper circuit <NUM> is configured to control statistics of the symbol groups rather than independently setting statistics of each symbol.

The multi-dimensional mapper, maps a group of Δ input symbols to Δ output symbols. There are M^Δ symbol combinations, where M is the symbol alphabet size. The larger Δ, the larger number of available combinations, and the more accurate generation of symbol statistics.

The constellation re-mapper circuit <NUM> can be implemented using a look-up table (LUT). A LUT is a block of memory locations indexed by an input address, to produce as an output a value stored at the indexed location.

For a multi-dimensional mapper, the LUT has M^ Δ locations, each storing N bits. The N symbol bits of the input symbol <NUM> identifies a particular constellation M and addresses a particular location in the LUT which returns N symbol bits of the output symbol <NUM>.

The LUT is pre-computed and stored. In some example the LUT is re-programmable. Re-programming can vary the size M^Δ. Alternatively or additionally, re-programming can change the value stored at each of the M^ Δ indexed locations.

The remapping can introduce a correlation between the odd and even symbol generated by the re-mapper. This correlation is an evident proof that re-mapping has been used and can be traced by looking at the statistical properties of the even and odd symbols at the transmitter output.

Kurtosis changes with entropy which is dependent upon information (data) rate. There is therefore a different LUT for each information (data) rate supported.

In some examples, the probabilistic amplitude shaping symbol source circuit <NUM> and the constellation re-mapper circuit <NUM> can be combined into a single circuit, for example, an application specific integrated circuit.

In some examples, the probabilistic amplitude shaping symbol source circuit <NUM>, the constellation re-mapper circuit <NUM> and the channel coder <NUM> can be combined into a single circuit, for example, an application specific integrated circuit.

In some examples, the probabilistic amplitude shaping symbol source circuit <NUM>, the constellation re-mapper circuit <NUM> and the modulator <NUM> (and optionally the channel coder <NUM>) can be combined into a single circuit, for example, an application specific integrated circuit.

The constellation re-mapper circuit <NUM> is used to reduce the kurtosis of all symbols with the smallest possible penalty in capacity.

Constrained optimization is used to create the mapping used by constellation re-mapper circuit <NUM>. The optimization seeks to maximize capacity while meeting a target level for kurtosis.

A search for the multidimensional map uses an algorithm designed to find the reversible permutation of M^ Δ elements which when applied to the input M symbols statistic gives a new M symbols statistic with the target kurtosis and minimum loss of information. The input of concatenated codeword of size N* Δ (alphabet size M^ Δ) to the map will produce an output concatenated codeword of size N* Δ.

This problem can be formulated as a global optimization problem in a large discrete search space, which counts ( <MAT>) possible combinations of reversible permutations. Several heuristic/metaheuristic algorithms exist to approximate global optimization that can be used for this task; for instance, the gradient descend or genetic algorithms can be used.

In one example, the algorithm starts from an initial M^ Δ map and applies a first permutation of k terms, generating randomly a new candidate map. For instance, the new candidate map only differs from the previous map for k=<NUM> terms.

Then the statistics of the symbols generated applying the new candidate map is computed. This includes the computation of the kurtosis and the capacity of the new candidate map after the permutation. Additional (or alternative) figures of merit can be used, such as the constellation energy.

The algorithm calculates using a cost function a single numerical term which represent the 'quality' of the new candidate map. The cost function has as variable input parameters the computed kurtosis of the candidate map and a computed capacity of the candidate map. For example, the cost function could be a weighted combination of a squared difference between actual kurtosis and a target kurtosis and a difference between actual capacity and maximum or target capacity.

If the candidate map improves over the currently preferred candidate map (the quality value of the new candidate map is higher) then the candidate map becomes the currently preferred candidate map otherwise it is rejected.

These steps are repeated until the stopping criteria are satisfied. This occurs when the permuted symbols of the preferred exhibit the target kurtosis and the desired capacity.

To allow faster convergence of the algorithm search for the desired solution, probabilistic techniques can be applied. In this case, the algorithm randomly selects a solution close to the current one, measures its quality, and then decides to move to it or to stay with the current solution based on either one of two probabilities between which it chooses on the basis of the fact that the new solution is better or worse than the current one.

The map can be found using a computer simulation with a symbolic and numerical language (for instance, MATLAB), which emulate the entire transmission chain, including noise. In this scenario the statistics can be estimated using symbolic mathematical tools.

The target kurtosis value for 64QAM is lower or equal to <NUM>.

The kurtosis target value can, for example, be selected based upon the alphabet of the modulation and selected for improved blind equalization at the receiver. In some examples, the probability-amplitude distribution is kurtosis-constrained having a kurtosis conditioned for blind equalization and a shape of the probability-amplitude distribution causes low-amplitude symbols to occur more frequently on average in the stream than high-amplitude symbols.

It will be appreciated from the foregoing that the apparatus <NUM> therefore comprises circuitry <NUM>, <NUM> configured, in response to receipt of input data <NUM>, to transmit a stream of symbols <NUM> of a constellation to a modulator <NUM> for modulation onto a carrier wherein a frequency of occurrence of symbols in the stream <NUM> is dependent upon a probability-amplitude distribution for symbols of the constellation, wherein the probability-amplitude distribution has a kurtosis less than a target value and a shape of the probability-amplitude distribution causes low-amplitude symbols to occur more frequently on average in the stream than high-amplitude symbols.

If the rate of the data bits <NUM> is variable, but the modulation rate is fixed, then different data rate will result in different information rates for the channel.

It is possible to transmit at different information rates (data rates) R (number of information bits per symbol) by changing the probability amplitude distribution P and using the same FEC code rate (c)<MAT> where H(P) is the entropy of P expressed in digital bits and m is number of bits per QAM symbol.

As the data rate R decreases, entropy decreases, and the probability amplitude distribution provided by the PAS symbol source circuit narrows increasing kurtosis.

As the data rate R increases, entropy increases, and the probability amplitude distribution provided by the PAS symbol source circuit broadens decreasing kurtosis.

The kurtosis depends on the statistical properties of the PAS symbols <NUM>, such as the signal Entropy H which in turn is a function of the distribution matcher information rate R. In particular, it is expected to increase when the information rate and signal entropy decreases for the same constellation type. Kurtosis is the result of the distribution matcher working at a selected information rate, and thus cannot be changed independently from the information rate, unless with use the constellation re-mapper circuit <NUM>.

In some examples, for example as illustrated in <FIG>, the apparatus <NUM> uses the constellation re-mapping circuit <NUM> irrespective of the data rate.

In one example, for example as illustrated in <FIG>, the stream of symbols <NUM> for transmission has a probability-amplitude distribution that has a kurtosis that is less than that for a probability-amplitude distribution of the symbols output by the PAS symbol source circuit <NUM> and is less than the target kurtosis value.

In other examples, for example as illustrated in <FIG>, the apparatus <NUM> uses the constellation re-mapping circuit <NUM> irrespective of the data rate. In this example, the stream of symbols <NUM> for transmission has a probability-amplitude distribution that has a fixed kurtosis that less than the target kurtosis value for all data rates.

In some examples, for example as illustrated in <FIG>, the apparatus <NUM> uses the constellation re-mapping circuit <NUM> only at lower data rates when a probability-amplitude distribution of the symbols output by the PAS symbol source circuit <NUM> is more than the target kurtosis value.

In one example, for example as illustrated in <FIG>, below a first data rate Ro, the stream of symbols <NUM> for transmission has a probability-amplitude distribution that has a kurtosis that is less than that for a probability-amplitude distribution of the symbols output by the PAS symbol source circuit <NUM> (dotted line) and is less than the target kurtosis value. The kurtosis can for example be at the kurtosis target value at a low data rate and reduce monotonically as the data rate increases towards the first data rate. Above the first data rate remapping is not performed and the stream of symbols <NUM> for transmission are the symbols output by the PAS symbol source circuit <NUM> which at these data rates has a probability-amplitude distribution with a kurtosis less than the target kurtosis value. The kurtosis reduces monotonically as the data rate increases beyond the first data rate. Consequently, kurtosis reduces monotonically before the first data rate, is re-set at the first data rate and then continues to reduce monotonically after the first data rate.

In another example, for example as illustrated in <FIG>, below a first data rate Ro , the stream of symbols <NUM> for transmission has a probability-amplitude distribution that has a fixed kurtosis (solid flat line) that is less than that for a probability-amplitude distribution of the symbols output by the PAS symbol source circuit <NUM> (dotted line) and is less than the target kurtosis value. Above the first data rate remapping is not performed and the stream of symbols <NUM> for transmission are the symbols output by the PAS symbol source circuit <NUM> which at these data rates has a probability-amplitude distribution with a kurtosis less than the target kurtosis value. The kurtosis reduces monotonically as the data rate increases beyond the first data rate. Consequently, kurtosis is fixed before the first data rate, is re-set at the first data rate and then continues to reduce monotonically after the first data rate.

The presence of a constant kurtosis with increasing data rate (<FIG>) or a stepwise increase in kurtosis with increasing data rate (<FIG>) is a measurable artefact in the optical channel <NUM>.

It will therefore be appreciated that in at least some examples the apparatus <NUM>, comprises.

The following table specifies an example of a mapping used to convert symbols <NUM> to symbols <NUM>.

In this example, Δ =<NUM>, and M=<NUM> (N=<NUM>). There are four amplitude levels, two bits, for X and four amplitude levels, two bits, for Y. Two binary distribution matchers are used for X, one for each bit. Two binary distribution matchers are used for Y, one for each bit. The data rate is R=<NUM>. Entropy <NUM> b/s for a code rate c=<NUM>.

In this example, p1=<NUM> & p2=<NUM>. The input symbols stats for the codes <NUM>, <NUM>, <NUM>, <NUM> are <NUM>, <NUM>, <NUM>, <NUM>. The kurtosis <NUM>.

The output symbols stats for the codes <NUM>, <NUM>, <NUM>, <NUM> are <NUM>, <NUM>, <NUM>, <NUM>. Kurtosis <NUM>.

In some examples, the probabilistic amplitude shaping symbol circuit <NUM> is configured to perform probabilistic amplitude shaping independently for an X amplitude of a symbol (real part of a symbol) and a Y amplitude of a symbol (imaginary part of a symbol).

In some examples, the probabilistic amplitude shaping symbol circuit is configured to perform probabilistic amplitude shaping independently for different polarizations.

<FIG>, <FIG>, <FIG> illustrate various examples of transmitter apparatus <NUM> as previously described.

The transmission chain is as described for <FIG>, although the channel FEC encoder <NUM> is absent in these examples. It can be present in other examples.

In these examples, the probabilistic amplitude shaping symbol source circuit <NUM> is provided by parallel multi-level distribution matchers <NUM>.

In these examples, the constellation re-mapper circuit <NUM> comprises parallel re-mapper LUTs <NUM>.

In <FIG>, the first multi-level distribution matcher <NUM> and re-mapper LUT <NUM> produces a stream of symbols that is used to produce real symbols (X amplitudes) of a complex M-QAM symbol. The second multi-level distribution matcher <NUM> and re-mapper LUT <NUM> produces a stream of symbols that is used to produce imaginary symbols (Y amplitudes) of the complex M-QAM symbol. The QAM modulator <NUM> combines the stream of real (X amplitude) symbol bits and the stream of imaginary (Y amplitude) symbol bits (with a stream of quadrant parity bits) to form a stream of M-QAM complex symbols.

In <FIG>, the division between the first and second multi-level distribution matchers <NUM> is not between real and imaginary but is between odd and even.

In <FIG>, the first multi-level distribution matcher <NUM> and re-mapper LUT <NUM> produces a stream of symbols. The even symbols in the stream of symbols are used to produce real symbols (X amplitudes) of a complex M-QAM symbol. The odd symbols in the stream of symbols are used to produce imaginary symbols (Y amplitudes) of the complex M-QAM symbol. The QAM modulator <NUM> combines the stream of real (X amplitude) symbol bits and the stream of imaginary (Y amplitude) symbol bits (with a stream of quadrant parity bits) to form odd symbols in a stream of M-QAM complex symbols.

The second multi-level distribution matcher <NUM> and re-mapper LUT <NUM> produces a stream of symbols. The even symbols in the stream of symbols are used to produce real symbols (X amplitudes) of a complex M-QAM symbol. The odd symbols in the stream of symbols are used to produce imaginary symbols (Y amplitudes) of the complex M-QAM symbol. The QAM modulator <NUM> combines the stream of real (X amplitude) symbol bits and the stream of imaginary (Y amplitude) symbol bits (with a stream of quadrant parity bits) to form even symbols in the stream of M-QAM complex symbols.

In <FIG>, the first multi-level distribution matcher <NUM> and re-mapper LUT <NUM> produces a stream of symbols. The even symbols in the stream of symbols are used to produce even real symbols (X amplitudes) of an even complex M-QAM symbol for a first polarization. The odd symbols in the stream of symbols are used to produce even real symbols (X amplitudes) of an even complex M-QAM symbol for a second polarization.

The second multi-level distribution matcher <NUM> and re-mapper LUT <NUM> produces a stream of symbols. The even symbols in the stream of symbols are used to produce even imaginary symbols (Y amplitudes) of the complex M-QAM symbol for the second polarization. The odd symbols in the stream of symbols are used to produce even imaginary symbols (Y amplitudes) of the even complex M-QAM symbol for the first polarization.

A third multi-level distribution matcher <NUM> and re-mapper LUT <NUM> produces a stream of symbols. The even symbols in the stream of symbols are used to produce odd real symbols (X amplitudes) of a complex M-QAM symbol for the second polarization. The odd symbols in the stream of symbols are used to produce odd real symbols (X amplitudes) of a complex M-QAM symbol for the first polarization.

A fourth multi-level distribution matcher <NUM> and re-mapper LUT <NUM> produces a stream of symbols. The even symbols in the stream of symbols are used to produce odd imaginary symbols (Y amplitudes) of the complex M-QAM symbol for the first polarization. The odd symbols in the stream of symbols are used to produce odd imaginary symbols (Y amplitudes) of the complex M-QAM symbol for the second polarization.

The QAM modulator <NUM> combines the odd and even complex symbols as a stream.

<FIG> is similar to <FIG> except that, for the third multi-level distribution matcher <NUM> and re-mapper LUT <NUM>, the even symbols in the stream of symbols are used to produce odd real symbols (X amplitudes) of the complex M-QAM symbol for the first polarization (not the second polarization) and the odd symbols in the stream of symbols are used to produce odd real symbols (X amplitudes) of the complex M-QAM symbol for the second polarization (not the first polarization).

A fourth multi-level distribution matcher <NUM> and re-mapper LUT <NUM> produces a stream of symbols. The even symbols in the stream of symbols are used to produce odd imaginary symbols (Y amplitudes) of the odd complex M-QAM symbol for the second polarization (not the first polarization). The odd symbols in the stream of symbols are used to produce odd imaginary symbols (Y amplitudes) of the complex M-QAM symbol for the first polarization (not the second polarization).

Referring to <FIG>, each re-mapper <NUM> produces an odd stream of symbols and even stream of symbols. There are therefore four sources of symbols that can be used, in any order, for.

Referring to <FIG> and <FIG>, each re-mapper <NUM> produces an odd stream of symbols and an even stream of symbols. There are therefore eight sources of symbols that can be used, in any order, for:.

<FIG> illustrates an example of a controller <NUM>. Implementation of a controller <NUM> may be as controller circuitry. The controller <NUM> may be implemented in hardware alone, have certain aspects in software including firmware alone or can be a combination of hardware and software (including firmware).

As illustrated in <FIG> the controller <NUM> may be implemented using hardware encoded or software encoded instructions that enable hardware functionality, for example, by using executable instructions of a computer program <NUM> in a general-purpose or special-purpose processor <NUM> that may be stored on a computer readable storage medium (disk, memory etc) to be executed by such a processor <NUM>.

As illustrated in <FIG>, the computer program <NUM> may arrive at the apparatus <NUM> via any suitable delivery mechanism <NUM>. The delivery mechanism <NUM> may be, for example, a machine readable medium, a computer-readable medium, a non-transitory computer-readable storage medium, a computer program product, a memory device, a record medium such as a Compact Disc Read-Only Memory (CD-ROM) or a Digital Versatile Disc (DVD) or a solid state memory, an article of manufacture that comprises or tangibly embodies the computer program <NUM>. The delivery mechanism may be a signal configured to reliably transfer the computer program <NUM>. The apparatus <NUM> may propagate or transmit the computer program <NUM> as a computer data signal.

Computer program instructions for causing an apparatus to perform at least the following or for performing at least the following:.

References to 'computer-readable storage medium', 'computer program product', 'tangibly embodied computer program' etc. or a 'controller', 'computer', 'processor' etc. should be understood to encompass specialized circuits such as field-programmable gate arrays (FPGA), application specific circuits (ASIC), signal processing devices and other processing circuitry.

As used in this application, the term 'circuitry' may refer to one or more or all of the following hardware circuitry:.

As a further example, as used in this application, the term circuitry also covers an implementation of merely a hardware circuit and its accompanying software and/or firmware.

The blocks illustrated in the <FIG>, <FIG> may represent steps in a method and/or sections of code in the computer program <NUM>. The illustration of a particular order to the blocks does not necessarily imply that there is a required or preferred order for the blocks and the order and arrangement of the block may be varied. Furthermore, it may be possible for some blocks to be omitted.

The term 'a' or 'the' is used in this document with an inclusive not an exclusive meaning. That is any reference to X comprising a/the Y indicates that X may comprise only one Y or may comprise more than one Y unless the context clearly indicates the contrary. If it is intended to use 'a' or 'the' with an exclusive meaning then it will be made clear in the context. In some circumstances the use of 'at least one' or 'one or more' may be used to emphasis an inclusive meaning but the absence of these terms should not be taken to infer and exclusive meaning.

Claim 1:
An apparatus (<NUM>), comprising:
circuitry, the circuitry comprising:
a probabilistic amplitude shaping symbol source circuit (<NUM>) configured, in response to receipt of input data (<NUM>), to produce a first stream of symbols (<NUM>) of a constellation for modulation, wherein:
the frequency of occurrence of symbols in the first stream (<NUM>) is dependent upon a probability-amplitude distribution for symbols of the constellation, wherein the probability-amplitude distribution is a probability-amplitude distribution that causes low-amplitude symbols to occur more frequently on average in the stream than high-amplitude symbols; and
a constellation re-mapper circuit configured, in response to receipt of the first stream of symbols, to:
map the first stream of symbols (<NUM>) to a second stream of symbols (<NUM>) of the constellation for modulation;
to produce the second stream of symbols (<NUM>) wherein the second stream of symbols has a lower kurtosis than the first stream of symbols; and
to transmit the second stream of symbols of the constellation to a modulator for modulation onto a carrier wherein:
a frequency of occurrence of symbols in the second stream is dependent upon a probability-amplitude distribution for symbols of the constellation, wherein the probability-amplitude distribution has a kurtosis less than a target value for 64QAM of <NUM> and a shape of the probability-amplitude distribution causes low-amplitude symbols to occur more frequently on average in the stream than high-amplitude symbols.