CA1217530A

CA1217530A - Block coded modulation system

Info

Publication number: CA1217530A
Application number: CA000451966A
Authority: CA
Inventors: George D. Forney, Jr.
Original assignee: Codex Corp
Current assignee: Motorola Solutions Inc
Priority date: 1983-04-14
Filing date: 1984-04-13
Publication date: 1987-02-03
Also published as: HK1008718A1; JPS59205860A; JPH0657021B2; EP0122805A2; EP0475464A2; EP0475464B1; EP0475464A3; US4597090A; EP0122805A3

Abstract

ABSTRACT OF THE DISCLOSURE
The invention relates to modulation systems for sending digital data over a band-limited channel in accordance with a code word corresponding to a plurality of modulation signal points drawn from a constellation of available signals. The constellation comprises groups having equal numbers of signals. The code word is selected from an available set of code words on the basis of a block of digital data, the code words belonging to the available set for the block being independent of the signal points cores-pounding to the code word selected for any other block. The system comprises an encoder for selecting said code word, the encoder being arranged so that the group from which at least one signal point for the block is drawn depends on the group from which at least one other signal point for the block is drawn. The system also comprises a modulator for sending the signal points over the channel. The signal points are two-dimensional, and the code words are 2m-dimensional, m being the number of signal points corresponding to each code word.

Description

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This invention relates to high-speed data transmission over band-limited channels, such as telephone lines.
Modulation systems for such channels commonly use two-dimensional carrier modulation, generically called double-sideband-quadra-ture-carrier (DSB-QC) modulation. Such modulation systems are discussed, for example, in United States patent 3,887,768 (Forney/Gallager), which also shows an implementation of such a system.
Conventional DSB-QC systems are used to send an integer number (N) of bits in each modulation interval of T seconds in a nominal bandwidth of l/T Ho. For example, some telephone line modems send 4 bits per modulation interval of 1/2400 sec. within a nominal bandwidth of 2400 Ho, thus achieving a 9600 bits per second (bus) data transmission rate. Modems for sending 6 bits per modulation interval to achieve 14,400 bus in a nominal 2400 Ho bandwidth are also available.
Such DSB-QC systems use signal constellations of ON
signals for sending N bits per modulation interval. A family of such constellations with signals arranged on a regular rectangular grid is described it Campopiano and Glazer, "A Coherent Digital Amplitude and Phase Modulation Scheme," IRE Transactions on Comma-nication Systems, Vol. SHEA, pp. 90-95, March 1962.
Drawings Figure 1 is a diagram of a family of five prior art signal constellations.
Figure 2 is a block diagram of modem apparatus.
Figure 3 is a diagram of a signal constellation in ~Z~7530 accordance with the preferred embodiment.
Figure 4 is a block diagram of the signal selection logic for the constellation of Figure 3.
Figure 5 is a table showing the correspondence between input bits and signal subsets.
Figure 6 is a state-transition diagram (trellis) for use with the constellation of Figure 3.
Figure 7 is a diagram of a family of four signal con-stylishness in accordance with an alternate embodiment.
Figures 8, 12, 15, 17, and 20 are block diagrams of signal selection logic in accordance with alternate embodiments.
Figures 9, 10, 11, 14, 16, and 19 are diagrams of signal constellations in accordance with alternate embodiments.
Figures 18 and 21 are diagrams of the relationships of signal quadruplets in the constellations of Figures 16 and 19, respectively.
Figures 13, 22, and 23 are trellises for use with alter-Nate embodiments.
Figure 1 shows the arrangement of signals in the Campopiano and Glazer constellations and the outer boundaries of those constellations for values of from 4 through 8. Table I
shows the so-called "required signal-to-noise ratio" P (defined in Campopiano et at. and in the Forney/Gallager patent) for the constellations of Figure 1. Each unit increase in corresponds to an increase in required slgnal-to-noise ratio of about a factor of two or 3.0 decibels (dub).

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Table I (Campopiano and Glazer) N S P (dub) 4 lo 10 10.0 32 20 13.0 6 64 42 16.2 7 128 82 19.1 8 256 170 22.3 N = number of bits sent per modulation interval S = number of signals in signal constellation P = required signal-to-noise ratio (dub) = P measured in decibels For higher transmission speeds, so-called coded module-lion techniques can provide improved resistance to noise and other channel impairments: that is, they can provide a reduction in required signal-to-noise ratio, or a so-called "coding gain".
In such coded systems, to send bits per modulation interval, signal constellations having more than ON signals are used, and coding is used Jo introduce dependencies between module-lion intervals so that the set of available signals from which a signal point can be selected in one modulation interval depends in general on the signal points selected for other modulation intervals.
In one coding technique for getting a coding gain (disclosed in Csajka et at, United States Patent 4,077,021, and Ungerboeck, "Channel Coding with Multilevel/Phase Signals," IEEE
Transactions on Information Theory, Vol. IT-28, pp. 55-67, January, 1982), the N bits appearing in each modulation interval are individually mapped into signal points selected from a con-stellation of 2~+1 signals. The signals in the constellation are organized into subsets such that the minimum distance between ~753~
two signals belonging to one subset is greater than the minimum distance between any two signals in the constellation. The selection of the signal point for each input bits is made to depend, in part, on the historical sequence of all previously selectee signal points, as represented by the state of a finite state device in the encoder. This so-called trellis coding effectively permits only certain sequences of signal points to be transmitted, and the coded historical information carried by every signal point is exploited at the receiver by a maximum likelihood sequence estimation technique (e.g., one based on the Viterbi algorithm, as described in Foreign, "The Viterbi Algorithm,"
Proceedings of the IEEE, Vol. 61, pp. 268-278, March, 1973.
A coding gain can also be realized using a block coded modulation system in which blocks of n input bits are sent in m modulation intervals, so that N=ntm bits are sent per modulation interval. For each block, m signal points are selected from a constellation having more than ON signals, a process which is equivalent to mapping each block into a code word selected from an available code word set of on code words arranged in 2m-dimen-signal space (called simply 2m-space), with the em coordinates of each code word, taken two at a time, defining the respective pair of coordinates in two-dimensional space of the m signal points to be selected. The code word set from which the code word for any block may be drawn is independent of the signal points selected for any other block. At the receiver, the decisions on which signal points were sent are based on the received signal points for each block (the so-called received word), preferably using maximum likelihood decoding.
One method of block coding involves using a code word set arranged on a finite portion of a densely packed infinite geometrical lattice in 2m-space; see, for example, Conway and Sloane, "Fast Quantizing and Decoding Algorithms for Lattice Qua-titers and Codes," IEEE Transactions on Information Theory, Vol. IT-2B, pp. 227-232, March 1982, and the references cited therein. A representative system of this type is disclosed in "Block Coding for Improved Modem Performance," a Canadian contra-button Tom XVII-No. 112) to Study Group XVII of the International Telegraph and Telephone Consultative Committee (C.C. IT ), March 1983, which describes an 8-space code for sending 4 bits per mod-lotion interval with an asymptotic coding gain of 3.4 dub over the encoded Campopiano and Glazer 16-signal constellation (defined by the No boundary in Figure 1).
The invention includes a family of block coded module-lion systems of progressively greater complexity exhibiting pro-gressively greater coding gain over encoded systems.
In general, in one aspect the invention features, in such block coded modulation systems, the improvement in which the signal constellation includes groups having equal numbers of sign nets, and the encoder is arranged so that the group from which at least one signal point for a block is drawn depends on the group from which at least one other signal point for the block is drawn, whereby a coding gain over encoded systems is achieved.
Thus, in accordance with a broad aspect of the invent lion, there is a provided a modulation system for sending digital lZ~753C~
data over a band-limited channel in accordance with a code word corresponding to a plurality of modulation signal points drawn from a constellation of available signals, said code word being selected from an available set of said code words on the basis of a block of said digital data, the code words belonging to said available set for said block being independent of said signal points corresponding to the code word selected for any other said block, said system comprising an encoder for selecting said code word, said constellation comprising groups having equal numbers of said signals, said encoder being arranged so that the group from which at least one said signal point for said block is drawn depends on the group from which at least one other said signal point for said block is drawn, and a modulator for sending said signal points over said channel, said signal points being Todd-tensional, said code words being 2m-dimensional, m being the numb bier of said signal points corresponding to each said code word, m being greater than two, each said block of said digital data comprising my bits.
In accordance with another broad aspect of the invention there is provided a modulation system for sending digital data over a band-limited channel in accordance with a code word cores-pounding to a plurality of modulation signal points drawn from a constellation of available signals, said code word being selected from an available set of said code words on the basis of a block of said digital data, the code words belonging to said available set for said block being independent of said signal points cores-pounding to the code word selected for any other said block, said lZ~7531~
system comprising an encoder for selecting said code word, said constellation comprising groups having equal numbers of said sign nets, said encoder being arranged so that the group from which at least one said signal point for said block is drawn depends on the group from which at least one other said signal point for said block is drawn, and a modulator for sending said signal points over said channel, said signal points being two-dimensional, said code words being 2m-dimensional, m being the number of said signal points corresponding to each said code word, each said block of said digital data comprising my bits, W being an integer repro-setting the number of bits of said digital data per signaling interval, m being an even integer no smaller than 2, said constel-lotion comprising 1.5x2N signals.
In accordance with another broad aspect of the invention there is provided a modulation system for sending digital data over a band-limited channel in accordance with a code word cores-pounding to a plurality of modulation signal points drawn from a constellation of available signals, said code word being selected from an available set of said code words on the basis of a block of said digital data, the code words belonging to said available set for said block being independent of said signal points cores-pounding to the code word selected for any other said block, said system comprising an encoder for selecting said code word, said constellation comprising groups having equal numbers of said sign nets, said encoder being arranged so that the group from which at least one said signal point for said block is drawn depends on the group from which at least one other said signal point for said - pa -~2~753~

block is drawn, and a modulator for sending said signal points over said channel, said signal points being two-dimensional, said code words being 2m-dimensional, m being the number of said signal points corresponding to each said code word, and wherein said encoder is further arranged so that my said signal points for each said block may be drawn from among all said signals in said constellation, and the one remaining said signal point for said block is drawn from a group that depends on the groups from which said m-1 signal points are drawn, and wherein said group from which said one remaining signal point is drawn depends on a single-parity-check code based on at least one of said bits.
In accordance with another broad aspect of the invention there is provided a modulation system for sending digital data over a band-limited channel in accordance with a code word cores-pounding to a plurality of modulation signal points drawn from a constellation of available signals, said code word being selected from an available set of said code words on the basis of a block of said digital data/ the code words belonging to said available set for said block being independent of said signal points cores-pounding to the code word selected for any other said block, said system comprising an encoder for selecting said code word, said constellation comprising groups having equal numbers of said sign nets, said encoder being arranged so that the group from which at least one said signal point for said block is drawn depends on the group from which at least one other said signal point for said block is drawn, and a modulator for sending said signal points over said channel, said signal points being two-dimensional, said - 6b -code words being 2m-dimensional, m being the number of said signal points corresponding to each said code word, and wherein said encoder is further arranged so that m-1 said signal points for each said block may be drawn from among all said signals in said constellation, and the one remaining said signal point for said block is drawn from a group that depends on the groups from which said m-1 signal points are drawn, and wherein said group from which said one remaining signal point is drawn is determined by said encoder based on a state-transition trellis for said block In accordance with another broad aspect of the invention there is provided a modulation system for sending digital data over a band-limited channel in accordance with a code word cores-pounding to a plurality of modulation signal points drawn from a constellation of available signals, said code word being selected from an available set of said code words on the basis of a block of said digital data, the code words belonging to said available set for said block being independent of said signal points cores-pounding to the code word selected for any other said block, said system comprising an encoder for selecting said code word, said constellation comprising groups having equal numbers of said sign nets, said groups each comprising subsets having equal numbers of said signals, said encoder being arranged 50 that the group and subset from which at least one said signal point for said block is drawn depends on the group and subset from which at least one other said signal point for said block is drawn, and a modulator for sending said signal points over said channel.
In accordance with another broad aspect of the invention - 6c -lZ17S30 there is provided in a modulation system for sending a block of digital data bits over a band-limited channel using a plurality of modulation signal points drawn from a two-dimensional constella-lion of available signals, the improvement wherein said constella-lion comprises a plurality of inner signals, and a plurality of outer signals located farther from the origin than said inner signals, one bit of said digital data determines whether any of said plurality of signal points will be drawn from said outer signals, and if an outer signal will be drawn, at least one other bit of said digital data determines which of said plurality of signal points will be an outer signal point.
In preferred embodiments, the signals are arranged (e.g., on a rectangular grid) so that the minimum squared distance between two signals belonging to the same group is greater than (e.g., twice the distance) between two signals belonging to different groups; the signal points are two dimensional, the code words are 2m-dimensional, m being the number of signal points corresponding to each code word, and each block of digital data has my bits; each block comprises a plurality of bits, and the groups from which the signal points for the block are drawn are determined by a single bit of the block; N is an integer m is an even integer no smaller than 2, and the constellation has 1.5x2N
signals; Nil r being an integer m is no smaller than 2, and the constellation comprises 2r+1 signals; the encoder is further arranged so that m-1 signal points for each block may be drawn from among all signals in the constellation, and the one - Ed -~L217530 remaining signal point for the block is drawn prom a group that depends (e.g., by means of a single-parity-check code or a state-transition trellis) on the groups from which the m-l signal points are drawn; m is 2; and N is 4 or 4 1/2.
In other embodiments, the groups each have subsets having equal numbers of signals, and the encoder is further arranged so that the subset from which at least one signal point for the block is drawn depends on the subset from which at least one other signal point for thy block is drawn; the signals are arranged (e.g., on a rectangular grid) so that the minimum squared distance between two signals belonging to one subset is greater than (e.g., twice the distance) between two signals belonging to different subsets within the same group; the subsets from which the signal points for a block are drawn are determined based on a plurality of bits representing less than all of the digital data of the block; N is an integer, m is at least 4, and the constellation has 2N+1 said signals; ~=r+l/2, r being an integer, m is an even integer no smaller than 4, and the constellation has 1.5x2r+l signals; the encoder is further arranged (e.g., by means of a single-parity-check code, a Hamming code, or a state-transition trellis) so that one signal point for each block may be drawn from among all signals in the constellation, m-2 of the signal points are drawn from groups that depend on the group from which that one signal point is drawn, and the single remaining signal point for the block is drawn from a subset that depends on the subsets from which the one signal point and the m-2 signal points are drawn; m is 4; and N is 4.

In other embodiments, the subsets each comprise classes having equal numbers of signals, and the encoder is further arranged so that the class from which at least one signal point for the block is drawn depends on the class from which at least one other signal point for the block is drawn; the minimum squared distance between two signals belonging to one class is greater than between two signals belonging to different classes within the same subset; each block has a plurality of bits, and the classes from which the signal points for the block are drawn are deter-mined based on a plurality of bits representing less than all of the digital data of the block; m is 8; is an integer and the constellation has 1.5x2N+1 signals; ~=r+l/2, r being an into-germ m is 8, and the constellation comprises 2r+2 signals.
In other embodiments, the classes each comprise sub-classes having equal numbers of signals, and the encoder is further arranged so that the subclass from which at least one signal point for the block is drawn depends on the subclass from which at least one other signal point for the block is drawn; the minimum squared distance between two signals belonging to one subclass is greater than (e.g., twice the distance) between two signals belonging to different subclasses within the same class; m is 12; N is an integer and the constellation has 2N+2 signals;
the subclasses from which the signal points for the block are drawn are determined by the encoder based on a Golly code applied to at least one of the bits; m is 12 and is 4; and the sub-classes from which the signal points for the block are drawn are determined based on a plurality of bits representing less than all 1~753~
of the digital data of the block.
In other embodiments, there are means for selecting at least one signal point of a block on the basis of less than all digital data in the block, the constellation has quadruplets each having four signals located at the same distance from the origin but separated by 90 intervals about the origin, the digital data comprises bits, and at least two of the bits are quadrant ally differentially encoded; the four signals belonging to each quad-replete are drawn from four different subsets; there is also a decoder arranged to decide which code word was sent based on maxim mum likelihood sequence estimation in accordance with the Viterbi algorithm; there is a demodulator having a decoder arranged to make tentative decisions about which signal point was sent prior to final decoding of all of the received signal points for a block, and the demodulator comprises adaptive control circuitry arranged to be responsive to the tentative decision; the signal points for each code word are drawn from the same group; the decoder is arranged to decide which code word was sent by first making a separate tentative decision based on code words whose signal points are drawn from each group, and thereafter a final decision based on the separate tentative decisions; selection of signal points for a block depends on a single-parity-check code based on at least one of the bits of the digital data; and -the decoder is arranged to decide which code word was sent by first making a tentative decision as to each signal point in the code word without regard to whether the parity check is satisfied, and accepting the tentative decisions as the final decision, either _ g .

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without changes if the parity check is satisfied, or after changing the least reliable one of the tentative decisions in the parity check is not satisfied.
In another aspect, the invention features, in a module-lion system for sending a block of digital data bits over a band-limited channel using a plurality of modulation signal points drawn from a two-dimensional constellation of available signals, the improvement in which the constellation has a plurality of inner signals, and a plurality of outer signals further from the origin than the inner signals, one bit of the digital data deter-mines whether any of the plurality of signal points will be drawn from the outer signals, and if an outer signal will be drawn, at least one other bit of the digital data determines which of the plurality of signal points will be an outer signal point.
In preferred embodiments, there are it signal points, the digital data comprises at least to bits, there are S inner signals and its outer signals, and t bits determine which signal point will be an outer signal point; to the block has 2N+1 bits and S is ON, N being an integer; I of the bits determine which outer signal is drawn, and N bits determine which inner signal it drawn; N is 4; t is 2; each block has 4N+l bits and S is ON, being an integer; and N-2 of the bits determine which outer signal is drawn and N bits determine which inner signal is drawn.
The 4-space, 8-space, 16-space, and 24-space modulation systems of the invention respectively exhibit asymptotic coding gains of about 1.5, 3.0, 4.5, and 6.0 dub over encoded systems that ~12~7530 send the same number of bits per modulation interval -through the same nominal bandwidth using DSB-QC modulation. The block coded systems of the invention are slightly inferior in performance to the best known prior art schemes (e.g., about .2 dub and .4 dub worse for 4-space and 8-space systems, respectively), but have substantial advantages in implementation. The invention uses constellations having relatively small numbers of signals. The encoding techniques are economical, simple, and easily imply-minted. Encoding delay is reduced because early signal points selected for a given block can be sent before encoding of the block has been completed. Decoding delay is small because received signal points can be finally decoded as soon as all signal points for a given block have been received. Useful tent-live decoding decisions can be made even sooner. Non-integral numbers of bits per interval can be sent. ~uadrantal differential encoding can be used to improve immunity to phase hits. Economic eel, simple, and easily implemented decoding techniques are made possible.
Other advantages and features of the invention will be apparent from the following description of the preferred embody-mint, and from the claims.
In the following section, the preferred embodiment (an 8-space system sending 4 bits per modulation interval and achieving an asymptotic coding gain of 3.0 dub) is described first.
Alternative embodiments thereafter described are other decoding and encoding techniques for 8-space block coded modulation systems; 8-space block coded systems for sending any integral ~2~7530 number of bits per modulation interval; multidimensional signal structures for sending a non-integral number of bits per module-lion interval (useful in both coded and encoded systems); 4-space, 16-space, 24-space, and 2m-space block coded systems; and other embodiments.
Structure and Operation -In the preferred embodiment (an 8-space block coded modulation system for sending 4 bits per modulation interval), a block of 16 bits is sent in each 4 modulation intervals using a signal constellation having 32 signals.
Referring to Figure 2, in transmitter 10 the bits appearing in an input bit stream 12 are grouped by serial/parallel converter 14 into 16-bit blocks 16 (i.e., n=16). Each block 16 is encoded by signal point selection logic 18 into a sequence of four (i.e., my 2-dimensional signal points 20 (which are then used for conventional DSB-QC modulation by modulator 22 for transmit-soon over channel 24).
Referring to Figure 3, the signal points are selected from a two-dimensional signal constellation 30 having 32 signals (i.e., twice the number of signals (16) needed for encoded module-lion at 4 bits per modulation interval, as with the signal con-stellation shown within the I boundary of Figure 1). The 32 signals are arranged in a rectangular grid with integer keyword-notes, each signal point having one odd and one even coordinate (in an arrangement like the Campopiano and Glazer constellation shown within the I boundary of Figure 1, but rotated by 45) and are divided into four disjoint subsets (A, Al, Boy By) each ~;~17530 having eight signals. The four subsets A, Al, Boy By are are arranged in two groups (A, comprising subsets A and Al, and B, comprising By and By) of two subsets each. The arrangement of signals in the subsets and groups is such that the minimum squared distance Dow) between any two signals on the plane (e.g., between an A signal and a B signal) is smaller than the minimum squared distance (Dow) between any two signals in one group (e.g., between an A signal and an Al signal, which is in turn smaller than the minimum squared distance Dow) between any two signals in the same subset (e.g., between an A signal and another A signal).
Signal selection logic 18 is arranged to select, for each block of input bits, a sequence of four signal points which are interdependent, i.e., the signal group and/or subset from which at least some of the signal points for a given block may be selected depends in part on at least one of the other signal points selected for that block. The code word set from among which the code word for a given block may be drawn is independent of the signal points selected for any other block.
Each of the 32 constellation signals can be uniquely specified by a total of 5 bits: one bit naming the group from which it is taken, one bit naming the subset within that group, and three bits naming which of the eight signal points within the named subset is to be selected. Thus the four signal points for a block could be uniquely specified by a total of 20 bits. However, the block has only 16 input bits on which to base the selections.
Referring to Figure 4, the group, subset, and signal ~217S30 point selection information needed to select each of the four signal points for each block it derived from the 16 input bits (bluebell) as follows.
Group selector 32 uses bit by (called the group bit) to determine the one group from which all four signal points for a given block is to be drawn. (The designation group refers to the group from which the first signal point is selected, and so on.) Group through group are therefore always the same group for a given block. For example, if blue all four signal points would be selected from group I; if bluely, all would be from group B.
Subset selector 34 uses bits by, by, and by to determine from which subsets of the selected group each of the four signal points is to be drawn. Bits by, by, and by respectively determine subsoil, subset, and subset. For example, in a block for which the signal points are from group A, if b2=0 the first signal point would be selected from subset A, otherwise (if b2=1) from Al.
Subset is a parity bit generated by subset selector 34 such that bits by, by, by and the parity bit (called subset bits) include an even number of 1 bits. Thus the subset bits can be viewed as coded bits derived by a (4, 3) single-parity-check code from the input bits b2-b4. The remaining bits (bs-bl6) taken three at a time (called signal point bits) specify which particular four signal points (signal pointl-signal point) are to be drawn by signal point selector 36 from the selected 8-point subsets. Thus group selector 32 and subset selector 34 assure that the group from which at least one of the signal points for a given block is selected depends on the group from which at least one of the other 12~7S30 signal points for the block is selected (thereby reducing, after the first signal point, the number of groups from which later signal points for a block can be selected), and to assure that the subset from which at least one of the signal points is selected depends on the subset from which at least one of the other signal points is selected (thereby reducing, after the first signal point, the number of subsets from which later signal points for a block can be selected).
Although the signal selection (coding) has been explained as the selection. of four successive signal points from a

2-dimensional signal constellation, it could also be viewed as an 8-space block coded modulation technique in which a set of 216 code words are arranged on a lattice in 8-space, and coding consists of selecting one of the code words for each block. The eight coordinates of the selected code words could then be taken two at a time to specify the four two-dimensional signal points to be used for carrier modulation.
The required signal-to-noise power ratio for the 32-point signal constellation of Figure 3 is P=10, the same as for the I
signal constellation of Figure 1. The minimum squared distance between code words in 8-space is 8, double the minimum squared distance (4) between signals in the No Figure 1 constellation, providing an asymptotic coding gain of a factor of 2, or 3.0 dub.
That the minimum squared distance between code words in 8-space is 8 can be seen as follows. Two different code words which have the same group bit and the same subset bits must differ in at least one of their respective four signal points. Because those two different signal points must be drawn from the same subset, and signal points in the same subset have a minimum squared distance of 8 on the two-dimensional plane, the two code words in 8-space likewise have a minimum squared distance of 8.
Similarly, two code words which have the same group bit but different subset bits must differ in at least two of their subset bits (because the subset bits have even parity). At least two of their respective four signal points must therefore differ in each case by a minimum squared distance of 4 (the minimum squared distance on the two-dimensional plane between two signal points chosen from a given group) for a total minimum squared distance of 8.
Likewise, two code words which have different group bits differ in all four of their signal points, and the minimum squared distance between two signal points on the two-dimensional plane is 2 for a total minimum squared distance of 8. Thus, in all possible cases, the minimum squared distance is 8.
The theoretical coding Cain of 3.0 dub is partially off set by the effect of the error event probability coefficient (see Foreign Viterbi article), which can be estimated as follows. In 8-space, each code word has up to 240 nearest neighbors (16 which have the same group and subset bits, 96 which have the same group bits but different subset bits, and 128 which have different group and subset bits). But because code words near the outer boundary of the code word set have fewer neighbors, the average code word only has about 180 near neighbors. The error event probability coefficient per modulation interval is therefore of the order of ~2~7~3~
180/4 = 451 corresponding to a fraction of a decibel of loss in performance for error probabilities of interest.
At the receiver, decoding of the four signal points into the corresponding bit stream is optimally done by maximum likely-hood decoding, i.e., picking the one code word which is most likely to have been sent, given the 8-space received word deter-mined by the four received two-dimensional signal points.
For each of the received two-dimensional signal points, the decoder tentatively finds (by conventional slicing in two-dimensional space) the closest (in Euclidian distance) signal from A group and the closest signal from the B group to the received signal point, noting, on a running basis, which of the coordinates of the tentatively selected signals for each group is least reliable (i.e., has the greatest apparent error). The sub-set bits of the four tentatively selected A signals are checked for parity. If parity checks, then the four tentatively selected A signals define the closest code word in the A group to the received word. If parity fails to check, the closest A code word is found by changing the least reliable coordinate to the keyword-Nate of the next closer A signal, which (because of the arrange-mint of the subset points in the constellation) will always cause a change of that signal from an A to an Al signal (or vice versa), and thus yield a code word of correct parity. An analog gout slicing and parity checking sequence finds the closest B code word to the received word.
The final decoding decision entails choosing whether the previously determined closest A code word or closest B code word ~:217530 is closer (in overall Euclidian distance) to the received keyword-notes in 8-space. The choice may be made by computing the sum of the squared coordinate differences between each of the two code words and -the received word and comparing the sums.
Decoding thus requires only slicing (twice) in each coordinate, storing (for A and B code words respectively) the location and magnitude of the largest apparent error so far, awoke-mutating the Euclidian distances for the best code words from each group, checking the parity of each best code word (and changing one coordinate in each code word, if needed, thereby changing the Euclidian distance for that word), and comparing the Euclidian distances for the resulting two best code words to select the better one. This decoding method may be readily and simply imply-minted (by conventional techniques) using a programmable MicroPro censor.
The block coded modulation system can be made trays-parent to 90 phase rotations between the received carrier and the transmitted carrier using quadrant Al differential coding on a block basis. The signal constellation of Figure 3 is arranged in quadruplets of signals having the same radii but separated by 90 phase differences, each quadruplet having one signal from each subset, so that a 90 clockwise rotation translates, within each quadruplet, the A signal to the By signal, the By signal to the Al signal, the Al signal to the By signal, and the By signal to the A signal; and analogous translations occur for 180 and 270 rotations. For example, the signals denoted 37 in Figure 3 form such a quadruplet. Referring to Figure 5, if each signal's subset ~2~7531~
and group bits are taken together as a 2-bit integer, 90 clock-wise rotation of any signal corresponds to an increment of 1 (module 4) in the value of that integer. Quadrant Al differential coding can then be accomplished by assigning the same three signal point bits to all four of the signals within each quadruplet, and coding the subset and group bits for each signal point in a code word by differential four-phase coding, using for example the subset and group bits of the last signal point of the previous code word as the reference.
If (Al, x2) denotes the differentially encoded subset and group bits of the last signal point of the previous code word, and (Ye, Ye) denotes the subset and group bits of a signal point in the current code word (before differential encoding), then (Al, I the differentially encoded subset and group bits of that signal point in the current code word, is determined by adding (Al, x2) to (Ye, Ye) as two-bit integers, module four. Because z2=x2 + Yo-yo the same differentially encoded group bit is used for all signal points in the block. At the receiver, assume that both the previous and the current code words are correctly decoded (except for a possible phase rotation of an integral multiple of 90). If the received bits are denoted (Al', x2') and (Al', Z2'), then (Yule Ye) is determined by subtracting (Al', x2') from (Al', Z2') as two-bit integers, module four. The resulting (Yule Ye) value will then be correct even if both (Al', x2') and (Al', Z2') have been advanced by p module 4 as a result of a p x 90 phase rotation.
Because decoding of a code word can be finished as soon as all four received signal points are determined, the decoding lZ~7~3~
delay and error propagation of this block coding system are strictly limited to four modulation intervals. Further, because tentative signal point decisions (made before all four signal points have been received) are only about 3 do less reliable than in the encoded 16-signal constellation of Figure 1, those tent-live decisions may be used to update adaptive equalizers and other tracking loops in the receiver without even waiting for all four signal points to be received. The tentative decisions are made by finding the closest signal on the constellation of Figure 3 to each received signal point, which can be accomplished by rotating the coordinates 45~ with respect to the signal constellation, followed by conventional slicing in each of the two rotated goof-donates. At the transmitter, transmission can begin as soon as five bits (by, by, by, by, by) nave been collected, thereby reducing the encoding delay which would exist for other kinds of block coding in which all of the bits of the block would have to be collected before encoding could begin.
Other Embodiments Other Decoding and Encoding Techniques for 8-Space Block Coded I Systems In another embodiment, maximum likelihood decoding can be accomplished by applying the Viterbi algorithm (in a manner described in Furriness Viterbi article, cited above).
Referring to Figure I, in order to apply the Viterbi algorithm, the encoding process can be illustrated by a finite-length 4-state trellis 40. The trellis shows from left to right the state of the encoder at five successive times, beginning with ISLES
a node 42 reflecting the time immediately before encoding of a new block begins and ending with a node 44 reflecting the time just after the encoding of that block has been completed. In between nodes 42, 44 are shown four modulation intervals separated by three time instants, each such instant being represented by a column of four nodes 46 corresponding to the four states which the encoded can occupy at that instant. Branches 48 from each node to the next nodes to its right correspond to modulation intervals and reflect transitions of the state of the encoder which can occur as a signal point is selected. Each branch 48 is labeled with a letter to indicate the subset corresponding to the signal point associated with that modulation interval. Each node 46 is marked with a two-bit value, the first bit reflecting the group bit for all signal points for that block (i.e., 0 = A, 1 = B), and the second bit reflecting the value of the accumulated parity check subset bit as of that time.
The encoder begins at a common node 42 (reflecting that the encoder's state does not depend on historical information).
From node 42 the encoder branches to one of four states depending on the group and subset bits of the first selected signal point.
For example, if the first selected signal point is from subset A, the next encoder state is at node 50. For the next two signal points, the encoder state stays in the half of the trellis defined by the group bit of the first signal point, but switches states (from one parity to the other) in accordance with the transitions shown. Only 16 of the 32 signals in the constellation are avail-able for selection of the second and third signal. The transition ~Z~7530 to the final node 44 is determined by the parity state of the encoder (e.g., an Al or By point is sent if the parity state is 1, otherwise an A or By point is sent). Only a subset of 8 of the 32 signals in the constellation are available for selection of the fourth signal point, and that subset depends on the first three selected signal points. Thus the paths through the trellis define the possible combinations of subsets from which the sequence of four signal points for each block can be chosen.
In decoding each received word, in accordance with the Viterbi algorithm, decisions are first made (using conventional slicing in two dimensions, with the coordinate axes rotated 45) of which of the eight signals in each of the four subsets is closest to each received signal point. A branch in trellis 40 (corresponding to each subset) is then associated with the closest signal in its corresponding subset, and the so-called metric of each branch is the squared distance between the received signal point and the corresponding closest signal associated with that branch. The remaining steps of the Viterbi algorithm decoding proceed as described in Furriness V turbo article, cited above. A
final decoding decision is possible after four modulation inter-vets, at which time all trellis branches converge to a single node 44.
In another embodiment, the input bits may be coded (using a Hamming code) into coded bits which are used to select the subsets from which the signal points may be selected.
Four bits in each block, blob, rather than being used in a group selector and subset selector (as shown in Figure 4) to select the group and subset from which each signal point is to be selected, are encoded into 8 coded bits z1-zg using the following conventional extended (8, I binary Hamming code (Hamming codes are discussed, e.g., in Berlekamp, Algebraic Coding Theory, McGraw-Hill, 1968):

sly = by Z2 = by + by z3 = by z4 = by + by us = by Z6 = by + by z7 = by by + by Z8 = by + by + by + by which can be generated using, for example, the following generator matrix (having minimum Hamming distance 4):

Bits Al Z8 are taken in pairs to select the subsets of Figure 3 from which the four signal points are to be drawn in accordance with the following table:
Bit Pair Subset 01 By B
11 Al The Hamming distance (do) between bit pairs is then equal to half the minimum squared distance between signals in the cores-pounding subsets, i.e., Dow, Boy = Dow, By) = do (Al, Boy = do (Al, By) = 2, corresponding to do (00, 01) = do (, 10) =

~Z17530 dew (11, 01) = do (11, 10) = 1; and do (A, Al) = do (By, By) =
4, corresponding to do (00, 11) = do ~01, 10) = 2.
The minimum squared distance between code words in 8-space is then I. For if two code words correspond to different Hamming coded bits ~l-Z8, then these bits differ in at least 4 places, which (because the Hamming distance is half the subset distance, as just described) implies that the squared distance between the code words is at least 8. Alternatively, two code words corresponding to the same Hamming coded bits Z1-Z8 have signal points all in the same subsets in each modulation interval;
in at least one interval the signal points must be different, and since they belong to the same subset the minimum squared distance between them is 8.
In other embodiments, the selection of the four subsets to be used in the four modulation intervals can be done by any method (in addition to the three already described) which produces minimum squared distance of 8 between code words (based on the distance properties of the subsets in 2-space). The 12 bits used to select the particular signal point for each modulation interval I (or the original 16-bit block) can be arbitrarily transformed.
The constellation can be any signal constellation that can be divided into four eight-point subsets with the required distance properties.
8-Space Block Coded Systems for Sending Any Integral umber of Bits per Modulation Interval In other embodiments, any number of bits (N) per module-lion interval can be sent by using a signal constellation on a 12~753g rectangular grid having 2N+l signals (e.g., constellations such as the ones shown in figure 1), divided into four equal sub-sets (each having 2N-l signals) and arranging the signals so that alternate signals fall respectively in groups A and B, and alternate signals within each group fall respectively in subsets (A, Al; Boy By). Four bits of each block of ON bits are used to select the four subsets by any method (including those described above) which assures a minimum squared distance between code words of do (where do is the minimum squared distance between signals in the constellation). The remaining 4 (N-l) bits are used to select which signals within the selected subsets are to be sent in each interval.
Table II shows the required signal-to-noise ratios P for values of lo from 3 through 7 using the constellations of Figure 1 with S signal points and the 8-space block coding system described above:
Table II
N S P (dub)

3 16 2.5 4.0

4 32 5 7.0 64 10.5 10.2 6 128 20.5 13.1 7 256 42.5 16.3 Compared with the encoded systems treated in Table I, about 3 dub of coding gain is achievable for any integer value of N, using a signal constellation of size 2N+1 rather than ON.
Multidimensional Signal Structures for Sending a Non-Integral Number of Bits per Modulation Interval In other embodiments, signal systems of multiple Nat ` ~Z~75~0 least four) dimensions are used for sending a non-integral number (~) of bits per modulation interval, where N is of the form of (r an integer, f a fraction between 0 and 1). Such systems send blocks of bits (each block longer than one modulation interval), and are efficient for use in encoded systems (as well as coded systems) because their required signal-to-noise ratio is only about fx3 dub more than for a comparable Figure 1 signal constella-lion for sending r bits per modulation interval, and because the number of signals in the constellation is about (l+f)2r.
Referring to Figure 7, signal constellations for use in sending ~=r+l/2 bits per modulation interval each comprise the or signals in the Nor constellation of Figure 1 called inner signals) plus an additional 2r-1 signals (called outer signals) generally arranged further from the origin, on the same grid, and with -the aye symmetries, as the inner signals.
To send blocks of 2r+1 bits in two modulation intervals, a first bit of -the 2r+1 bits determines whether any outer signal is to be sent. If not, the remaining or bits taken r at a time determine the two inner signals for the two intervals. If so, a second bit determines whether the outer signal will be sent in the first or second modulation interval, r-l bits determine which outer signal, and the remaining r bits which inner signal is sent in the other interval. The average power required to send both signal points will be 3/4 the average power required to send inner signals (which art sent on average 3/4 of the time) plus 1/4 the average power required to send outer signals (which are sent on average 1/4 of the time).

~Z~7~30 For example 4, 1/2 bits per modulation interval (r=4) can be sent in 2 modulation intervals using the constellation having 24 signals (within the I 1/2 boundary of Figure 7) divided into an inner group of 16 signals, arranged as shown in Figure 1 I and an outer group of 8 signals, with the arrange-mint of signals exhibiting quadrant Al symmetry.
Referring to Figure 8, in the signal selection logic, bit by is used by outer point selector 60 to determine whether none or one of the two selected signal points will be an outer signal. If one will be, then bit by is used by selector 60 to determine which interval it will be in, and bits b3-bg are used by signal point selector 62 to select the outer and inner signals as shown in Figure 8. If neither signal point is to be an outer signal, then bits bobs and b6-bg are used by signal point selector 62 to select respectively the two inner signals.
The average power required to send inner points is 10.0, and to send outer points 26Ø The required signal-to-noise ratio is therefore 14 (11.5 dub), which is 1.5 dub more than the require-mint for sending 4 bits per interval using the 16-signal constel-lotion of Figure 1.
Table III shows the numbers of constellation signals Sand the required signal-to-noise ratios P for sending half-integer numbers (between 4 1/2 and 7 1/2) of bits per interval using this system with the constellations of Figure 7:
Table III
N S P (dub) 4 1/2 24 14 11.5

5 1/2 48 28.5 14.5 1/2 96 57 17.5 7 1/2 192 116 20.6 12~753 In other embodiments, any number of bits per interval of the form N=r+2~t (t an integer) can be sent by adding to inner signals (consisting of the or signals in an encoded signal con-stellation), outer signals (consisting of it signals located as close to the origin as possible consistent with maintaining an overall rectangular grid exhibiting quadrant Al symmetry). The input bits are taken in blocks of try and determine it signal points for transmission. Of the try bits in each block, one bit determines whether any of the signal points should be an outer signal. If so, then t additional bits are used to determine in which of the it intervals an outer signal will be sent, r-t bits determine which outer signal will be sent, and r(2t-1) bits, taken r at a time, are used to select the inner signals to be sent in the other 2t-1 intervals. If none of the signal points is to be an outer signal, the remaining try bits, taken r at a time, are used to select the inner signals to be sent in the it intervals. The required signal-to-noise ratio for such a block coding system is about 3 x 2-t dub greater than for an encoded system in which a signal constellation having or signals is used to send r bits per interval.
For example, referring to Figure 9, to send 4 1/4 bits per interval, in 17-bit blocks each 4 intervals long (r=4, to a signal constellation having 16 inner signals and 4 outer signals is used. One input bit determines whether any of the four output signal points will be an outer signal. If so, two further bits determine in which of the four intervals the outer signal will be sent, two further bit select that outer signal, and 12 bits, lZ~7~;30 taken 4 at a -time, select the three inner signals. If no outer signal is to be sent, the remaining 16 bits, taken 4 at a time, are used to select the four inner signals. The required signal-to-noise ratio to send 4 1/4 bits per interval is 7/8x10+1/8x26=12 ~10.79 dub) or .79 dub more than to send 4 bits per interval in an encoded system. (In this case, the outer signals might be moved to the axes to save a little more power, if maintenance of the grid is not required.) In other embodiments, a system that sends of bits per modulation interval using a constellation having S signals can be extended to one sending ~'=r+f+2~t bits per interval using a constellation having (1+2-t)S signals. To the S original signals are added its additional signals (assuming its is an integer, and a multiple of 4 if quadrant Al symmetry is to be maintained). These additional signals are further segmented into subgroups in the same proportion as in the unextended system;
i.e., each subgroup is 2-t as large as in the unextended system, and can therefore be specified by t fewer bits. For example, if in the unextended system there are S=1.5x2r signals divided into or inner signals and 2r-1 outer signals, then in the extended system the its additional points are divided into art additional inner signals and 2r-t-1 additional outer signals.
Input bits are grouped into blocks of 2tr+2tf+l bits (assuming 2tf is an integer) and used to determine it signal points. One of these bits determines whether any of the add-tonal its signals is to be used; if no-t, the remaining trough) bits are used to determine the it signal points using the unextended system; if so, then t bits are used to determine which interval uses an additional signal and the remaining 2t(r+f)-t are used to determine which signals, using the unsex-tended system, except that in the selected interval one of the additional signals is used and is selected with t fewer bits because the corresponding subgroups are all a factor of 2-t smaller.
In this iterative fashion, systems that send N=r+f bits where f is any binary fraction of the form f=2-tl+2-t2+...+2-tX, tl--t2 c... c to can be built up, with signal constellations of S=2r(1+2-t~ +2~t2)...(1+2-tk~
signals (provided that tl+t2+...+tk is less than or equal to r, or to r+2 if quadrant Al symmetry is desired).
4-Space Block Coded Systems In other embodiments, 4-space block coded modulation systems send an integer number (N) of bits per modulation interval using signal constellations of the type of Figure 7, or a half-integer number (N=r+l/2) bits per interval using constellations of the type of Figure 1.
To send a half-integer number of bits per interval, 2r+1 bits are grouped into a block which determines two signal points from a constellation having 2r+l signals. The constellation is divided into two groups A and B of equal size, the two groups respectively containing alternate signals. For example, in Figure 10 a constellation having 32 signals is divided into two 16-signal lZ~753~
groups. As another example (not using a rectangular grid), figure 11 shows an 8-signal (8-phase) constellation divided into two 4-signal groups. The minimum squared distance between signals belonging to the same group (d2(A,A)-d2(B,B)) is a least twice as great as the minimum squared distance between signals belonging to different groups (dubbed). In Figure 10 (and in any rectangular grid structure), d2(A,A)=2d2(A,B); in Figure 11 d2(A,A)=3.4d2(A,B) .
One bit from each block (-the group bit) determines whether both signal points will be from group A or both from group B. The remaining or bits taken r at a time determine which of the or A points or B points is selected for each interval. The minimum squared distance between code words must be at least do, because (a) between two code words that have different group bits (e.g., one group A code word and one group B code word), there must be a squared distance of at least dubbed in each of the two intervals, while (b) between two code words with the same group bit, there must be a squared distance of at least d2(A,A)=d2(B,B)~2do in one interval. However, the power used is only that required for sending Al bits per interval encoded with a minimum squared distance of do, which is about the same as for an encoded system sending r bits per interval with a minimum squared distance of do. Thus, 4-space coding allows sending r+l/2 bits per interval at about the same required signal-to-noise ratio as an encoded system sending r bits per interval.
An embodiment using the Figure 10 32-signal constella-12~753~

lion to send 4 1/2 bits per modulation interval is as follows.
Referring to Figure 12, in the signal selection logic, the first input bit (by) is used by group selector 40 to determine the groups (group and group) from which both signal points will be selected, input bits by through by are used by signal point selector 42 to select signal point from the named group, and input bits by through by to select signal point from the named group.
The minimum squared distance between code words is 4, and the required signal-to-noise ratio P is 10, just as or the 16-signal encoded Figure 1 constellation. Thus 4 1/2 bits per interval can be sent with the 4-space coded system at the same P
as 4 bits per interval encoded.
At the receiver, the received code word (i.e., the two received signal points for each block) may be decoded by deter-mining the most likely two A signal points to have been sent and the most likely two B signal points to have been sent (using 2-dimensional slicing in each modulation interval), and choosing the pair (i.e., the code word) which has the minimum squared distance to the received word.
In other embodiments, decoding is accomplished by making a tentative decision (in each modulation interval) of the most likely signal point to have been sent in that interval (using two-dimensional slicing with the two-dimensional coordinates being rotated 45~ with respect to the signal constellation). If the tentative decisions in both intervals are of signal points in the same group, those tentative decisions become the final decision.

lZ~753~) Otherwise, the tentative decision having the least reliable goof-donate is changed to the next nearest signal point (which will be from the other group), and the new tentative decision becomes the final decision.
In other embodiments, maximum likelihood decoding can be accomplished by applying the Viterbi algorithm. Referring to Figure 13, the encoding process can be characterized by a finite-length 2-state trellis that determines the groups from which signal points are selected in two successive intervals in a manner analogous to Figure 6. In this case Viterbi algorithm decoding is essentially equivalent TV the first decoding method for 4-space codes described above.
Table IV shows the numbers of signals in the constella-lion and the required signal-to-noise ratios for sending half-integer numbers (between 3 1/2 and 7 1/2) of bits per interval using this 4-space coding system.
Table IV

S P (dub) 3 I 16 5 7.0 4 1/2 32 10 10.0 5 1/2 64 21 13.2

6 1/2 128 41 16.1

7 1/2 256 85 19.3 By dividing the groups of signals of the 4-space block coded system into subsets yin the manner previously described), quadrant Al differential coding on a block basis can be used in the same manner previously described for the 8-space system.
To send an integral number bits per interval using 4-space coding, the signal constellations for sending ~1/2 bits per interval described earlier in the section headed Multidimen-lZ~7530 signal Signal Structures for Sending a Non-Integral Number of Bits .
per Modulation Interval, are used. In two dimensions, these constellations have 1.5~2N signals, as shown in Figure 7, for example. Because they are based on a rectangular grid, they may be divided into A and B groups of equal size (1.5x2~-l signals in each) by assigning alternate signals to -the two groups. The minimum squared distance between signals in the same group is twice the minimum squared distance between signals in different groups. In coding, one input bit (the group bit) from a block of lo OW input bits determines whether A or B signals are to be used in both of two intervals. A second input bit determines whether an outer signal is to be used; if not, the remaining 2(N-l) bits (taken N-l at a time) determine the two inner signals of the appropriate group; if so, a third bit determines which interval will contain an outer signal, I bits select which outer signal of the appropriate group, and the remaining I bits select the inner signal of the appropriate group in the other interval. For the same reasons as before, the minimum squared distance between code words is do. The coding gain is about 1.5 dub. Any of the previously described decoding methods can be used.
To send 4 bits per interval, the 24-signal Figure 7 constellation is rotated 45 and divided into A and B groups as well as into inner and outer signals, as shown in Figure 14.
Input bits are grouped into blocks of 8 bits each and each block determines two signal points. Referring to Figure 15, in the signal selection logic, one bit (by), the group bit in each block, is used by group selector 50 to specify the group from which both ~7530 signal points are to be selected, and a second bit (by) it used by outer point selector 52 to specify whether one of the signal points should be an outer signal. (At most one of the signal points for each block may be an outer signal.) If neither signal point is to be an outer signal, the remaining 6 (i.e., I
bits (b3-bg) taken three at a time are used by signal point selector 54 to determine which of the eight inner signals of the selected group is to be selected for each modulation interval. If one signal point is to be an outer signal, the third bit, by, determines which of the two signal points is to be an outer signal and the bits b4-bg select the outer and inner signal as shown in Figure 15.
The average power required to transmit inner signals is Pinner = 5, and to transmit outer signals is Pouter = 13.
On average, half of the time the two signal points will both be inner signals, and the other half of the time one will be an inner signal and the other an outer signal. The average power required to send both signal points for a block is therefore 1/2(2 x Pinner) + punier + Pouter or 14- The required signal-to-noise ratio is then 7 (8.45 dub) per interval compared with 10 (10 dub) required for sending encoded signal points using the 16-point signal constellation of Figure 1, a 1.55 dub coding gain.
Theoretically, as shown by Conway and Sloane, the best known 4-space code for sending 4 bits per interval has a required signal-to-noise ratio of 6.75 (8.29 dub). The present embodiment which has a required signal-to-noise ratio of 7 (8.45 dub) provides ~2~7S3~
a simpler coding system at a cost of only 0.16 dub compared with the best known code for the same parameters. Further, because the code word set has quadrant Al symmetry, quadrant Al differential coding may be used (as described above) to obtain immunity to phase rotations of multiples of 90.
Table V shows the numbers of constellation signals S and the required signal-to-noise ratios P for sending integer numbers (from 4 to 7) of bits per interval using this system.
Table V
N S P (dub) 4 24 7 8.45 48 14.25 11.54 6 96 28.5 14.55 7 192 58 17.6 It can be seen that a coding gain of about 1.5 dub is obtained for all values of N with a signal constellation only 1.5 times the size of that needed for an encoded system.
Further 8-Space Block Coded Systems The 4-dimensional signal structures with 2-dimensional constellations like those of Figure 7 can also be used in 8-space block coded systems by dividing the signals into subsets A, Al, Boy By in the manner described above. The resulting systems send half-integer numbers of bits per modulation interval with about 3.0 dub coding gain over the encoded 4-dimensional signal struck lures. Table VI is an extension of Table II showing some of thesehalf-integer systems in relation to the integer systems previously described.

753C~
Table VI
S P dub 3 16 2.5 4.00 I 24 3.5 5.45 4 32 5.0 7.00 4l~2 48 7.125 8.54 64 10.5 10.22 5~2 96 14.1~5 11.51 16-Space Block Coded Systems In other embodiments, 16-space block coded modulation systems are used to send an integer number of bits per module-lion interval using Figure 7-type constellations, or half-integer numbers using Figure l-type constellations.
The constellation must be further partitioned into eight equal-sized classes Ago, Awl, Aloe All, Boo, Bolt s10~ sly after first being divided into equal-sized groups A and B and dividing each group into equal-sized subsets A, Al, and Boy By. At each step the division is made by taking alternate signals. Further-more, subsets are divided into two classes with opposite sub-scripts; e.g. A is divided into Ago and All, Al into Awl and Aloe and so forth. The result is that if the minimum squared distance between constellation signals is do=d2(A,B), the minimum squared distance between signals in classes in the same group is 2d2 times the Hamming distance (do) between their sub-scripts, e.g., Dow, Aloud; dwelled; and the minimum squared distance between signals in the same class is do. figure 16 shows the 48-signal constellation of Figure 7 rotated 45 and partitioned in this way.
In coding, one input bit of a block of I input bits determines the group (A or B) from which all 8 signals points will 12~7~3C~

be selected. Eleven input bits are coded in a conventional (16, 11) Hamming code (with minimum Hamming distance equal to 4; see, e.g. Berlekamp, cited above) to give 16 coded bits, which are taken two at a time to determine the class subscripts for each signal point. The remaining 8N-12 input bits select which signals points are used; if is an integer, by using the 4-dimensional system signal constellations of 1.5x2~-1 signals and taking the bits 2~-3 at a time to determine pairs of signal points as desk cried above; if N it a half-integer (N=r+l/2), then by using 2-dimensional signal constellations of 2r+2 signals and taking bits ~-3/2=r-1 at a time to determine signal points from the selected classes.
The minimum squared distance of such 16-space block code words is 8d2. Two code words with different group bits must differ by at least do in all 8 signal points. Two code words with the same group bit but different Hamming coded bits must differ in at least four class subscripts, so their squared disk lance is at least 4x2do. Two code words with the same group bit and Hamming coded bits have all signal points in the same classes, but must have in at least one interval two different signal points in the same class, which must therefore have squared distance do. The minimum squared distance between code words is therefore 8 times (or 9 dub) better than that of the encoded constellation, but that constellation could without coding send 1 1/2 more bits per interval than it can with coding, which is equivalent to about 4.5 dub, so the net coding gain is about 4.5 dub.

12~L7S30 An embodiment using the Figure 16 48-signal constella-lion to send 4 bits per interval is as follows.
Referring to Figure 17, 32 input bits are used to select the 8 output signal points as follows. One bit is used by group selector 70 to determine the group (A or B) from which all signal points will be selected. Eleven bits are coded by jamming coder 72 using a conventional (16,11) extended Hamming code (with Hamming distance DH=4) into 16 coded bits. Taken in pairs, the 16 coded bits are used by class selector 74 to specify the sub-scripts of the classes from which the eight output signal points are respectively selected. The remaining 20 bits (taken 5 at a time) are used by signal point selector 76 to select pairs of signal points of appropriate classes, where each class comprises 6 signals, 4 inner and 2 outer, and selection between inner and outer signals is made in the same manner as previously described.
The required signal-to-noise ratio is 3.5625 or 5.5dB, a 4.5dB gain over the encoded I constellation of Figure 1.
Table VII shows the number of signals S in the cons tell-anion and the required signal-to-noise ratio P for sending integer and half-integer numbers (between 2 1/2 and 6 1/2) of bits per interval in blocks of 8 intervals each, using the 16-space coding system described above and the signal constellations of Figures 1 and 7.

lZl~S30 Table VII
N S P (dub) 2 lt2 16 1.25 1.0 3 24 1.75 2.5 3 1/2 32 2.5 4.0 4 48 3.56 5.5 4 1/2 64 5.25 7.2 96 7.13 8.5 5 1/2 128 8.25 10.1 6 192 14.5 11.6 6 1/2 256 21.25 13.3 In the signal constellations for the 16-space block coding systems, each signal is a member of a quadruplet of signals at the same radius from the origin and separated by intervals of 90 about the origin. (For example, the points marked 60 and 62 in Figure 16 form such quadruplets.) Because all such quadruplets are of one or the other forms shown in Figure 18, quadrant Al differential coding (in the manner previously described) can be used with -the group bit and an appropriate one of the class sub-script bits (in this case the second for A and the first for Inverted in place of the group and subset bits used in 8-space coding, but otherwise as previously described.
24-Space Block Coded Systems In other embodiments, 24-space block coded systems can be used to send an integer number of bits per interval using Figure l-type constellations, or a half-integer number of bits per interval using Figure 7-type constellations.
To send N bits per interval, N an integer, a constella-lion of 2~+2 signals on a rectangular grid (such as one of those of Figure 1) is used. The constellation is divided into 16 subclasses Aijk~ Bilk (i,j,k-0 or 1) by further subdivi-lZ~L753~) soon of the classes Air and Bit defined in the previous section by taking alternate signals in accordance with a pattern defined below. The minimum squared distance between signals in different groups is then do between signals in the same group but different subsets is do, in the same subset but different classes is 4d2, in the same class but different subclasses is do, and finally between signals in the same subclass is 1 do .
Referring to Figure 19, the 64-signal constellation of Figure 1 has been partitioned into 16 4-point subclasses (for use with a 24-space block coding system). In general, a partition of any rectangular-grid constellation into subclasses using the pattern of subscripts shown in Figure 19 may be used.
In coding, one input bit of a block of 12N bits deter-mines the group (A or B) from which all 12 signal points will be selected. Twelve input bits are coded in the (24, 12) Golly code (which has minimum Hamming distance 8; see, e.g., Berlekamp, cited above) to give 24 coded bits, which are taken two at a time to give the (i, j) subscripts of the subclass from which each signal point is to be selected. Eleven input bits determine the k sub-scripts for the first 11 points, and the Thea k subscript is chosen so that the 12 k subscripts have even parity (an even number are equal to 1) for A code words and odd parity for B code words. The remaining 12N-24 bits are taken N-2 at a time to select one of the 2N-2 signals of the selected subclass for each of the 12 intervals.
With this construction, the minimum squared distance ~217~30 between code words is Dow, where do is the minimum squared distance between signals in the constellation. This is 12 dub better than the encoded constellation having 2N+2 signals, but that constellation without coding could send 2 bits per interval more than with this coding, which is equivalent to about 6 dub, so the net coding gain is about 6 dub.
One embodiment, using the Figure 19 constellation to send 4 bits per interval, is as follows. Referring to Figure Z0, 48 input bits are used to select 12 output signal points as shown.
One bit is used by group selector 80 to determine the group (A or B) from which all signal points will be selected. Twelve bits are coded by Golly coder 82 using a conventional (24, 12) Golly code (with Hamming distance do into 24 coded bits. Taken in pairs, the 24 coded bits are used by class selector 84 to specify the (i, j) subscripts of the 12 signals to be selected. Eleven bits and the group bit are used by subclass selector 86 to specify the k subscripts of the 12 points, with 11 subscripts equal to the corresponding bits, and the Thea subscript a parity check on these 11 bits plus the group bit. The remaining 24 bits, taken two at a time, are used by signal point selector 88 to select one of the four signals in each selected subclass.
The required signal-to-noise ratio with this system is 2.625 or 4.2dB, 5.8dB better than with the encoded No constella-lion of Figure 1.
Table VIII shows the numbers of signals in the constella-lion and the required signal-to-noise ratios P for sending integer numbers N (between 2 and 6) of bits per interval in blocks of 12 intervals, using the 24-space coding described above and the signal constellations of Figure 1. (The constellations of Figure 7 could be used to send half-integer numbers of bits per interval with similar coding gain.) Table VIII
_ S P (dub)_ 2 16 .625 -2.0 3 32 1.255 1.0 4 64 2.625 4.2 128 5.125 7.1 6 256 10.625 10.3 In the signal constellations for these 24-space block coded modulation systems, each signal is a member of a quadruplet of signals of the same radius and separated by phase intervals of 90. (For example, the signals marked 90, 92, 94, and 96 in Figure 19 form such quadruplets, respectively.) All such quadrup-lets are of one of the four types shown in Figure 21, and there-fore quadrant Al differential coding (in the manner previously described) can be used with the group bit and an appropriate sub-script bit (in this case the first for A; and the second for B, inverted) in place of the group and subset bits used in 8-space coding, but otherwise as previously described.
Block Coded Systems of Other Numbers of Dimensions In other embodiments, block coded systems in 2m-space can be derived from the 4-space and 8-space block coded systems.
The 4-space system can be extended to a 2m-space system that sends N-l/m bits per interval or Mel bits per block (for any m greater than or equal to 2) if N is an integer, or for any even m greater than or equal to 2 if N=r+l/2 is a half-integer number.

1~17530 In the former case the signal constellation of size ON is used, and in the latter the constellation of 1.5x2r signals divided into inner and outer signals as previously described. In either case the constellation it divided into A and B groups as previous-lye described. The encoder is arranged 50 that in the first m-l intervals, either A or B signals may be sent, but in the last interval, the group must be constrained to satisfy a parity condo-lion (e.g., that the total number of A signals be even). For example, the groups may be selected by a two-state trellis encoder 13 operating according to the m-interval trellis of Figure 22.
Otherwise selection of signals within the groups is as previously described. Decoding may be by the Viterbi algorithm The minimum squared distance between code words remains do in this system because of the parity condition and the fact that the minimum squared distance between signals within the same group is at least twice the minimum squared distance between signals in different groups. However, the number of bits trays-milted per interval is now only l/m less than in an encoded system using the same constellation, so that the coding gain is approxi-mutely 3(1-1/m) dub, which approaches 3 dub for m large. For example, for my an 8-space coded modulation system with coding gain of about 2.25 dub is obtained.
The 8-space system can be extended to a 2m-space system that sends N bits per interval for any m greater than or equal to 4 and any integer I, using a signal constellation of ON+
signals divided into 4 equal-sized subsets as previously described. The encoder is arranged so that all m points are I
chosen from the same group, the first m-l points being from either subset of that group, and the final point being chosen from a subset selected to satisfy a parity condition (e.g., that the total number of 1 subscripts be even). For example, the subsets may be selected by a 4-state trellis encoder that operates accord-in to the m-interval trellis of Figure 23. Otherwise selection of signals within subsets is as previously described. Decoding may be done using the Viterbi algorithm using the Figure 23 trellis.
The minimum squared distance between code words remains do in this system because m is greater than or equal to 4, there is a parity condition on subsets combined with the between-subset distance of do, and there is a within-subset distance of do. The number of bits per interval is N for any value of m, so the coding gain remains about 3 dub. However, fox m greater than 4 the number of near neighbors of any code word is reduced since code words of the opposite group have distance at least moo and are thus no longer near neighbors, reducing the error event probability coefficient by about a factor of 2.
other Embodiments Other embodiments are within the following claims. For instance, the input bits can undergo any transformation which does not result in loss of information (such as scrambling, permute-lion, or binary linear transformations) before being encoded. The digital data to be sent can be in characters other than bits. The output signal points or indeed their coordinates can be arbitrary-lye permuted without loss of coding gain. The coordinates can be ~Z~7~
used individually in one-dimensional modulation systems (such as single side-band or vestigial-sideband modulation systems), or in groups of appropriate dimension in multidimensional modulation systems. other signal constellations (including ones not based on rectangular grids) can be used provided that their signals can be partitioned into equal-sized groups (subsets, classes, subclasses) having at least approximately the desired distance properties described earlier. Any 2m-dimensional orthogonal linear trays-formation of output signal points may be made without affecting relative distance properties in 2m-space. Quadrant Al differential coding can be done using any method such that 90 rotations of code words change only quadrant Al phase bits, and such phase bits are encoded differentially (mod 4) with reference to any previous-lye transmitted quadrant Al phase bits. Multidimensional signal constellations for any binary fractional numbers of bits per interval as described herein, may be used with the block coded modulation systems provided that parameters are consistent.

- I -

Claims

THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE
PROPERTY OR PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:

1. A modulation system for sending digital data over a band-limited channel in accordance with a code word corresponding to a plurality of modulation signal points drawn from a constell-ation of available signals, said code word being selected from an available set of said code words on the basis of a block of said digital data, the code words belonging to said available set for said block being independent of said signal points corresponding to the code word selected for any other said block, said system comprising an encoder for selecting said code word, said constellation comprising groups having equal numbers of said signals, said encoder being arranged so that the group from which at least one said signal point for said block is drawn depends on the group from which at least one other said signal point for said block is drawn, and a modulator for sending said signal points over said channel, said signal points being two-dimensional, said code words being 2m-dimensional, m being the number of said signal points corresponding to each said code word, m being greater than two, each said block of said digital data comprising mN bits.

2. The improvement of claim 1 wherein said signals are arranged in said constellation so that the minimum squared dis-tance between two said signals belonging to the same said group is greater than between two said signals belonging to different said groups.

3. The improvement of claim 1 wherein said signals of said constellation are arranged on a rectangular grid.

4. The improvement of claim 1 wherein the minimum squared distance between two said signal points belonging to the same said group is twice the minimum squared distance between two said sig-nal points belonging to different said groups.

5. The improvement of claim 1 wherein each of said block comprises a plurality of bits, and the said groups from which said signal points for said block are drawn are determined by a single said bit of said block.

6. A modulation system for sending digital data over a band-limited channel in accordance with a code word corresponding to a plurality of modulation signal points drawn from a constella-tion of available signals, said code word being selected from an available set of said code words on the basis of a block of said digital data, the code words belonging to said available set for said block being independent of said signal points corresponding to the code word selected for any other said block, said system comprising an encoder for selecting said code word, said constellation comprising groups having equal numbers of said signals, said encoder being arranged so that the group from which at least one said signal point for said block is drawn depends on the group from which at least one other said signal point for said block is drawn, and a modulator for sending said signal points over said channel, said signal points being two-dimensional, said code words being 2m-dimensional, m being the number of said signal points corresponding to each said code word, each said block of said digital data comprising mN bits, N being an integer represent-ing the number of bits of said digital data per signaling inter-val, m being an even integer no smaller than 2, said constellation comprising 1.5x2N signals.

7. The improvement of claim 1 wherein N=r+1/2, r being an integer, and said constellation comprises 2r+1 signals.

8. The improvement of claim 1 wherein said encoder is further arranged so that m-1 said signal points for each said block may be drawn from among all said signals in said constella-tion, and the one remaining said signal point for said block is drawn from a group that depends on the groups from which said m-1 signal points are drawn.

9. A modulation system for sending digital data over a band-limited channel in accordance with a code word corresponding to a plurality of modulation signal points drawn from a constell-ation of available signals, said code word being selected from an available one of said code words on the basis of a block of said digital data, the code words belonging to said available set for said block being independent of said signal points corresponding to the code word selected for any other said block, said system comprising an encoder for selecting said code word, said constellation comprising groups having equal number of said signals, said encoder being arranged so that the group from which at least one said signal point for said block is drawn depends on the group from which at least one other said signal point for said block is drawn, and a modulator for sending said signal points over said channel, said signal points being two-dimensional , said code words being 2m-dimeneional, m being the number of said signal points corresponding to each said code word, and wherein said encoder is further arranged so that m-1 said signal points for each said block may be drawn from among all said signals in said constellation, and the one remaining said signal point for said block is drawn from a group that depends on the groups from which said m-1 signal points are drawn, and wherein said group from which said one remaining signal point is drawn depends on a single-parity-check code based on at least one of said bits.

10. A modulation system for sending digital data over a band-limited channel in accordance with a code word corresponding to a plurality of modulation signal points drawn from a constell-ation of available signals, said code word being selected from an available set of said code words on the basis of a block of said digital data, the code words belonging to said available set for said block being independent of said signal points correspond-ing to the code word selected for any other said block, said system comprising an encoder for selecting said code word, said constellation comprising groups having equal numbers of said signals, said encoder being arranged so that the group from which at least one said signal point for said block is drawn depends on the group from which at least one other said signal point for said block is drawn, and a modulator for sending said signal points over said channel, said signal points being two-dimensional, said code words being 2m-dimensional, m being the number of said signal points corresponding to each said code word, and wherein said encoder is further arranged so that m-1 said signal points for each said block may be drawn from among all said signals in said constellation, and the one remaining said signal point for said block is drawn from a group that depends on the groups from which said m-1 signal points are drawn, and wherein said group from which said one remaining signal point is drawn is determined by said encoder based on a state-transition trellis for said block.

11. The improvement of claim 6 wherein m is 2 and N is 4.

12. A modulation system for sending digital data over a band-limited channel in accordance with a code word corresponding to a plurality of modulation signal points drawn from a constell-ation of available signals, said code word being selected from an available set of said code words on the basis of a block of said digital data, the code words belonging to said available set for said block being independent of said signal points corres-ponding to the code word selected for any other said block, said system comprising an encoder for selecting said code word, said constellation comprising groups having equal num-bers of said signals, said groups each comprising subsets having equal numbers of said signals, said encoder being arranged so that the group and subset from which at least one said signal point for said block is drawn depends on the group and subset from which at least one other said signal point for said block is drawn, and a modulator for sending said signal points over said channel.

13. The improvement of claim 12 wherein said signals are arranged in said constellation so that the minimum squared dis-tance between two signals belonging to one said subset is greater than between two said signals belonging to different said subsets within the same said group.

14. The improvement of claim 12 wherein said signal points are 2-dimensional, said code words are 2m-dimensional, m being the number of said signal points corresponding to each said code word, and each said block of said digital data comprises mN bits.

15. The improvement of claim 12 wherein said signals of said constellation are arranged on a rectangular grid.

16. The improvement of claim 12 wherein the minimum squared distance between two said signal points belonging to the same said subset is twice the minimum squared distance between two said signal points belonging to different said subsets.

17. The improvement of claim 12 wherein each said block comprises a plurality of bits, and the subsets from which said signal points for said block are drawn are determined based on a plurality of bits representing less than all of said digital data of said block.

18. The improvement of claim 14 wherein N is an integer, m is at least 4, and said constellation comprises 2N+l said signals.

19. The improvement of claim 14 wherein N=r+1/2, r being an integer, m is an even integer no smaller than 4, and said con-stellation comprises 1.5x2r+1 signals.

20. The improvement of claim 14 wherein said encoder is further arranged so that one said signal point for each said block may be drawn from among all said signals in said constellation, m-2 of said signal points are drawn from groups that depend on the group from which said one signal point is drawn, and the single remaining said signal point for said block is drawn from a subset that depends on the subsets from which said one signal point and said m-2 signal points are drawn.

21. The improvement of claim 20 wherein said subset from which said single remaining signal is drawn depends on a single-parity-check code based on at least one of said bits.

22. The improvement of claim 20 in which said subsets from which said signal points for said block are drawn are determined by said encoder based on a state-transition trellis for said block.

23. The improvement of claim 20 wherein said subsets and said groups from which said signal points for said block are drawn are determined by said encoder based on a Hamming code applied to at least one of said bits.

24. The improvement of claim 14 wherein m is 4.

25. The improvement of claim 20 wherein m is 4.

26. The improvement of claim 18 wherein m is 4 and N is 4.

27. The improvement of claim 12 wherein said subsets each comprise classes having equal numbers of said signals, and said encoder is further arranged so that said class from which at least one signal point for said block is drawn depends on the class from which at least one other said signal point for said block is drawn.

28. The improvement of claim 27 wherein said signals are arranged in said constellation so that the minimum squared distance between two signals belonging to one said class is greater than between two said signals belonging to different said classes with-in the same said subset.

29. The improvement of claim 27 wherein said signal points are 2-dimensional, said code words are 2m-dimensional, m being the number of said signal points corresponding to each said code word, and said block of said digital data comprises mN bits.

30. The improvement of claim 27 wherein said signals of said constellation are arranged on a rectangular grid.

31. The improvement of claim 27 wherein the minimum squared distance between two said signal points belonging to the same said class is twice the minimum squared distance between two said sig-nal points belonging to different said classes.

32. The improvement of claim 27 wherein each said block comprises a plurality of bits, and the classes from which said signal points for said block are drawn are determined based on a plurality of bits representing less than all of said digital data of said block.

33. The improvement of claim 29 wherein m is 8.

34. The improvement of claim 33 wherein N is an integer and said constellation comprises 1.5x2N+1 said signals.

35. The improvement of claim 33 wherein N=r+1/2, r being an integer, and said constellation comprises 2r+2 signals.

36. The improvement of claim 29 or 32 wherein said classes from which said signal points for said block are drawn are deter-mined by said encoder based on a Hamming code applied to at least one of said bits.

37. The improvement of claim 33 wherein N is 4.

38. The improvement of claim 27 wherein said classes each comprise subclasses having equal numbers of said signals, and said encoder is further arranged so that said subclass from which at least one signal point for said block is drawn depends on the subclass from which at least one other said signal point for said block is drawn.

39. The improvement of claim 38 wherein said signals are arranged in said constellation so that the minimum squared dis-tance between two signals belonging to one said subclass is greater than between two said signals belonging to different said subclasses within the same said class.

40. The improvement of claim 38 wherein said signal points are 2-dimensional, said code words are 2m-dimensional, m being the number of said signal points corresponding to each said code word, and said block of said digital data comprises mN bits.

41. The improvement of claim 38 wherein said signals of said constellation are arranged on a rectangular grid.

42. The improvement of claim 38 wherein the minimum squared distance between two said signal points belonging to the same said subclass is twice the minmum squared distance between two said signal points belonging to different said subclasses.

43 The improvement of claim 38 wherein each said block comprises a plurality of bits, and the subclasses from which said signal points for said block are drawn are determined based on a plurality of bits representing less than all of said digital data of said block.

44. The improvement of claim 40 wherein m is 12.

45. The improvement of claim 44 wherein N is an integer and said constellation comprises 2N 2 said signals.

46. The improvement of claim 40 or 43 wherein said sub-classes from which said signal points for said block are drawn are determined by said encoder based on a Golay code applied to at least one of said bits.

47. The improvement of claim 44 wherein N is 4.

48. The improvement of claim 1, 12 or 27 further comprising means for selecting at least one said signal point of said block on the basis of less than all said digital data in said block.

49. The improvement of claim 38 further comprising means for selecting at least one said signal point of said block on the basis of less than all said digital data in said block.

50. The improvement of claim 6, 9 or 10 further comprising means for selecting at least one said signal point of said block on the basis of less than all said digital data in said block.

51. The improvement of claim 1, 12 or 27 wherein said constellation comprises quadruplets each having four said signals, the signals belonging to each said quadruplet being located at the same distance for the origin but separated by 90° intervals about said origin, said digital data comprises bits, and at least two of said bits are quadrantally differentially encoded.

52. The improvement of claim 38 wherein said constellation comprises quadruplets each having four said signals, the signals belonging to each said quadruplet being located at the same dis-tance from the origin but separated by 90° intervals about said origin, said digital data comprises bits, and at least two of said bits are quadrantally differentially encoded.

53. The improvement of claim 6, 9 or 10 wherein said con-stellation comprises quadruplets each having four said signals, the signals belonging to each said quadruplet being located at the same distance from the orgin but separated by 90° intervals about said origin, said digital data comprises bits, and at least two of said bits are quadrantally differentially encoded.

54. The improvement of claim 1, 12 or 27 wherein said con-stellation comprises quadruplets each having four said signals, the signals belonging to each said quadruplet being located at the same distance from the origin but separated by 90° intervals about said origin, said digital data comprises bits, and at least two of said bits are quarantally differentially encoded and wherein said groups each comprise subsets having equal numbers of said signals and said four signals belonging to each quadruplet are drawn from four different said subsets.

55. The improvement of claim 38 wherein said constellation comprises quadruplets each having four said signals, the signals belonging to each said quadruplet being located at the same dis-tance from the origin but separated by 90° intervals about said origin, said digital data comprises bits, and at least two of said bits are quadrantally differentially encoded and wherein said groups each comprises subsets having equal numbers of said signals and said four signals belonging to each quadruplet are drawn from four different said subsets.

56. The improvement of claim 1, 12 or 27 further comprising a decoder arranged to decide which code word was sent based on max-imum likelihood sequence estimation in accordance with the Viterbi algorithm.

57. The improvement of claim 38 further comprising a decoder arranged to decide which code word was sent based on maximum like-lihood sequence estimation in accordance with the Viterbi algorithm.

58. The improvement of claim 6, 9 or 10 further comprising a decoder arranged to decide which code word was sent based on maximum likelihood sequence estimation in accordance with the Viterbi algorithm.

59. The improvement of claim 1, 12 or 27 further comprising a demodulator having a decoder arranged to make tentative decisions about which signal point was sent prior to final decoding of all of the received signal points for a block, and said demodula-tor comprises adaptive control circuitry arranged to be responsive to said tentative decision.

60. The improvement of claim 38 further comprising a demodu-lator having a decoder arranged to make tentative decisions about which signal point was sent prior to final decoding of all of the received signal points for a block, and said demodulator comprises adaptive control circuitry arranged to be responsive to said tent-ative decision.

61. The improvement of claim 6, 9 or 10 further comprising a demodulator having a decoder arranged to make tentative decisions about which signal point was sent prior to final decoding of all of the received signal points for a block, and said demodulator comprises adaptive control circuitry arranged to be responsive to said tentative decision.

62. The improvement of claim 1, 12 or 27 wherein said digital data comprises bits and the selection of said signal points for a block depends on a single-parity-check code based on at least one of said bits.

63. The improvement of claim 6, 10 or 38 wherein said digital data comprises bits and the selection of said signal points for a block depends on a single-parity-check code based on at least one of said bits.

64. The improvement of claim 1, 12 or 27 wherein said digit-al data comprises bits and the selection of said signal points for a block depends on a single-parity-check code based on at least one of said bits, further comprising a decoder arranged to decide which said code word was sent by first making a tentative decision as to each signal point in said code word without regard to whether said parity check is satisfied, and accepting said tentative decisions as the final decision, either without changes if said parity check is satisfied, or after changing the least reliable one of said tentative decisions if said parity check is not satisfied.

65. The improvement of claim 38 wherein said digital data comprises bits and the selection of said signal points for a block depends on a single-parity-check code based on at least one of said bits, further comprising a decoder arranged to decide which said code word was sent by first making a tentative decision as to each signal point in said code word without regard to whether said parity check is satisifed, and accepting said tentative decisions as the final decision, either without changes if said parity check is satisifed, or after changing the least reliable one of said tentative decisions if said parity check is not satisfied.

66. In a modulation system for sending a block of digital data bits over a band-limited channel using a plurality of modula-tion signal points drawn from a two-dimensional constellation of available signals, the improvement wherein said constellation comprises a plurality of inner signals, and a plurality of outer signals located farther from the origin than said inner signals, one bit of said digital data determines whether any of said plur-ality of signal points will be drawn from said outer signals, and if an outer signal will be drawn, at least one other bit of said dig-ital data determines which of said plurality of signal points will be an outer signal point.

67. The improvemnt of claim 66 wherein there are 2t said signal points, said digital data comprises at least t+l bits, there are S said inner signals and 2-tS said outer signals, and t said bits determine which of said plurality of signal points will be an outer signal point.

68. The improvement of claim 67 wherein said signals of said constellation are arranged on a rectangular grid.

69. The improvement of claim 68 wherein said constellation comprises quadruplets each having four said signals, the signals belonging to each said quadruplet being located at the same dis-tance from the origin but separated by 90° intervals about the origin of a signal plane, and at least two of said bits are quad-rantally differentially encoded.

70. The improvement of claim 67 wherein t=1.

71. The improvement of claim 70 wherein said block comprises 2N+1 said bits and S is 2N.

72. The improvement of claim 71 wherein N-1 of said bits determine which said outer signal is drawn, and N of said bits determine which inner signal is drawn.

73. The improvement of claim 70 wherein N is 4.

74. The improvement of claim 67 wherein t is 2.

75. The improvement of claim 74 wherein said block comprises 4N+1 bits and S is 2N,N being an integer.

76. The improvement of claim 74 wherein said block comprises 4N+3 bits and S is 1.5x2N,N being an integer.

77. The improvement of claim 75 wherein N-2 of said bits determine which said outer signal is drawn and N of said bits determine which inner signal is drawn.