CA1306543C - Partial response channel signaling systems - Google Patents

Abstract

Partial Response Channel Signaling Systems Abstract of the Disclosure Apparatus for generating a (running digital sum) sequence of digital signals XX and/or a (partial response coded) sequence of digital signals yk, k = 1, 2, ..., such that Yk = XK + xK-L, L an integer, in which the yk signals are a sequence in a given modulation code, In one aspect, the signals Yk are chosen to be congruent to coset representatives specified in accordance with a modulation code, a plurality of code constellations are used, and at least one constellation includes both a point with a positive sum of coordinates and one with a negative sum of coordinates. In another aspect, the signals xk are chosen to be congruent to a sequence of alternate (precoded) coset representatives. In other aspects, the Yk alphabet signals are evenly spaced, and a selectable, e.g., an optimal, tradeoff between Sx and Sy is made. An N-dimensional modulation code is generated as a sequence of one-dimensional signals. A
maximum likelihood sequence estimation decoder reconstructs the estimated running digital sum, and generates a signal whenever the estimated running digital sum is outside a permissible range. In another aspect, the decoder includes a modified maximum likelihood sequence estimator adapted to find MQ partial decoded sequences, where Q is the number of encoder states, and M is an integer.

Description

13~ 3 12~2V
~artial Response Channel Si~nalin~ SYstems Backqround of the Invention This invention relates to modulation coding and s partial response systems In modulation coding, symbols are encoded as signals drawn from a constellation in such a way that only certain seguences of signals are possible.
In recent years, a number of kinds of tr~llis-type modulation codes have been developed and applied (e.g., in modems) to realize coding gains of 3 to 6 dB over high-signal-to-noise-ratio, band-limited channels such as voice grade telephone channels.
Early trallis codes were due to Ungerboeck (Cjsaka et al., U.S. Patent No. 3,877,768; Ungerboeck, "Channel Coding with Multilevel/Phase Signals," IEEE
Transactions on Information Theory, Vol. IT-28, pp.
55-67, January, 1982). Ungerboeck's codes for sending n bits per symbol are based on 4-subset or 8-subset partitions of one-dimensional (PAM) or two-dimensional (QAM) 2n+l-point signal constellations, combined with a rate-l/2 or rate-2~3 linear binary convolutional code that dQterminQs a sQquence of subsets. A further set of "uncoded" bits then determines which signal points 25 within the specified subsets are actually sent. The partition and the code are designed to guarantee a certain minimum squared distance d2in between permissible sequences of signal points. Even after giving effect to the power cost of~an expanded signal constellation (a factor of four (6 da) in one dimension, or a factor of two ~3 dB) in two dimensions), the increase in minimum squared distance yields a coding :13~6~'~3 2 6041~172 gain that ranges from about a factor of two (3 dB~ for simple codes up to a factor of four (6 dB) for the most complicated codes, for values of n that may be as large as desired.
Gallager Canadian Patent Application S.N. 473,150 discussed in Forney et al, "Efficient Modulation for ~and-Limited Channels," IEEE J. Select. Areas Commun., Vol. SAC-2, ~pp. 632-647, 1984) devised a multidimensional trellis code based on a 1~-subset partition of a four--dimensional signal constellation, combined with a rate-3/4 convolutional code. The four-dimensional subset is determined by selectiny a pair of two-dimensional subsets, and the points of the four-dimensional signal constellation are made up of pairs of points from a two-dimensional signa7 constellation. With only an 8-state code, a d2min of four times the uncoded minimum sequence diætance can be obtained, while the loss due to expanding the signal constellation can be reduced to about a factor of 21/2 (1.5 dB), yielding a net coding gain of the order of 4.5 dB. A similar code was designed by Calderbank and Sloane ("Four-dimensional Modulation With an Eight-State Trellis Code", AT&T Tech. J., Vol. 64, pp. 1005-1018, 1985 U.S. Patent No. 4,581,601).
Wei Canadian Patent Application S.N. 504,573, now issued as Canadian Patent No. 1,248,182 devised a number of multidimensional codes based on partitions of constellations in four, eight, and sixteen dimensions, combined with rate-(n-1)/n convolutional codes. His multidimensional constellations again consist of sequences of points from two-dimensional constituent constellations. The codes are designed to minimize two-dimensional constellation expansion, to obtain pr ~

13~1651~3 performance (coding gain) versus code complexity over a broad range, and for other advantages such as transparency t,o phase rotations. Calderbank and Sloane, "New Trellis Codes", IEEE Trans. Inf. TheorY~ to appear 5 .~arch, 1987, "An Elghc-dimensional Trellis Code," Proc.
IrrE, Vol. 74, pp. 757-759, 19~6 have also devised a variety of multidimensional trellis codes, generally with similar performance versus complexity, more constella~ion expansion, but in some cases fewer states, lQ All of the above codes are designed for channels in which the principal impairment (apart from phase rotations~ is noise, and in particular for channels with no intersymbol interference. The implicit assumption is that any intersymbol interference introduced by the actual channel will be reduced to a negligible level by transmit and receive filters; or, more specifically, by an adaptive linear equalizer in the receiver. Such a system is known to work well if the actual channel does not have severe attenuation 20 within the transmission bandwidth, but in the case of severe attenuation ("nulls" or "near nulls") the noise power may be strongly amplified in the equalizer ("noise enhancement").
A well-known technique for avoiding such "noise enhancement" is to design the signaling system for controlled intersymbol interference rather than no intersymbol interference. The best-known schemes o this type are called "partial response" signaling schemes 5Forney, "Maximum Likelihood Sequence Estimation of Digital Se~uences in th~ Presence of Intersymbol Interference," IEEE Trans. Inform. TheorY, Vol. IT-18, pp. 363 3?~ 1972).

In a typical (one-dimensional) partial response scheme, the desired output Yk at the receiver is designed to be the difference of two successive inputs x~, i.e., y~ = xk - xk_l, rather than Yk =
x~. In sampled-data notation usi~g tAe delay opera.or D, this means that the ~esired ou~put sequence y~3) equals x(D)(l-D) rather than x(D); this is t~us called a ~l-D" partial response system. Because the spectrum of a discrete-tims channel with impulse response l-D has a null at zero frequency (DC~, the combination of the transmit and receive filters with the actual channel likewise must hav~ a DC null to achieve this desired response, On a channel which has a null or a near null at DC, a receive equalizer designed for a l-D desired response will introduce less noise enhancement than one designed to produce a perfect (no intersymbol interference) response, Partial response signaling is also used to achieve other objectives, such as reducing sens~tivity to channel impairments near the band ed~e, easing filtering requirements, allowing for pilot tones at the band edge, or reducing adjacent-channel interference in frequency~division multiplexed systems.
Other types of partial response systems include a l+D system which has a null at the Nyquist band edge, and a l-D system which has nulls at both DC and the Nyquist band edge. A quadrature (two-dimensional) partial responsQ system (QPRS) can be modeled as having a two-dimensional complex input; the ~complex) response l+D results in a QPRS-~system which has nulls at both the upper and lower band edges in a carrier-modulated (QAM) bandpass system. All of thesQ partial responsQ systems are closely related to one another, and schemes for one ~3S~5'~3 are easily adapted to another, so one can design a system for the 1-D response, say, and easily extend it to the others.
Calderbank, Lee, and Mazo ("Baseband Trellis Codes ~ith A Spec~ral ~ull at Zero"; submitzed to I~EE

~rans. Inf. Theory) have proposed a scheme to construc~

trellis-coded sequences that hav~ spectral nulls, particularly at DC, a problem that is related tG the design of partial response systems, even though its lo objectives are in general somewhat different.
Calderbank et al. have adapted known multidimensional trellis codes with multidimensional signal constellations to produce signal sequences with spectral nulls by the following technique, The multidimensional signal constellation has twice as many signal points as are necessary for the non-partial-response case, and is divided into two equal size disjoint subsets, one of multidimensional signal points whose sum of coordinates is less than or equal to zero, the other whose sum is 2a greater than or equal to zero. A "running digital sum'`
(RDS) of coordinates, initially set to zero, is adjusted for each selected multidimensional signal point by the sum of its coordinates. If the current RDS is nonnegative, then the current signal point is chosen 25 from the signal subset whose coordinate sums are less than or equal to 2ero; if the RDS is negative, then the current signal point is chosen from the other subset.
In this way the RDS is kept bounded in a narrow range near zero, which is known to force the signal sequence 3Q to have a spectral~null at DC. At the same time, however, the signal points are otherwise chosen from the subsets in the same way as they would have been in a non-partial-responsa system: the expanded multidimensional constellation is divided into a certain number of subsets with favorable distance properties, and a rate-(n-l)/n convolutional code determines a sequence of the subsets such that the minimum squared 5 distance between sequences is ~uaran~eed ~o be at least dmin. The coding gain is reduced by ~lae constellation doubling (by a factor of 21/2, or 1.5 da~ in four dimensions, or by a factor of 2 1/4, or 0.75 dB in eight), but otnerwise similar performance is achieved as in the non-partial response case with similar code complexity.
Summar~ of the Invention One general feature of the invention is generatin~ a sequence of digital signals xk and/or a sequence of digital signals y~ ~the sequence Yk being in accord with a given modulation code), k - 1, 2, ..., such that the relationship between the xk signals and Yk signals is Yk = Xk ~ Xk L' ~
an integer. An encoder selects J signals Yk, J > 1, ~Yk~ Yk+l' Yk+J_l) to be congruent to a sequence of J coset representatives c~ (modulo M), M
an integer, specified in accordance with the given modulation code, the J symbols being chosen from one of a plurality of J-dimensional constellations, the choice be~ng based on a previous xk " k' ~ k. At least one of the constellations includes both a point with a positive sum of coordinates, and another point with a negative sum of coordinates. The encoder is arranged so that the signals x~ have finite variance Sx.
Another general feature of the invention is that the encoder selects the signals xk to be congruent to a sequence of alternative coset representatives CX (modulo M), where ~3(~

Ck = Ck ~ C'k L ~modulo M), in the case where Yk Xk + Xk_L, cl~ = c~ + c'~_~ (modulo M), in the case where Yk = Xk ~ ~k-L
Another general feature of the inven~ion is that the Yk signals fall within an alphabet of possible Yk signals that are spaced apart within the alphabet evenly by a spacing ~, and the encoder causes the sequence Yk to have a variance Sy less than 2So and the sequence xk to have a variance Sx not much greater than Sy/4(Sy - S0), S0 being approximately the minimum signal power required to represent n bits per signal with a ~-spaced alphabet.
Another general feature of the invention is that the encoder causes the xk and Yk signals to have any selected variances Sx and Sy within pxedetermined ranges.
In preferred embodiments, the ranges are controlled by a parameter ~, Sx is approximately So/(l _ ~2), and Sy is approximately 2So/~l + ~).
Another general feature of the invention is apparatu~ for generating a sequence in a given N-dimenslonal modulation code, by generating a sequence o~ one-dimensional signals based on coded and uncoded bits, the modulation code being based on an N-dimensional constellatîon partitioned into subsets associated with the code, the subsets each representing a plurali~y of N-dimensional signals, the apparatu comprisinq an encoder for deriving, for each N-dimensional symbol, a set of N, M-valued one-dimensional coset representatives ck corresponding to congruence classes of each of the N coordinates ~3~65~3 (modulo M) of the symbol, each coset representative designating a subset of one-dimensional values in a one-dimensional constellation of possible coordinate values for each of the N dimensions, each one-dimensional signal in the seq~er.ce ~ei.~ ~elec ed from the possible coordinate values based on uncoded bits.
In preferred embodiments, either the xk or Yk sequence may be delivered as an output; L = l;
Yk = Xk ~ Xk-L; the code may be a trellis code or a lattice code; M may be 2 or 4 or a multiple of 4 or 2 + 2i; J may be 1 or the same as the number of dimensions : in the modulation code; k' = k - l; J is 1 and each constellation is a one-dimensional range of values centered on ~xk 1' < B < 1, preferably n, o; there are a finite set of (e.g., two non-disjoint) J-dimensional constellations; Yk and xk may be real valued or complex valued.
Another general feature is a decoder for decoding a sequence Zk = Yk + nk~ k = 1~2~
into a decoded sequence Yk, where the sequence of signals Yk is such that ~a) the sequence is from a given modulation code: (b) the running digital sum X Y~ + Yk_l + Yk_2 + -- has finite variance Sx; (c) the signals Yk fall in a predetermined permissible range dependent on Xk, k' < k; and the sequence nk represents noise. A range violation monitor reconstructs the estimated running digital sum Xk Yx +~Yk-l + - ~ compares the decoded seque~ce Yk with a predetermined permissibla range ~ased on the estimated running digital sum x~, k' <
k, and generates an indication whenever the Yk is outside the permissible range.

1306~3 9 ~0412-1723 ~ nother general feature of the inventlon 1~ a clecofler for decoding a sequence Zk ~ Yk ~ nk, k = 1, 2, ..., where the sequence of signals Yk is such that (a) the sequence is ~rom a given modulation code, the code bein~ capable of being generated by an encoder with a finite number Q of states; (b) Yk = Xk ~ Xk L~ L an in~eger, where the sequence xk has finite variance Sx, and the sequence nk represents noise, comprisiny a modified ma~irnum likelihood sequence estimator adapted to find MQ partial decoded sequences, up to some time K, one sequence for each combination of the finite number Q of states and each of a finite number M of integer-spaced values modulo ~1, such that each sequence (a) is in the code up to the time K; (b) corresponds to the encoder being in a given state at the time K; ~c) corresponds to a value of xK at the time K that is congruent to a given one of the values, modulo M.
~ he invention adapts ~nown modulation codes, particularly trellis codes, for use in partial response systems to achleve the same kinds of advantages that trellis codes have in non-partial response systems -- notably, substantial coding gains for arbitrarily large numbers n of bits/symbol with reasonable decoding complexlty. The invention also enables the design of trellls codes for partial response systems in such a way as to achieve both a relatively low input signal power Sx and a relatively low output power Sy, and permits smoothly trading off these two quantities against each other. Furthermore, higher-dimensional trellis codes can be adapted for use in partial response systems which are inherently lower-dimensional.

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~3~)65~3 9a 60~12-1723 According to a broad aspect of the invention there is provided apparatus for generating a sequence of digital signals xk and/or a sequence of digital signals Yk, k = 1, 2, ..., such that the sequence of Yk signals is a partial-response-coded sequence derived from the sequence of xk signals, said signals Yk being a sequence in a given mod~la~ion code, sald apparatus comprising a cose~ selector for generating coset representatives ck in accordance with said given modulatlon code; and an encoder for sele~ting J said signals Yk, J ~ Yk, Yk+1,...~Yk~ 1) to be congruent to a sequence of J coset representatives ck (modulo AN), AN being an N-dimensional lattlce, N being a positive integer, said J signals being chosen from one of a plurality of NJ-dimensional constellatlons, said choice being based on a previous xk,7k'< k, at least one of said plurality of NJ-dimensional constellations comprising both a point with a positive sum of coordinates and another point with a negative sum of coordinates, said encoder being arranged so that said signals xk have finite variance Sx.
According to another broad aspect of the invention there is provided apparatus for generating a sequence of digital signals Xk, and/or a sequence of digital signals Yk, k = 1, 2, ..., such that the sequence of Yk signals is a partial-response-coded sequence derived from the sequence of xk signals, said signals Yk being a sequence in a given modulation code, said apparatus comprising a coset selector for generating coset representatives ck in accordance with said given modulation code;

~.~

~O~S~3 sb 60412-17~3 a generator of a sequence of alternati~e coset representatives Ck' chosen so that the sequence of coset representatives ck i~ a partial-response-coded sequence derived from the sequence of Ck' signals, and an encoder for selecting said signals xk to be congruent t~ a sequence of alternative coset representatives ck" where the congruence is modulo M if said coset representativès ck are real, M being an integer, and modulo ~N if said ck signals are N-dimensional, AN being an N-dimensional lattice, N heing an integer.
According to another broad aspect of the invention there is provided apparatus for generating a sequence of digital signals Xk and~or a sequence of digital signals Yk, k = 1, 2, ..., such that the sequence of Yk signals is a partial-response-coded sequence derived ~rom the sequence of xk signals, sald xk and Yk sequences having variances Sx and Sy, said symbols Yk being a sequence in a g~ven modulation code, said apparatus comprlsing means for receiving an input signal; and an encoder responsive to sald receiving means for generating said xk and/or Yk signals such that the ratio of variance S to y variance Sx is selectable within a predetermined range.
Accordlng to another broad aspect of the invention there is provided apparatus for generating a sequence of digital signals Xk and/or a sequence of digital signals Yk, k = 1, 2, ..., capable of representing n bits per signal, such that the relationship k Yk is Yk = Xk t xk_L, L an integer, said x and Y
signals having variances Sx and Sy, said Yk signals falling within ,. .
f~ ~ :~

13~6~3 9c 60~ 1723 an alphabet of possible Yk slgnals that are spaced apart within said alphabet evenly by a spacing ~, said apparatus comprising means for receiving an input signal having n bits per signal;
and an encoder responsive to said receiving means for generating said se~uence Yk and said sequence xk such tha~ said sequence Yk has a variance Sy less that 2S0 and said sequence xk has a variance Sx not much greater than Sy2/~(Sy~S0), S0 being approximately equal to the minimum signal power required to represent n bits per signal with a ~-spaced alphabet.
According to another broad aspect of the invention there is provided apparatus for generating a sequence in a given N-dimensional modulation code by ~enerating a sequence of one-dimensional signals, N being a positive number, said modulation code being based on an N-dimensional constellation partltioned into subsets associated with said code, said subsets each containing N-dimensional signal points, the choice of said subset being based on coded bits and uncoded blts of said slgnal points, said apparatus ~omprising means for receiving an input signal and generating the coded blts and the uncoded bits therefrom; and an encoder for deriving from said coded and uncoded bits, for each said N-dimensional symbol, a set of N, M-valued one-dimensional coset representatives ck corresponding to congruence classes of each of the N coordinates (modulo M~, M being a positive number, each coset represen~ative designating a subse~ of one-dimensional values in a one-dimensional constellation of ~, ~L306S~3 9d 6~412-1723 possible coordinate values for each of said N dimenslons, each said one-dimen~ion signal in said sequence being selected from said possible coordinate values based on uncoded bits.
According to another broad aspect of the invention there is pro~ided in a decoder for decodin~ a sequence Zk ~ Yk + nk~ k =
1, 2, ..., into a decoded sequence Yk, where the sequence of signals Yk is such that ~a) said sequence is from a given modulation code;
(b) the running diqital sum xk 3 Yk 1 ~ Yk 2 + ~ has finite variance Sx;
(c) said signals Yk fall in a predetermined permissible range dependent on Xk', k' < k; and the sequence nk represents noise, a range violation monitor comprising:
a means for reconstructing the estimated running digital sum k Yk + Yk_1 + ..,, and a means for comparing said decoded sequence Yk with said predetermined permissible range based on said estlmated running digital sum Xk,, k' ~ k, and for generating an indication when said Yk is outside said permissible range.
According to another broad aspect o~ the invention there is provided a decoder for decoding a sequence xk ~ Yk + nk~ k = 1, 2, ..., where sequence nk represents noise and the sequence of signals Yk is such that (a) said sequence is from a given modulation code, said code being capable of being generated by an encoder with a finite 13~6S ~3 9e 60412-1723 number Q of s~ates;
(b~ Yk ~ Xk i xK_L, L an integer, where said sequence xk has finite variance Sx, and the sequence nk represents noise, comprising a means for receiving the sequence Zk; and a modifled maximum likelihood sequence estimator responsive to the receiving means, said estimator being adapted to find MQ
partial decoded sequences up to some time K, where M, Q, and K are positive finite numbers, one such said sequence for each combination of said finite number Q of states and each of a finite number M of integer-spaced values modulo M, such that each said sequence la~ is in said code up to said time K;
(b) corresponds to said encoder being in a given said state at said time K;
(c) corresponds to a value of xk at said time K that is congruent to a given one of said values, modulo M.
Other advantages and features will become apparent from the following description of the preferred embodiments, and from the claims.
B

~3~6~ }3 Description of the Preferred Embodiments We first briefly describe the drawinqs.
Dr awinqs Fiqure 1 is a block diagram of a l-D partial s response channel~
r igure ~ is a bloc:~ d:iagram o~ an encode~ for an 8-state Ungerboeck code.
~ igure 3 is a signal constellation for the Ungerboeck code partitioned into 8 subsets.
Figure 4 is a block diagram of an equivalent encoder f or the Ungerboeck code.
Figure 5 is a blocX diagram of a generalized N-dimensional trellis encoder.
Figure 6 is a block diagram of a modified Figure 5, based on coset representatives.
Figure 7 is a block diagram of an equivalent one-dimensional encoder.
Figure 8 is a block diagram of a generalized N-dimensional trellis encoder.
Figure 9 is a block diagram of a generalized encoder with coset precoding.
Figure 10 is a block diagram combining Figures 8 and 9.
FigurQ 11 is a block diagram of an encoder with RDS eedback a~d coset pracoding.
Figures 12, 13, 14 are alternative embodiments of Figure 11.
Figures 15, 16, 17 are block diagrams of three equivalent filtering arrangements.

3~ Figure lB is a block diagram of a generalized decoder.
Figures 19, 20, 21 are block diagrams of alternative encoder~.

13Q65'13 Figure 22 is an alternative signal constellation.
Figure 23 i5 a block diagram of an encoder for use with the constellation of Fi~ure 22.
Figures 24, 25, 26 are bl~c~ diagrams of thrae eguivalent encoders.
Figure 27 is a blocX diagram of an alternative decoder.
Figure 28 is a schematic diagram of an expanded signal constQllation, Figure 29 is a rhombus for use with the constel].ation of ~igure 23.
Figure 30 is a block diagram of a twa-dimensional RDS feedback encoder.
Figure 31 is a diagram of the dimensions of a rhombus for uæe with the constellation of Figure 28.
Figure 32 shows a pair of constellations.
Figure 33 shows two disjoint constellations.
Structure and OPeration Referring to Figure 1, the invention includes a technique for generating signal sequences to be used as inputs for a partial response channel 10, for example a one-dimensional (real) l-D partial respons~ baseband system with a null at DC. (Later we shall indicate briefly how to modify such a design for other types of partial response systems.) Each output signal Zk f such a system is given by Zk Yk + nk' where the nk sequence (n(D)) represents noise, and the Yk sequence (y(D)) is a partial-response-coded (PRC) sequence defined by Yk Xk - Xk-l' where the xk seguence (x(D)) is the sequence of 13~6S~3 channel inputs. Because Xk = X~C_ 1 + Yk ' the xk sequence can be recovered from the PRC sequence by forming a runnin~ digital sum of the Yx values S (given an initial value for t;~e ~ sequence), -hus ~e call the ck sequence tne ~DS sequence. he sampie variances of the RDS sequence x(D~ and the PRC sequence y(D) will be denoted as Sx and Sy~ respectively.
The discrete-time partial response l-D
k epresented by block 12) is a composite of the responses of a chain of transmit filters, an actual channel, receive filters, equalizers, samplers, etc., desiqned in a conventional way to achieve a composite partial response l-D with the noise power P (of the noise sequence n(D)) being small relative to the PRC
power Sy~ We shall thus want to send a relatively large number n of bits per channel input. A detector (not shown) operates on the noisy PRC sequence z(D~ to estimate x~D) (or equivalently y(D), since there is a one-to-one relationship between them). If the detector is a maximum likelihood sequence estimator, then, to first order, the objective is to maximize the minimum squared distance dmin between permissible PRC
seguences y(D).
~n some applications, the design constraint will simply be to minimize the sample variance Sx of the RDS (input) sequence. In others. the constraint will be on Sy~ In still other applications, there will be an effective power constraint somewhere in the 3a middle of the composite filter chain, so that it will be desirable to keep both Sx and Sy small, and in fact to provide for a smooth design tradeoff between them.

13~5'}3 A related problem is the design of sequences with spectral nulls, e.g., a null at zero frequency (DC). There the objective may be to design sequences y(D~ that can represent n bits per sample, that have a spectral null, that have as small a sample ~ariance Sv as possible, but that also have a large minimum squared distance dmin between possible y(D) sequences. A
common auxiliary objective is to keep the variation of the running digital sum (RDS) of the y(D) sequence lQ limited as well, for systems reasons. Because the running digital sum sequence x~D) is, e.g., y(D)/(l-D), and its sample variance Sx is a measure of its variation, the present invention may also be applicable to the design of sequences with spectral nulls.
A number of design principles are useful in achieving our objectives. The first princip.e is to design th~ input (RDS) sequence x(D~ so that the output (PRC) seguence y(D), taken N values at a time, is a sequence of N-dimensional signal points belonging to subsets of an N-dimensional constellation determined by a known N-dimensional trellis code. Then the minimum squared distance dmin between PRC sequences will be at least the dmin guaranteed by the trellis code.
Furthermore, a maximum likelihood sequence estimator for the trelli8 code can be easily adapted for use with this system, and while perhaps not optimum, it will achieve the same effective dmin for essentially the same decoding complexity as for the same trellis code in a non-partial response system.
An illustrati~e embodiment of the present invention is based on a known 8-state 2-dimensional trellis code similar to that of Ungerboeck as described in the article cited above, which uses a 128-point two-dimensional constellation to send 6 bits per 13(~6S~3 (two-dimensional) signal. ~This is also similar to the code used in CCITT Recommendation V.33 for a 14.4 kbps data modem.) Figure 2 sho~s the encoder 20 for this code. For each six-bit symbol 21 delivered from a data source 23, 2 o~ the 6 in?ut bits t~ encoder 20 en~er a rate-2/3, 8-state convolutionaL encoder 22. Ihe 3 output bits of this encoder are used in a subset selector 24 to select one of 8 subsets of a 128-point signal constellation, illustrated in Figure 3; there are 16 points in each subset (points in the eight subsets are labelled A through H respectively) The remaining 4 ~'uncoded bits" 26 (Figure 2) are used in a signal point selector 28 ta select from the chosen subset the ~two-dimensional) signal point to be transmitted. The code achieves a gain in dmin of a factor of 5 (7 dB) over an uncoded system, but loses about 3 d~ in using a 128-point rather than a 64-point constellation, so the net coding gain is about 4 d~.
One-dimensional form of the 2-dimensional Unqerboeck code The sequence of symbols xk sent over the channel is one-dim~nsional in a l-D baseband partial response system. It is helpful (though not essential), therefore, to transform known trellis codes into one-dimensional form. There are two aspects to this transformation: first, to characterize the two-dimensional subsets as compositions of constituent one-dimensional subsets, and second, to characterize the finite two-dimensional constellation as a composition o constituent one-dimensional constellations. We now show 3Q how this decomposition is done for the illustrative twa-dimensional Ungerboeck code, and then indicate how it may be done in the general case of an N-dimensional trellis cod0.

~306S~3 The first step is to notice that each of the eight two-dimensional subsets A, B, ..., can be viewed as the union of two smaller two-dimensional subsets, say Ao and A1, Bo and Bl, etc., where each of the 16 smaller subsets can be character~zed as follo-~s Let the possible values ~f each c~ordir.ate of a signai point be partitioned into four classes a, b, c, d; then each of the smaller two-dimensional subsets consists of the points whose two coordinates are in a specified pair of classes. A convenient mathematical expression for this decomposition arises if we scale Figure 3 so that signal points are ane unit apart in each dimension (and the coordinates of each point are half-integers); then the classes a, b, c, d are equivalence classes (modulo 4), and each of the 16 sets Ao, Al, Bo~ ... are the points whose two coordinates are congruent to a given pair (x, y) modulo 4, where x and y may each take on one of the four values {a, b, c, d~, e.g. {~1/2, _3/2~. These four values are called 2Q (one-dimensional) 'coset representatives'. The points of the constellation of Figure 3 have been labeled with Os and ls to show one possible arrangement of the 16 subsets. For example the Go point 29 has coordinates x ~ 5/2, y - 9/2, and its coset representatives are (5/2, 9/2) modulo 4 or (-3/2, 1/2).
We may now modify Figure 2 as follows.
Referring to Figure 4, the three output bits of the encoder 22 plus one of the uncoded bits 30 are used as inputs to subsQt selector 32, which selects one of 16 3Q subsets based on the faur input bits, with uncoded ~it 30 selecting between Ao and Al, or 30 and Bl, etc., accarding to which of the original 8 subsets is salected by the three convolutionally encoded bits produced by encoder 22. In effect, encoder 22 and bit 131;~6S'~3 30 represent an 8-state rate-3/4 encoder, with the output selecting one of 16 subsets, although the set of possible si~nal point sequences has not changed. Next, designate each of these 16 smaller subsets by a pair of one-dimensional coset representatives 34, one ~or eac~
coordinate, ~here each coset represen.ative ck ~.ay takQ on one of four ~alues. The pair o~ coset representatives is denoted (clk, c2k).
An aspect of the invention is the observation that all of the good codes cited above -- i.e., those of Ungerboeck, Gallager, Wei, and Calderbank and Sloane --can be transformed in the same way. That is, any of these ~-dimensional trellis codes can be generated by an encoder that selects one of 4N subsets, where the subsets are specified by N 4-valued one-dimensional coset representatives, corresponding to congruence classes of each coordinate ~modulo 4). In some cases it is only necessary to use 2N subsets specified ~y N
2-valued one-dimensional coset representatives (e.g., 1_1/2}) corresponding to congruence classes of each coordinate (modulo 2); e.g., for Ungerboeck's 4-state 2D
code, Gallager's 8-state 4D code (and the similar code of Calderbank and Sloane), Wei's 16-state 4D code and 64-state ~D code, etc. Also, we have observed that many 25 good lattice code8 can be transformed in thiS way; e.g., the Schlafli lattice D4 and the Gosset lattice E~
can be represented by seguences of 4 or ~ two-valued one-dimensional coset representatives (modulo 2); the ~arnes-Wall lattices ~16 and ~32 and the Leech lattice ~24 can be represented by four-valued one-dimensional coset representatives (modulo 4).

13(~S~3 A general orm for all of these codes is shown in Figure 5. The encoder is N-dimensional and operates once for every N siqnals to be sent over the channel.
In each operation, p bits enter a binary encoder c 33 s and are encoded into p + r coded bits. I'hese coded bi.
select ( in selector 35) one of 2?~r subsets of an N-dimensional signal constellation (the subsets corresponding to the 2P~r cosets of a subla~tice ~l of an N-dimensior.al lattice A, the constellation being lo a finite set of 2n+r points of a translate of the lattice ~, such that each subset contains 2n p points). A further n - p uncoded bits selects ~in selector 37) a signal point from the selected subset.
Thus the code transmits n bits for every N-dimensional symbol, using a constellation of 2n+r N-dimensional signal pointsN The encoder C and the lattice partition A/~' ensure a certain minimum squared distance d2in between any two signal point sequences that belong to a possible subset seguence.
The observation above ~about the transformability of all good codes) is the result of the mathematical obse~vation that for all of the good trellis and lattice codes cited, the lattice 4ZN of N-tuples of integer multiples of 4 is a sublattice of thQ lattice ~' (and in some cases 2ZN is). Then, for somQ integer q, ~' is the union of 2g cosets of 4zN in A', The practical effect of this observation is that, provided that n ~ g + p, we can take the p + r coded ~its plus q uncoded bits into a subset selector that selects one of 2q+p+r cosets of 4ZN in ~, and further that these cosets can be identified by a sequence of N 4-valued one-dimensional coset representatives (clk, c2k, ...~cNk), Cjk represent integer-spaced equivalence classes ~3~6~

tmodulo 4), Thus already Figure 5 can be modified as shown in Figure 6. In this modification, we assume that the 2n+r-point signal constellation divides evenly into 2q~p+r subsets, each containing the same number 5 of signal points (2n q ?).
~ he illustrative ungereoecX code emDodi~.ent is an example in which N = 2, ~ = z2, ~,~ = 2RZ2, p = 2, p + r = 3, ~ = 1, and n = 6.
The second StQp i~ ta decompose the 10 con5tQllation into constituent one-dimensional constellations. For the constellation of Figure 3, each coordinate can take on one of 12 values which can be grouped as 8 'inner points' (e.g., t~l/2, ~3/2, ~S/2, ~7/2}) and 4 'outer points' (e.g., {~9/2, ~11/2~), as suggested by boundary 31 in Figure 3.
There are 2 inner points and 1 oute~ point in each of the four one-dimensional e~uivalence classes (e.g., the class whose coset representative is +1/2 contains the two inner points +1/2 and -7/2, and the outer point 9/2, because these three points are congruent to +1/2 (modulo 4). Given a coset representative, therefore, it is only necessary to specify whether a point is an inner point or an outer point and, i~ an inner point, which of the two inner points it is, ~his can be done with two bits, say blk ~' inner or outer) and b2k (a which inner) (or with one three-valued parameter ak).
WQ may say that the pair (blk, b2k) is a range identifying parameter ak, which takes on one of three values, indicating the following three ranges (a) from 0 to 4 ~inner point, positive);
(b) from -4 to 0 (inner point, negative);
(c) from -6 to -4 and from 4 to 6 (outer point).

~306S43 The fact that each range spans a portion of the real line of total width 4 that contains exactly one point congruent to any real number (modulo 4) means that the range-identifyinq parameter a~ plus the coset representative CX specify a u~ ai~nal ~oi.~-, Cor any value of Ck.
The signal point selector 36 of Figure 4 can then be decomposed as follows. Referring to Figure 7, three uncoded bits ~0 enter a range-identifying parameter selection element 42 for each pair of coordinates. One uncoded bit determines whether any outer point is sent. If so, a second bit then determines which coordinate will contain the outer point, and the third bit selects which inner point in lS the oth~r coordinate. If not, ~oth coordinates are inner points, and the second and third bits select which inner point in each coordinate. Thus, in sum, element 42 maps the three uncoded input bits 40 into two pairs of output bits 44 al = (bll, bl2), 2 2Q (b21~ b22), with each pair of bits used to determine one coordinate in conjunction with the corresponding coset representative, cl or c2, generated by coset representative pair selector 46. Thus the whole encoder has been reduced to a form in which each coordinate xk (48) is selected (in a coordinate sQlector 50) by 4 bits, two representing ck and two representing ak ' (blk, b2k).
All constellations commonly used with the above-cited codes can be decomposed in this way. The 30 principles are similar to those discussed in my U.S. ~j~
Patent 4,597,090 and in Forney et al., "Efficient Modulation..." cited above, where N-dimensional constellations were built up from constituent 2-dimensional constellations; a similar buildup from 13Q65~3 2-dimensional constituent constellations ~as used by Wei in conjunction with trellis codes in his ~h.6-. Pàtent Application, cited above.
The general form of encoder for N-dimensional codes is shown in Figure 8. For every ~ c~or~inates, p bits ,1 enter an encoder ~2 and ~ ~ r coded oi~s ~ are produ~ed; these plus q uncoded bits 56 enter a selector 58 which selects a sequence of N coset representatives c~(60)i the remaining n-p-q uncoded bits 62 are transformed (in a selector 64) into a sequence of range-identifying parameters ak (66) which together with the ck determine (in a signal point selector 68) a sequence of N signal point values xk (70) by a signal point selection function f(ck, ak) which operates on a one-dimensional basis. In general, the range-identifying parameter ak determines a subset of the real line (one-dimensional constellation) of width (measure) 4 which contains exactly one element congruent to any possible ck value (modulo 4), and the function f(ck, ak) selects that element. For all codes cited, the coset representative alphabet may be taken as four integer-spaced values (modulo 4); for some codes, the coset representative alphabet may be taken as two integer-spaced valuQs (modulo 2) (in which case the ranges are of width 2). The sizQ of the ak alphabet is as large as necQssary to send n bits per N
coordinates. The signal point sequences generated by this form of encoder are generally the same as those in the original code, and in particular, are separated by 30 the same minimum squared distance d2in as the ~
original code.

~30~ 3 Cose~ Precodinq N-dimensional signal point sequences generated by known good trellis codes, wh~n serialized to one-dimensional signal points, cannot in general be used as inputs to the partial response c~a.~nel of Fi~ure 1 without degradation of dmin (Decause of intersy~boi interference). However, a technique which we call coset precoding allows the adaptation of these known codes to partial response systems without increase of Sx or deg~adation of d2in. The general technique is illustrated in Figu~e 9.
wo use the same convolutional encoder 52 as used by the known trellis code, preerably in the form of Figure 8. The p + r coded output bits 54, rather than selecting a subset directly, are converted ~as in Fiqure 8) in a subset selector/serializer 70 into a sequence ck of N one-dimensional coset representatives cl, ...~CN, corresponding to the subset that would be selected in a non-partial response system, These 2Q coset representatives are then 'precoded' (in a precoder 72) into an alternativQ (or 'precoded') coset representative sequence Ck' (74), where Ck = Ck-1 + Ck (modulO 4) (In the cases where it is possible to use modulo 2 coset representatives, this precoding can be done modulo 2.) Thus the precoded coset representative sequence 74 is a running digital sum modulo 4 (or 2) of the ordinary coset representative sequence. Precoded coset representatives Ck' can then be grouped N at a time in grouper 75 to specify tin signal point selector/serializer 76) an N-dimensional subset; a signal point can then be selected (based on the uncoded bits 78) in the usual way; and the resulting signal 1306~3 point can be sent out as a sequence x(D) of N
one-dimensional sîgnals xk over the partial response channel (in the same order as they were precoded~
Note that if the ck are half-integers, then tAe Ck' alternate bet~een ~o sets of 4 val~es, o~e set displaced by l/2 from ~he otAer. ~Ais has on'y a minor effec~; we can, for example, "dither" alternating coordinates XX by +l/4 and -1/4 so as to accommodate this periodicity, Alternatively, -~e may let the ck alphabet be inte~er-valued, Q . g., {O, l, 2, 3}; then the Ck' are always from the same alphabet, e.g., (~1/2, ~3/2}, These offsets of c~' or CX do not affect the dmin of the code.
If the encoder is in the form of Figure 8, then Figure 9 can be put in the form of Figure lo, where the same hlocks do the same things. In particular, since we have characterized the function f(ck, ak) as one that selects the unique element congruent to ck in a range identified ~y ak, it does not matter if the precoding changes the ckl alphabet from the ck alphabet; indeed, the (modulo 4) in the precoder is unnecessary in principle, though possibly useful in practice.
With either Figure 9 or Figure lO, it can be shown that the PRC sequence Yk ' Xk ~ Xk 1 has elements that are congruent to ck (modulo 4), so they fall in th~ subsets of the original trellis code, and therefore have at least the same dmin. The RDS
sequence xk has the same average energy Sx as in the original trellis code if the Ck' alphabet is the s~e as the ck alphabQt; sven if not, approximate equality still holds. (In the illustrative embodiment, the average energy per coordinate is 10.25, with ~3~65~3 integer-spaced signals.) If the ck are integers, the Xk are independent, identically distributed random variables, and thus (a) Sy = 2Sx;
(b) the spectrum o~ the RDS seq~er.ce ~Xk} is flat (white) within its Nyquist band;
(c) the spectrum of the PRC sequence {Yk} is the same as that of the partial response channel.
Even if the ck are not integers, these statements are still approximately true.
Coset precoding can be modified for other kinds of partial response systems as follows. For a l+D
(one-dimensional) partial response system, use the same system except with ck_l' subtracted rather than added in precodQr 72, so that ck = ck ~ cx_l (modulo 4). For a l-D system, replace the delay element D by a delay element DL, so that Ck' = ck L~ + Ck For a l+D two-dimensional system, use two l+D precoders in parallel, with pairs of outputs from the subset selector~serializer as inputs, and with the two outputs determining the real and imaginary (in-phase and quadrature) parts of thQ two-dimensional signal point to be transmitted.
RDS Feedback Depending on the application, it may be desirable to reduce the average energy Sy of the PRC
sequence, at the cost of increasing the average~enerqy Sx of the RDS sequence. This will also tend to flatten the PRC spectrum, while raisinq the low~frequency content of the RDS spectrum. Ju tesen, "Information Rates and Power Spectra of Digital Codes", IEEE Trans. Inform. TheorY, Vol, IT-28, pp. 457-472, 13065~}3 1982, has introduced the notion of a "cutoff frequency~' fO below which the PRC spectrum is small and above which it tends toward flatness, and has shown tha~ fO
is approximated by fO - (Sy/2SX)f~, where f~J

5 is the Nyquist band edge frequency.) A general .~ethod of per^orming this tradeo~L
while maintaining the dmin of the trellis code in the PRC sequences is to augment the encoder of Figures 9 or lo as follows.
The PRC sequence may be computed from the RDS
sequence; for the l-D channel, each PRC signal is just Yk = Xk ~ Xk 1' Referring to Fig. 11, we may let the signal point selector 80 base each x~ on xk 1 (by feeding xk back through a delay element 82) as well as on the current precoded coset representative Ck' and on the range-identifying parameter ak, in such a way that large PRC values Yk (calculated in summer 84) are avoided. As long as the signals xk are still chosen to be congruent to the Ck' (modulo 4), the signals Yk will be congruent to the ck ~modulo 4), and therefore will preserve the d2in of the trellis code. (Note that although the idea is to precompute the PRC value Yk to keep it small, what is actually ed back is the previous RDS value xk 1' 50 that we call this RDS feedback.) For the illustrative embodiment, this could work as follows. As already noted, the normal selection function f(ck, ak) of selector 80 can be characterized by saying that the 8 inner points are the 8 half-integer values lying in the range f ~m -4 to +4, while the 4 outer points are the 4 half-integer values lying in the range from -6 to -4 and +4 to +6. We can vary the inner point range and outer point range as a function of xk 1' as long as the inner point range 13~6S~3 spans 8 signal points, 2 from each equivalence class, while the outer point range spans 4 signal points, 1 from each equivalence class.
A general way of doing this is to translate all ranges by a translation variaDle ~(x~_l) w:~ich is a function of xk 1 That is, in the illustra~ive embodiment, the inner point range is modified to be from -4 + R(x~ 1) to 4 + R(xk_l), and the outer point range to be from -6 + R(xk l) to -4 + R(xk_l) and from 4 + R(xk_l) to 6 + R(xk_l).
The function R(xk l) should be generally increasing with xk l so as to reduce the Yk. We have been able to show that the optimum choice is ~(Xk l) = ~Xk l~ where n is a parameter in the range o < n < 1~ When n = O, the RDS feedback through element 82 disappears and Figure ll reduces to coset precoding as in Figure lO. With this choice, if S0 is the value of Sx in the ordinary case (B = 0), then it is approximately true that (a) Sx ' So/(l - n );
(b) Sy - 2So/( 1 + B);
(c) The spectrum Sx(f) of the RDS sequence is proportional to 1/(1 - 2ncOS ~ + n ) ~ where ~ =
lrf /fN;
(d) The spectrum Sy(f) of the PRC sequence is proportional to 2(1 - cos~)/(l - 2n cos~ + B );
the "cutoff frequency" fO is (l-n)fN.
(e) The xk are limited to the range form -M/2(1+B) to M/2(1-B), and the Yk are limited to the 30 range -M to M, if the range of coordina~es in the orignal code is from -M/2 to M~2.
As B approaches 1, Sy approaches S0, and Sy(f) approaches a flat spectrum with a sharp null at DC. Meanwhile, Sx becomes large and Sx(f) 13~65~3 approaches a l/(l-D) spectrum, except that it remains finite near DC. We have been able to show that this is the best possible tradeoff bet~een Sx, Sy~ and S0.
Figures 12, 13, 14 sho~ three equivalent ways of generating .~ and/or y~ based o~ c,;, a,~, a.~d ~k 1 Figure 12 corresponds most closely ~o ~igure 11.
In Figure 12, the feedback varia~le c'k_l in coset precoder ~2 is replaced by xk_l, since c'k-l -Xk 1 tmodulo 4), and only the value of c'k (modulo 4~ is used in selector 80. R~ak) denotes the range identified by ak, and R(xk 1) represents the range translation variable introduced by RDS feedback. Since Yk k Xk-l - c ~ - xk_l (modulo 4) and C'k ~ Ck + xk_l (modulo 4), Yk ~ Ck (modulo 4).
Figures 13 and 14 are mathematically equivalent to Figure 12, in the sense that if they have the same starting value xk 1 and the same sequence of inputs (Ck, ak), they will produce the same sets of outputs (xk~ Yk). In Figure 13, Yk is chosen as the unique element congruent to ck in the range R(ak) +
R(xk 1) ~ Xk 1~ and xk is determined from Yk as Yk + Xk 1' so Xk _ c k ~ Ck + Xk 1 (modulo 4), and is the unique element in the range R(ak) + R(xk_l) congruent to c'k (modulo 4). In Figure 14, an innovations variable ik is chosen as the unique element congruent to cl'K _ ck + Xk 1 ~ R~xk_l) (modulo 4) in the range R(ak), and xk is determined from ik as xk = ik +
R(xk_l), so xk - c k + R(Xk-l) ~ ck~
(modulo 4), and is the unique element in the range R(ak) + R(xk_l) congruent to c'k (modulo 43.
Figure 12 combines the delay element in the precoder with the delay element necessary for RDS feedback, and 13(:~65~3 is most useful if xk is the desired output and the C'k are always from the same alphabet, e.g. {+1/2, ~3/2~. Figure 13 eliminates the precoder altogether, and is most useful if Yk is the desired output and the 5 c,~ are always from the same alphabet, e.g {~ 1/2, + 3/2~. Figure 1~ takes tAe range transla~ion variable R(xx 1) outside of the selector, so that the ik are always chosen from the same range (the union of all R(ak)); the innovations sequence i(D) is 10 approximately a sequence of independent identically distributed random variables ik (ignoring the mi~or variations induced by the c'k congruence constraint) and this auxiliary sequence can be useful if a white (spectrally flat) sequence deterministically related to x~D) or y(D) is desired.
Figures 15, 16, 17 illustrate three equivalent filtering arrangements for use with the x(D), y(D), and i(D) seguences of Figures 12, 13, 14. In Figure 15, the RDS sequence x(D) is filtered in a transmit filter 2Q HT(f) before being transmitted (as signal s(t)) over the actual channel ~not shown). In Figure 16, the PRC
sequence y(D) is filtered in a transmit filter H'T(f) whose response is equivalent to that o a cascade of a l/(l-D) sampled-data filter and HT(f); since y(D) has a DC null, it does not matter that the response of l/(l-D) is infinite at DC (particularly if HT(f) also has a DC null). In Figure 17, the innovations sequence i(D) is filtered in a transmit filter H"T(f) whose response is equivalent to that of a cascade of a l/(l-BD) sampled-data filter an~d HT(f); this is equivalent to Figures 15, 16 if R(xk 1) = Bxk l;
otherwise, the equivalent sampled-data filter is the filter corresponding to xk = ik + R(xk 1)' in 13~6S~3 general nonlinear. Any one of these equivalent forms may be preferable depending on HT(f), R(xk 1)~ and the implementation technology being used.
Certain modifications of the above RDS feedback systems may b~ desirable in prac~ ce For examDle, may ~e desirable to change ~he .~or,~ of tne rar~ges R(ak) from those used when R(xk 1) = - For example, in the illustrative embodiment, a simply implemented form of RDS feedback is as follows: when xk 1 is positive, let Yk be chosen as usual in the range from -4 to 4 if ak indicates an inner point, but if ak indicates an outer point, let Yk be the number congruent to c~ in the range from -4 to -8; when Xk 1 is negative, use the range from 4 to 8 for outer points. Then (a) the range of the PRC sequence Yk is limited to -7 1/2 to +7 1/2, rather than -11 to 11 when there is no RDS feedback;
(b) the PRC variance Sy is reduced to 13.25 from 20.5, a reduction of 1.9 dB, and about 1.1 d~
above S0 ~ 10.25;
(c) thQ mean Of Yk is - 3/2 if xk 1 is positive, and + 3/2 if xk 1 is negative, so that the RDS sequence tends to remain in the neighborhood of zero. While it is difficult to compute Sx exactly, it follows from the facts that E[ykxk_l] = Sy/2 and E~Yk Xk-l]
- -(3/2) E[¦xk 1l] that the mean of the absolute value of xk is Sy/3 = 4.42, so that the RDS sequepce xk is fairly well bounded. (With no RDS feedback, the mean of the absolute value of xk is 2.75);

13065'~3 (d) the variance of the Yk, given xk 1~ is S0 - 11, about 0.3 dB higher than the S0 = 10.25 possible with no RDS feedback. The minimum possible S.c for s~ = 13 . 25, sO = ll, is S:c ~ 19.5, correspcndi~ SO n - 0.66 since Sx = Slx¦ 2x must be greater than (4.42) ~ ls.s, so with this simple method we achieve less than the optimal spectral tradeoffi (e) every possible Yk is associated with a unique pair (Ck, ak). As we shall discuss in more detail below, this means that a decoder need not keep track of an estimated running digital sum of the estimated PRC sequence, and that there will be no error propagation in the decoder.
In summary, this simple method does not achieve the best power tradeoff between Sx and Sy~ but does effectively limit not only Sy but also the peak values of Yk, keeps the RDS sequence xk fairly well bounded, and avoids error propagation at the receiver.
Thus these methods allow for trading off of Sx versus Sy (i) from the unconstrained case, where the x~ sequence is uncorrelated, Sx has the same energy S0 as is necessary to send n bits per symbol in the non-partial-response case, and Sy = 2Sx, (ii) almost to the case where the Yk sequence is uncorrelated, Sy = S0, and Sx becomes very large.
These tradeoffs are possible for all trellis and lattice 30 codes cited. ~

~3~6S~3 Decodin~The above methods succeed in generating PRC
sequences that belong to a known good code, and therefore have a d2in at least as great as that of 5 the code.
Referring to rig. 1~, a suitaoie detector for the noise received PRC sequence z(Dt = y(D) + n(D) is therefore a maximum likelihood sequence estimator (Viterbi algorithm) for the known good code, adapted as follows:
(a) A first step of decoding may be, for each noisy received PRC value Zk = Yk + nk~ for each of the four classes of real numbers congruent to the four one-dimensiQnal coset representatives Cjk (modulo 4), j = 1,2,3,4, find (block 92) the closest element Yjk in each class to Zk' and its 'metric' mjk ~ (Yjk ~ Zk) (squared distance from z~);
(b) In a cods based on an N-dimensional lattice partition ~/~', a second step of decoding may be, for each of the 2P+r cosets of ~' in ~, to find the best (lowest metric) of the 2q cosets of 4ZN whose union is that coset of ~', by summing the respective metrics of the constituent one-dimensional metrics mjk and comparing these sums (block 94);
(c) Decoding can then proceed in the usual manner (block 96), using as a metric for each coset of ~' the best metric determined in step (b).
The decod~r will ultimately produce an estimate of the sequence of cosets of A', which can be mapped to a sequence of estimated coset representatives Ck, which can be mapped to the 13Q6S'~3 corresponding Yk, from which the original ak and x~ can be recovered if desired (block 98). These last steps require that the decoder keep tra~k o~ the running digital sum x,~ l of ~he est imates Yx~
since tne ~RC sequences are i~ ~he Xnown code, the error probabili~y of this decoder will be at least as good as that of the known code, in the se~se of achieving at least the same effective d2in.
However, because the PRC sequences are actually only a subset of the known code sequences, such a decoder is not a true maximum likelihood sequence estimator for the PRC sequences. As a result, it may occasionally decode to a sequ~nce which is not a legitimate PRC seque~ce.
Legitimate PRC sequences must satisfy the two following additional conditions:
(a) A legitimate finite PRC sequence y(D) must be divisible by l-D; i.e., the sum of its coordinates must be zero;
(b) the range constraints imposed by the signal paint selector must be satisfied for all Yk (or equivalently xk or ik), based on the reconstructed values of the RDS xk 1' If this decoder makes a normal decoding error, corre8ponding to a short period of wrong coset QStimates followed by correct coset estimates, it is possible that the corresponding finite PRC error sequence will have a running digital sum other than zero. This will cause a persistent error in the decoder's estimated running 3a digital sum xk~ which may lead to occasional mapping errors back to the Yk, ak, and ultimately Xk, even though the cosets ck are correct, for as long as the error in the RDS estimate persists.

:~3~6S43 The decoder must therefore continually monitor (block 99) whether the range constraints in the reconstructed Yk and xk are satisfied. If they are not, then it knows that its estimated RDS x,~_l is incorrect; is should adjus~ by t;~e ~ini.~r~m a.~o~nt necessary for the range cons~rai,~t to be sa~is'ied, assuming that the coset sequencQ c~ is correct. With probability 1, this will eventually result in resynchronization of the estimated RDS to the correct value, and normal decoding can resume. However, there may be a considerable period o~ error propagation.
Avoidance of Error ProPaqation We now give a general method of avoiding error propagation at the receiver. The method works best when the signal constellation consists of all points in ~
within an N-cube, but is not restricted to that case.
It may be regarded as a generalization of the principles of eaxlier forms of precoding (modulo M) for use with coded sequences.
The basic idea is that each possible PRC value Yk should correspond to a unique (Ck, ak) value, when the code can be formulated in one-dimensional form as in Figure 7; or, more generally, that each group of N
Yk values should correspond not only to a unique sequence of N ck values but also to a unique set of uncoded bits, if a general N-dimensional signal point selector is used as in Figure 6. Then the inverse map from decoded Yk to coded and uncoded bits is independent of the decoder's estimate of the running 3Q digital s~, so that (a) the decoder need not keep track of the RDS;
(b) error propagation does not occur.
Thus in Figure 18, block 99 can be eliminated.

~3~5'~3 Figure 19 shows how this may be done where the code can be formulated in one dimensional form, as in the illustrative embodiment. From ck and a~, a signal point selector selects a value s~ (cx, 5 a~) as in Figure 8. rn tne illustrative embodiment, s~ ta~es on one of 12 val~es, ~amely, t.~e half-integral values in the range from -6 to 6. In general, Sk will take on one of the values from an integer-spaced alphabet in a range of width M; we denote this range by Ro~ Then, as in Figure 13, Yk is selected as the unique number congruent to Sk ~modulo M) in the range Ro + R(xk_l) xk_l o where R(xk 1) is an RDS feedback translation variable, and xk 1 is the previous RDS signal point. The current RDS xk is computed as Yk + Xk 1~
Figures 20 and 21 are equivalent methods of generating xk and/or Yk from the sequence sk such that Yk ~ Sk (modulo M), analogous to Figures 12 and 14. In Figure 21, an innovations variable ik is generated which is more or less white and uniformly distributed OvQr the range Ro~ so that its variance S0 is approximately M2J12; thus S0 ~ 12 for th0 illustrative embodiment, a penalty of about 0.7 dB over the value of 80 ~ 10.25 achievable with no RDS
2S feedback. As in Figures 12, 13, 14, all three sequences xk~ Yk, and ik carry the same information, and as in Figures 15, 16, 17, any can be used as the input to a filter which shapes the spectrum for transmission.
The penalty in the innovations variance is 3~ elimi~ted if the original code coordinates are uniformly distributed over a range Ro; i.e., if the original constellation is bounded by an N-cube with side Ro~

~3~ÇiS~3 As an illustrative embodiment with a square constellation, we use the same two-dimensional 8-state Ungerboeck encoder as in Figure 2, except with the 12~-point constellation o~ Figure 22 rather than that of 5 Figure 3. The constellation cons~s~s of alternate ~oints from the conventional 2~6-~oint 16 .x 16 constellation; thus the coordinates have the 16 half-integral values { ~ 1/2, ~ 3/2, ..., _ 15/2}, but with the restriction that the sum of the two coordinates must be an even integer (0, modulo 2).
The minimum squared distance between signal points is thus 2, rather than 1; and the d2in of the code is lo, rather than 5. The variance of each coordinate is now 21.25 rather than 10.25, which after scaling by 2 is a loss of 0.156 dB relative to the Figure 3 constellation, since the cross is more like a circle than is the square. (In lattice terminology, we are now using the 8-way lattice partition RZ2/4Z2 rather than æ /2RZ ).
It will be observed that now each of the eight subsets corresponds to a uniqua pair of coset representatives (cl, c2) modulo 4, such that cl +
C2 ~ 0 (modulo 2). Therefore, the three coded bits of Figure 2 determine a pair of coset representatives directly in sub8et selector 24, rather than with the aid of an uncoded bit as in Figure 4. The four uncoded bits then select one of the 16 points in the selected subset. In this case, the uncoded bits may simply be taken two at a time to determine one of the four ranges 3Q -8~to -4, -4 to 0, 0 to 4, or 4 to 8. This is conveniently expressed by letting the two-bit range identifying parameters (al, a2) each represent one of the four values {_2, _6}; then the coordinate s,election function is simply Sk = f(ck, ak) - ck ~3C~5'~3 + ak. Note that the possible values for sk are the 16 half-integral values in the range Ro from -8 to 8, of width M = 16.
Conventional precoding may then be don~ ~odulo 5 16. The entire encoder is illustr2t~d in Fi~re 23.
The RDS value xk is the SUIT sk ~ ~sk-l (m 16). In this case the xk values are essentially independent identically distributed (white) random variables, and Yk = Xk - Xk-l ~ Sk (mo To obtain spectral tradeoffs via RDS feedback as in Figures 12, 13, 14, let sk continue to represent the desired congruence class of Yk (modulo 16), and let R(xk 1) be an RDS feedback variable as in Figures 12, 13, 14, ideally equal to ~xk 1 Then Figures 24, 25, 26 show three equivalent methods of obtaining qU ces xk and/or Yk = Xk ~ xk_l such that Yk ~ Sk (modulo 16) and Sx and Sy have the desired tradeoff, gi~en S0 = 21.25. Here Ro is the range from -~ to 8.
In this case the innovations variable ik has variance S0 ~ 162/12 - 21.33, essentially the same as the variance of each coordinate in Figure 22, so that there is no penalty beyond the 0.16 dB involved in using Fig. 22 rather than Fig. 3.
As already noted, the decoder need not keep track of the RDS, because, given the estimated PRC
sequence Yk, the ck~ ~k~ and ultimately the original input bit sequence are uniquely determined.
However, if the decoder does keep track of the estimated 3Q ~DS and th~ corresponding ranges that the Yk should fall into, it can detect that an error has occurred whenever the decoded Yk falls outside the estimated S~3 - 36 ~

range. Even if not used for error correction, such range violation monitoring can yield an estimate of decoder error rate.
Auqmented Decoders ~ ~rue ~aximum li~elihood sequence estimator would take into account rhe e~tire state of the encoder and channel, which in general will include the value of the RDS xk 1 (~he channel state) as well as the state of the encoder C. Such a decoder would achieve the true dmin of th~ PRC sequences, and would be free of error propagation. However, because xk l in general takes on a large number of values, in principle possibly an infinite number with RDS feedback, such a decoder may not bo practical. In addition, to achieve the true dmin may require an essentially infinite decoding delay because the code/channel combination becomes quasi-catastrophic when n is large, as we shall sxplain more fully below.
It may be worth considering augmenting the decoder to at least achieve the true dmin of the code, however. 3ecause all finite PRC sequences are divisible by l-D, all finite-weight error sequences must have even weight. Thus, the true dmin is always even. In the illustrativQ embodiment, the true dmin i8 actually 6, not 5.
A general method for achieving the true dmin in such cases while only doubling the effective numbQr of states in the decoder is as follows. LQt the decodQr split each stata of the encode~ C into two, one corresponding to an even RDS and one to an odd. During decoding, two sequences then merge into the same state only if their estimated RDS
has ths same value (modulo 2). Thus, it becomes imposSible for two sequences differing by an odd-weight 13(36S~

error seguence to merge, so that the effective d2in is the weight of the minimum even-weight error sequence in the original code. Further, if there is a decoding error that res~lts in a persistent estimated RDS error, as discussed above, that error must ~e at least 2, so it will tend ~o be detected sooner.
Tha decoder o~ Figure 18 can be used, modified only as shown in Figure 27. For most codes, each of the subsets of the signal constellation (cose~s of ~' in ~? will contain points all of which have a sum of coordinates which is either even or odd. For example, in Figure 3, four of the eight subsets contain points whose coordinate sum is O (modulo 2), and 4 contain points whose coordinate sum is 1 (modulo 2). Thus the metric of each subset (coset of ~' in ~) can be determined as before in blocks 92 and 94; the maximum likelihood sequence estimator 196 is then modified to find the best sequence of cosets that (a) is in the code, and ~b) has a running digital sum congruent to 2Q zero (modulo 2). The decoded coset sequence is mapped back to Yk and xk in block 98 as before, with adjustment of xk l by block 99 if necessary (adjustments will now be by multiples of 2).
There is a drawback to this technique, however, in addition to the doubling of the decoder state space.
Two sequences may differ by an odd-weight error sequence followed by a long string of zeros (no differences).
The decoder may then ~ollow parallel pairs of states in the decoder trellis for a very long time, without 3~ re~$olving the ambiguity. This 'quasi-catastrophic' behavior can ultimately be resolved by th~ maximum likQlihood sequence estimator only by a range violation 13G6S'~3 - 3~ ~

due to the differing RDS parity on the two paths. Thus, the decoding delay required to achieve the true d2in may be very large.
For this ~eason, it Will generally be preferable simply .o chcse a~. encoder c .~ith ~-~ice the number of states, and use an unaugmented decoder for c.
For example, there is a 16-state 2-dimensional Ungerboeck code with d2in = 6; even though it may have a somewhat larger error coefficient ~han the 8-state code with an augmented 16-state decoder, we believe that in practice it will be preferable.
It may be worth mentioning that PRC sequences drawn from the 4-state two-dimensional Ungerboeck code also have a true d2in of 6, since that code has dmin ' 4, with the only weight-4 error sequences being single coordinate errors of magnitude 2, which are also not divisible by l-D. A 16-state decoder which keeps track of the RDS modulo 4 can achieve this dmin. However, in this case not only is the code quasi-catastrophic, but also the error coefficient is large, so again it would seem that the ordinary 16-state 2D Ungerboeck code would be preferable.

As mentioned earlier, a complex (or quadrature) partial response system (QPRS) may be modeled as a 1 + D
sampled-data filter operating on a complex-valued RDS
sequence x(D) to produce a complex-valued PRC sequence y~D~ - (1 + D) x(D); i-e-~ Yk ' -Xk + X-k-l used with double-sideband quadrature amplitude 3~} modulation over a bandpass channel, such a system results in nulls at both band edges, fc ' fN~
where fc is the carrier frequency and fN ~ 1/2T is the width of a single Nyquist band.

S~-~3 When N is even and 4ZN is a sublattice of ~', as is the case with all good codes previously mentioned, then we can adapt a kncwn good code for use in a QPRS system by using essentially the same principles as before. .~ coset of 4z~ can be specified by N/2 complex-valued coset representatives Ck, ~here coset represQntatives take on on~ of 16 possible values, corresponding to 4 integer spaced values (modulo 4) for the real and imaginary parts of Ck, respectively. The general picture of Figure 8 then holds, except that coset selector 58 and range-identifying parameter sQlector 64 select N/2 complex-valued coset representatives ck and range-identifying parameters _~, and the signal point selector operates once per quadrature signal and puts out complex-valued signals Xk. Coset precoding as in Figure 9 is done by forming the complex-valued precoded coset c'~ _ ck -Ck 1' (modulo 4~ onco per quadrature symbol. RDS
feedback as in Figures 11, 12, 13 is done by using a function R(ak) that identifies a region of complex space of area 16 that contains exactly one element from any coset of 4z2, and a complex-valued translation variable R(xk_~), ideally equal to nxk 1~ In the cases where 2Z or 2RZ is a sublattice of A', precoding can be done modulo 2 or 2 + 2i, respectively, and R(ak) can identify a region of area 4 or a containing exactly one element from any coset of 2Z~
or 2RZ , respectively.
Hiqher-Dimensional SYstems We have shown embodiments in which coordinates of N-dimensional symbols are formed on a signal-by-signal (one or two-dimensional) basis, with signal-by-signal feedback of the previous RDS value Xk 1~ Similar kinds of performance can be obtained by 130~3 systems which select signals on a higher-dimsnsional basis. In ~uch systems, the precoded coset representatives must be grouped as in Figure 9 so as to select subsets in the appropriate dimension, signal points then selected i.~ that di"ension, and t.~e coordinates then serialized again for transmission over the channel. If the order of cosets is maintained, then such a system retains the property that the PRC
sesuenc~s are from the given code and have the specified dmin. In such a system it may ~e more natural to do (RDS) feedback on a higher-dimensional basis rather than on each signal.
N-Dimensional Codes Although representation of codes in one-dimensional form is desirable, it is not essential.
In this section we show how codes may be generated directly in N dimensions. In certain forms, the N-dimensional code is entirely equivalent to its one-dimensional counterpart. In other forms, simplified 2Q emhodiments may be obtained.
Again, we shall use for illustration the 8-state 2-dimensional Ungerboeck-type code of Figure 2, with the 2-dimensional 128-point constellation of Figure 3. In this constellation, recall that each coordinate takQs on values from the alphabet of the 12 half-integral values in the range from -6 to 6; the two-dimensional constellation uses 128 of the 144 possible pairwise combinations of elements of this alphabet.
As a first step, we expand the signal constellation to an infinite number of values, as follows. Let the expanded constellation consist of all pairs of numbers that are congruent to some point in the original (Figure 3) constellation (modulo 12). Thus the 13C165'13 points in the expanded constellation consist o~ pairs of half-integral values. If we regard the original constellation as a cell bounded by a 12 x 12 square 98, then the expanded constellation consists of the infinite repeti~ion of this cell t;~roughout 2-s?ace, as indicated diagrammatically in Figure 28. No~e that eac;~ cell contains only 128 of the 144 possible points; there are 4 x 4 'holes' 99 in the expanded constellation.
The key property of this expanded constellation 101 is that if we place a 12 x 12 square anywhere in the plane (with sides oriented horizontally and vertically), the square will enclose exactly 128 points, one congruent to each of the points in the original constellation. An even more general statement is true:
if we place a rhombus 102 with horizontal width 12 and vertical height 12 (see Figure 29~ anywhere in the plane, it too will enclose l2a points, one congruent to each point in the original constellation.
Referring to Figure 30, we may now implement RDS feedback on a two-dimensional basis as follows. Let Xk 1 represent the running digital sum of all Yk pr~vious to the current (two-dimensional) symbol. Let R(xk 1) now denote a region of the plane corresponding to a 12 x 12 rhombus as in Figure 29, with both the shape and the location of the rhombus possibly depending k-l et ~Yo,k, Yo,k+l) denote the point in the original constellation that would be selected (in selectors 104, 105) by the three coded bits and four uncoded bits according to the unconstrained code (Figure 2). Then (in selector 106) select (Yk, Yk+l) as the unique point in the two-dimensional expanded constellation that lies within the region R(xk 1) and is congruent to (Yo~k~ Y0~k+l) ( 13~)6~'~3 12); these will be the two coordinates Yk. We can obtain (xk~ xk+l) from Xk = Yk ~ Xk-l' Xk+
Yk+l + Xk as shown, we now show that this two-dimensional system can produce the same outputs as the one-dimensional RDS
feedback system (modulo 12) shown earlier, with the optimal one-dimensional RDS feedback variable R(xk 1) = Bxk 1 Referring to Figure 31, in one dimension, given xk 1' Yk is chosen as the unique value in the range Ro + Bx~_l - x~_l congruent to Sk (modulo 12), where we now recognize that Sk is congruent to YO k. Hence, one coordinate of tho rhombus used in the two-dimensional system can be taken to lie in the same width-12 range. Then, given xk 1 and Yk, and thus also Xk = Yk + Xk-l' Yk+l is chosen as the unique value in the range Ro - (l-B)xk = Ro - (l-~)Yk ~ B)xk 1 that is congruent to sk+l ~ Yo k+l (modulo 12). Thus, Yk~l lies in the range Ro - (l-~)xk_l (same as Yk), shifted by 2Q -(l-~)yk.
Thus, by proper choice of rhombus, we can emulate the performance of a one-dimensional (modulo 12) RDS feedback system with a two-dimensional system. It will thus have the same advantagQs~ including avoidance of error propagation and near-optimal tradeoff between Sx, Sy and SO, and the same disadvantages, notably the increase in SO to 12 over the 10.25 otherwise possible"
We can choose other two-dimensional RDS
feedback variables (regions) to further simplify implementation, and achieve other advantages, at the cost of suboptimal power tradeoffs. For example, a system almost identical to the simplified one-dimensional system described earlier results if we 1306~ 3 let R(xk 1) be the square 120 of side 12 centered at (-2, -2) when xk 1 is positive, and the square 122 centered at (+2, +2) when x~ 1 is negative. Thus ~e use one of the two constellations 124, 126 shown schematically in Fi~ure 32 As in the previous one-dimensional system, inner points are always chosen from the same set regardless of XX 1~ but outer points are varied so as to bias Yk in a positive or negative direction. The ranges of the Yk are strictly limited from -7 1/2 to 7 1/2. rn fact, this system is identical to the earlier simplified system, except that Yk+l is chosen on the basis of xk_l, rather than Xk. In practice, all the measures of performance and spectrum will be very 1~ similar.
Another variant yields a system akin to that o~
the Calderbank, Lee, and Mazo type. A CLM-type system uses an expanded signal constellation with twice the ordinary number of signal points, divided into two disjoint constellations, one to be used when xk 1 is positive, and the other when xk 1 is negative. Figure 33, for example, shows a 16 x 16 square constellation divided into two disjoint constellations 110, 112 of 128 points each, æuch that each such constellation divides evenly into 8 subsets of 16 points each. One constellation consists of points the sum of whose coordinates is positive or zero and is used when xk 1 is negative; the other consists of points whose coordinate sums are negative or zero and is used when xk_l is positive. In two dimensions, doubling the constellation size doubles Sy and thus does not yield a favorable power tradeoff; however, in higher dimensions the penalty due to the use of two disjoint constellations is less.

13~65~3 - 4~ -These ideas can be generalized to N dimensions, as follows. If there is a one-dimensional formulation of the code as in Figure 8, using modulus M, then an N-cube of side M completely surrounds the N-dimensional constellation, and the resul.ing cell can be replicated to cover N-space wishout compromising the minimum squared distance between code sequences that are congruent to original code sequences modulo M. Then we may use an N-dimensional RDS feedback function R(xk 1)' where for all xk_l, R(xk_l) is a reg of N-space of volume ~ that contains exactly one point in each equivalence class of N-vectors modulo M, in an N-dimensional analogue of Figure 30.
Other embodiments are within the following claims.

Claims (36)

1. Apparatus for generating a sequence of digital signals Xk and/or a sequence of digital signals Yk, k = 1, 2, ..., such that the sequence of Yk signals is a partial-response-coded sequence derived from the sequence of xk signals, said signals Yk being a sequence in a given modulation code, said apparatus comprising a coset selector for generating coset representatives Ck in accordance with said given modulation code; and an encoder for selecting J said signals Yk, J ? 1, (Yk, Yk+1,...,Yk+J-1) to be congruent to a sequence of J coset representatives ck (modulo .DELTA.N), .DELTA.N being an N dimensional lattice, N being a positive integer, said J signals being chosen from one of a plurality of NJ-dimensional constellations, said choice being based on a previous Xk,, k'<k, at least one of said plurality of NJ-dimensional constellations comprising both a point with a positive sum of coordinates and another point with a negative sum of coordinates, said encoder being arranged so that said signals Xk have finite variance Sx.

2. Apparatus for generating a sequence of digital signals Xk and/or a sequence of digital signals Yk, k = 1, 2, ..., such that the sequence of Yk signals is a partial-response-coded sequence derived from the sequence of xk signals, said signals Yk being a sequence in a given modulation code, said apparatus comprising a coset selector for generating coset representatives Ck in accordance with said given modulation code;
a generator of a sequence of alternative coset representatives ck, chosen so that the sequence of coset repre-sentatives ck is a partial-response-coded sequence derived from the sequence of Ck' signals, and an encoder for selecting said signals xk to be con-gruent to a sequence of alternative coset representatives ck', where the congruence is modulo M if said coset representatives ck are real, M being an integer, and modulo .DELTA.N if said ck signals are N-dimensional, .DELTA.N being an N-dimensional lattice, N being an integer.

3. Apparatus for generating a sequence of digital signals Xk and/or a sequence of digital signals Yk, k = 1, 2, ..., such that the sequence of Yk signals is a partial-response-coded sequence derived from the sequence of xk signals, said xk and Yk sequences having variances Sx and Sy, said symbols Yk being a sequence in a given modulation code, said apparatus comprising means for receiving an input signal; and an encoder responsive to said receiving means for generating said xk and/or Yk signals such that the ratio of variance Sy to variance Sx is selectable within a predetermined range.

4. The apparatus of claim 1 wherein the relationship between the xk signals and Yk signals is Yk = Xk ? Xk-L, L an integer.

5. The apparatus of claim 2 wherein the relationship between the xk signals and Yk signals is Yk = Xk ? Xk-L, L an integer and wherein Ck' = Ck - c' k-L (modulo M), in the case when Yk = Xk + xk-L, Ck' = ck + c'k-L (modulo M), in the case where Yk = Xk - Xk-L.

6. Apparatus for generating a sequence of digital signals Xk and/or a sequence of digital signals Yk, k = 1, 2, ..., capable of representing n bits per signal, such that the relationship between xk and Yk is Yk = Xk ? Xk-L, L an integer, said xk and Yk signals having variances Sx and Sy, said Yk signals falling within an alphabet of possible Yk signals that are spaced apart within said alphabet evenly by a spacing .DELTA., said comprising means for receiving an input signal having n bits per signal; and an encoder responsive to said receiving means for generating said sequence Yk and said sequence xk such that said sequence Yk has a variance Sy less than 2SO and said sequence xk has a variance Sx not much greater than Sy2/4(Sy-SO), SO being approximately equal to the minimum signal power required to represent n bits per signal with a .DELTA.-spaced alphabet.

7. The apparatus of claim 6 wherein said sequence Yk is a sequence in a given modulation code.

8. The apparatus of claim 3 wherein the relationship between the xk signals and Yk signals is Yk = Xx + Xk-L, L an integer.

9. The apparatus of claim 8 wherein said xk and Yk signals sequences are capable of representing n bits per signal, said Y
signals fall within an alphabet of possible Yk signals that are spaced evenly by an amount .DELTA., said predetermined range is control-led by a parameter .beta., and Sx is approximately S0/(1-.beta.2), and Sy is approximately 2S0/(1+.beta.), S0 being approximately the minimum signal power required to represent n bits per symbol with a .DELTA. -spaced alphabet in accordance with said code.

10. Apparatus for generating a sequence in a given N-dimensional modulation code by generating a sequence of one-dimensional signals, N being a positive number, said modulation code being based on an N-dimensional constellation partitioned into subsets associated with said code, said subsets each containing N-dimensional signal points, the choice of said subset being based on coded bits and uncoded bits of said signal points, said appara-tus comprising means for receiving an input signal and generating the coded bits and the uncoded bits therefrom; and an encoder for deriving from said coded and uncoded bits, for each said N-dimensional symbol, a set of N, M-valued one-dimensional coset representatives ck corresponding to con-gruence classes of each of the N coordinates (modulo M), M being a positive number, each coset representative designating a subset of one-dimensional values in a one-dimensional constellation of possible coordinate values for each of said N dimensions, each said one-dimension signal in said sequence being selected from said possible coordinate values based on uncoded bits.

11. The apparatus of claim 4, 5, 6 or 8 further comprising an output at which said sequence Yk is delivered.

12. The apparatus of claim 4, 5, 6 or 8 further comprising an output at which said sequence xk is delivered.

13. The apparatus of claim 4, 5, 6, or 8 wherein L is 1.

14. The apparatus of claim 4, 5, 6, or 8 wherein the relationship between the xk signals and Yk signals is Yk = Xk -xx-L, L an integer,

15. The apparatus of claim 4, 5, 7, 8, or 10 wherein said modulation code is a trellis code.

16. The apparatus of claim 4, 5, 7, 8, or 10 wherein said modulation code is a lattice code.

17. The apparatus of claim 4, 5, or 10 wherein M is 2.

18. The apparatus of claim 4, 5, or 10 wherein M is 4.

19. The apparatus of claim 4, 5, or 10 wherein M is a multiple of 4.

20. The apparatus of claim 4 wherein J is 1.

21. The apparatus of claim 4 wherein J is the same as the number of dimensions in said modulation code.

22. The apparatus of claim 4 wherein k' is k-l.

23. The apparatus of claim 4 wherein J is 1 and each said constellation is a one-dimensional range of values centered on .beta.3Xk-1, O < .beta. < 1.

24. The apparatus of claim 23 wherein .beta. > 0.

25. The apparatus of claim 4 wherein there are a finite set of said J-dimensional constellations.

26. The apparatus of claim 25 wherein there are two said J-dimensional constellations.

27. The apparatus of claim 4, 5, 6, or 8 wherein Yk and Xk are real valued.

28. The apparatus of claim 4, 5, 6, or 8 wherein Yk and Xk are complex valued.

29. The apparatus of claim 4 or 5 wherein Yk and xk are complex valued and wherein M is 2+2i, i= ?=?.

30. The apparatus of claim 4 wherein at least two or said J-dimensional constellations are not disjoint.

31. In a decoder for decoding a sequence Zk = Yk + nk, k =
1, 2, ..., into a decoded sequence Yk, where the sequence of signals Yk is such that (a) said sequence is from a given modulation code;
(b) the running digital sum xk = Yk 1 + Yk- 2 + ...
has finite variance Sx;
(c) said signals Yk fall in a predetermined permis-sible range dependent on Xk,, k' < k; and the sequence nk represents noise, a range violation monitor comprising:
a means for reconstructing the estimated running digital sum xk = Yk + Yk-1 + ..., and a means for comparing said decoded sequence Yk with said predetermined permissible range based on said estimated running digital sum xk,,k' < k, and for generating an indication when said Yk is outside said permissible range.

32. A decoder according to claim 31, in which said estimated running digital sum ?k is adjusted based on said indica-tion so that ?k will be inside said permissible range.

33. A decoder according to claim 32, in which said adjust-ment is by the minimum possible amount such that ?k falls inside said permissible range.

34. A decoder for decoding a sequence xk = Yk + nk, k =
1,2, ..., where the sequence of signals Yk is such that (a) said sequence is from a given modulation code, said code being capable of being generated by an encoder with a finite number Q of states;

(b) Yk = Xk + Xk-L, L an integer, where said sequence Xk has finite variance Sx, and the sequence nk represents noise, comprising a modified maximum likelihood sequence estimator adapted to find MQ partial decoded sequences, up to some time K, one such said sequence for each combination of said finite number Q of states and each of a finite number M of integer-spaced values modulo M, such that each said sequence (a) is in said code up to said time K;
(b) corresponds to said encoder being in a given said state at said time K;
(c) corresponds to a value of xk at said time K
that is congruent to a given one of said values, modulo M.

35. A decoder as in claim 34 wherein M is 2.

36. A decoder as in claim 34 wherein M is 4.