DE3805582A1 - CODING AND DECODING DEVICES FOR CHANNELS WITH PARTIAL CHARACTERISTICS - Google Patents

Abstract

Apparatus for generating a (running digital sum) sequence of digital signals xk and/or a (partial response coded) sequence of digital signals yk, k = 1,2,???, such that yk=xk+/-xk-L, in which the yk signals are a sequence in a given modulation code comprises means for selecting a set of congruent 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 are 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. <IMAGE>

Description

The invention relates to modulation coding and Systems with a transmission or response behavior, which is of a partial nature and hereinafter referred to as "partial characteristic" is referred to (based on the English term "partial response", abbreviated PR). Codings with partial characteristics are abbreviated referred to as PRC.

Modulation coding uses symbols as signals encoded in such a way from a constellation that only certain sequences of signals are drawn possible are.

Various types of modulation codes have been used in recent years of the so-called trellis type and applied (e.g. in modems) to encode gains from 3 to 6 dB over channels with a high signal-to-noise ratio and limited bandwidth such as B. Telephone channels to he aim.

The first trellis codes go back to Ungerboeck (See U.S. Patent No. 3,877,768 to Cjsaka et al Work by Ungerboeck "Channel Coding with Multilevel / Phase Signals ", published in IEEE Transactions on Information Theory, volume IT-28, pages 55-67, January 1982).  

Ungerboecksche codes for sending n bits per symbol are based on section subdivisions (so-called "partitions") of one-dimensional (PAM) or two-dimensional (QAM) signal constellations of the number of points n ^{n +1} in four or eight subsets, combined with a linear binary "convolutional" code (also called "convolutional code") at "rate" 1/2 or rate 2/3, which determines a sequence of subsets. Another set of "uncoded" bits then determines which signal points are actually sent within the specified subsets. The section division (partition) and the code are constructed in such a way that a certain value for the square of the smallest distance d ² _min (square of the Hamming distance) between permissible sequences of signal points is ensured. Even if one uses the increased power of an extended signal constellation (a factor of four or 6 dB in one dimension or a factor of 2 or 3 dB in two dimensions), the increased value of the square of the Hamming distance leads to a coding gain that (dB 6) ranges from about a factor of two (3 dB) for simple codes up to a factor of four for the kompliziertesten codes, namely for arbitrarily large values of n.

Another multi-dimensional trellis code designed by Gallager is based on a partition of a four-dimensional signal constellation in 16 subsets, combined with a rate 3/4 convolutional code (see U.S. Patent Application No. 5,777,044 dated February 6, 1984 and the paper) by Forney et al. "Efficient Modulation for Band-Limited Channels", published in IEEE J. Select. Areas Commun., Volume SAC-2, 1984, pages 632-647). The four-dimensional subset is determined by selecting a pair of two-dimensional subsets, and the points of the four-dimensional signal constellation are formed by pairs of points from a two-dimensional signal constellation. With just an 8-state code, ad ² _min can be achieved, which is four times the Hamming distance of the uncoded sequence, while the losses caused by the expansion of the signal constellation to about a factor of ^2½ ( i.e. 1.5 dB) can be reduced, which results in a resulting coding gain of the order of 4.5 dB. A similar code was designed by Calderbank and Sloane (see "Four-dimensional Modulation With An Eight-State Trellis Code", AT&T Tech. J., Vol. 64, pages 1005-1018, 1985, and US Pat. No. 45 81 601).

Various other multidimensional codes designed by Wei are based on partitions of constellations in four, eight and sixteen dimensions, combined with convolutional codes of the rate (n -1) / n (cf. U.S. Patent Application No. 7 27 398 of 25 April 1985). Here, too, the multi-dimensional constellations consist of sequences of points drawn from two-dimensional partial constellations. The codes are designed to minimize the expansion of two-dimensional constellations in order to maintain a good ratio of quality (coding amplification) to the complexity of the code over a wide range and to achieve further advantages such as transparency against phase shifts. Various multidimensional trellis codes were also designed by Calderbank and Sloane, which generally have a similarly good ratio of quality to complexity, a larger constellation expansion, but in some cases have fewer states (cf. "New Trellis Codes", IEEE Transactions on Information Theory , March 1987 and "An Eight-dimensional Trellis Code", Proc. IEEE, Volume 74, 1986, pages 757-759).

All of the codes mentioned above are constructed for channels, which is mainly affected by noise are (in addition to phase rotations), especially for channels without Intersymbol interference. This implies that that each Inter symbol interference due to send and receive filters or special by an adaptive linear equalizer in the receiver  a negligible amount is reduced. Such one System is known to work well when the real channel no serious attenuation within the transmission bandwidth Has; however, in the case of strong damping (zeros or approximate zeros in the transfer function) can greatly increase the noise power in the equalizer become ("noise enrichment").

A known technique for avoiding such "noise enrichment" is to align the signal transmission system to control intersymbol interference instead of eliminating intersymbol interference. The most well-known schemes for this are signal transmissions with so-called "partial characteristics" (partial response, see work by Forney "Maximum Likelihood Sequence Estimation of Digital Sequence in the Presence of Intersymbol Interference", published in IEEE Transactions on Information Theory, volume IT-18, 1972 , Pages 363-378). In a typical (one-dimensional) system with partial characteristics, the intended intended output y _k at the receiver is the difference between two consecutive inputs x _k , so y _k = x _k - x _{k -1} applies instead of y _k = x _k . In the usual notation for data samples using the delay or runtime operator D , this means that the desired output sequence y (D) is equal to x (D) (1- D) instead of x (D); in this case one speaks of a system with a ′ ′ 1– D ′ ′ partial characteristic. Since the spectrum of a discrete-time channel with the impulse response 1- D has a zero at zero frequency (direct current), the combination of the transmit and receive filters with the real channel must also have a zero at direct current in order to obtain this desired characteristic. On a channel that has a zero or approximate zero in DC, a receive equalizer designed for desired (1- D) characteristics causes less noise enrichment than an equalizer that is designed to achieve a perfect response (no intersymbol interference).

The signal transmission with partial characteristics is also used to achieve other goals: it diminishes e.g. B. close to sensitivity to channel interference the band edge, it lowers the demands on the Filtering, it allows the transmission of pilot tones on the band edge, and it reduces adjacent channel interference in frequency division multiplex systems.

Other types of systems with partial characteristics are e.g. B. the (1+ D) system, which has a zero at the Nyquist band edge, and the (1- D ² ) system, which has both DC and Nyquist band edges. A quadrature (two-dimensional) system with partial characteristics (QPR system) can be designed as a system with a two-dimensional complex input; the (complex) answer 1+ D is obtained in a QPR system that has zeros on the upper and lower band edges in a carrier-modulated (QAM) bandpass system. All of these partial response systems are closely related, and the schemes for one of these systems can easily be adapted to another; so you can a system z. B. construct for (1+ D) characteristic and easily expand it to the other characteristics.

Calderbank, Lee, and Mazo have proposed a scheme for constructing sequences in trellis coding that have zeros in the spectrum, especially with DC; a problem that is related to the construction of partial response systems, even though the goals here are generally somewhat different (see the work "Baseband Trellis Codes with A Spectral Null at Zero", submitted by IEEE Transactions on Information Theory). Calderbank et al. Have designed known multidimensional trellis codes with multidimensional signal constellations so that signal sequences with spectral zeros are generated, using the method described below: the multidimensional signal constellation has twice as many signal points as in the case of the non-partial characteristic are necessary, and is divided into two equally large disjoint subsets, one of which comprises multidimensional signal points for which the sum of the coordinates is less than or equal to zero, while the other subset comprises such points where the sum is 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 non-negative, the current signal point is selected from the subset that has the signal points with the coordinate sums less than or equal to zero; if the RDS is negative, the current signal point is selected from the other subset. In this way, the RDS is kept close to zero within a narrow range, which is known to force the signal sequence to have a spectral zero in the case of direct current. At the same time, however, the signal points are otherwise selected from the subsets in the same way as would occur in a system with a non-partial characteristic: the expanded, multi-dimensional constellation is divided into a certain number of subsets with favorable properties with regard to distance, and a convolutional code that Rate (n -1) / n determines a sequence of the subset such that the square of the smallest distance between the individual sequences is certainly at least equal to d ² _min . The coding gain is constellation doubling reduced (by a factor of 2 ^½ or umd 1.5 dB in four dimensions, or by a factor of 2 ^¼ or 0.75 dB in eight dimensions), but otherwise a similar quality is achieved as in the In the case of non-partial characteristics with similar code complexity.

A general feature of the present invention is that a sequence of digital signals x _k and / or a sequence of digital signals y _k (which is in accordance with a given modulation code) is generated, where k = 1, 2,. . ., so that the relationship y _k = x _k ± x _kL exists between the x _k signals and the y _k signals, where L is an integer. An encoder selects J signals y _k , where J is 1, so that ( y _k , y _{k +1} , _... , Y _{k + J -1} ) is congruent with a sequence of J subgroup actors c _k (modulo M ) , with M equal to an integer, determined in accordance with the given modulation code. The J symbols are selected from one of several J -dimensional constellations, the choice being based on a previous x _k ′, with k ′ < k . At least one of the constellations contains both a point with a positive coordinate sum and another point with a negative coordinate sum. The encoder is designed so that the signals x _{k have} finite variance S _x .

According to a further general feature of the invention, the encoder selects the signals x _k so that they are congruent with a sequence of alternative subgroup actors c _k (modulo M) , the following being true:

c ′ _k = c _k - c ′ _kL (modulo M) , in the event that
y _k = x _k + x _{k - L} ;
c ′ _k = c _k + c ′ _kL (modulo M) , in the event that
y _k = x _k - x _{k - L.}

Another general feature of the invention is that the y _k signals lie within an alphabet of possible y _k signals that are evenly spaced Δ from one another within the alphabet. The encoder ensures that the sequence y _{k has} a variance S _y that is less than 2 S ₀, and that the sequence x _{k has} a variance S _x that is not much greater than S ² _y / 4 ( S _y - S ₀), where S ₀ is approximately the minimum signal power required to represent n bits per signal with an alphabet that has the distance Δ .

According to yet another feature of the invention, the encoder causes the signals x _k and y _{k to have} any chosen variance S _x and S _y within predetermined ranges.

In preferred embodiments, the ranges are controlled by a parameter β , where S _{x is} approximately equal to S ₀ / (1- β ² ) and S _y approximately equal to 2 S ₀ / (1 + β ) .

Another general feature of the invention is an arrangement for generating a sequence in a given N- dimensional modulation code by generating a sequence of one-dimensional signals based on coded and uncoded bits. The modulation code is based on an N -dimensional constellation which is divided into subsets in association with the code, each subset representing a large number of N -dimensional signals. The arrangement has an encoder which, for each N -dimensional symbol, selects a set of N M -value one-dimensional subgroup members c _k which correspond to congruence classes of each of the N coordinates (modulo M) of the symbol, each subgroup member having one Subset of one-dimensional values in a one-dimensional constellation of possible coordinate values for each of the N dimensions is determined, and each one-dimensional signal is subsequently selected from the possible coordinate values on the basis of uncoded bits.

In preferred embodiments, the following features can be implemented: either the x _k sequence or the y _k sequence can be supplied as an output variable; L = 1; y _k = x _k - x _kL ; the code can be a trellis code or a lattice code; M can be 2 or 4 or a multiple of 4 or 2 + 2 i ; J can be equal to 1 or the same size as the number of dimensions in the modulation code; k ′ = k -1; J equals 1 and each constellation is a one-dimensional range of values grouped around the center β x _{k -1} , where 0 β <1, preferably β <0; there is a finite set (e.g. two non-disjoint) J -dimensional constellations; y _k and x _k can be real values or complex values.

The invention further relates to a decoder for decoding a sequence z _k = y _k + n _k , with k = 1, 2,. . ., into a decoded sequence y _k , whereby the following applies to the sequence of signals y _k : firstly, the sequence comes from a given modulation code, secondly has the running digital sum

x _k = y _k + y _{k -1} + y _{k -2} +. . .

finite variance S _x , and third, the signals fally _k  into a predetermined one allowable range ofx _k 'Depends onk′ <k  is. The sequencen _k  represents noise Pedal monitor reconstructs the estimated running digital sum _k = _k + _{k -1}+. . ., compares the decoded episode _k  with a predetermined allowable range on the Basis of the estimated current digital sum _k ' With k′ <k, and delivers an ad whenever _k  except is within the permissible range.

Another general feature of the invention is a decoder for decoding a sequence z _k = y _k + n _k with k = 1, 2,. . ., where the following applies to the sequence of the signals y _k : firstly, the sequence comes from a given modulation _code which can be generated by means of an encoder with a finite number Q of states, secondly, y _k = x _k ± x _kL , where L is an integer and where the sequence x _{k has} finite variance S _x . The sequence n _k represents noise. The said decoder contains an estimation device for modified sequences of maximum probability in order to find a number MQ of decoded partial sequences in an interval up to a certain time K , one for each combination of the finite number Q of states and modulo M for each of a finite number M of integer spaced values, so that each sequence is in code until time K , second corresponds to the situation that the encoder is in a given state at time K , and third corresponds to the situation that x _k has a value at time K that is congruent with a given example of the modulo M values.

With the present invention, known modulation codes, particularly trellis codes, are prepared for use in systems with a partial characteristic in order to achieve the same types of advantages that trellis codes have in systems with non-partial characteristics, namely considerable coding gains for arbitrarily large ones Number n of bits per symbol with decoding complexity remaining within reasonable limits. The invention also allows the construction of trellis codes for systems with partial characteristics in such a way that one can manage with both relatively low input signal power S _x and with relatively low output power S _y , and it is also possible to weigh these two values fluently. In addition, higher-dimensional trellis codes can be set up for use in systems with partial characteristics that naturally have fewer dimensions.

Further advantages and features of the invention emerge from the Claims, as well as from the description below, in the preferred embodiments as examples are explained:

Fig. 1 is a block diagram of a channel with (1- D) partial characteristic;

Fig. 2 is a block diagram of an encoder for an 8-state code according to Ungerboeck;

Fig. 3 shows a signal constellation for the Unger boeck code, divided into eight subsets;

Figure 4 is a block diagram of an equivalent encoder for the Ungerboeck code;

Fig. 5 is a block diagram of a generalized N-dimensional trellis encoder;

Fig. 6 is a block diagram of a modified encoder of Fig. 5, which operates on the basis of subgroup performers;

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 subgroup precoding;

Fig. 10 is a block diagram for the combination of Figs. 8 and 9;

Fig. 11 is a block diagram of an encoder with a feedback of the running digital sum (RDS feedback) and coset precoding;

Figures 12, 13 and 14 are alternative embodiments of the encoder of Figure 11;

Figures 15, 16 and 17 are block diagrams of three equivalent filter arrangements;

Figure 18 is a block diagram of a generalized decoder;

Fig. 19, 20 and 21 are block diagrams of alternative encoder;

Fig. 22 shows an alternative signal constellation;

Fig. 23 is a block diagram of an encoder for use with the constellation of Fig. 22;

Figures 24, 25 and 26 are block diagrams of three equivalent encoders;

Fig. 27 is a block diagram of an alternative decoder;

Fig. 28 is a schematic illustration of an expanded signal constellation;

Figure 29 shows a rhombus for use with the constellation of Figure 28;

Figure 30 is a block diagram of a two-dimensional encoder with RDS feedback;

Figure 31 shows the dimensions of a rhombus for use with the constellation of Figure 28;

Fig. 32 shows a pair of constellations;

Fig. 33 shows two disjoint constellations.

According to FIG. 1, the invention uses a technique for generating signal sequences that serve as input signals for a channel 10 with Partialcharakteristik.

As an example, a baseband system with a one-dimensional (real) "1- D " partial characteristic is dealt with, which has a zero point in direct current. (A later description will briefly describe how one can modify such a construction for other types of systems with partial characteristics.) Each output signal z _k of such a system is given by the equation

z _k = y _k + n _k ,

where the n _k sequence (n (D)) represents noise and the y _k sequence (y (D)) is a sequence encoded with a partial characteristic (PRC sequence). This latter sequence is itself defined by the equation

y _k = x _k - x _{k -1} ,

where the x _k sequence (x (D)) is the sequence of the channel's input signals. Because of

x _k = x _{k -1} + y _k

the x _k sequence can be recovered from the PRC sequence by forming a running digital sum (RDS) of the y _k values (specifying an initial value for the x _k sequence); therefore, the x _k sequence is referred to below as the RDS sequence. The variances of the samples of the RDS sequence x (D) and the PRC sequence y (D) are referred to as S _x and S _y .

The discrete-time partial characteristic 1- D (represented by block 12 ) is a combination of the characteristics of a chain of transmit filters, an actual channel, receive filters, equalizers, samplers, etc., which are constructed in a conventional manner to produce a combined partial characteristic 1- To obtain D in which the noise power P (the noise sequence n (D)) is small compared to the PRC power S _y (ie the power of the sequence coded with partial characteristics). It is therefore desirable to send a relatively large number n of bits per channel input. A detector (not shown) processes the noisy PRC sequence z (D) to determine x (D) (or, equivalently, the sequence y (D) since there is a reversibly unique relationship between the two sequences). If the detector is a maximum likelihood sequence estimator, the first goal is to maximize the squared minimum distance (Hamming distance square) d ² _min between allowable PRC sequences y (D) .

In some applications, the design constraint may simply be to minimize the variance S _{x of} the samples of the RDS sequence (input sequence). In other cases, the requirements can relate to the size S _y . In still other cases, the design requirement may relate to the effective performance somewhere in the middle of the combined filter chain, so it is desirable to keep both S _x and S _y small and to provide a smooth balance between these sizes.

A related problem is the construction of sequences whose spectra have zeros, e.g. B. a zero at zero frequency (direct current). The goal here can be to construct such sequences y (D) which can represent n bits per sample, which have a zero in the spectrum, which have the smallest possible sample variance S _y , but on the other hand a large value of the square of the Hamming distance d ² _min between possible y (D) sequences. A common secondary goal is to keep the change in the running digital sum (RDS) of the y (D) sequence also limited, for system reasons. Since the RDS sequence x (D) z. B. is equal to x (D) / (1- D) and their sample variance S _{x is} a measure of the RDS change, the present invention can also be applied to the construction of sequences with spectral zeros.

Several principles can be applied to achieve the above goals. The first principle is to construct the input or RDS sequence x (D) so that the output or PRC sequence x (D) , when taking N values at a time, is a sequence of points in an N -dimensional Signal space (so-called "N -dimensional signal points"), which belong to subsets of an N -dimensional constellation, which is determined by a known N -dimensional trellis code. Then the square of the Hamming distance d ² _min between PRC sequences will be at least equal to the d ² _min value that is guaranteed by the trellis code. In addition, an estimation facility for the highest likelihood sequence in the trellis code can easily be adapted for use in the present system and, if not optimal, will achieve the same effective d ² _min value for substantially the same complexity of decoding as in the same case Trellis codes in a system without partial characteristics.

An exemplary embodiment of the present invention relies on an eight-state, two-dimensional trellis code, similar to the Ungerboeck code described in the work cited above. The code discussed here uses a two-dimensional constellation with 128 points to send 6 bits per (two-dimensional) signal. (This is also similar to the code that is to be used in CCITT recommendation V.33 for a data modem with 14.4 kbit / s.) FIG. 2 shows the encoder 20 for this code. For each 6-bit symbol 21 that is supplied from a data source 23 , two of the six input bits of the encoder 20 enter a convolutional encoder 22 , which operates at the rate 2/3 and thus has eight states. The three output bits of this encoder are used in a subset selector 22 to select one of eight subsets of a 128 point signal constellation shown in Figure 3; each subset therefore has 16 points (points in the eight subsets are each labeled with one of the eight letters A to H ). The remaining four "uncoded bits" 26 ( Fig. 2) are used in a signal point selector 28 to select the (two-dimensional) signal point to be sent from the selected subset. The code provides a gain corresponding to a factor of 5 ( i.e. 7 dB) compared to an uncoded system in d ² _min , but about 3 dB is lost by using a constellation with 128 points instead of 64 points, so that the resulting coding gain is approximately equal to 4 dB.

One-dimensional form of the two-dimensional Ungerboeck code

The sequence of the symbols x _k sent over the channel is one-dimensional in a baseband system with (1- D) partial characteristics. It is therefore helpful (although not necessary) to transform known trellis codes into one-dimensional form. This transformation has two aspects: first characterization of the two-dimensional subsets as compositions of one-dimensional constituent subsets and secondly characterization of the finite two-dimensional constellation as a composition of one-dimensional constituent constellations. It is now to be shown how this decomposition takes place for the example of the two-dimensional Ungerboeck code, and then it is shown how this can be done in the general case of an N -dimensional trellis code.

The first step is to determine that each of the eight two-dimensional subsets A, B ,. . . can be regarded as the union of two smaller two-dimensional subsets, namely A ₀ and A ₁, B ₀ and B ₁, etc., each of the 16 smaller subsets can be characterized as follows: The possible values of each coordinate of a signal point are in four Classes a, b, c, d divided; each of the smaller two-dimensional subsets then consists of points, the two coordinates of which are in a prescribed pair of classes. A convenient mathematical expression for this decomposition can be obtained by dividing Figure 3 so that signal points are spaced one unit 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 A ₀, A ₁, B ₀,. . . are the points whose two coordinates are congruent with a given pair (x, y) modulo 4, where x and y can each take one of the four values { a, b, c, d }, e.g. B. {± 1/2, ± 3/2}. These four values are called (one-dimensional) "sub-group actors". The points of the constellation according to FIG. 3 are additionally designated with index numbers 0 and 1 in order to show a possible arrangement of the 16 subsets. So z. B. the G ₀ point 29 coordinates x = 5/2, y = 9/2, and its subgroup actors are (5/2, 9/2) modulo 4 or (-3/2, 1/2).

The Fig. 2 can now be modified in the following manner. According to FIG. 4, the three output bits of the encoder 22 plus one of the uncoded bits 30 are used as input variables for the sub-set selector 32 which selects one of the 16 subsets due to the four input bits, wherein the uncoded bit 30, the selection between A ₀ and A ₁ or B ₀ and B ₁, etc., depending on which of the original eight subsets are selected by the three convolutionally coded output bits of the encoder 22 . Effectively, encoder 22 and bit 30 represent a rate 3/4 eight-state encoder, with the output selecting one of the 16 subsets, although the amount of possible signal point sequences has not changed. Next, each of the 16 smaller subsets is determined by a pair of one-dimensional subgroup cast members 34 , one for each coordinate, where each subgroup cast member c _{k can take} one of four values. The pair of subgroup actors is denoted by (c _{1 k} , c _{2 k} ).

It is an aspect of the present invention that all of the good codes mentioned above, ie the codes of Ungerboeck, Gallager, Wei and Calderbank and Sloane, can be transformed in the same way. That is, each of these N -dimensional trellis codes can be generated by an encoder that selects one of 4 ^N subsets, the subsets being determined by N four-valued one-dimensional subgroup members, corresponding to congruence classes of each coordinate (modulo 4). In some cases, it is only necessary to use 2 ^N subsets, which are determined by N two-valued one-dimensional subgroup members (e.g. {± 1/2}), corresponding to congruence classes of each coordinate (modulo 2); this applies e.g. For example, for the four-state 2 D code by Ungerboeck, the 8-state 4D code by Gallager (and similar code to Calderbank and Sloane), further the 16-state code, and the 64 D 4 -Status-8 D code according to Wei etc. In addition, it was found that many good lattice codes can be transformed in the manner described; so z. B. the Lattice D 4 code according to Schäfli and the Lattice E ₈ code according to Gosset represent by following four or eight two-valued one-dimensional subgroup actors (modulo 2), and the Lattice codes Λ ₁₆ and Λ ₃₂ according to Barnes -Wall, as well as the Lattice code Λ ₂₄ according to Leech can be represented by four-valued one-dimensional subgroup actors (modulo 4).

A general form of an encoder for all of these codes is shown in FIG . The encoder is N- dimensional and processes N signals to be sent over the channel at a time. With each operation, p bits enter a binary encoder C 33 and are encoded into p + r coded bits. These coded bits select (in selector 35 ) one of 2 ^{p + r} subsets of an N -dimensional signal constellation (the subsets correspond to the 2 ^{p + r} subgroups of a subgrid Λ ′ and an N -dimensional grid Λ , the constellation being a finite one Set of 2 ^{n + r} points of a parallel shift of the grid Λ , so that each subset contains 2 ^np points). Further np uncoded bits (in selector 37 ) select a signal point from the selected subset. Thus, the code sends n bits for each N -dimensional symbol using a constellation of 2 ^{n + r} N -dimensional signal points. The encoder C and the lattice subdivision (grating subdivision) Λ / Λ ′ ensure a specific value for the square of the minimum distance d ² _min between any two signal point sequences which belong to a subset of possible sequences.

The above-mentioned finding (about the transformability of all good codes) is the result of the mathematical finding that for all the good trellis and lattice codes mentioned, the lattice 4 Z ^N of N tuples, integer multiples of 4, is a sublattice of the lattice Λ ′ is (and in some cases it is 2 Z ^N ). Then for any integer q the grid Λ ′ is the union of 2 ^q subgroups of 4 Z ^N in Λ ′. The practical effect of this finding is that, provided that n q + p, the p + r coded bits plus q uncoded bits can be taken into a subset selector that is one of 2 ^{q + p + r} subgroups of 4 Z ^N in Λ selects, and that these subgroups can also be identified by a sequence of N four-valued one-dimensional subgroup members (c _{1 k} , c _{2 k} , _... , C _Nk ), where c _{jk represent} integer-spaced equivalence classes (modulo 4). Thus, already the Fig. 5 can be modified, as shown in FIG. 6. With this modification it is assumed that the signal ^{constellation} consisting of 2 ^{n + r} points can be divided equally into 2 ^{q + p + r} subsets, each of which contains the same number of signal points (2 ^nqp ).

The exemplary embodiment of the Ungerboeck code is a code in which N = 2, Λ = Z ², Λ ′ = 2 RZ ², p = 2, p + r = 3, q = 1 and n = 6.

The second step is to split the constellation into four one-dimensional constituent constellations. For the constellation according to FIG. 3, each coordinate can assume one of 12 values which are divided into eight "inner points" (e.g. {± 1/2, ± 3/2, ± 5/2, ± 7/2} ) and four "outer points" (e.g. {± 9/2, ± 11/2}) grouped, as indicated by the boundary line 31 in FIG. 3. There are two inner points and one outer point in each of the four one-dimensional equivalence classes (e.g. the class whose subgroup cast member 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 with +1/2 (modulo 4). Given a subgroup actor, it is therefore only necessary to state whether a point is an inner or an outer point and, if it is an inner point, which of the two inner points it is. This can be done with two bits, namely a bit b _{1 k} for indicating whether the inner or outer point, and a bit b _{2 k} for indicating which inner point (another possibility is the use of a trivalent parameter a _k ).

It can be said that the pair (b _{1 k} , b _{2 k} ) is an area-identifying parameter a _k that can take one of three values to indicate the following three areas:

• 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).

The fact that each area comprises a part of the real line with the total width 4 , which contains exactly one point which is congruent with any real number (modulo 4), means that the area-identifying parameter a _k plus the subgroup actor c _k uniquely determines a unique signal point for any value of c _k .

The signal point selector 36 according to FIG. 4 can then be dismantled in the following way. According to FIG. 7, three uncoded bits 40 enter an element 42 for selecting the area-identifying parameter for each pair of coordinates. An uncoded bit determines whether any outside point is sent. If so, a second bit determines which coordinate contains the outer point and the third bit selects which inner point has the other point. If no outer point is sent, both coordinates are inner points, and the second and third bits select which inner point is in each coordinate. The element 42 thus maps the three uncoded input bits 40 into two pairs of output bits 44 , namely a ₁ = (b ₁₁, b ₁₂) and a ₂ = (b₂₁, b ₂₂), each pair of bits being used to have a coordinate in To determine connection with the corresponding sub-group actor c ₁ or c ₂, which actors are generated by the voter 46 . The entire encoder is thus reduced to a form in which each coordinate x _k ( 48 ) is selected by 4 bits (in a coordinate selector 50 ), 2 bits representing the actor c _k and 2 bits the parameter a _k = (b _{1 k} , b _{2 k} ).

All constellations that are usually used with the codes mentioned above can be broken down in this way. The principles are similar to those described in US Pat. No. 4,597,090 and in Forney's aforementioned work, inter alia, "Efficient Modulation..." are described and according to which N -dimensional constellations are constructed from two-dimensional constituent constellations. A similar set-up of two-dimensional constituent constellations has been used by Wei in connection with trellis codes (cf. Wei US patent application mentioned above).

The general form of an encoder for N -dimensional codes is shown in FIG. 8. For every N coordinates p bits 51 enter an encoder 52 and p + r coded bits 54 are generated; these bits plus q uncoded bits 56 arrive at a selector 58 which selects a sequence of N subgroup actors c _k ( 60 ); the remaining npq uncoded bits 62 are transformed (in a selector 64 ) into a sequence of area-identifying parameters a _k ( 66 ) which, together with c _k (in a signal point selector 68 ), determine a sequence of N signal point values x _k ( 70 ), with the help of a signal point selection function f (c _k , a _k ) that works on a one-dimensional basis. In general, the area-identifying parameter a _{k determines} a subset of the real line (one-dimensional constellation) with the width (dimension) 4, which contains exactly one element that is congruent with any possible c _k value (modulo 4), and the function f (c _k , a _k ) selects this element. For all the codes mentioned, the alphabet of the subgroup actors can be taken as four integer-spaced values (modulo 4); for some codes, the alphabet of the subgroup actors can be taken as two integer-spaced values (modulo 2) (in the latter case, the areas have a width of 2). The a _k alphabet is as large as necessary to send n bits for every N coordinates. The signal point sequences sent by this type of encoder are generally the same as the sequences in the original code, in particular the square of the Hamming distance d ² _{min is} the same as in the original code.

Subgroup precoding

The sequences of N -dimensional signal points produced by known good trellis codes, if they are brought into the form of a series of one-dimensional signal points, cannot generally be used as inputs for the channel having a partial characteristic according to FIG. 1 without the d ² _min value to deteriorate (due to inter-symbol disturbance). However, a technique referred to herein as "subgroup precoding" allows these known codes to be prepared for systems with partial characteristics without increasing S _x or worsening d ² _min . This technique is generally illustrated in FIG. 9.

The same convolutional encoder (convolutional encoder) 52 as used in the known trellis code can be used, preferably in the form shown in FIG. 8. However, the p + r coded output bits 54 do not select a subset directly, but rather become (as in the case of Fig. 8) in a subset selector / serializer 70 in a sequence c _k from N one-dimensional subgroup actors c ₁,. . ., c _N converted, according to the subset that would be selected in a system without partial characteristics. These subgroup performers are then "precoded" (in a precoder 72 ) to obtain an alternative (or "precoded") sequence c _k 'of subgroup performers ( 74 ), where

c _k ′ = c _{k -1} ′ + c _k (modulo 4)

(In cases where it is possible to use modulo 2 subgroup actors, this precoding can be done with modulo 2). Thus, the precoded sequence 74 of subgroup performers is a running digital sum modulo 4 (or 2) of the usual sequence of subgroup performers. The precoded subgroup actors c _k 'can then be combined in a grouper 75 into groups of N copies in each case in order to define an N -dimensional subset (in a signal point selector / serial generator 76 ); then a signal point can be selected in the usual way (based on the uncoded bits 78 ) and the resulting signal point can be transmitted as a sequence x (D) of N one-dimensional signals x _k over the channel having the partial characteristic (in the same order as precoded).

If the actors c _{k are} half integers, then the actors c _k ′ alternate between two sets of four values, one set being offset by 1/2 from the other set. This has only an insignificant effect; you can e.g. B. alternating coordination x _k by the width +1/4 and -1/4 back and forth "tremble" to take this periodicity into account. Alternatively, the c _k alphabet can be left with integer values, e.g. B. {0, 1, 2, 3}, then the actors c _k 'are always from the same alphabet, z. B. {± 1/2, ± 3/2}. These offsets of c _k ′ or c _k do not affect the d ² _min value of the code.

If the encoder is in the form of FIG. 8, FIG. 9 can be brought into the form of FIG. 10, where the same blocks do the same thing. Since f (c _k , a _{k) has} been characterized as a function that selects the only element congruent with c _k in a range identified by a _k , it does not matter if the precoding encodes the c _k ′ alphabet from the c _k -Alphabet changes; in principle, the modulo-4 operation in the precoder is actually unnecessary, although it may also be useful in practice.

With reference to FIG. 9 or FIG. 10 it can be shown that the PRC sequence y _k = x _k - x _{k -1 has} elements that are congruent with c _k (modulo 4), so that they are in the subsets of the original trellis code and therefore have at least the same d ² _min value. The RDS sequence x _k has the same mean energy S _x as in the original trellis code if the c _k ′ alphabet is the same as the c _k alphabet; even without this requirement there is still approximate equality. (In the exemplary embodiment, the mean energy per coordinate is 10.25, for signals which are spaced by an integer.) If c _k are integers, then the parts x _{k are} independent, identically distributed random variables, so that

• a) S _y = 2 S _x ;
• b) the spectrum of the RDS sequence { x _k } within the Nyquist band is flat or flat ("white"spectrum);
• c) the spectrum of the PRC sequence { y _k } is the same as the spectrum of the channel with partial characteristics.

Even if c _{k are} not integers, the above statements are still approximate.

The subgroup precoding can be modified for other types of systems with partial characteristics in the following way. The same system is used for a system with a (1+ D) partial characteristic (one-dimensional partial characteristic), except that c _{k -1} 'is subtracted instead of added in the pre-encoder 72 , so that c _k ' = c _k - c _{k -1} '(modulo 4). For a (1- D ^L ) system, the delay _element D is replaced by a delay _element D ^L , so that c _k ′ = c _kL ′ + c _k . For a two-dimensional (1+ D) system, two (1+ D) precoders are used in parallel, the inputs of which are acted upon by two outputs of the subset selector / series generator and the two outputs of which are the real part and the imaginary part (in-phase -Component and Quadraturkom component) of the two-dimensional signal point to be determined.

Feedback of the current digital sum (RDS feedback)

Depending on the application, it may be expedient to reduce the average energy S _{y of} the PRC sequence and to accept an increase in the average energy S _{x of} the RDS sequence. This also tends to flatten the PRC spectrum while increasing the low frequency content in the RDS spectrum. Justesen has introduced in his work "Information Council and Power Spectra of digital codes" (IEEE Transactions on Information Theory, Vol IT-28, 1982, pages 457-472) the notion of a "cut-off frequency" f ₀ below which the PRC spectrum is small and above which it tends to be flat, and he has shown that f ₀ can be approximated by f ₀ S ( S _y / 2 S _x) f _N , where f _{N is} the frequency of the Nyquist band edge.

A general method for implementing a compromise of the above type while maintaining the d ² _min -value of the trellis code in the PRC sequences is to expand the encoder of Fig. 9 or Fig. 10 in the following manner.

The PRC sequence can be calculated from the RDS sequence; for the (1- D) channel, each PRC signal is exactly y _k = x _k - x _{k -1} . According to FIG. 11, the signal point selector 80 can be operated in such a way that it bases every x _k on x _{k -1} (by feeding back x _k via a delay element 82 ), as well as on the current precoded subgroup actors c _k ' and on the area-identifying parameter a _k , such that large PRC values y _k (calculated in the summing circuit 84 ) are avoided. As long as the signals x _{k are} still selected as congruent with c _k ′ (modulo 4), the signals y _{k are} congruent with c _k (modulo 4), and therefore the d ² _min value of the trellis code is preserved. (Although the idea is to precalculate the PRC value y _k to keep it small, it is the previous RDS value x _{k -1} that is actually being fed back, so this process is referred to here as "RDS feedback " designated.)

For the exemplary embodiment, this could be done in the manner described below. As previously mentioned, the normal dialing function f (c _k , a _{k) of} the selector 80 can be characterized by saying that the eight inner dots are the eight half values of integers ranging from -4 to +4 while the four outer dots are the four half values of integers that range from -6 to -4 and from +4 to +6. One can now change the area of the inner points and the area of the outer points as a function of x _{k -1} , as long as the area of the inner points comprises eight signal points, two from each equivalence class, while the area of outer points comprises four signal points, respectively one from each equivalence class.

A general way of realizing this idea is to shift all areas by a shift variable R (x _{k -1} ), which is a function of x _{k -1} . That is, in the embodiment discussed here, the area of the inner points is modified to extend from -4+ R (x _{k -1} ) to 4+ R (x _{k -1} ), and the area of outer points is modified that it ranges from -6+ R (x _{k -1} ) to -4+ R (x _{k -1} ) and from 4+ R (x _{k -1} ) to 6+ R (x _{k -1} ).

The function R (x _{k -1} ) should generally increase with increasing x _{k -1 in} order to reduce the y _k in this way. It could be shown that the optimal choice for this function is R (x _{k -1} ) = β x _{k -1} , where β is a parameter in the range 0 β <1. In the case of β = 0, the RDS feedback via element 82 disappears, and the coding in FIG. 11 is reduced to the subgroup precoding, as is done according to FIG. 10. If S ₀ denotes the value of S _x in the usual case β = 0, then the following applies approximately with the above choice:

• a) S _x = S ₀ / (1- β ² );
• b) S _y = 2 S ₀ / (1+ β );
• c) the spectrum S _x (f) of the RDS sequence is proportional to 1 / (1-2 β cos R + β ² ), where R = π f / f _N ;
• d) the spectrum S _y (f) of the PRC sequence is proportional to 2 (1-cos R / 1 (1-2 β cos R + β ² ); the "cut-off frequency" f ₀ is (1- b ) f _N ;
• e) the signals x _k are limited to the range from - M / 2 (1+ β ) to M / 2 (1- β ), and the signals y _k are limited to the range from - M to M if the Range of coordinates in the original code extends from - M / 2 to M / 2.

When β goes to 1, S _y goes to S ₀, and S _y (f) takes the form of a flat spectrum with a sharp zero at DC. Meanwhile, S _x becomes large, and S _x (f) approaches a 1 / (1- D) spectrum, only that it remains finite near DC. It could also be shown that this is the best possible compromise between the quantities S _x , S _y and S ₀.

Figures 12, 13 and 14 show three equivalent ways to generate x _k and / or y _k based on c _k , a _k and x _{k -1} . FIG. 12 most closely corresponds to FIG. 11.

In FIG. 12, the feedback variable c _{'k -1} replaced coset precoder 72 by _k x _-1, because c' _{k -1} x _k ≡ _-1 (modulo 4), and only the value of c _'k ( modulo 4) is used in voter 80 . R (a _k ) denotes the range identified by a _k , and R (x _{k -1} ) represents the range shifting variable introduced by the RDS feedback. Since y _k = x _k - x _{k -1} ≡ c ′ _k - x _{k -1} (modulo 4) and c ′ _k ≡ c _k + x _{k -1} (modulo 4), y _k ≡ c _k (modulo 4 ).

The encoders of Figures 13 and 14 are mathematically equivalent to the encoder of Figure 12 in the sense that they provide the same sets of output signals (x _k , y _k ) when they start with the same initial value x _{k -1} and receive the same sequence of input signals (c _k , a _k ) . In the case of FIG. 13, the only element chosen for y _{k is} congruent with c _k in the range R (a _k ) + R (x _{k -1} ) - x _{k -1} , and x _k is determined from y _k according to the relationship x _k = y _k + x _{k -1} , so that x _k ≡ c ′ _k ≡ c _k + x _{k -1} (modulo 4); it is the only element in the range R (a _k ) + R (x _{k -1} ) that is congruent with c ′ _k (modulo 4). In the case of FIG. 14, the only element chosen as an "innovation variable" i _{k is} that which is congruent with c ′ ′ _k ≡ c _k + x _{k -1} ≡- R (x _{k -1} ) (modulo 4) in the region R. (a _k ) , and x _k is determined from i _k according to the relationship x _k = i _k + R (x _{k -1} ), so that x _k ≡ c ′ ′ _k + R (x _{k -1} ) ≡ c _k ′ (modulo 4), it is the only element in the range R (a _k ) + R (x _{k -1} ) that is congruent with c ′ _k (modulo 4). In the case of Fig. 12, the delay element in the precoder is combined with the delay element necessary for the RDS feedback, and it is very expedient if x _{k is} the desired output and the quantities c ' _{k are} always of the same alphabet, e.g. B. {± 1/2, ± 3/2}. In the case of Fig. 13, the precoding is completely omitted and it is very useful if y _k the desired output and the values c _k always come from the same alphabet, e.g. B. {± 1/2, ± 3/2}. In the case of FIG. 14, the range-shifting variable R (x _{k -1} ) is obtained outside the selector, so that the values for i _{k are} always chosen from the same range (the union of all R (a _k )) ; the innovation sequence i (D) is approximately a sequence of independent, identically distributed random variables i _k (ignoring the small changes introduced by the constraint of congruence at c ′ _k ), and this additional auxiliary sequence can be useful if one “white” (having a flat spectrum) sequence is desired, which is deterministically related to x (D) or y (D) .

The Fig. 15, 16 and 17 show three equivalent filtering arrangements for use with the sequences x (D), y (D) and i (D) of FIGS. 12, 13 and 14. In the case of FIG. 15, the RDS sequence x (D) in a transmission filter H _T (f) ge filtered before it (as signal s (t)) is sent over the actual channel (not shown). In the case of Fig. 16, the PRC sequence y (D) is filtered in a transmission filter H ' _T (f) , the characteristics of which are equivalent to the characteristics of a cascade of a 1 / (1- D) filter for sampled data and the filter H _T (f) . Since y (D) has a zero at DC, it does not matter that the characteristic of the 1 / (1- D) filter at DC is infinite (especially if H _T (f) also has a zero at DC). In the case of Fig. 17, the innovation sequence i (D) is filtered in a transmission filter H '' _T (f) , the characteristics of which are equivalent to the characteristics of a cascade of a 1 / (1- β D) filter for sampled data and the filter H _T (f) . This is equivalent to Figures 15 and 16 if R (x _{k -1} ) = β x _{k -1} ; otherwise, the equivalent sample filter (filter for sampled data) is the filter corresponding to the function x _k = i _k + R ( x _{k -1} ), which is generally non-linear. Depending on H _T (f) , R ( x _{k -1} ) and the technology used for the implementation, one or the other of these equivalent forms may be preferable.

In practice, certain modifications to the RDS feedback systems described above may be desirable. So it can e.g. B. be useful to change the shape of the areas R (a _k ) compared to those shapes that are used when R ( x _{k -1} ) = 0. An easy to implement form of RDS feedback is e.g. B. the following: if x _{k -1 is} positive, y _{k is} chosen as usual in the range from -4 to +4 if a _k indicates an inner point; however, if a _k indicates an outer point, the number is taken for y _k that is congruent with c _k in the range from -4 to -8. If x _{k -1 is} negative, the range from 4 to 8 is used for outer points. The following then applies:

• a) the range of the PRC sequence y _k is limited to the range from -7½ to + 7½, instead of extending from -11 to +11, as in the case of no RDS feedback;
• b) the variance S _{y of} the PRC sequence is reduced from 13.25 to 20.5, which corresponds to a decrease of 1.9 dB and is approximately 1.1 dB above S ₀ = 10.25;
• c) the average of y _k is equal to -3/2, if x _k-1 is positive, and negative x _{k -1} + 3/2 it is, so that the RDS sequence has the tendency ready cash in the post to stay from zero. While the exact calculation of S _{x is} difficult, it follows from E [ y _k x _{k -1} ] = S _y / 2 and from E [ y _k x _{k -1} ] = - (3/2) E [| x _{k -1} |] that the mean of the absolute value of x _k is S _y / 3 = 4.42, so that the RDS sequence x _k is kept within reasonable limits. (Without RDS feedback, the mean of the absolute value of x _k is 2.75.);
• d) the variance ofy _k  givenx _{k -1} is equal to S₀ = 11, i.e. about 0.3 dB higher than that without RDS Feedback possible valueS₀ = 10.25. The least possible value ofS _x  in the caseS _y = 13.25 and S₀ = 11S _x ≃19.5 whatβ Corresponds to ,60.66. There S _x =S _|x| +E [|x|] ² is, mustS _x  be larger than (4.42) ²≃19.5; so with this simple Achieve less method than the optimal compromise the spectra;
• e) every possible y _k is assigned to a single pair (c _k , a _k ) . As will be explained in more detail below, this means that a decoder need not follow an estimated running digital sum of the estimated PRC sequence and that there is no error propagation in the decoder.

In summary, this simple method does not provide the best performance compromise between S _x and S _y , but it effectively limits not only S _y , but also the peak values of y _k , keeps the RDS sequence x _k within reasonably good limits and prevents error propagation in the receiver.

The methods described above thus allow a compromise between S _x and S _y over a wide range. The compromise possibility ranges from the unrestricted case that the x _k sequence is uncorrelated, where S _{x has} the same energy S ₀ as is necessary to send n bits per symbol in a system without a partial characteristic, and where S _y = 2 S _x is. The compromise possibility then extends almost to the case that the y _k sequence is uncorrelated, where S _y = S ₀ and S _x becomes very large. These compromises are possible for all the trellis and lattice codes listed above.

Decoding

The methods described above are successful in generating PRC sequences (code sequences with partial characteristics) belonging to a known good code, and therefore they have ad ² _min values that are at least as large as that of the code.

According to FIG. 18, a suitable detector for the PRC sequence z (D) = y (D) + n (D) received under noise is a device that estimates the sequence of maximum probability for the known good code (Viterbi algorithm ), in the following way:

• a) a first step of decoding can consist in for every high-noise recipient PRC valuee.g. _k =y _k +n _k  and for each of the four Classes of real numbers that are congruent with the four one-dimensional subgroup representativesc _jk   (modulo 4) are withj= 1, 2, 3, 4, which is the value e.g. _k  closest element _jk  to find as well whose "metric"m _jk =( _jk -e.g. _k ) ^2nd, that is Square of the distance frome.g. _k  (Block92).
• b) In the case of a code based on an N -dimensional Lattice subdivision Λ / ein ′, a second step of decoding can be to find the best (i.e. the smallest) for each of the 2 ^{p + r} subgroups from Λ ′ to Λ Find metric) copies of those 2 ^q subgroups of 4 Z ^N whose union is the relevant subgroup of Λ ′ by summing up the relevant metrics of the one-dimensional constituent metrics m _jk and comparing these sums (block 94 ).
• c) The decoding can then be done in the usual way to be continued (block96) by as a metric for each subgroup ofΛ′ Uses the best metric is how it was determined in step b). The decoder ultimately generates an estimate of the Sequence of subgroups ofΛ'Which in one episode valued subgroup performer _k  pictured which can then be translated into the appropriate _k   are depicted, from which, if desired Original valuesâ _k  and _k  can be recovered (Block98). These last steps require that The decoder keeps track of the current digital total _{k -1} of the estimates _k .

Since the PRC sequences are encrypted in the known code, the probability of error of this decoder is at least as "good" as that of the known code, in the sense that at least the same effective d ² _min value is achieved. However, since the PRC sequences are actually only a subset of the sequences of the known code, such an encoder is not an actual estimator for estimating the maximum probability sequence for the PRC sequences. Therefore, it may occasionally happen that a sequence is decoded that is not a legitimate PRC sequence. Legitimate PRC episodes must meet the two additional conditions below:

• a) a legitimate finite PRC sequence y (D) must be divisible by 1- D , ie the sum of its coordinates must be zero;
• b) The range limits set by the signal point selector must be imposed on everyone _k  (or equivalent to all _k  or _k ) suffice based on the reconstructed values of the elements _{k -1}  the RDS episode.

If the decoder in question is a normal De coding error, corresponding to a short period incorrect subgroup estimates followed by correct ones Subgroup estimates, it is possible that the corresponding finite erroneous PRC sequence an ongoing Has digital sum other than zero. this will a persistent error in the estimated current digital total _{k -1} of the decoder cause what can that occasionally errors back in the values _k , â _k  and _k  are mapped, even if the subgroups _k  are correct. This situation lasts as long as that Error in the RDS estimate persists.

The decoder must therefore constantly monitor (block99), whether the range conditions in the reconstructed values _{<; I 34210 00070 552 001000280000000200012000285913409900040 0002003805582 00004 34091TA <k} and _k  be respected. This is not the case, then the decoder knows that the RDS value _{k -1} is wrong and that he value that around Must adjust the minimum amount that is required to meet the range conditions, provided  that the subgroup episode _k  correct is. With the true In the end, this will probably become 1 with the post-syn Chronization of the estimated RDS to the correct value reached, and normal decoding can be resumed will. However, error propagation can occur over a considerable period of time.

Avoidance of error propagation

Below is a general method of how to avoid error propagation in the receiver. The method works best when the signal constellation consists of all points in the Λ within an N -dimensional cube, but it is not limited to this case. The method can be seen as a generalization of the principles of previous forms of pre-coding (modulo M) for use with coded sequences.

The basic idea is that every possible PRC value y _k  one(c _k ,a _k )-Value should match if the code is in one-dimensional form like inFig. 7 formulated can be, or, more generally, that every group ofN y _k -Not just a single episode fromN c _k -Values should match, but also a single sequence of uncoded bits, if a general one N-dimensional signal point selector as inFig. 6 is used. Then the reverse mapping from decoded _k -Values in coded and uncoded bits independent depending on the estimation of the running digital sum so that

• a) the decoder does not have to follow the RDS (ie no necessary tracking with the RDS), and
• b) no error propagation takes place.

Thus, block 99 can be omitted in FIG. 18.

FIG. 19 illustrates how this can be done if the code can be formulated in one-dimensional form, as in the case of the present exemplary embodiment. From c _k and a _k , a signal point selector selects a value s _k = f (c _k , a _k ) , as in the case of FIG. 8. In the exemplary embodiment, s _k assumes one of twelve values which are half the values of the whole Numbers range from -6 to -6. Generally speaking, s _{k will take} one of the values from an integer spaced alphabet within a range of width M ; this area is denoted by R ₀. As in the case of FIG. 13, that number is chosen for y _k which is the only congruent with s _k (modulo M) in the range R ₀ + R (x _{k -1} ) - x _{k -1} of width M , where R (x _{k -1} ) is a shift variable in the RDS feedback and x _{k -1 is} the previous RDS signal point. The current RDS value x _k is calculated as the quantity y _k + x _{k -1} .

FIGS. 20 and 21 illustrate equivalent methods of generating x _k and / or y _k from the sequence s _k such that y _k ≡ s _k (modulo M), similar to FIGS. 12 and 14. Referring to Figure . 21 is an innovation variable i _k generated, the "white" is more or less and is uniformly distributed over the range R ₀ so that its variance S ₀ approximately ² M / 12. S ₀≃12 thus applies to the exemplary embodiment, that is to say a disadvantage of approximately 0.7 dB compared to the value S ₀ = 10.25, which can be achieved without RDS feedback. As in the case of FIGS. 12, 13 and 14, all three sequences x _k , y _k and i _k carry the same information, and as in the case of FIGS. 15, 16 and 17, each of these sequences can be used as an input signal for a filter , which gives the spectrum the desired shape for transmission.

The disadvantage in the innovation variance is eliminated if the original code coordinates are evenly distributed over an area R ₀, ie if the original constellation is within the limits of an N -dimensional cube with the side length R ₀.

As an embodiment with a quadratic constellation, the same two-dimensional Ungerboeck encoder with eight states as in the case of FIG. 2 is used, except that the constellation consisting of 128 points according to FIG. 22 is used instead of that according to FIG. 3. The constellation contains every second point (ie the "alternating" points) of the conventional 256 point (16 × 16) constellation, so the coordinates have the 16 half integer values {± 1/2, ± 3/2,. . ., ± 15/2}, but with the restriction that the sum of the two coordinates must be an even number (0, modulo 2). The square of the minimum distance between signal points is therefore 2 instead of 1, and the d ² _min value of the code is 10 instead of 5. The variance of each coordinate is now 21.25 instead of 10.25, which after taking into account the Rated factor 2 means a loss of 0.156 dB compared to the constellation according to FIG. 3 because the cross is more similar to a circle than the square. (Discussed in the terminology of the Lattice code, is now the aft-mesh division [8-way partition Lattice] RZ ^2/4 Z ² is used instead of Z ^2/2 RZ ^2).

It should be noted that each of the eight subsets now corresponds to a single pair of subgroup actors (c ₁, c ₂) modulo 4), so that c ₁ + c ₂ = 0 (modulo 2). Therefore, the three coded bits according to FIG. 2 determine a pair of subgroup performers directly in the subset selector 24 , ie not with the aid of an uncoded bit, as in the case of FIG. 4. The four uncoded bits then select one of the 16 points in the selected subset. In this case, the uncoded bits can simply be taken two at a time to determine one of the four ranges -8 to -4, -4 to 0, 0 to 4, or 4 to 8. This can be expressed simply by having each of the area-identifying 2-bit parameters (a ₁, a ₂) represent one of the four values {± 2, ± 6}; the coordinate selection function is then simply s _k = f (c _k , a _k ) = c _k + a _k . It should be noted that the possible values for s _{k are} the 16 half values of the integers in the range R ₀ from -8 to 8, which has the width M = 16.

A conventional modulo 16 precoding can then be carried out. The entire encoder is shown in Fig. 23. The RDS value x _k is the sum s _k + x _{k -1} (modulo 16). In this case, the x _k values are essentially independent, identically distributed (white) random variables, and y _k = x _k - x _{k -1} ≡ s _k (modulo 16).

In order to achieve compromises between the spectra using RDS feedback as in the case of FIGS. 12, 13 and 14, s _{k should} continue to represent the desired congruence class of y _k (modulo 16), and R (x _{k -1} ) should be one RDS feedback variable as in the cases of FIGS . 12, 13 and 14, which is ideally equal to β x _{k -1} . Figures 24, 25 and 26 show then three equivalent methods for generation of sequences x _k and / or y _k = x _k -. X _{k -1} such that y _k ≡ s _k (modulo 16), and that S _x and S _y given S ₀ = 21.25 have the desired compromise ratio. R ₀ here is the range from -8 to 8.

In this case, the innovation variable i _{k has} a variance S ₀≃16² / 12 = 21.33 which is substantially the same as the variance of each coordinate in Fig. 22, so that there is no disadvantage of more than 0 , 16 dB reduction that results from using FIG. 22 instead of FIG. 3.

As already mentioned, the decoder does not have to be the RDS follow because given the estimated PRC sequence _k  the values _k ,â _k  and finally the original input bit sequence are clearly determined. If the decoder, however the track with the estimated RDS and the corresponding Areas in which the _k -Values should fall then he can feel the occurrence of an error in the  more if the decoded _k  outside the estimated range falls. Even if it's not for error correction Such monitoring of the area can be used transmission an estimate of the error rate of the decoder deliver.

Extended decoder

A real estimator for the sequence of maximum probability would have to take into account the overall condition of the encoder and channel, which generally includes the value of the RDS x _{k -1} (channel condition) and the condition of the encoder C. Such a decoder would achieve the true d ² _min value of the PRC sequences and would be free from error propagation. However, since x _{k -1} generally takes on a large number of values, in principle possibly an infinite number with RDS feedback, such a decoder would hardly be practicable. In addition, an essentially infinite decoding delay would be necessary to achieve the true d ² _min value because the combination of code and channel becomes quasi-catastrophic when n is large, as will be explained in more detail below.

However, it may be worthwhile to expand the encoder so that at least the true d ² _min value of the code is achieved. Since all finite PRC sequences are divisible by 1- D , all error sequences that have finite weight must have an even weight. Thus the true d ² _min value is always even. In the exemplary embodiment, the true d ² _min value is actually 6 and not 5.

A general method for achieving the true d ² _min value in such cases, whereby the effective number of states in the decoder is only doubled, is as follows: the decoder splits each state of the encoder C into two, one corresponding to an even RDS and one corresponding to an odd RDS. During the decoding, two episodes only appear in the same state if their estimated RDS has the same value (modulo 2). This makes it impossible for two sequences to be merged that differ in an error with an odd weight, so that the effective d ² _min value is equal to the weight of the sequence with the smallest even-weighted error in the original code. Furthermore, a decoding error, which should lead to a persistent error in the estimated RDS in the manner described above, should be at least weight 2 so that it can be sensed earlier.

The decoder canFig. 18 used after it has only been modified as described inFig.  27 is shown. For most codes, each of the subsets the signal constellation (subgroups ofΛ' in Λ) Contain points where everyone has the sum of their coordinates is even or odd. So included e.g. B. in the constellationFig. 3 four of the eight Subsets of points whose total coordinates are 0 (modulo 2), and four subsets contain points whose Coordinate sum is 1 (modulo 2). Everyone's metric Subset (subset ofΛ′ IndΛ) can therefore in blocks92 and94 determine as before; the estimator 196 for the maximum probability sequence is then modified to be the best sequence of Subgroups finds which is first in the code and second a running digital sum (RDS) congruent with 0 (modulo 2). The decoded subgroup sequence is in the block98 as before back in _k  and _k  pictured, if necessary with an adjustment of _{k -1} in the block99  (The adjusting changes are made here in multiples of 2).

In addition to doubling the space of the decoder states, this technique has another disadvantage. It can happen that two sequences differ by an error sequence with an odd number, followed by a long chain of zeros (no differences). The decoder then follows parallel pairs of states in the trellis grating of the decoder for a very long time without resolving the ambiguity. In the end, this "quasi-catastrophic" behavior can only be resolved by the facility which estimates the consequence of maximum probability under one area violation, because of the different RDS parity in the two paths. Thus, the decoding delay required to achieve the true d ² _min value can be very large.

For this reason, it will generally be preferable to simply take an encoder C with twice the number of states and use a non-expanded decoder for C. There are e.g. B. a two-dimensional Ungerboeck code with 16 states and d ² _min = 6; although this code can bring a somewhat larger error coefficient than the 8-state code with an extended 16-state decoder, it should be advantageous in practice.

It should be mentioned that the PRC sequences drawn from the two-dimensional 4-state code according to Ungerboeck also have a true d ² _min value of 6, since in the said code d ² _min = 4, the only error sequences of the weight 4 are individual coordinate errors of size 2, which are also not divisible by 1- D . A 16-state decoder that holds 4 tracks with RDS modulo can achieve this d ² _min value. In this case, not only the code quasi-catastrophic, but also the error coefficient is large, so that the ordinary 16-state seems preferable 2 D code to Ungerboeck here.

Quadrature systems

As already indicated above, a system with a complex (or quadrature) partial characteristic (QPR system) can be emulated as a (1+ D) sample filter that processes a complex RDS sequence x (D) to form a complex PRC sequence to generate y (D) = (1+ D) x (D) ; that is y _k = x _k + x _{k -1} . Such a system, when used in conjunction with a double sideband quadrature amplitude modulation over a bandpass channel, leads to zeros on both band edges f _c ± f _N , where f _{c is} the carrier frequency and f _N = 1/2 T the width of a single Ny quistband.

If N is an even number and 4 Z ^{N is} a sublattice of Λ ′, as in the case of all the good codes mentioned above, then a known good code can be made up for use in a QPR system by applying essentially the same principles as above . A subgroup of 4 Z ^N can be determined by N / 2 complex subgroup performers c _k be, the subgroup performers each taking one of 16 possible values, corresponding to four integer-spaced values (modulo 4) for the real part and for the imaginary part of c _k . The general picture of FIG. 8 then applies, except that the subgroup selector 58 and the selector 64 selecting the area-identifying parameter then select N / 2 complex subgroup performers c _k and area-identifying parameters a _k and the signal point selector once per Quadrature signal acts and outputs complex signals x _k . The subgroup precoding is carried out, as in the case of FIG. 9, by forming the complex precoded subgroup c ' _k ≡ c _k - c ' _{k -1} (modulo 4) once per quadrature symbol. An RDS feedback occurs as in the case of FIGS . 11, 12 and 13 by using a function R (a _k ) which identifies an area of the complex space of area 16 which contains exactly one element from any subgroup of 4 Z ² , and using a complex displacement variable R ( x _{k -1} ), which ideally equals β x _{k -1} . In cases where 2 Z ² or 2 RZ ^{2 is} a sublattice of Λ ', the precoding can be carried out with modulo 2 or with modulo 2 + 2 i , and R ( a _k ) can cover an area of area 4 or 8 identify which contains exactly one element from any subgroup of 2 Z ² or 2 RZ ² .

High-dimensional systems

Embodiments have previously been dealt with in which the coordinates N -dimensional symbols are formed signal by signal (one- or two-dimensional) and the previous RDS value (running digital sum) x _{k -1} signal by signal is fed back. Comparable performance can be achieved with systems that select signals on a higher dimensional basis. In such systems, the pre-encoded subgroup cast members must be grouped as in Fig. 9 so that they select subsets in the appropriate dimension, then signal points must be selected in that dimension, and then the coordinates must be brought back into series form to broadcast it over the channel. If the order of the subgroups is maintained, such a system retains the property that the PRC sequences come from the given code and have the specified d ² _min value. In such a system, it may be more natural to do the (RDS) feedback on a higher dimensional basis, rather than individually, signal by signal.

N- dimensional codes

Although the representation of codes in one-dimensional form Such a representation does not have to be expedient.  

The following section is intended to show how codes can be generated directly in N dimensions. In certain forms, the N -dimensional code is completely equivalent to its one-dimensional counterpart. Simplified designs can be achieved in other forms.

Here too, the two-dimensional 8-state code according to Ungerboeck as in the case of FIG. 2 is used for illustration purposes, namely with the two-dimensional 128-point constellation according to FIG. 3. It should be remembered that in this constellation each coordinate Takes values from the alphabet of the twelve half values of integers in the range from -6 to 6; the two-dimensional constellation uses 128 of the 144 possible pair combinations of elements of this alphabet.

As a first step, the signal constellation is expanded to an infinite number of values, as follows. The expanded constellation is intended to include all pairs of numbers that are congruent with any point in the original ( Fig. 3) constellation modulo 12. Thus, the points in the expanded constellation consist of pairs of half-values of whole numbers. If one considers the original constellation as a cell that is delimited by a 12 by 12 square 98 , then the expanded constellation consists of the infinite repetition of this cell through the entire two-dimensional space, as is indicated schematically in FIG. 28. It should be noted that each cell contains only 128 of the 144 possible points, there are 4 by 4 "holes" 99 in the expanded constellation.

The key property of this expanded constellation 101 is as follows: if you place a 12 by 12 square anywhere in the plane (with the sides of the square oriented horizontally and vertically), this square encloses exactly 128 points, each of the points being the original constellation is congruent with one of the points in the square. An even more general statement applies: if one places a rhombus 102 with the horizontal width 12 and the vertical height 12 (see FIG. 29) somewhere in the plane, it also encloses 128 points, each point of the original constellation being congruent with one this point is.

According to the Fig. 30 will now be given an RDS feedback on two-dimensional basis, as follows. The variable x _{k -1} is to represent the running digital sum (RDS) of all those y _k that appeared before the current (two-dimensional) symbol. The size R (x _{k -1} ) is now to denote a region of the plane which corresponds to a 12 by 12 rhombus as in FIG. 29, both the shape and the location of the rhombus being dependent on x _{k -1 as} far as possible . (y _{0, k} , y _{0, k +1} ) shall denote the point in the original constellation that would be selected by the three coded bits and the four uncoded bits according to the unrestricted code ( FIG. 2) (in the voters 104 , 105 ). Then (in the selector 106 ) with (y _k , y _{k +1} ) the only point in the two-dimensional extended constellation is selected which lies within the range R (x _{k -1} ) and is congruent with (y _{0, k} , y _{0 , k +1} ) (modulo 12); these are the two coordinates (y _k ) . One can get (x _k , x _{k +1} ) from x _k = y _k + x _{k -1} , x _{k +1} = y _{k +1} + x _k , as shown.

It can now be shown that the two-dimensional system can produce the same output variables as the one-dimensional system with RDS feedback (modulo 12) described above, namely with the optimal one-dimensional RDS feedback variable R (x _{k -1} ) = b x _{k -1} . According to FIG. 31, in one dimension, given x _{k -1,} the only value in the range R ₀ + β x _{k -1} - x _{k -1 is} chosen for y _k , which is congruent with s _k (modulo 2), now it can be seen that s _{k is} congruent with y _{0, k} . It can thus be assumed that a coordinate of the rhombus used in the two-dimensional system lies in the same region of the width 12. Given x _{k -1} and y _k and therefore also with x _k = y _k + x _{k -1} , the only value in the range becomes y _{k +1}
R ₀ - (1- β ) x _k = R ₀ - (1- β ) y _k - (1- β ) x _{k -1}
chosen, the congruent is s _{k +1} = y _{0, k +1} (modulo 12). Thus y _{k +1} lies in the range R ₀- (1- β ) x _{k -1} (just like y _k ) , shifted by - (1- β ) y _k .

With the right choice of rhombus, you can achieve the same performance with a two-dimensional system as with a one-dimensional (modulo 2) RDS feedback system. One thus has the same advantages, namely the avoidance of error propagation and the almost optimal compromise between S _x , S _y and S ₀; one also has the same disadvantages, namely the increase of S ₀ to 12 compared to the otherwise possible value of 10.25.

Other two-dimensional RDS feedback variables (ranges) can be chosen to further simplify implementation and achieve other benefits, but at the cost of non-optimal performance compromises. So you get z. B. a system that is almost identical to the simplified one-dimensional system described above, if for R (x _{k -1} ) in the case of a positive value of x _{k -1} the square 120 with the side length 12 and its center at (- 2, -2) and, in the case of a negative value of x _{k -1,} the square 122 with its center at (+2, +2). As schematically illustrated in FIG. 32, one of the two constellations 124 and 126 is used in each case.

As with the one-dimensional system described above, inner points are always chosen from the same set, regardless of x _{k -1} , but outer points are chosen to vary, so that y _{k is} "deflected" in a positive or a negative direction. The ranges of y _k are strictly limited from -7½ to 7½. In fact, this system is identical to the simplified system described above, except that y _{k +1 is} chosen based on x _{k -1} instead of x _k . In practice, all measures in terms of performance and spectrum are very similar.

Another variant leads to a system that is related to systems of the CLM type (system type according to Calderbank, Lee and Mazo). A CLM system uses an extended signal constellation with twice the number of signal points as usual and a division into two disjoint constellations, one of which is used when x _{k -1 is} positive and the other when x _{k -1} is negative. Fig. 33 shows an example of a constellation in a 16 by 16 square, which is divided into two disjoint constellations 110 and 112 with 128 points each, such that each of these constellations is divided equally into eight subsets of 16 points each . A constellation consists of points for which the sum of their coordinates is positive or zero, and is used when x _{k -1 is} negative; the other consists of points for which the coordinate sums are negative or zero, and is used when x _{k -1 is} positive. In two dimensions, the doubling of the constellation size leads to the doubling of S _y , so that a favorable compromise in performance is not achieved; with higher dimensions, however, the disadvantage that results from using the two disjoint constellations is less great.

The above considerations can be generalized to N dimensions as follows. If the code is formulated one-dimensionally, as in the case of FIG. 8, using the modulos M , then an N -dimensional cube with the side length M completely surrounds the N -dimensional constellation, and the resulting cell can be multiplied by the N -dimensional Fill in space without jeopardizing the square of the minimum distance between code sequences that are congruent with the original modulo M code sequences. One can then use an N -dimensional RDS feedback function R (x _{k -1} ), where for all x _{k -1} the area R (x _{k -1} ) is an area of the N -dimensional space with the volume M ^N that contains exactly one point in each equivalence class N -dimensional vectors modulo M , in an N -dimensional analogue of FIG. 30.

Further embodiments are also in the range of Claims.

Claims (35)

1. Arrangement for generating a sequence of digital signals x _k and / or a sequence of digital signals y _k , with k = 1, 2,. . ., so that between the x _k signals and the y _k signals the relationship y _k = x _k ± x _{k - L} applies, with L an integer, the signals y _{k being} a sequence in a given modulation code by an encoder for selecting J signals from the signals y _k , where J is 1 and (y _k , y _{k -1} , _... y _{k + J -1} ) congruent with a sequence of J subgroup performers c _k ( modulo M) and M is an integer and said subgroup actors are specified according to the given modulation code, and the J symbols are selected from a constellation of a plurality of J -dimensional constellations, and this choice is based on a previous x _k 'Takes place, with k '< k , and wherein at least one of said constellations contains a point with a positive sum of its coordinates and another point with a negative sum of its coordinates, and wherein the encoder is arranged so that the signals x _k end have variance S _x . 2. Arrangement for generating a sequence of digital signals x _k and / or a sequence of digital signals y _k , with k = 1, 2,. . ., so that the relationship y _k = x _k ± x _{k - L} exists between the x _k signals and the y _k signals, where L is an integer and the signals y _{k is} a sequence in a given modulation code by an encoder for selecting the signals x _k such that they are congruent with a sequence of alternative subgroup actors c ′ _k (modulo M) , where: c ′ _k = c _k - c ′ _kL
(modulo M) , if y _k = x _k + x _{k - L} ′
c ′ _k = c _k + c ′ _kL
(modulo M) , in the case that y _k = x _k - x _{k - L} ′ c _{k is} a subgroup actor, which is specified according to the modulation code. 3. Arrangement for generating a sequence of digital signals x _k and / or a sequence of digital signals y _k , where k = 1, 2,. . . and where n bits can be represented per signal, such that the relationship y _k = x _k ± x _{k - L} exists between x _k and y _k , where L is an integer and the signals x _k and y _{k are} a variance S _x or S _y and where the signals y _k fall into an alphabet of possible y _k signals which are evenly spaced within the alphabet by a distance Δ , characterized by an encoder which causes the sequence y _{k to have} a variance S _{y has} less than 2 S _O and the sequence x _{k has} a variance S _x that is not much larger than S ² _y / 4 (S _y - S ₀), where S ₀ is approximately the minimum signal power required to represent n bits per signal with an alphabet that has the distances Δ . 4. Arrangement according to claim 3, characterized in that the sequence y _{k is} a sequence in a given modulation code. 5. Arrangement for generating a sequence of digital signals x _k and / or a sequence of digital signals y _k , where k = 1, 2,. . . is such that the relationship y _k = x _k ± x _{k - L} exists between the signals x _k and the signals y _k , where L is an integer and the sequences of the signals x _k and y _{k are} variances of S _x and S _y , and wherein symbols y _{k are} a sequence in a given modulation code, characterized by an encoder which causes the signals x _k and y _{k to have} arbitrarily selectable variances S _x and S _y within predetermined ranges. 6. Arrangement according to claim 5, characterized in that the signal sequences can represent n bits per signal and that the signals y _k fall in an alphabet of possible y _k signals which are evenly spaced by a dimension Δ and that areas by a parameter β are controlled and that S _{x is} approximately equal to S ₀ / (1- β ² ) and S _{y is} approximately equal to 2 S ₀ / (1+ b ) , where S ₀ is approximately the signal power that is required at least by n bits per symbol in an alphabet that has the spacing Δ , according to the said code. 7. Arrangement for generating a sequence of signals in a given N-dimensional modulation code by generating a sequence of one-dimensional signals, the modulation code being based on an N -dimensional constellation which is divided into subsets which are assigned to the code and each of which is N - contains dimensional signal points, the subset being selected on the basis of coded bits and uncoded bits of the signal points, characterized by an encoder which, for each N -dimensional symbol, contains a set of N M -value one-dimensional subgroup actors from the coded and uncoded bits c _{k is derived} according to congruence classes of each of the N coordinates (modulo M) , each subgroup actor denoting a subset of one-dimensional values in a one-dimensional constellation of possible coordinate values for each of the N dimensions, each of said one-dimensional signals in the sequence from the possible coordination ns values is selected on the basis of uncoded bits. 8. Arrangement according to claim 1, 2, 3 or 5, characterized by an output where the sequence y _{k is} output. 9. Arrangement according to claim 1, 2, 3 or 5, characterized by an output where the sequence x _{k is} output. 10. The arrangement according to claim 1, 2, 3 or 5, characterized in that L = 1. 11. The arrangement according to claim 1, 2, 3 or 5, characterized in that the relationship between the signals x _k and the signals y _{k is} described by y _k = x _k - x _{k - L} 'where L is an integer. 12. The arrangement according to claim 1, 2, 3, 5 or 7, characterized in that the modulation code is a trellis code is. 13. Arrangement according to claim 1, 2, 3, 5 or 7, characterized in that the modulation code is a lattice code is. 14. Arrangement according to claim 1, 2, 3, 5 or 7, characterized in that M is 2. 15. The arrangement according to claim 1, 2, 3, 5 or 7, characterized in that M is 4. 16. The arrangement according to claim 1, 2, 3, 5 or 7, characterized in that M is a multiple of 4. 17. The arrangement according to claim 1, characterized in that J is equal to 1. 18. The arrangement according to claim 1, characterized in that J is a number which is equal to the number N of dimensions in the modulation code. 19. The arrangement according to claim 1, characterized in that k 'is equal to k -1. 20. The arrangement according to claim 1, characterized in that J is 1 and that each constellation is a one-dimensional range of values with the central value β x _{k -1} , with 0 β <1. 21. The arrangement according to claim 2, characterized in that β <0. 22. The arrangement according to claim 1, characterized in that a finite set of the J -dimensional constellations is provided. 23. The arrangement according to claim 22, characterized in that two J -dimensional constellations are provided. 24. The arrangement according to claim 1, 2, 3 or 5, characterized in that y _k and x _{k have} real values. 25. Arrangement according to claim 1, 2, 3 or 5, characterized in that y _k and x _{k have} complex values. 26. The arrangement according to claim 25, characterized in that M is 2 + 2 i . 27. The arrangement according to claim 1, 2, 3 or 5, characterized in that at least two of the J -dimensional constellations are non-disjoint. 28. Arrangement for decoding a sequence z _k = y _k + n _k , with k = 1, 2,. . ., into a decoded sequence of signals y _k , which is such that

• a) is the result of a given modulation code;
• b) the running digital sum x _k = y _{k -1} + y _{k -2} +. . . has a finite variance S _x ;
• c) the signals y _k fall within a predetermined permissible range which depends on x _k , with k ′ < k ,

and being the resultn _k  Represents noise by an area violation control device, which is the estimated current digital total _k = _k + _{k -1}+. . . reconstructed and which the decoded sequence _k  with the predetermined allowable Area based on the estimated current digital sum _k 'Compares withk′ <k is and which provides an ad when _k  outside of permissible range. 29. Decoder according to claim 28, characterized in that the estimated running digital sum _k  on the Adjust the basis of the display so that _k  in the permissible range falls. 30. Decoder according to claim 29, characterized in that that the adjustment is made to the minimum necessary is over _k  in the permissible range bring. 31. Decoder for decoding a sequence x _k = y _k + n _k , with k = 1, 2,. . ., the sequence of the signals y _{k being} such

• a) that the sequence is from a given modulation code that can be generated by an encoder with a finite number Q of states;
• b) that y _k = x _k ± x _{k - L} , where L is an integer and where the sequence x _{k has} a finite variance S _x ,

and wherein representing the sequence n _k noise, characterized by means for estimating the maximum likelihood sequence in modified form to find up to a certain time K a number MQ partially encoded sequences, respectively of a Q for each combination of a finite number of states and modulo M for each of a finite number M of integer spaced values such that each of said sequences

• a) up to the said time K is in the code,
• b) corresponds to the case that the encoder is in a given state at time K ,
• c) corresponds to the case that x _k at time K has a value that is congruent with a given example of the said values modulo M.

32. Decoder according to claim 31, characterized in that M is 2. 33. Decoder according to claim 31, characterized in that M is 4.