GB2523999A - Method and apparatus for improved QAM constellations - Google Patents

Abstract

A method of determining non-uniform constellation positions of a two-dimensional Quadrature Amplitude Modulation scheme (NU-QAM) is provided. The scheme has words of n coded bits mapped to each constellation point, for a signal to be transmitted over a channel in a system using a forward error corrector (FEC). The method selects a signal to noise ratio (SNR) appropriate for the channel and the forward error corrector and then constrains the positions of at least some constellation points to have n-fold symmetry, where n is an integer > 4, e.g. n = 8 or 16. The method then determines the positions of the constellation points that maximise a measure of channel capacity at the selected SNR. The constraint to n-fold symmetry simplifies calculations whilst ensuring appropriate constellations positions are selected.

Description

I

Method and Apparatus for Improved QAM Constellations

BACKGROUND OF THE INVENTION

This invention relates to encoding and decoding transmissions encoded according to QAM modulation schemes, and to methods for determining constellations for such schemes. The invention is particularly suited, but not limited, to digital television standards such as DVB-T and DVB-T2.

Reference should be made to the following documents by way of

background:

[1] ETSI Standard ETS 300 744, Digital Broadcasting Systems for Television, Sound and Data Services; framing structure, charnel coding and modulation for digital terrestrial television, 1997, the DVB-T Standard.

[21 European Patent Application 1221793 which describes the basic structure of a DVB-T receiver.

[3] FRAGOULI, C, WESEL, R D, SOMMER, D, and FETTWEIS, G P, 2001. Turbo codes with nonuniform constellations. IEEE International Conference on Communications, ICC 2001.

[41 W020131117883 British Broadcasting Corporation. Method and Apparatus for Improved QAM Constellations.

[51 ZOLLNER, J and LOGHIN, N, 2013. Optimization of High-order Non-uniform QAM Constellations. IEEE International Symposium on Broadband Media Systems arid Broadcasting, 2013.

Quadrature amplitude modulation (QAM) is a modulation scheme that operates by modulating the amplitudes of two carrier waves, using the amplitude-shift keying (ASK) digital modulation scheme or amplitude modulation (AM) analog modulation scheme. The two carrier waves, usually sinusoids, are out of phase with each other by 90° and are thus called quadrature carriers or quadrature components -hence the name of the scheme. The modulated waves are summed, and the resulting waveform is a combination of both phase-shift keying (P5K) and amplitude-shift keying (ASK), or (in the analog case) of phase modulation (PM) and amplitude modulation.

By representing a transmitted symbol (a number of bits also referred to as a word) as a complex number and modulating a cosine and sine carrier signal with the real and imaginary parts (respectively), the symbol can be sent with two carriers on the same frequency. As the symbols are represented as complex numbers, they can be visualized as points on the complex plane. The real and imaginary axes are often called the in phase, or laxis and the quadrature, or Q-axis. Plotting several symbols in a scatter diagram produces the constellation diagram. The points on a constellation diagram are called constellation points, each point representing a symbol. The number of bits conveyed by a symbol depends upon the nature of the QAM scheme. The number of points in the constellation grid is a power of 2 and this defines how many bit.s may be represented by each symbol. For example, 16-QAM has 16 points1 this being 2 giving 4 bits per symbol. 64QAM has 64 points, this being 26 giving 6 bits per symbol or word. 256-QAM has 256 point, this being 28 giving 8 bits per symbol or word.

Upon reception of the signal, a demodulator examines the signal at points in time, determines the vector represented by the signal and attempts to select the point on the constellation dagram which is closest (in a Euclidean distance sense) to that of the received vector. Thus it will demodulate incorrectly if the corruption has caused the recewed vector to move closer to another constellation point than the one transmitted. The process of determining the likely bit sequences represented by the QAM signal may be referred to as demodulation or decoding.

An example digital terrestrial television transmitter is shown in Figure 1, as will be described in greater detail later, and a corresponding receiver is shown in Figure 2. Th.e coding arrangement within the transmitter indudes a QAM mapper 46 arranged to map symbols to QAM constellation points. The system uses Orthogonal Frequency Division Multiplex (OFDM) transmission. All data carriers in one OFDM frame are modulated using either QPSK. 16-QAM or 64-QAM constellations. The constellations used are shown in figures 9a to 9c o? the standard.

It is known to use QAM constellations that are non-uniform in spacing.

This may be referred to as non-uniform QAM (abbreviated to NUQAM herein). In the paper by FRAGOULI, C, WESEL, R D, SOMMER, D, and FETTWEIS, G P, referred to above, a non-uniform CAM scheme is discussed. Further non-uniform QAM schemes are discussed in W020131117883 of British Broadcasting Corporation also referenced above. An example non-uniform QAM constellation is shown in Figure 3.

The publication by ZOllner and Loghin referenced above, proposes non- uniform constellations which abandon the constraint of QAM as a variables-separable grid. This approach showed an improvement over a range of SNRs.

The resulting constellations have a genuinely 2-D, non-variables-separable form which tends to be more nearly circular rather than square. In principle the positions of all the points in the 2-D constellation should be optimised, leading to a very much larger number of optimisation variables. In a constellation of L points, this would require optimisation of each individual point (a case they call "L-NUC"). To reduce the optimisation complexity, the constellations are treated as quadrant symmetric, a constraint which they call "QNUC" or "Quadrant-symmetric" NUC. In this approach is assumed that the whole constellation retains left-right and up-down symmetry, which implies that the design process has been reduced to that of designing just one quadrant, with the others being symmetrical copies obtained by using both axes as a mirror. The complexity of the optimisation is reduced because the number of free variables is roughly one quarter of the totally free case.

SUMMARY OF THE INVENTION

The improvements of the present invention are defined in the independent claims below, to which reference may now be made. Advantageous features are set forth in the dependent claims.

The present invention provides an encoding! decoding method, an encoder! decoder and transmitter or receiver for use in the method. In addition, the invention provides a method for determining QAM constellations.

We have appreciated that the prior methods for determining QAM constellations to use in transmission schemes do not appropriately consider the actual channel conditions of a broadcast system. In particular, we have appreciated that known non-uniform QAM constellations of prior systems are not optimised and that the basis for selecting QAM parameters can be improved.

In broad terms, the invention provides a method of determining non-uniform constellation positions of a two dimensional modulation scheme in which symmetry is imposed as a constraint on the positions of at least some of the constellation points that is a higher order of symmetry than the traditional quadrant symmetry used in existing schemes. Whilst the constraint of quadrant symmetry reduces the complexity of the initial determination of constellation positions, we have appreciated that improvements can be made.

The constraint that at least some constellation points have greater than quadrant symmetry may be conveniently referred to as N-fold symmetry where N is greater than 4. Preferred choices are 8-fold symmetry or 16-fold symmetry, or more generally N being a power of 2. Other symmetry values may be possible.

The important point to note is that we have appreciated a departure from the historic implicit quadrant 4-fold symmetry is both possible and in some circumstances desirable.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention wll be described in more detail by way of example with reference to the accompanying drawings, in which: Fig. I is a schematic diagram of a known DVB transmitter to which the invention may be applied; Fig. 2 is a schematic diagram of a known DVB receiver to which the invention may be applied; Fig. 3 shows a non-uniform 16-QAM constellation as descnbed in the DVB-T standard; Fig. 4 is a diagram showing the Shannon capacity of a channel; Fig. 5 is a diagram showing the CM capacity of a channel in comparison to Shannon capacity assuming the use of various uniform QAM constellations; Fig. 6 is a diagram showing the BICM capacity of a channel in comparison to Shannon capacity assuming the use of various uniform QAM constellations; Fig. 7 shows the shortfall in B1CM capacity from Shannon capacity for various uniform QAM constellations; Fig. 8 is a plot of calculated BICM capacity against QAM outer-point distance for a selected SNR showing a maximum capacity at a specific outer-point distance; Fig. 9 is a plot of BICM capacity gain for non-uniform 16-QAM constellations optimised at each SNR; Fig. 10 is a plot of 16-QAM outer-point position against the SNR for which such outer-point positions optimise the BICM capacity; Fig. 11 is a plot of BICM capacity shortfall from Shannon capacity against selected SNR for various QAM orders for both uniform and optimised non-uniform cases; Fig. l2is a plot of constellation-point positions against the SNR for which the positions are optimised for 64 NUQAM; Fig. 13 is a plot of constellation-point positions against the SNR for which the positions are optimised for 256 NUQAM; Fig. 14 is a plot of BICM shortfall fivm Shannon limit showing uniform QAM and NUQAM at selected SNRs; Fig. 15 is a plot of constellation-point positions against the SNR for which the positions are optimised for 1024 NUQAM; Fig. 16 is a plot of constellation-point positions for 256 QAM for which the B1CM capacity is optimised; Fig. 17 is a plot of BICM shortfall from Shannon limit showing uniform QAM. 256-NUQAM and condensed 256 QAM at selected SNRs; Fig. 18 is a plot of BICM shortfall from Shannon limit showing uniform QAM, 1024-NUQAM and condensed 1024 QAM at selected SNRs; Fig. 19 shows the BICM capacity shortfall for higher orders of NUQAM: FIg. 20 shows various symmetry constraints; FIg. 21 shows the improvement over 16-NUQAM when using 8-fold symmetry; FIg. 22 shows a comparison of 4-fold symmetry and 8-fold symmetry with 16-NUQAM; FIg. 23 shows a further comparison of 4-fold symmetry and 8-fold symmetry; Fig. 24 shows determined spot positions for 64-NU-8SM and 64-NU-4SM; Fig. 25 compares both the bit mapping and spot positions of the four-fold symmetric 64-NU-4SM with those of conventional uniform 64-QAM FIg. 26 shows the results for a a 64-NU-85M scheme with 8-fold symmetry condensed to 56 or48 points; Fig. 27 shows how two condensations A and B of 256-N U-I6SM are appropriate at two different SNRs by demonstrating which points either naturally tend to merge or nearly do so; FIg. 28 shows the evolution of spot positions with SNR for two different condensations from 64-N U-8SM, to 56 and 48 points respectively; Fig. 29 compares the results of a wide range of schemes, as a plot of BICM capacity shortfall vs. design SNR; Fig. 30 is a dose-up view of Fig. 29 for easier comparison of the schemes proposed; Fig. 31 shows how spot positions for a scheme with 8-fold symmetry suggest a hybrid where most points have 16-fold symmetry; Fig. 32 puts into context the results for the hybrid schemes suggested by Fig. 31.

DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT OF THE INVENTION

DVB Tmnsmltter A known transmitter will first be described to which the invention may be applied to provide context. Such transmitters are known to the skilled person.

Within the following description the embodiment of the present invention provides a new method for deriving the constellations to be used in the mapper described below and a new transmitter using such constellations.

The transmitter of Figure 1 receives video (V), audio (A), and data (D) signals from appropriate signal sources via inputs 12 and these are applied to an MPEG-2 coder 14. The MPEG-2 coder includes a separate video coder 16, audio coder 18 and data coder 20, which provide packetised elementary streams which are multiplexed in a programme multiplexer 22. Signals are obtained in this way for different programmes, that is to say broadcast channels, and these are multiplexed into a transport stream in a transport stream multiplexer 24. The output of the transport stream multiplexer 24 consists of packets of 188 bytes and is applied to a randomiser 26 for energy dispersal, where the signal is combined with the output of a pseudo-random binary sequence (PRBS) generator received at a terminal 28. The randomiser more evenly distributes the energy within the RF (radio frequency) channel. The signal is now applied to a channel coding section 30 which is generally known as the forward error corrector (FEC) and which comprises four main components, namely: an outer coder 32, an outer interleaver 34, an inner coder 36, and an inner intedeaver 38.

The two coding stages 32, 36 provide a degree of redundancy to enable error correction at the receiver. The two interleaving stages 34, 38 are necessary precursors for corresponding de-interleavers at a receiver so as to break up bursts of errors so as to allow the error correction to be more effective.

The outer coder 32 is a Reed-Solomon (RB) coder, which processes the signal in packets of 188 bytes and adds to each packet 16 error protection bytes.

This allows the correction of up to 8 random erroneous bytes in a received word of 204 bytes. This is known as a (204, 188, P8) Reed-Solomon code. This is achieved as a shortened code using an RS (255, 239, t=8) encoder but with the first 51 bytes being set to zero.

The outer interleaver 34 effects a Fomey convolutional interleaving operation on a byte-wise basis within the packet structure, and spreads burst errors introduced by the transmission channel over a longer time so they are less likely to exceed the capacity of the RS coding. After the interleaver, the nth byte of a packet remains in the nth byte position, but it will usually be in a different packet. The bytes are spread successively over 12 packets, so the first byte of an input packet goes into the first output packet, the second byte of the input packet is transmitted in the second output packet, and so on up to the twelfth. The next byte goes into the first packet again, and every twelfth byte after that. As a packet contains 204 bytes, and 204 = 12 x 17, after the outer interleaving a packet contains 17 bytes that come from the same original packet The inner coder 36 is a punctured convolutional coder (PCC). The system allows for a range of punctured convolutional codes, based on a mother convolutional code of rate 1/2 with 64 states. The data input is applied to a series of six one-bit delays 40 and the seven resultant bits which are available are combined in different ways by two modulo-2 adders 42,44, as shown. These adders provide the output of the inner coder in the form of an X or 01 output and a V or 02 output, the letter 0 here standing for the generator sum. The X and V outputs are combined into a single bit stream by a serialiser 45.

The puncturing is achieved by discarding selected ones of the X and V outputs in accordance with one of several possible puncturing patterns. Without puncturing, each input bit gives rise to two output bits. With puncturing one of the following is achieved: Every 2 input bits give 3 output bits Every 3 input bits give 4 output bits Every 5 input bits give 6 output bits Every 7 input bits give B output bits Returning to Figure 1, the inner interleaver 38 in accordance with the standard is implemented as a two-stage process, namely bit-wise interleaving followed by symbol interleaving. Both are block based. First, however, the incoming bit stream is divided into 2, 4 or 6 sub-streams, depending on whether QPSK (quadrature phase shift keying), 16-QAM (quadrature amplitude modulation), or 64-QAM is to be used, as described below. Each sub-stream is separately bit interleaved and all the streams are then symbol interleaved.

The bit inteileaver uses a bit interleaving block size which corresponds to one-twelfth of an OFDM symbol of useful data in the 2k mode and 1/48 of an OFDM symbol in the 8k mode.

The symbol interleaver maps the 2, 4 or 6-bit words onto 1512 or 6048 active carriers, depending on whether the 2k or 8k mode is in use. The symbol interleaver acts so as to shuffle groups of 2, 4 or 6 bits around within the symbol.

This it does by writing the symbol into memory and reading out the groups of 2, 4 or 6 bits in a different and permuted order compared with the order in which they were written into the memory.

The groups of 2, 4 or 6 bits (referred to as coded bits, symbols or words) are applied to a mapper 48 which quadrature modulates the bits according to QPSK, 16-QAM or 64-QAM modulation, depending on the mode in use. (QPSK may also be represented as 4-QAM.) The constellations are shown in Figure 9 of the standard. It will be appreciated that this requires 1, 2 or 3 bits on the X axis and 1, 2 or 3 bits on theY axis. Thus while reference has been made to 2, 4 or 6 bits in the shuffling process, in fact the shuffling is applied to 1, 2 or 3 bits in the real part and 1, 2 or 3 bits in the imaginary part.

The signal is now organized into frames in a frame adapter 48 and applied to an OFDM (orthogonal frequency-division multiplexer) coder 50. Each frame consists of 68 OFDM symbols. Each symbol is constituted by 1705 carriers in 2k mode or 6817 carriers in 8k mode. Using the 2k mode as an example, instead of transmitting 1705 bits sequentially on a single carrier, they are assembled and transmitted simultaneously on 1705 carrIers. This means that each bit can be transmitted for much longer, which, together with the use of a guard interval, avoids the effect of multipath interference and, at east in 8k mode, allows the creation of a single-frequency network.

The duration of each symbol, the symbol period, is made up of an active or useful symbol period, and the guard interval. The spacing between adjacent carriers is the reciprocal of the active symbol period, thus satisfying the condition for orthogonality between the carriers. The guard interval is a predefined fraction of the active symbol period, and contains a cyclic continuation of the active symbol.

The frame adapter 48 also operates to insert into the signal pilots, some of which can be used at the receiver to determine reference amplitudes and phases for the received signals. The pilots include scattered pilots scattered amongst the 1705 or 6817 transmitted carriers as well as continual fixed pilots.

The pilots are modulated in accordance with a PRBS sequence. Some other carriers are used to signal parameters indicating the channel coding and modulation schemes that are being used, to provide synchronization, and so on.

The OFDM coder 50 consists essentially of an inverse last Fourier transform (FFT) circuit 52, and a guard interval inserter circuit 54. The construction of the OFDM coder will be known to those skilled in the art.

Finally, the signal is applied to a digital to analogue converter 56 and thence to a transmitter front end' 58, including the transmitter power amplifier, and is radiated at radio frequency from an antenna 60.

DVB Receiver A known receiver will also be described for completeness. The embodiment of the invention modifies the demapping so as to allow the constellation scheme according to the invention to be correctly decoded.

In the receiver 100 an analogue RF signal is received by an antenna 102 and applied to a tuner or down-converter 104, constituting the receiver front end, where it is reduced to haseband. The signal from the tuner is applied to an analogue-to-digital converter 106, the output of which forms the input to an OFDM decoder 108. The main constituent of the OFDM decoder is a fast Fourier transform (FF1) circuit, to which the FFT in the transmitter is the inverse. The FFT receives the many-carrier transmitted signal with one bit per symbol period on each carrier and converts this back into a single signal with many bits per symbol period. The existence of the guard interval, coupled with the relatively low symbol rate compared with the total bit rate being transmitted, renders the decoder highly resistant to multipath distortion or interference.

Appropriate synchronisation is provided, as is well-known to those skilled in the art In particular, a synchronising circuit will receive inputs from the ADC 106 and the Ffl 108, and will provide outputs to the F and, for automatic frequency control, to the tuner 104.

The output of the OFDM decoder 108 is then applied to a channel equalizer 110. This estimates the channel frequency response, then divides the input signal by the estimated response, to output an equalised constellation.

Now the signal is applied to a circuit 112 which combines the functions of measurement of channel state, and demodulation or demapping of the quadrature modulated constellations. The demodulation converts the signal back from QPSK. 16-QAM, or 64-QAM to a simple data stream, by selecting the nominal constellation points which are nearest to the actual constellation points received; these may have suffered some distortion in the transmission channel.

At the same time the circuit 112 estimates the likelihood or level of certainty that the decoded constellation points do in fact represent the points they have been interpreted as. As a result a likelihood or confidence value is assigned to each of the decoded bits. The output of the metric assignment and demapping circuit 112 is now applied to an error corrector block 120 which makes use of the redundancy which was introduced in the forward error corrector 30 in the transmitter. The error corrector block 120 comprises: an inner deinterleaver 122, an inner decoder 124, in the form of a soft-decision Viterbi decoder, an outer deinterleaver 126, and an outer decoder 128.

The inner deinterleaver 122 provides symbol-based deinterleaving which simply reverses that which was introduced in the inner interleaver 38 in the transmitter. This tends to spread bursts of errors so that they are better corrected by the Viterbi decoder 124. The inner deinterleaver first shuffles the groups of 2, 4 or6 real and imaginary bits within a symbol (that is, 1, 2 or 3 of each), and then provides bit-wise deinterleaving on a block-based basis. The bit deintesleaving is applied separately to the 2,4 or 6 sub-streams.

Now the signal is applied to the Viterbi decoder 124. The Viterbi decoder acts as a decoder for the coding introduced by the punctured convolutional coder 36 at the transmitter. The puncturing (when used) has caused the elimination of certain of the transmitted bits, and these are replaced by codes indicating a mid-value between zero and one at the input to the Viterbi decoder. This will be done by giving the bit a minimum likelihood value. If there is no mirimum likelihood code exactly between zero and one, then the added bits are alternately given the minimum values for zero and for one. The Viterbi decoder makes use of the soft-decision inputs, that is inputs which represent a likelihood of a zero or of a one, and uses them together with historical information to determine whether the input to the convolutional encoder is more likely to have been a zero or a one.

The signal from the Viterbi decoder is now applied to the outer deinterleaver 126 which is a convolutional deinterleaver operating byte-wise within each packet. The deinterleaver 126 reverses the operation of the outer interleaver 34 at the transmitter. Again this serves to spread any burst errors so that the outer coder 128 can better cope with them.

The outer decoder 128 is a Reed-Solomon decoder, itself well-known, which generates 188-byte packets from the 204-byte packets received. Up to eight random errors per packet can be corrected.

From the Reed-Solomon outer decoder 128 which forms the final element of the error corrector block 120, the signal is applied to an energy dispersal removal stage 130. This receives a pseudo-random binary sequence at an input 132 and uses this to reverse the action of the energy dispersal randomiser 26 at the transmitter. From here the signal passes to an MPEG-2 transport stream -n dernultiplexer 134. A given programme is apphed to an MPEG-2 decoder 136; other programmes are separated out as at 138. The MPEG-2 decoder -136 separately decodes the video, audio and data to provide elementary streams at an output 140 corresponding to those at the nputs 12 on Figure 1.

Modulation Orders Conventional uniform rectangular modulation such as in DVB-T and DVB-T2 uses Gray coded hit mapping to represent every symbol in the consteflation.

As already mentioned, the DVB-T2 specffies a particular constellation.

The number of coded bits required to represent each constellation point depends on the constellation size as shown in Table 1.

Constellation Number of bits, Bit ordering QPSK 2 { bitO bitl} 16-QAM 4 {hitObiti hit2hit3} 64-QAM 6 { bitO bitl bit2 bit3 bit4 bit5} 256-QAM 8 { bitO biti bit2 bit3 bit4 bit5 bit6 bit7} 1024-QAM 10 { bitO biti bit2 bit3 bit4 bit5 bitS bit7 bit8 4096-QAM 12 { bitO biti hit2 bit3 bit4 bit5 bit6 bit7 hitS Table 1: Bit ordering and required bits for different constel!ation size The Improvement of W02013/117883 The technique of this disclosure derives the degree of non-uniformity or ratio of outer point to inner point positions by considering the SNR of the channel.

in order to understand the improvement, some background theory will first he described.

As is known to the skilled person, the theoretical "maximum capacity" (the maximum possible data throughput) is defined in a paper by Shannon in 1948 as the capacity C (in bitis) of a channel of band W (Hz) perturbed by added white thermal noise whose average power is N when the transmitted signals have an average power P is given by (equation 1): (;=W.k,g2 The above capacity formula defines the maximum capacity of a single band-limited channel with added white Gaussian noise (AWGN). We have appreciated that there are assumptions: that the performance of the channel is limited solely by the AWGN, there is no other degradation and that the noise is A\NGN. Furthermore, there is an assumption regarding the random Gaussian-distributed nature of the signals themselves. However, the DVB signals use consteHations and not theoretical random signals. In the context of DVB, we have more specific practical circumstances we have to apply. The fact that QAM uses a sequence of consteflations means that the signal sent now has some discrete distribution. Even after adding channel noise, the resulting received-signal distribution wifi not, and cannot, be Gaussian, so the optimum capacity of the classic formula cannot be attained, whatever the coding we choose to apply. We have appreciated that a better approach to optimisation is needed.

VVe can make use of the more general mutual inFormation formula; the mutual information (K Y) between the transmitted signal x and the received signal y to give a definition of the capacity we seek (equation 2): i(XY) / / v, !og* Using the above formula allows alLernative measures of actual channel capacity to be derived, such as: (i) the Coded Modulation (CM) capacity in which we assume a particular constellation alphabet is used but place no restraint on cleverness' in using it; (ii) Bit-Interleaved Coded Modulation (BICM) capacity in which we assume coded data bits (from some FEC code) are suitably interleaved and mapped in a particular way to the points of a particular constellation.

Coded Modulation (CM,) capacity We suppose that we transmit constellation symbols selected from an alphabet of possibilities. Thus there wiH be specific discrete values x1 of x to be transmitted. We therefore have to modify the mutual information formula so that it contains an integral over y (the received signal, made continuous by the added noise) and summations over the discrete Things are easiest for the classical rectangular QAM constellations, since these can be treated as two orthogonal 1-dimensional constellations, each having one-half the total capacity. Suitable care must of course be taken when relating the noise variance on each axis to the SNR and the total signal power'.

If one constellation axis has n positions (e.g. 8 in 64-QAM), the coded modulation capacity may be derived as (equation 3): I(X;Y) = / Eoi2E,Lsv 11=1 ?I=l 4=1 A graph showing the calculated CM capacity for various uniform QAM orders with SNR is shown in Figure 5. As can be seen, each larger constellation has greater CM capacity but the gulf from unconstrained Shannon capacity increases with SNR.

Bit-Inteifeaved Coded Modulation (BICM) capacity We suppose that we transmit constellation symbols, just as in CM above.

However, we are to a degree now specific about how we come to transmit these symbols. We assume that coded bits (the form of forward error coding generating them being unspecified, except that a binary code is assumed) are mapped to the constellation points in one of the many familiar ways. For a simple example, we can assume that 16-QAM with Gray coding is in use. Each constellation has 4 coded bits mapped to it, 2 to each of the independent axes. We may suppose that the constellation positions (on one axis) are {-3, -1, +1, +3), mapped as follows: position -i +x MSB o a i z LSB z a o z Suppose the MSB is a 1. That means the point transmitted will be either +1 or +3, depending on the state of the LSB. What we have to assume is that the bits mapped to a particular constellation point are independent, and that each bit is as likely to be a Dora 1. So now, if the MSB is transmitted as a 1, then the PDF of the received signal p(ytransmitted MSB is 1) will have two equal-height peaks at y = +1 and y = +3. (This compares with the single peak in P&Ixi) that arose in the CM calculation). We can then work out the capacity of each bit level separately by applying the mutual-information formula to each one (noting that level's mapping), and finally take the total capacity to be the sum of these bit-level capacities.

The capacity of a bit b may be expressed as (equation 4): capacity of bit b = ±J,P(b 1,y)1og3 We assume equiprobable Os and Is are transmitted, so that P (b is 1) = P (b isO) = 1/2. Then p (b isO, y) = p(ylb isO) P(b isO) = p (yb is 0)/2, and similarly for p (b is 1, y). Puthng these in, writing the log of the fraction as the difference of two logs, expanding and regrouping we get the following form, convenient for numerical integration, for the capacity of bit b (equation 5): i ((;M1b18012 MO)tpIbisl)log2p(ylbial)) Now, assuming the channel adds A\M3N having variance cx2to each axis, we can substitute expressions for the conditional probabilities, this time assuming the other constellation bits are equiprobable (equation 6): p(yjbieO)= !Epftzj) and similarly for p (yb is 1). Finally, as before, but expressed using the alphabet concept, we also substitute (equation 7):

-__

n n1j, V'io To get the BICM capacity for the QAM constellation we do this calculation for each of the bits and sum their capacities. In practice this means calculating the capadty of one axis and doubhng it. The BICM capacity we calculate in this way is certainly a valid upper limit for the use of a bit interleaved single code.

As can he seen from the equation for capacity of a bit (equation 4) and the substitutions for conditional probabilities (equations 6, 7), the BICM capaaty of a channel is a function of AWGN and hence a function of SNR. A graph of the BICM capacity wfth SNR for various uniform QAM orders is shown in Figure 6. As can be seen, as SNR increases, the QAM sizes take turns to have greatest BICM capacity but gulf from the unconstrained Shannon theoretical limit grows as before. For example, 640AM is the leader around 12 dB SNR, while 256-QAM is best around 18 dB, with 1024-QAM taking over above 23 dli The shortfall of the BICM calculation of capacity from the unconstrained Shannon theoretical limit can be seen in Figure 7. This visibly confirms each order takes turns as best, and that the gulf grows with SNR. 1r

The improvement of W02013!117883 appreciated that QAM is not Gaussian and that known fixed non-uniform QAM constellations are deficient.

The improvement resides in the idea of adapting the non-uniformity of the CAM constellation in order to maximise the capacity, in particular the BICM capacity, at some particular design' SNR, and adapting it again at every other SNR.

We may draw a distinction between design SNR (the SNR for which the capacity is optimised) and the operational SNR actuaUy experienced by any particular receiver. A system for broadcasting h.as one transmitter and many receivers, usually with no return signalling. In this case the same signal format must be sent to all receivers. In such a situation it would be appropriate to choose a design SNR for the system, namely the SNR at which some aspect of the system is optimised. Preferably, the design SNR corresponds to the SNR likely to be experienced by a receiver at the edge of the intended coverage area. Other receivers within the coverage area may well experience an appreciably better SNR. Optimising for the design SNR will in this case optimise the capacity br the worst-placed receiver. Other receivers having a higher operational SNR will receive the very same signal; while they therefore gain no capacity advantage from their greater SNR, they will nevertheless receive an equally satisfactory result as will have been achieved for the worst case. Although in phnciple these particular receivers could be sent a signal with higher capacity, that would only be at the cost of losing service to receivers at edge of intended coverage. The "design" SNR in the embodiment is thus that predicted for the worstplaced receiver for which coverage is n..tended; it is then assumed that aH receivers will enjoy this same SNR or better in practice, and thus all will perform satisfactorfly.

By being able to optimise capacity for the design SNR then the highest capacity which it is possible to deliver to all simultaneously is achieved.

In principle, an alternative embodiment could be a one4o-one 2-way ink.

in which case the design SNR may be adapted based on the actu& SNR experienced at a receiver; the receiver could report back to the sender what SNR it is experiencing for the time being. In principle the transmitter can then adapt the transmission to achieve the best result. Existing systems might periaps switch QAM orders in such a situation. n principle, such a system embodying the present invention they could instead adapt the positions of the constellation points to maximise the capacity at the current SNR, so that the design and operational SNR are one and the same.

The improvement will first be explained with reference to 16-QAM. This presents a simple case to examine, precisely because there is very little that can be changed. If we consider that uniform 16-QAM uses positions {-3,-1, +1, +3}.

then we can make a nonwniform version having positions +l,+y}, using one parameter y (the ratio of the outer point position to the inner point position).

For any particular SNR, using the equations discussed above or calculations based upon them, we can plot the BICM capacity as a function of y and hence find the BICM optimum for one SNR. This is shown in Figure & Note that the two vertical gridlines correspond toy = 3 (left), for uniform QAM, and y 361 (right), which is a fixed value determined by other methods. We see that for this SNR the optimum y in fact lies between these two, and that there is a very modest improvement in capacity.

The process can easily be repeated for other SNRs, and doing so we find the optimum y depends on the SNR. We can then find the optimum y and resulting BICM capacity for each SNR.

The chosen approach to the calculation is to use numerical optimisation.

PotentiaUy, the relationship between the optimum y and the SNR could he expressed as a function and the value of y determined analyUcafly. For example, if BICM capacity coLdd be easily expressed as f(y), then the position of the maxima could he solved by differentiation. However, as the method is applied to higher orders, the calculation becomes more complex. As explained later, for higher orders there are more parameters, for example 7 parameters for 256-QAM. so that the function to be solved becomes differentiating with respect to each parameter in turn and solve for example df/do = 0, df/d3 = 0 and so on. In view of the complexity, instead the preferred approach is numerical optimisation.

The embodiment described uses the known Matheniatica program and its "NMaximize" command; this uses a multiplicity of numerical optimisation techniques, which, in essence maximise the function f(o,yO,E,(q) by varying each of the parameters (o.,y,ö,s,(,ffl. 1r

The results are shown in Figure 9. The solid curve shows the BIOM capacity improvement (compared with uniform 16-QAM) for the single, fixed non-uniform consteUation having y 3.61 produced by the known methods. The points show the BICM capacity improvement for non-uniform QAM which is optimised for each particular SNR using the improvement. We had reasoned that per-SNR optimisation would be better, and this s confirmed by the plotted points.

They show that the old' method was close to optimum for SNRs in the range say 6 to 9 dB, but elsewhere the per-SNR optimisation is dearly better. Of course, the benefits are smafi. as we might expect. Eventuafly at high SNR there is no longer any advantage for non-uniformity, just as predcted. The new method also shows a striking improvement at low SNR, towards 0 dB and below.

Fig. 10 shows as expected that for high SNR the optimum y tends towards 3, returning the constellation to uniformity. It has its peak value (rather greater than that of th.e known method) around 7 dB SNR, below which it drops again. Wien the SNR is low enough (below about 1 dB) y converges to 1, so that the constellation has collapsed from 16-to 4-QAM, and its LSB now has zero capacity. This explain.s the apparent advantage of non-uniformity at very low SNRs, as shown in Fig. 9: the advantage is actually simply that of 4-QAM over uniform 16-QAM at ow SNR. As can be seen, the optimised outer constellation point position y tends to 3 at high SNR, this being the uniform QAM position since the inner-point position is taken as 1 giving a uniform spacing of 2. At low SNR values the consteUation outer point position reduces below 3 and so is most compressed' in the sense that the outer-point and inner-point positions are closer to one another. Around 7 dB, the outer-point position y is a maximum of around 3.8 meaning that the outer--point is stretched" in the sense that the outer-point and inner-points are further away.

To extend our optim!sat!on method to higher-order constellations is easy in principle, but computationaVy chaflenging. We have to define more parameters over which to optimise the BICM (or indeed CM) capacity, and these muftiply alarmingly. We label the assumed consteDation points on one axis as foflows: 16-QAM:-{-y,--1, +l,+y} -64-QAM:-{-y,--3--a,-1. +1,-i-o,413,±v} 256-QAM:-frn,-<-O,-v,-B,-o,-1. +1 to+13,+y+O.+c+(.+n} so that 16-QAM has I parameter, 64-QAM has 3 and 256-QAM has 7.

1024-QAM, would have 15 parameters. We can even extend this to 4096 QAM with 31 parameters and 16384 QAM with 63 parameters. With this number of parameters we no longer have any option of using plots to find maxima. Instead, we use numerical optiniisation.

The BICM capacities achieved are iflustrated in Figure 11. The results for uniform QAM consteflations are reproduced as dashed lines, while the corresponding results for the per-SNR optimised non-uniform QAM constellations are shown in the same tone and labelled, as sohd lines with plot points. We see, as already concluded, that the non-uniform 16-QAM improves slightly on uniform 16-CAM in the expected SNR range from roughly 6 to 11 dB It also gives an improvement at low SNR by converging on the uniform-4-QAM curve (because here it is in effect collapsing down to 4-QAM). Non-uniform 64-QAM and 256-QAM are rather more interesting: they give much larger improvement compared with their uniform versions. This is perhaps not surprising, as there is very little scope to optimise the simple 16-QAM constellation, but these larger constellations have more parameters to adjust. Their results also converge on the results for lower QAM orders at low SNR.

We get more insight by looking at the optirnised consteflation positions, see Figure 12 for 64-QAM and Figure 13 for 256-QAM.

Figure 12 shows how the optimum consteVation-spot positions vary when the BICM of 64-QAM is optimised at different SNRs. The grid lines at vafties{1, 3, 5, 7} remind us where they woLild ie in conventional uniform 64-QAM.

Remember that for simpHcity the innermost positions have been kept at ±1 so as to minimise the number of parameters to be optimised. We see that at high SNR the positions are indeed converging towards the uniform-QAM values {1 3, 5, 7}.

At low SNR, 7 dB, we see that it has fuRy converged to non-uniform 16-QAM. In between those two extremes, we see first a somewhat squashed consteflation at lower SNRs, then an expanded one where all of {u, , v} exceed the uniform-QAM values before they reduce again.

Figure 13 shows how the optimum constellation-spot positions vary when the BIOM of 256-QAM is optimised at different SNRs. Referring back to the capacity plots of Figure 11, we note that optimised non-uniform 256-QAM offers quite signfficant benefits over both uniform 256-QAM and optimised non--uniform 64-QAM for SNRs above say 13 dB SNR, while still offering more modest benefits over optimised non-uniform 64-QAM below that. Figure 13 shows several different regions. At very high SNR, the constellation tends to approach the uniform 256-QAM constellation. Around say 20dB SNR, we see the constellation is stretched out, the most in the outer positions. As the SNR reduces below that we see the constellation becoming compressed, and as the SNR decreases, some points begin to merge, and maybe dc-merge and re-merge with others. It is possible that this slightly confused behaviour is an artefact of the numerical optimisation, or of the existence of multiple solutions.

E.g. by simply changing the initial conditions of the optimisation a different, anomalous result in terms of positions can be achieved for 8 dB SNR, without changing the capacity achieved.

Nevertheless, it is clear that the constellation does in effect shrink its number of points as the SNR reduces, going from 256-QAM, down to ultimately becoming non-uniform 16--QAM at about 7 dB SNR. in many places we have essentiaHy 144-QAM, but with dfferent points pairing to produce it at different SNRs; around 16dB we have essentiafly 196-QAN. Interestingly, at no point does it seem to coUapse fully to 64-QAM. The most important thing is that these messy hybrids do achieve greater capacity at the SNRs for which they are optimised than more normal' QAM consteflations do.

Further Improvement of W02013/117883 It becomes computationafly complex to compute the outer-point ratios for higher order consteUations, and potentiafly computationafly infeasible. A further improvement was made, from the above analysis, that within certain SNR ranges it is possible to reduce the complexity of c&cuiation by computing ratios for fewer than the full set of 2 points of an n-order QAM constehation and then using this calculation as an approximation for the full QAM constellation.

Consider again the shortfafl in capacity of BICM in comparison to the Shannon limit (as previously shown in Figure 11) extended to include calculations of some of the SNR values for 1024 and 4096 QAM as shown in Figure 14. As before, the dashed lines denote uniform QAM, while the corresponding results for the per-SNR optimised non-uniform QAM (NUQAM) constellations are shown in the same tones and labelled, as sohd lines with plot points.

The improvement gained by non-uniform 1024-QAM over uniform 1024-QAM in the SNR range from 15 to 20 dB is very substantial, and sufficient to put 1024-N UQAM in the lead over the previously-favoured 256-NUQAM. This is despite the fact that un/form 256-QAM has better capacity than un/form 1024-QAM in this range. (The natural' range of application of uniform 1024-QAM comes at higher SNRs). Indeed, at best the shortfall from the unconstrained Shannon limit is reduced to as little as 0.123 bitisymbo at 16.5 dB SNR. The gain over 256-NUQAM increases further at higher SNRs, but the shortfall now increases too, suggesting that higher orders of NUQAM would now take over as the best choice such as the 4096-QAM shown. Th.e shortfall curve has some curious detail; although the shortfall is mininiised at about 16.5 dB SNR, there are other points where the curvature changes sign, as if there are different zones of behaviour.

The results of computing the per-SNR ratio optimised constellation positions for 1024 QAM are shown in Figure 15. As previously noted, the inner point is deemed to be at position unity, so that all of the other positions are expressed as a ratio to 1. Recall that uniform QAM has a spacing of 2 giving a sequence: 1,3,5,7,9, 11, 13, 15, 17, 19,21,23,25,27,31,33. At high SNR (above 24 dB) all the constellation points are distinct, so we do indeed have a genuine 1024-NUQAM constellation, albeit at first they are somhat compressed together. As the SNR increases, this compression turns to expansion of the constellation, which reaches its maximum extent around 27 dB.

As the SNR gets very high we see clear signs of the positions converging on the uniform-QAM positions that are denoted in the Figure by the horizontal gridlines at{1,3,5. ..29,31}.

In the middle zone (roughly 20 to 24 dB) we see that some of the spots have virtually converged. So in this range we could consider that we have something like' 576-QAM.

* a is nearly merged with the fixed position I * and y have nearly merged at about 3 * 6 and c have nearly merged at about 5 * t and qare close in value * 8 and i are fairly close initially, and K and A less so, the remainder being well distinct throughout.

In the lower-SNR zone (roughly 15.5 to 17.5 dB) we see that more of the spots have virtually converged. So in this range we could consider that we have something like' 256-, 400-and 484-QAM by turns.

* a, andy are nearly merged with the fixed position I * 6 and c have nearly merged at about 3, and and q are also nearly merged at a slightly greater value * $ and are nearly merged * K and A are very dose * p and v are distinct but faiuly dose, while and o remain well distinct However, these descriptions are better thought of as tendencies-by-way-of-explanation; the points do all remain distinct (albeit you have to look to several decimal places in some cases). Note that our optimisation of 1024-NUQAM at 16.5 dB (the best result in terms of shortfall from Shannon) has a clear capacity advantage over 256-N UQAM even though we can observe it to be virtually' converged to 256-QAM. The per-SNR optimised 1024-NUQAM positions we have obtained do rather tend towards only 256-N UQAM at the bottom of the SNR range examined, yet the calculated BICM capacity appears appreciably better than was achieved when we directly optimised 256-NUQAM (as shown in Figure 14).

Calculations may be performed to confirm that gradually reducing the number of constellation points by merging those positions that are very close anyway does, as expected, reduce the corresponding theoretical BICM capacity -but not by a very great deal, even when the number of positions is reduced to the point where the constellation has only 256 positions, the same number as 256-QAM. Yet 1024 QAM at low SNR where it has only 256 positions still produces a better capacity than of 256-QAM. This apparent conundrum can be darified by considering the way the calculations are performed. In the previous work to optimise 256 QAM, we started with 8 bits Gray-mapped to the 256 QAM positions, and optimised that state of affairs. In the current work, we started with 1024 bits Gray-mapped to the 1024 QAM positions, and optimised that different situation. It so happens that in certain SNR ranges some of the positions were very dose, and if we progressively merge them we do eventually end up with a constellation with 256 positions. However, it is a different scenario in that 10 bEts are still mapped to that constellation, albeit that we have very badly weakened some of them by the merging of positions. No bit is totally eliminated.

We have therefore shown that performing calculations to derive constellation positions using fewer than a full 2 points of a given QAM order gives sufficiently accurate constellation positions for the full order, at least in an appropriate SNR range. The full order when used in a broadcast system gives improved capacity over a lower order. We will use the name Condensed QAM for this approach, and propose a notation like 1024-256-ConQAM for the case where we start from 1024-QAM Gray mapping (carrying in this case 10 coded bitisymboi) but flier-ge (or "condense) some of the positions before optimisation so that we end up with (in this exanipe) 256 distinct points. The number of points to which the consteflation is condensed need not be a power of 2. Furthermore a name Uke 1024-256-ConQAM is not enough to specify a scenario uniquely.

because you might choose different ways to merge down to the same number of states before optimisation.

We wifl first consider the example of condensing 256-QAM. 256-NUQAM is a good place to start since we can try many optimisations fairly quickly. The somewhat messy' behaviour of the optirnised positions with design SNR leads us into some comphcation, as there is no one condensation pattern that is likely to be universaUy applicable. See Figure 16 which shows: * above say 17dB SNR all the points are distinct so no condensed version would work weD * roughly from lOto 17 dBwe have a -1 * roughly from 11 to 14dB we have {a -I j3 y} *roughlyati0dBwehave{cu.* 1,5-) * hdow 10dB we have (a v' 5 This cads us to try several ConOAM variants, imposing these condensations before optimisation: * 256-196-ConQAM. imposing simply a -I *256-144-A-ConQAM, imposing{a --.

* 256-144-B-ConQAM, imposing {o -. y, S -.

* 256-144-C-ConQAM, imposing {a I S -} Figure 17 shows the calculated BICM shortfaD for each of these variants of 256 NUQAM. The calculations are performed by imposing the conditions above and then computing the optimum positions of the merged variables using a numerical approach based on the equations 4 to 7 above. As predicted, the different versions perform best in certain SNR ranges. As expected, the less-- condensed 256-196-ConQAM performs weD upto 17 dB, while 256-144-AConQAM works well from say 10.5 to 15.5 dB. 256-144--B--ConQAM is best 2$ below 10dB (but thUs off very quickly above), whe 256-144CConQAM essentiafly devised just for 10dB is indeed the best there, fafling off both above and below 10dB. In summary, improvements can be made if you pick the right flavour of 256-ConQAM to match the SNR desired. Nevertheless, with the right choice, 256-ConQAM indeed essentiafly matches the capacity of its parent 25S-NUQAM. whfle having fewer states to calculate Figure 18 shows the BICM capacity shortfall of various condensations for 1024 NUQAM. This includes the same curves as F!gure 17, particularly noting the curve for 256-N UQAM and additionafly showing 1024-NUQAM as weD as the i:oliowing condensabons: * 1024-324-ConQAM, with (a--. 1, --* 1,y -, 1, ö -. c, (-q, S *--* K--A} 1024-256-ConQAM, with (ci -1j3 -÷ l.y -÷ 1, 5-q, c -. - 9-, K Below 18dB SNR the 1024-324-ConQAM gets close to 1024-NUQAM, while the more condensed 1024-256-ConQAM only does so below 16.5 dB. Both are very close indeed at 15 dB, the lowest value for which we have an optimised 1024-N UQAM result. For still lower SNRs the two condensations essent!ally match. At higher SNRs (above 18 dB) these ConQAMs perform appreciably worse than the parent NUQAM, just as we would expect from observing Fig. 2; less-aggressive condensations would be needed here.

The concepts may be extended to ever higher QAM orders.

The Present Improvement We have appreciated that the non-uniform QAM consteUations derived accorduig to the approach described above appear to have a imitation at ow SNR values. We have therefore appreciated further techniques by which yet further improved consteflation.s may be derived. The improvements may be embodied in systems and methods previously described herein, in particular an encoder and decoder of Figures 1 and 2 operating using an improved constellation The improvements may use the various measure of capacity calculations previously described such as CM capacity and 31CM capacity, and the type of analysis previously described.

In order to understand the improvements, we first summarise findings in relation to varous orders of NUQAM.

We have already noted that with NUQAM we take advantage of imposing full variables-separabifity so that we reaDy have o independent, orthogonal NUPAMs. We also further impose simple symmetry of the NUPAM about its centre (the other axis) and so we greatly simplify the process of optimisation. This simplification has made possible the optirnisation of reaUy rather large NUQAM consteflations. All these NUQAM constellations, by virtue of being in effect NUPAM by NUPAM are square in shape, with all the points lying on a rectangular, albeit non-uniform grid. This constraint appears to cause limitations at lower SNRs, or for very small constellations. 2';

4U. .J Figure 19 shows the 31CM capacity shortfall against signal to noise ratio for increasing higher orders of NUQAM. Around 11 dB signal to noise ratio, all higher orders of NUQAM appear to tend to the same capacity shortfall which s worse than that for appreciably higher or lower design SNRs. This may be referred to as a "pessimum' in the sense that there is a shortfall from what would be hoped in terms of performance.

We have appreciated that the arrangements proposed so far are typically rectangular and axes-separable, and furthermore inherently have 2-fold symmetry along each axis, or 4-fold symmetry. In the case of traditional QAM arrangements, uniform, a type of rectangular 4-fold symmetry is provided. Also, in the case of NUQAM arrangerie..rits discussed, 4-fold axes-separable symmetry is provided. We have appreciated that these conditions represent constraints that may impact the maximum capacity. The points have typicaVy been constrained to lie on a rectangular (albeit nonuniform) grid of rows and columns with axis-independent mapping. One-half of the coded bits determine on which column the point lies, according to the x-axis mapping, while the other half determine on which row the point hes. according to the y-axis mapping.

The symmetry on one axis for 16 QAM is shown by the positions of outer points having symmetrical values as foflows: 16-QAM:-{--y;--i, +l.+y} The same apphes to the other axis so that there are two axes of symmetry, namely 2-fold symmetry.

We have appreciated that differing constraints relating to the ayout and symmetry of constellation points can affect the maximum capacity available at a given SNR. We have also appreciated that some constraints are needed to avoid computationally expensive calculations of unconstrained variables. The present improvement proposes non-uniform non-rectangular constellations but having a symmetrical constraint of many-fold symmetry (greater than 4 fold, quadrant symmetry). We will refer to this as non-uniform X-fold symmetric modulation or NU-XSM for short. A non-uniform scheme having 8-fold symmetry will therefore be described as NU-BSM.

The modulation schemes using constellations that have many fold symmetry are two dimensional in the sense that constellation points lie on Lwo axes, the point being discoverable in relation to each axis at the receiver. The scheme is a quadrature amplitude modulation scheme in the broad sense that a transmitted signal has an in-phase and a quadrature component. However, unlike previous QAM schemes; there is no quadrant symmetry provided.

Accordingly, whst such schemes may be referred to as QAM schemes, and this would he correct in the general sense of the term, we wifi refer to the new proposed schemes as many fold symmetry schemes or more simply by the notation noted above.

The types of symmetry are discussed below for the avoidance of doubt.

in any given many fold symmetric scheme, at least a pair of points, and preferably multiple pairs of points are symmetrical about an wds of symmetry, A point lying on an axis of symmetry could technically fall within the symmetry scheme, but this is not the intended constraint. Instead, a many fold symmetric constraint is one in which at east pairs of points have the n-fold symmetry.

Figure 20 shows three proposed X-fold symmetrical constraints that may be used as part of numerical analysis to determine the position of constellation points that maximise measure of channel capacity. Figure 20a shows 4-fold symmetry in which each quadrant is reflected in each axis. This is the symmetry that has been used in prior rectangular QAM and non-uniform schemes. Figure 2b shows a proposed many fold symmetric extension to this concept in which 8-fold symmetry is provided. In this scheme an axis of reflection at 45c: to each of the major axes is provided. Figure 2c shows 16-fold symmetry with an axis of reflection at 22.5° to each major axis. The use of this additional symmetry has benefits. For example, it greatly reduces the complexity of numerical optimization needed to find the optimum positions of points based on the constraint of maximising a measure of channel capacity. It also provides a different SNR performance in comparison to other sot. enies which may be useful at certain SNR values.

In general, the naming convention adopted is x--NU-ySM, wherein: * x is the number of points in the consteflation, a power of two * NU denotes Non-Uniform * y denotes the symmetry, e.g. 4 for fourfold symmetry * SM denotes Symmetrical Modulation (i.e. the modulation constellation has y-fold symmetry) Once we abandon the square (if nonuniform) grid of prior NUQAM schemes we have more poEnts whose position must be determined, and since their position now has to be specified in two dimensions each point accounts for vo free real variables instead of one. Consider the case with no constraint at all, which we could define as call L-NUC. It has L points whose position should be freely adjusted, thus requiring 2L real variables to specify them, and of which just one of them can be arbitrarily chosen as say 1. So we have an optimisation involving 2L -1 real variables. Now imagine we impose s-fold symmetry,L e. in our notation L-NU-sSM The number of real variables to be optimised is thereby reduced to: i) This potenfially provides a considerable reduction in computation.

Figure 21 shows the improvement compared with rectangular 16-NUQAM when imposing 8-Fold symmetry to prov!de 16-NU-8SM. As can be seen, irom an SNR around Ii dB and lower, the curve departs from the 16-NUQAM curve and provides a lower shortfall in capacity. However, at high SNR, this additional constraint actually provides slightly worse shortfafl in capacity.

Figure 22 shows the shortfall for 16-NIJ-4SM namely 4-fold symmetry for 16-NUQAM. As can be seen, this continues to provide an improvement at low SNR, but matches the equivalent 16-NUQAM at high SNR.

Figure 23 shows a further comparison of 64-NU-4SM and 64-NU-BSM. At ow SNRs the two diferent amounts of symmetry match, but at high SNR values the 4-fold symmetry arrangement actually beats the 8-fold symmetry arrangement. Below 15 dB there is negligible penalty for imposing the tighter symmetry constraint of 64-NU-8SM. Above that we see behaviour similar to the 16-point case; 64-NU-BSM eventually under-performs 64-NUQAM while 64-NU- 4SM appears to converge towards matching the 64-N UQAM or UQAM performance. Although you can't tell from the plot, even at the high SNR of 22 dB, 64-NU-4SM does retain a very tiny advantage over 64-LJQAM. 01 course this is of mostly academic interest since for this SNR you would choose to use something else We conclude that the constraint of 4-fold symmetr'j of otherwise unconstrained, non-rectangular, non-variables separable quadrants provides an improvement over traditional rectangular consteflations having the same number of points, both uniform QAM and non-uniform QAM. Applying higher orders of symmetry as a constraint gives results that, in useful ranges of SNR, are scarcely nferior to -4SM, and thus also share -4SM's performance advantage over UQAM & NUQAM. whfle offering certain practical advantages over -4SM. Higher orders make design easier (fewer variables), and as we go on to show, may simplify receivers.ln general, many-fold symmetry prov!des improvements over existing NUQAM at various orders and symmetry values. The improvement may generafly be considered as a many-fold symmetry constraint on non-rectangular constellation positions with the symmetry factor selected according to Pie desired SN ft Figure 24 shows the determined spot positions for 64-N U-8SM and 64-NU-4SM. The nature of the 4-fold symmetry constraint results in each quadrant being a reflection of the corresponding quadrant about each major axis. The general appearance is of rings and radials', although this is somewhat deceptive: closer scrutiny of the results shows: the radials' aren't radials, they don't pass through the origin * furthermore. the rings' aren't precise rings either The low-SNR cases show that 64-NU-8SM collapses to 16-NU-BSM there. Furthermore there are clear signs at SNRs below say 14dB that th.e innermost points are tending to merge, which suggests there is scope for condensation, discussed later. At higher SNR, the inner points re-align so as to depart further from a rings and radials' model.

The way 64-NU-4SM evolves with design SNR is very interesting. At 12 dB the constellation appears very closely similar to 64-*NU*-8SM, and its capacity matches closely too the capaciLies start to diverge only at 13 dB, and only marginally there. We may assume that 64-NU-4SM and 64-NU-8SM are indistinguishable for all SNRs below 12 dB. However, the appearance of this constellation is very different at higher SNRs. As for the 16-point case, 64NU- 4SM first passes through a kind of rugby-ball' phase (at least for the inner points of the constellation). Unlike 16-NU-4SM though, at the highest SNR it does not tend to a square NUQAM layout at all, even though its BICM capacity at 22 dB only quite marginally exceeds that of 64-UQAM (and both are very dose indeed to the limiting capacity of 6 bit/sym of a 64-point constellation).

What seems to be happening is this. At very high SNRs, as code rates tend to 1, we know we expect the best solution to become in some sense close-packed so that the minimum distance is maximised. If constrained to square constellations, this implies (as we have found) that NUQAM tends to UQAM at high SNR. If constrained to a uniform rectangular grid of a particular spacing, then we know that with larger constellations it is possible to reduce the mean power slightiy by taking some points from the corners and placing them alongside the centres of the sides to make a constellation that is more nearly circular overall while retaining a completely uniform spacing.

The nature of the symmetry can be better understood by showing the entire constellation for 64 non-uniform QAM with 4-fold symmetry, namely 64-N U- 4SM at a relatively low signal to noise ratio of SNR = 14 dB. The left hand side of Figure 25 shows standard uniform 640AM for comparison. As is known to the skilled person, 64 QAM is a 6 bit scheme gMng a total of 64 constellation positions. The mapping of the bit positions to corresponding groups of constellation points is shown by the red (black) and blue (gray) shading. As shown for standard uniform QAM on the left hand side of Figure 25, bit number I of a 6 bit sequence is mapped to the top and bottom half of the constellations, for example value I being the upper half of the constellation and valueD being the lower half. Similarly, bit number 2 is mapped to left and right with value I being the right hand side and value 0 being the left hand side. Further bit positions are mapped as shown so that for any particular constellation the bit sequence of 6 bits in unique.

In contrast, the 4-fold symmetry 64 non-uniform non-rectangular arrangement 64-NU-4SM is shown on the right hand side of Figure 25. In a similar fashion, the bit mapping for bits I and 2 are top and bottom half and left and right halves of the constellation. Thereafter, the mapping is visibly less similar to the uniform QAM arrangement. For example, due to the change in shape, bit number 3 has some slight similarity to the uniform QAM arrangement with upper and lower portions representing one bit value and verticafly central left and right portions representing a different bit value for that bit. However, this bit has more similarities for the positions for PSK or APSK, except that the points do not lie on orderly rings and radials at regular angles. Bit numbers 4, 5 and 6, though, have visibly rather different mappings due to the very different shape. in this example, the greatest uncertanty wifl be potentially in relation to hit number 5, where the innermost points are closely adjacent and so could represent a 1 or a 0. However, as the scheme uses a forward error corrector this ambiguity can be resolved within the error correction scheme, just as was the case for various NUQAMs in which some points became very dose or indeed essentiaHy indistinguishabie at some design SNRs.

We have appreciated that the many-fold symmetrical schemes can also be condensed in the same manner as the CONQAM schemes discussed above.

Figure 26 shows the results for condensations of an eight-fold symmetric 64-N U- 8SM scheme to 56 or 48 points respectively. As can be seen. 64-56-CON-8SM works very well but 64-48-CON -SSM does not perform as well as th.e SNR increases. This can be explained by referring again to Figure 25 in which we see that the positions of innermost poinLs become dose to one another at ow SNR vaiues but are further apart at higher SNR values. We can choose to intentionally condense the scheme by constraining the position of at least one point in a symmetry sector to equal the position of at least one other point, in particular having the two innermost points in each symmetry sector constrained to be at the same position.

The gentler condensation, 64-56-Con-8SM, performs essentially as well as 64-N U-8SM over the whole range of design SNRs for which 64-N U-8SM is useful. The more forced condensation of 64-48-Con-8SM means that it only approximates the uncondensed capacity at 8 dB SNR and below. Nevertheless, although its divergence from the 64-56-Con-8SM or uncondensed is marked above 9 dB, that doesn't necessarily mean it is without merit or potential use.

Even at 12 dB, where the resLilts diverge qu!te dramatically, 64-48-Con- 8SM comfortably beats all square constellations, even those of absurd complexity. MIMO receivers can be simphfled if the use of condensed constellations reduces the number of points. MIMO-receiver approaches can vary, but a complexity proportional to aL east the square of the number of actual points is often quoted. On that basis, the benefit of condensing to 48 points from 64 is a reduction in complexity by factor (4k' 2 64) / = 05625, which would be well worth having and s a lot better than using 256-NUQAM, whose capacity in any case remains inferior up to 13dB SNR.

The evolution of the spot positions of these two constellations with design SNR is illustrated in Fig 28. We may note that at 7dB the 64--48-Con-8SM has in effect further collapsed to just 24 points while still offering a small capacity benefit over the 16 points of 16-NU-8SM.

The many--fold symmetrical schemes may be condensed in a variety of ways. The main example. just mentioned, is by intentionally setting the innermost points such that pairs of points that would othevise be closely spaced are co-located at the same position. In one example, a single pair of points could be constrained to be at the same position and the remaining points constrained according to the symmetry requirement of the scheme, such as 4-fold, 8-fold or 16-fold and so on. Other possibilities include merging specific pairs ol points within each symmetry sector. For example, with 8-fold symmetry, there are two symmetrical sectors per quadrant we can choose to constrain 2 points to be at the same position in each such symmetry sector. We could constrain two pairs of points to be at two positions in each symmetry sector. The concept may be extended to e'ier higher orders of symmetry constraining 1, 2 or more pairs of points so as to provide a differing condensation schemes for the capacity calculation and designed SNR. In some oases, it is appropriate for more than pairs to be merged to a single point, for example, 4 points could be condensed to a single point.

Forcing the two innermost points in each octant of an 8-fold symmetry scheme to merge leads to 7 distinct points per octant, and so 56 points in total in the whole constellation. This we can call 64-56-Con-8SM. The number of free real variables in the optimisation is reduced to 13. If we merge the next two nearest points in each octant as well we have 6 distinct points per octant, and so 48 points in total in the whole constellation, now 64-48-Con-8SM. The number of free real variables in the optimisation is reduced to 11. Of course in this case we had the luxury of being able to decide the appropriate condensation when we already had results for the full, uncondensed constellation, so we could be confidentofthemworking.

We will now give a specific example of many-fold NUQAM condensation with respect to Figure 27.

The condensations shown in Figure 27 are selected for different signal to noise ratios. The upper half shows a design SNR of 10 dB, the lower half shows a design SNR of 13 dB, both schemes being for 16-fold symmetry 256 non-uniform QAM identified as 256-NU-168M. At SNR=lOdB, the choice of points that could be condensed (condensation A) is visually apparent by the graphical representation of the constellation positions of Figure 27. As can be seen for condensation A, points 1, 2, 9 and 10 would merge together well as would points 3, 4, 11 and 12. In addition, points 6 and 13 merge together and points 7 and 16 merge together. It is likely that a condensation in which these points are set to be at the same position as each other prior to determining the positions of all the points subject to the constraint of 16-fold symmetry will work well. Similarly, for condensation B, points 1,2, 9 and 10 would appear to merge well and points 3, 4, 11 and 12 would appear to merge well. Attempting to merge points 7 and 16 or points 6 and 13 would now appear less likely to provide an appropriate result.

On the other hand, points 5 and 13 and points 6 and 14 could be candidates which could be forced to be merged to the same position.

The comparison of results for the condensations A and B for 16-fold symmetric 256-NU-I6SM are shown in Figures 29 and 30. The scale of the Figures are expanded in comparison to previous figures and the origin is no longer set to zero for ease of demonstrating the results. As can be seen in Figure 29, between 12 and 14 dB, both of the condensations A and B shown in Figure 27 provide a usefLil gain in capacity. Figure 30 gives a further magnified view of the results of Figure 29 and show how condensation B (shown with diamonds) shows improvement over condensation A (shown with stars) above an SNR value arcund 12 dB.

Better resufts are achieved by the 256-pt consteflations and their condensations from 7.5 to 14 dB, although some of the set underperform the 64-pt family at some SNRs. The very best resuLts generally appear to be at 13 dB SNR, where the best of the set is the uncondensed hybrid 256-NU-BH16SM (some points constrained to 8-fold symmetry and others to 16-fold). At this 13 dB design SNR the order of performance from best to worst (although we are noting trivial capacity differences of only a few miflibit/sym here) is: * 256-NU-SH 1 6SM (uncondensed hybrid) 256-NU--16SM (uncondensed. 16-fold symmetry) * 256-128-con-8H16SM-B tied with 256-128-con-8SM * 256-128-con-16SM-B The tie noted in the third bullet suggests that at this SNR, the hybrid 8H16SM essentially matches the non-hybrid -8SM (which we have not explicitly computed), since their condensations match (if we assume the condensation penalty is similar for each, wh!ch may be arguable). The capacity gap between the first and third is similar to that between the second and fourth, and illustrates the order of the condensation penalty. Furthermore, this condensation penalty slightly exceeds the penalty for fully imposing 16-fold symmetry on the 8-fold symmetric constellation.

Looking at the wider SNR range we see that all 256-pt derivatives are starting to fail above the 13dB optimum. The -B condensation to 128 pts works best at higher SNRs while the -A condensation to 128 pts is better at lower SNRs, with the crossover around 11.5 dB. The greater rigidity of imposing -16SM symmetry does not work so well at lower SNRs, losing out to 64-pt constellations below about 10.5 dB, rather than collapsing to 64-pts. The hybrid -8H16SM arrangement holds on better, but still loses to 64-pt constellations below 8 dB while a more--condensed version of -*8SM does slightly better still, but nevertheless even this does not simply collapse to match 64pts, but rather underperforms it We have appreciated that the further variations may provide improved results. One of the reasons behind the condensation schemes is that they provide capacity results that may equal the corresponding non-condensed scheme, whilst significantly reducing the overhead in computing the constellation positions at the design stage. There can also be a practical advantage in simplifying the calculations that a receiver has to make in decoding, simply because there are fewer candidate transmitted positions to consider.

We have appreciated a further arrangement for devising constellation positions which we will refer to as a "hybrid" arrangement in which most points follow one many-fold symmetry scheme, but at least some points do not follow the same many-fold constraint. Instead, these points may be left unconstrained or, preferably in this hybrid arrangement; with a symmetry constraint different from the main many-fold constraint. An example of this approach is in Figure 31 which demonstrates how a hybrid scheme may be devised in which most points follow a 16-fold symmetry constraint, except the inner points, which are constrained to have 8-fold symmetry. W will refer to this scheme according to the notation 256-8-H I8SM.

Figure 31 actually shows the positions (within one octant) of the spots of a 256-128-Con-8SM constellation, optimised for 13dB SNR, these spots following the design constraint of 8-fold symmetry. It demonstrates that most of these points in practice also reflect naturally about a 22.5° axis, i.e. follow 16-fold symmetry, by plotting as open circles the reflections of the points shown as dark squares. In nearly all cases these circles align with the points marked as light filled circles, showing that 16-fold symmetry could equally have been applied to these points. Just one pair of points do not naturally have this property, so in the proposed hybrid scheme these are left to follow only an 8-fold symmetry constraint (reflecting about a 45° axis) while all other points have 16-fold symmetry imposed, i.e. reflect about a 22.5° axis. An example of the results of such a hybrid scheme, and where in the SNR range it is appropriate is shown in Figure 32.

Receiver Processing Some explanation of the improvement gained using the embodiments of the invenflon may be made by considering the operation of the receiver. A recewer using soft decisions calculates what are known as LLRs, log likehhood--ratios. In knowing what voltage y has been received: the receiver then needs to infer from that information the likelihoods that a 0 or 1 has been transmitted, and the log of their ratio is taken as the soft-decision metric fed to the FEC decoder (error corrector block 120 of Figure 2).

The use of a logarithmic form is convenient, because multiplication of probabihties can be achieved by simple addition: e.g. in implementations of a Viterbi decoder. For simple 2evel signalling (as in 4-QAM) it is easy to show that the LLR is a linear function of voltage y, having a slope proportional to the (linear) SNR. Things get more complicated with higher orders of QAM. A very high SNR the LLR now takes a piecewise linear form, but this becomes more curvy at lower SNRs. The overall gain' still varies with SNR, just as for 4-QAM. It can therefore be useful to consider a normahsed metric, where the LLR has been divided by the SNR, when comparing the metrics calculated at different SNRs, This makes it easier to compare degrees of curviness, and note any movements of the decision boundaries (zerocrossings) as the SNR changes. The vertical grdiines are at. the constellation-spot positions.

At some constellation positions (values of voltage), the lower significant bits (LSB, LSB+1 and LSB+2) provide no contribution. However, when those lower significant bits are at higher voltages (relating to non-merged states) they provide a contribution. We can see that as we go to higher-order BICM-optimised NUQAMs (or their well-chosen ConQAM derivatives) the LSBs become weaker', having dead-zones' in their metrics where they contribute little. Clearly they become in a sense part-time': when the high-significance bits cause non-merged states to be occupied, they have something to offer; when merged states are occupied the LSBs become powerless. In effect it is very like puncturing.

Punctured codes are used as a way to have a family of FEC codes that cover a range of code rates. A good mother code having a low code rate is used as a starting point. When a code of higher rate is needed, that implies that fewer coded bits can he transmitted for a given number of input uncoded bits. One way to achieve this is simply to omit to send some of the coded bits that the mother code has generated. This is done in a systematic pattern known to both transmitter and receiver and is known as puncturing the code. At the receiver, dummy bits are fed to the FEC decoder in those locations in the sequence where the punctured bits were never transmitted, so that the decoder receives the same number as were originafly generated. However, these added dummy bits are marked as erasures, so that the decoder knows not to attach any significance to them. The marking-as-erased simply means that the soft-decis!on metric is set to zero (in effect I have zero confidence in the accuracy of this hit).

Now consider what is happening as we adopt higher-order NUQAM consteflations. We find that BiCM-capacity optimisation eaves some of the consteflation points very close together indeed (and in ConQAM they are deliberately co-located). The consequence is that the receiver metric for the affected bits (the LSB. and some others, depending on the consteUation) is very fiat and equal to zero (or essentiaHy so) for a range of positions around the (nearly) merged positions. So when the received signal is in this range, the soft-decision information is as good as marking an erasure.

The difference between th!s and puncturing is very small. In puncturing, a coded bit is punctured because of where it happens to fall in the coded sequence in relation to the prearranged puncturing pattern. In NUQAM, the essentially-erased bits suffer this fate as a consequence of being mapped at a weak level (eg. the LSB) in a symbol where the high-significance b!ts happen to take a combination which determines that the weak bit' in question is mapped to a (nearly) merged state. But some of the time the same weak bit' is mapped to a constellation position that is well-separated from its neighbours, and then it does make a contribution to the capacity. Suppose a particular application needs to transmit a payload whose uncoded bit rate is equivalent to 6 bit/symbol. Suppose also that we use a particular FEC code of rate 1/2. So it generates 12 coded bits per transmission symbol. We could map all of these coded bits to 4096-NUQAM (or a ConQAM derivative).

To make for easy numbers, suppose the mapping is such that the 2 LSBs are erased' say 1⁄2 of the time, and 2 next-to-LSBs are erased say 1/4 of the time. On average then 10.5 coded bits are received unerased per symboL so the effective code rate becomes 6 ÷ 10.5 = 4/7. if instead we used say 256-QAM (arid assume no flat spots in the metdc), then we can send B coded bits per symbol, and the effective' code rate becomes 6 ÷ B = aM, a rather higher rate: in this case achieved by traditional puncturing. Perhaps by avoiding explicit puncturing, and letting it happen as an incidental yet integral part of the mapping/demapping process of hugh-order NUQAM, we are in some way helping the BICM work more effectively. Similar reasoning applies when condensing non-square constellations as discussed above.

Receiver Benefit with Many Fold Symmetry 1r We can also consider a benefit at the receiver side with using the many fold symmetric schemes. Complexity of calculating LLR metrics is a factor to consider in choosing systems. Rectangular NUQAM or C0nQAM, used without rotation and in a 8150 context, has the advantage that the LLRs are in effect functions of the position in one dimension, as the axes are variables separable.

The complexity is to a degree dependent on the number of points in one dimension, ia the square root of the number of points in the constellation. The number of coded bits determines the number of LLR functions that have to be evaluated, and this goes as the Log2 of th.e parent constellation size. Once rotation s applied to NUQAM/ConQAM then this simplification s lost and the LLRs become functions of position in 2-D.

The non-square constellations outlined here always require the LLRs to be evaluated as a function of 2-D position. just the same as rotated NUQAM/ConQAM. However, there are some saving graces: * non-square constellations give good, oFten the best BICM capacity when using much smaller constellations than for NUQAM/ConQAM, thus reducing both the number of points to be taken into account and the number of LLRs to evaluate * if we were to store pre-computed LLR results in a ookup tabe, then we can find some savings because of the inherent symmetries in the LLR, symmetries that can be enhanced (and the storage reduced) where higher-order symmetries than the foLirfold of prior arrangements give acceptable resufts. This seems to be the case in the SNR range for each non-square consteflation where it is most useful (ic. avoiding very high or very ow design SNRs for that consteHation size).

mposing higher degrees of m!rror symmetry has therefore been used as a way to reduce design-stage computation time and may a'so reduce receiver storage 1 ookup tabhes are used for LLRs. A hybrid 8/16-fdd symmetry was used to simpify design of 256-point consteflations but in this case there is probaby no advantage for receivers over 8-fod symmetry. A noteworthy resuft is that non-square consteflations of 256 points, perhaps condensed to the order of 128 points, signihcanfly Improve the BCM capacity at 13dB compared with NUQAMs/ConQAMs of even higher ev&s of complexity, SUghtly fewer points

Claims (9)

1. CLMMS1. A method of determining non-uniform consteflation positions of a o dimensional modulation scheme, the scheme having words of n coded bits mapped to each consteUation point, for a signal to be transmitted over a channel in a system using a fonNard error corrector (FEC), comphsing: selecting a signal to noise ratio (SNR) appropriate for the channel and the fcard error corrector; constraining the positions of at east some constellation points to have n fold symmetry, where n is an integer>4; and determining the positions of the constellation points that maximise a measure of channel capacity at the selected SNR.
2. 2. A method according to claim 1, wherein n is a power of 2.
3. 3. A method according to claim I or 2, comprising constraining the positions of all points to have n-fold symmetry.
4. 4. A method according to claim I or 2, comprising constraining the position of at least two constellation points to the same position as each other when deteniüning the positions of the constellation points that maximise the measure of channel capacity.
5. 5. A method according to claim 4, wherein th.e constellation points that are constrained to be at the same position as each other are innermost points.
6. 6. A method according to claim 1 or 2, comprising leaving innermost points unconstrained when determining the positions of the constellation points that niaximise the measure of channel capacity.
7. 7. A method according to claim I or 2, comprising constraining the positions of most constellation points to have n fold symmetry and other points to have m fold symmetry, where m is an integer < n.
8. 8. A method according to any preceding claim, wherein n = Born = 16.
9. 9. A method of encoding or decoding a non-uniform two dimensional modulated signal using a constellation scheme, the scheme having words of n coded bits mapped to each constellation point, for a signal to be transmitted over a channel in a system using a forward error corrector (FEC), comprising encoding or decoding using constellation positions, wherein the constellation positions of the mapping scheme are determined by the method of any preceding claim.18. A transmitter for transmitting a non-uniform two dimensional modulated signal of the type having a a modulation scheme with words of n coded bits mapped to each constellation point, for a signal to be transmitted over a channel the transmitter having a forward error corrector (FEC), and comprising: a mapper unit arranged to receive words of n coded bits, and encode these onto the one or more carriers wherein the mapper unit comptises constellation positions of the mapping scheme that have been determined by: selecting a signal to noise ratio (SNR) appropriate for the channel and the forward error corrector; constraining the positions of at least some constellation points to have n-fold symmetry, where n is an integer >4; and determining the positions of the constellation points that maximise a measure of channel capacity at the selected SNR.19. A transmitter according to claim 18, wherein n is a power of 2.20. A transmitter according to claim 18 or 19, wherein the mapper is arranged to constrain the positions of all points to have n-fold symmetry.21. A transmitter according to claim claim 18 or 19, wherein the mapper is arranged to constrain the position of at least two constellation points to the same position as each other when determining the positions of the constellation points that maximise the measure of channel capacity.22. A transmitter according to claim claim 21, wherein the constellation points that are constrained to be at the same position as each other are innermost points.23. A transmitter according to dairn dairn 18 or 19, wherein the mapper is arranged to leave innermost points unconstrained when determining the positions of the consteuation points that maximise the measure of channel capacity.24. A transmitter according to claim claim 18 or 19, comprising constraining the positions of most constellation points to have n fold symmetry and other points to have m fold symmetry, where m is an integer c n.25. A transmitter according to claim 18 or 19; wherein n = 8 or n = 16.26. A receiver for receiving a non-uniform two dimensional modulated signal of the type having a modulation scheme with words of n coded bits mapped to each constellation point, for a signal to be transmitted over a channel the transmitter having a forward error corrector (FEC), and comprising: a de-mapper unit arranged to receive one or more carriers and to decode these to words of n coded bits from each constellation point wherein the de-mapper unit comprises consteflation positions of the mapping scheme that have been determined by: selecting a signal to noise ratio (SNR) appropriate for the channel and the forward error corrector; constraining the positions of at east some constellation points to have n-fold symmetry, where n is an integer > 4; and determining the positions of the constellation points that maximise a measure of channel capacity at the selected SNR.27. A receiver according to claim 26, wherein n is a power of 2.28. A receiver according to claim claim 26 or 27; wherein the de-mapper is arranged to use positions of all points to having n-fold symmetry.29. A receiver according to claim claim 18 or 19, wherein the de-mapper is arranged to use positions of at east two constellation points at the same position as each other.30. A receiver according to claim claim 26, wherein the consteflation points that are constrained to he at the same position as each other are innermost points.31. A receiver according to claim claim 26 or 27, wherein the de-mapper comprises innermost points that are unconstrained when determining the positions of the consteHation points that maximise the measure of channel capacity.32. A receiver according to claim 26 or 27. wherein the demapper uses positions of most constellation points to have n fold symmetry and other points to have m fold symmetry, where m is an integer c n.33. A receiver according to claim 26 or 27, wherein n = 8 or n = 16.