NL8800474A - CHANNEL SIGNALING SYSTEM OPERATING WITH PARTIAL RESPONSE. - Google Patents

Description

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Partial response channel signaling system.

The invention relates to modulation encoding and partial response systems.

In modulation encoding, symbols are encoded as signals taken from a constellation in such a way that only certain signal strings are possible.

In recent years, a number of types of grating-type modulation codes have been developed and applied (eg in modems) to realize encoding gains of 3 to 6 dB over band-limited channels with a high signal-to-noise ratio, such as voice-rated telephone channels -10 lity.

The early grating codes have been proposed by Ungerboeck (Cjsaka et al., U.S. Patent 3,877,768; Ungerboeck, "Channel Coding with Multi level / Phase signals", IEEE Transactions on Information Theory, Vol. IT-28, biz. 55-67, January 1982). Ungerboeck's 15 codes for sending n bits per symbol are based on divisions into 4-subsets or 8-subsets of one-dimensional (PAM) or two-dimensional (QAM) constellations of 2n + 1 signal points, combined with a linear binary convolution code with 1/2 or 2/3 input / output bit ratio, which determines a series of subsets. A further set of "uncoded" bits then determines which signal points are actually broadcast within the specified subsets. The partition and code are designed to provide a

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certain minimum squared distance dcm7-n to ensure between the allowable ranges of signal points. Even if the power cost of an expanded signal constellation (factor 4 (6 dB) in one dimension, or factor 2 (3 dB) in two dimensions) is taken into account, the increase in the minimum squared distance results in an encoding gain ranging from about a factor of 2 (3 dB) for simple codes up to a factor of 4 (6 dB) for the most complicated codes, for values of n that can be as large as desired.

Gallager (U.S. Patent Application S.N. 577,044, filed February 6, 1984, and discussed in Forney et al., "Efficient Modulation for Band-Limited Channels", IEEE J. Select Areas Common. Vol. SAC-2, biz.

632-647, 1984) designed a multidimensional grating code based on a 16 subset partition of a four-dimensional signal constellation, combined with a convolutional code with a 3/4 input / output bit ratio. The four-dimensional subset is determined by selecting a pair of two-dimensional subsets, and the points of the four-dimensional signal constellation are manufactured as the pairs of points of a two-dimensional signal constellation. With only an 8 state encoding, a d2mjn of 4.5 times the uncoded minimum string spacing can be obtained, because the loss due to the expansion of the signal constellation can be reduced by a factor of 21 ^ 2 (1.5 dB) leading to a net encoding gain of the order of 4.5 dB. A similar code was designed by Calderbank and Sloane ("Four-dimensional Modu-10 Lation With An Eight-State Trellis Code", ATT Tech. J., Vol. 64, Biz. 1005-1018, 1985; US Patent 4,581,601).

Wei (U.S. Patent Application SN 727,398, filed April 25, 1985) designed a number of multidimensional codes based on partitions of constellations in four, eight, and sixteen dimensions combined with convolutional codes with (n-1) / n input / output bit ratio . Also, multidimensional constellations consist of series of points of two-dimensional constituent constellations. The codes are designed to minimize the two-dimensional constellation expansion, to achieve performance (encoding gain) as a function of code complexity over a wide range, and further advantages such as transparency for phase rotations. Calderbank and Sloane "New Trellis Codes", IEEE Trans.

Inf. Theory, to be released in March 1987; An Eight-dimensional Trellis Code, Proc. IEEE, Vol. 74, biz. 757-759, 1986 also designed a number of multidimensional grating codes, generally with similar performance-complexity ratio, more constellation expansion, but less states in some cases.

All of the above codes are designed for channels in which the fundamental interference factor (apart from phase rotations) is noise, and in particular for channels without mutual symbol interference. The implied assumption is that any mutual symbol interference introduced by the actual channel will be reduced to a negligible level by send and receive filters; or more particularly by an adaptive linear equalizer in the receiver. Such a system is known to work well if the actual channel does not exhibit severe attenuation within the transmission bandwidth, but in the case of a strong attenuation ("zeros" or "near zeros"), the noise power in high be amplified in the smoother ("noise increase").

A technique known per se to avoid such "noise increase" 40 is to design the signaling system for a controlled mutual symbol interference rather than no mutual symbol interference. The best known schemes of this type are known as the "partial response signaling schemes" (Forney, "Maximum Likelihood Sequence Estimation of Digital Sequences in the Presence of 5 Intersymbol Interference", IEEE Trans-Inform. Theory, Vol. IT-18, biz .

363-378, 1972).

In a typical (one-dimensional) partial response scheme, the desired output value y ^ is designed at the receiver as the difference of two consecutive input values x ^, ie y ^ = x ^ -10 xj (_i, instead of y ^ = X | <. In the sampled data notation, using the delay operator D, this means that the desired output sequence y (D) equals x (D) (1D) instead of x (D), thus becoming a "1-D" partial response system Since the spectrum of a discrete time channel with impulse response 1-D has a zero at the zero frequency (DC), the combination of the transmit and receive filters with the actual channel should similarly have a zero at the zero frequency to obtain this desired response On a channel that has a zero or near zero at the zero frequency, a receive equalizer designed for a desired 1-D response will introduce less noise gain than one designed to provide a perfect response ( no mutual symboo l interference).

Partial response signaling is also used to achieve other objectives, such as reducing the sensitivity to channel disturbances near the band edge, easing the requirements imposed on filters, allowing pilot tones at the band edge, or reducing interference through adjacent channels in frequency multiplex systems.

Other types of partial response systems include a 1 + D

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array that has a zero at the Nyquist band edge, and a 1-D * · 30 array that has zeros at both the zero frequency and at the Nyquist band edge. A (two-dimensional) quadrature system with partial response (a QPRS system) can be designed with a two-dimensional complex input; the (complex) response 1 + D results in a QPRS system that has zeros at both the top and bottom band edges 35 in a carrier modulated band pass system (a QAM system).

All of these partial response systems are closely related, and schemes for one can easily be adapted for the other, for example, one can design a scheme for a 1-D response and easily extend it to the others, for example.

40 Calderbank, Lee, and Mazo ("Baseband Trellis Codes with a Spectral Null, b b. C 0 4 7 4« 4 at Zero "; offered to IEEE Trans. Inf. Theory) have proposed a scheme to use lattice-coded strings construct spectral zeros, especially at the zero frequency, a problem related to designing systems with partial response, even though the objectives are generally somewhat different. Calderbank and others have adapted known multi-dimensional grating codes with multidimensional signal constellations to produce signal strings with spectral zeros by the following technique. The multidimensional signal constellation has two 10 times as many signal points as needed for the non-partial response case, and is divided into two equally sized sub-sets, one of the multidimensional signal points whose sum of coordinates is less than or equal is to zero and the other whose sum is greater than or equal to zero. A "running digital sum" 15 denoted in the following by (RDS) of coordinates, initially set to zero, is updated for each selected multi-dimensional signal point using the sum of its coordinates. If the current RDS is not negative then the current signal point is selected from the signal subset whose coordinate sums are less than or equal to zero; if the RDS is negative then the current signal point is selected from the other subset. In this way, the RDS is kept limited within a narrow range near zero, which, as is known, forces the signal sequence to have a spectral zero at the zero frequency. At the same time, however, the signal points are otherwise selected from the subsets in the same manner as they would be selected in a non-partial response system: the expanded multidimensional constellation is divided into a given number of subsets with favorable spacing properties and a convolutional code with (n) / n input / output bit ratio defines a series of subsets such that the minimum squared distance between the series is guaranteed to be at least d2m-jn.

The encoding gain is reduced by the constellation doubling (by a factor of 21 ^ or 1.5 dB in four dimensions, or by a factor of 21/4 or 0.75 dB in eight dimensions), but otherwise a similar performance is obtained as in the case of non-partial response with the similar code complexity.

Overview of the invention

A general feature of the invention is the generation of a series of digital signals x | <and / or a series of digital signals 40 y ^ (the series y | <corresponds to a particular modulation code). ; de), k = 1, 2, ... such that the relationship between the signals and y ^ signals is equal to yk = ± X | <_t_, L with L an integer. An encoder selects J signals where J ^ -1, (yk, yk + 1. — yk-hl-1) which are congruent to a series of J 5 group set representatives (modulo M), with M an integer, which group set representatives are specified in accordance with said given modulation code, which J symbols are selected from one of a number of J-dimensional constellations, which choice is based on a previous x ^ ', with 10 k' <k. At least one of the mentioned constellations contains both a point with a positive sum of the coordinates and another point with a negative sum of the coordinates. The encoder is arranged such that said signals Xk have a finite variance Sx.

Another general feature of the invention is that the encoder selects the signals X {/ congruently from a set of alternative group set representatives c'k (modulo M) where c'k = cfc - c'k_L (modulo M) in the case y | <= x ^ + Xk_L 'c'k = ck + C'k-L (modulo M) in case yk = 20 xk - Xk + L'

Another general feature of the invention is that the yk signals fall within an alphabet with possible yk signals with equal distances equal to the value Δ, that the encoder ensures that the series yk has a variation Sx smaller than 2So and the series 25 Xk has a variation Sx not much greater than S2y / 4 (Sy-5o), where Sq is approximately the monimal signal power required to represent n bits per symbol within an alphabet spaced equal to Δ.

Another general feature of the invention is that the encoder ensures that the Xk and yk signals have randomly selected variances sx and Sy within predetermined ranges.

In preferred embodiments, the trajectories are controlled by a parameter β, where Sx is approximately equal to Sq / (1-32), and Sy is approximately equal to 2Sq / (1 + 0).

Another general feature of the invention is an apparatus for generating a sequence in a given N-dimensional modulation code by generating a sequence of one-dimensional signals based on coded and unencoded bits, which modulation code is based on an N-dimensional constellation, divided into subsets-40s associated with the code, which subsets each have c (./ r * f.

6 * N-dimensional signal points comprise an encoder for deriving, for each said N-dimensional symbol, a set of N, M-valued one-dimensional group set representatives corresponding to the congruent classes of each of the N coordinates (modulo M ) of the symbol, wherein each group set representative indicates a subset of one-dimensional values in a one-dimensional constellation of possible coordinate values for each of said N dimensions, and each one-dimensional signal in said series is selected from the possible coordinate values based on unencrypted bits.

In preferred embodiments, either the γ or γ series may be provided as an output series; L = 1; yk = xr -de code can be a grating code or a grid code; M can be equal to 2 or 4 or a multiple of 4 or 2 + i; J can be 1 or 15 equal to the number of dimensions in the modulation code; k '= k-1; J is 1 and each constellation is a one-dimensional range of values centered on 3Xk-1.0 <3 <1, preferably $ greater than 0; there is a finite group of (for example, two intersecting) J-dimensional constellations; y ^ and xj <can be real values or complex values.

Another general feature relates to a decoder for decoding a sequence = yk + nk, k = 1, 2, ... »in a sequence yk such that: (a) said sequence has a given modulation code; (b) the running digital sum Xk = yk-1 + yk-1 25 + yk_2 + ··· has a finite variant! e Sx; (c) the signals yk fall in a predetermined allowable range depending on Xk1, k '<k; and the series nk represents noise. A trajectory exceedance monitor reconstructs the estimated running digital sum Xk = $ k + yk-1 + ··· »compares the decoded sequence yk 30 with the predetermined allowable trajectory based on said estimated running digital sum x ^, k '<k , and generates an indication if said ^ is outside said allowable range.

Another general feature of the invention relates to a decoder for decoding a sequence z ^ = y | <+ nk, k = 1, 2, ..., wherein the sequence of signals yk is that (a) the string is from a given modulation code, which code is suitable for being generated by an encoder with a finite number of states equal to Q; (b) y ^ = X | <+ Xk_L »with L a 9th-40 integer, in which the series Xk has a finite variance Sx, and, 86 0 04 7 4 7 ί represents the series 1¾ noise, comprising a modified with maximum probability operating sequence estimator adapted to find MQ partially decoded sequences, up to a certain time K, a sequence for each combination of said finite number Q states and each of a finite number M of values modulo M with spacings, those integers are such that each string (a) in said code is up to time K; (b) corresponds to said encoder that is in a certain state at said time K; (c) corresponds to a value of xk at said time K which is congruent with one of the given values modulo M.

The invention adapts known modulation codes, especially trailing codes, for use in partial response systems to achieve the same kind of benefits that grating codes have in systems with non-partial response, in particular significant coding gains for arbitrarily large numbers n bits / symbol with reasonable decoding complexity. The invention also makes it possible to design grating codes for systems with partial response in such a way that both a relatively low input signal power Sx and a relatively low output signal power Sy are achieved, and allows for a balanced compromise between these two quantities from each other. Furthermore, higher dimensional grating codes can be adapted to use in systems with partial response, which are inherently lower dimensional.

Other advantages and features will become apparent from the following description of preferred embodiments and from the claims.

Description of the Preferred Embodiments

First of all, we will briefly describe the drawings.

30 Drawings

Figure 1 is a block diagram of a channel with 1-D patile response.

Figure 2 is a block diagram of an encoder for an 8-state Ungerboeck code.

Figure 3 is a signal constellation for the Ungerboeck code, divided into 8 subsets.

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

Figure 5 is a block diagram of a generalized N-dimensional 40-degree grating encoder.

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Figure 6 is a block diagram of a modified figure listed on group collection representatives.

Figure 7 is a block diagram of an equivalent one-dimensional unit.

Figure 8 is a block diagram of a generalized N-diagonal grating encoder.

Figure 9 is a block diagram of a generalized code <with group set precoding.

Figure 10 is a block diagram showing Figures 8 and 9.

Figure 11 is a block diagram of an encoder with RDS link and group set precoding.

Figures 12, 13, 14 are alternative embodiments of Figure 11.

Figures 15, 16, 17 are block diagrams of three equivalent configurations.

Figure 18 is a block diagram of a generalized deco.

, ^ ___ J5s ^ figures 19, 20 ^ 2j ^ zfjn block diagrams of alternative c - "7 .. 20 units.

Figure 22 is an alternative signal constellation.

Figure 23 is a block diagram of an encoder for use. constellation from figure 22.

Figs 6 are pale sweet. tri-equivalences 25 deer units.

Figure 27 is a block diagram of an alternative decoder. Figure 28 is a schematic diagram of an expanded needle constellation.

- ....._. - - Figure 29 is a diamond for use with the constellation from 30 28.

Figure 30 is a block diagram of a two-dimensional encoding with RDS feedback.

Figure 31 is a diagram of the dimensions of a diamond for use with the constellation of Figure 28.

35 'Figure 32 shows a few constellations.

Figure 33 shows a pair of non-intersecting structures 7 ♦ 9

Structure and operation

With reference to Figure 1, the invention includes a technique for generating signal arrays to be used as input signals for a channel 10 with partial response, for example, a one-dimensional (actual) 1-D baseband system with partial response and a zero at the zero frequency. (We will briefly explain later how such a design can be modified for other types of systems with partial response). Each output signal zk of such a system is given by 10 zk = yk + nk in which the sequence nk (n (D)) represents the noise and the sequence yk (y (D)) is an encoded partial response (PRC) sequence defined by ^ k = xk "xk-l 15 where the series xk (x (D)) is the series of channel input signals. Since ί ^^ Ι + ^ k, the series iq ^ yrorden can be recovered from the PRC series by the current digital sum of the yk values (assuming an initial value for the series of the series xk, the RDS series (running digital sum) is thus obtained. The sample variance of the RDS series x (D) and of the PRC series y (D) zul ^ ixj ^ e ^ with Sx and

The partial response in discrete time 1-D (vertical ^ 25 hlok ^ 42 ^ -J, s_ de sjmipnqesf.eldp of the responses of a chain varies ^ gtw - ^^^ mission filters, first channel, receive filters, smoothers, bemon - ^ = v *, Nv ^ starers, etc., designed on ^ erTliüTTveïftitm ^ to obtain a composite partial response 1-D where the noise power P (of the noise series n (D)) is small relative to the PRC power 30 Sy So we want to send a relatively large number of n bits per channel input A detector (not shown) functions on the noise-like PRC series z (D) to estimate x (D) (or equivalent ^ y (D), because there is a one-to-one relationship.) Alstte 'detector consists of an estimation unit for estimating the series with maxfMTe' 'where-35' probability, then the aim at first instance is to maximize the mirrfmaje squared distance d ^ mi * n between the allowable PRC series y (D).

In some applications, the design condition will simply be minimizing the sampling variance Sx of the RDS (in-40 gang) series. In others, the condition will rest on Sy. In other * .8860474 \ * 10 re applications, there will be an effective power limitation somewhere in the middle of the composite filter chain, so it will be desirable to keep both Sx and Sy small, and in fact a balanced design compromise between the two to provide.

A related problem is designing series with spectral zero points, e.g. a zero point at the zero frequency (direct voltage). In that case, the aim may be to design arrays y (D) which can represent n bits per sample, which have a spectral zero point, which have the smallest possible variance Sy, but which also have a large minimum squared distance d2mjn between possible y (D) strings. A common further objective is also to limit the variation of the running digital sum (RDS) of the y (D) series for system reasons. For example, because the running digital sum sequence x (D) is equal to y (D) / (1D) and its sample variance Sx is a measure of its variation, the present invention can also be applied to designing series with spectral zero points .

A number of design principles can be applied to achieve the objectives. The first principle is to design the (RDS) input series x (D) such that the (PRC) output series y (D), starting from N values at a time, is a series of N-dimensional signal points belonging to subsets of a N-dimensional collection determined by a known N-dimensional lattice code. Then the minimum squared distance d ^ min between PRC arrays will be at least equal to the d ^ m-jn guaranteed by the grid code. Further, a maximum probability raster code sequence unit can be easily adapted to use in this system, and, while perhaps not optimally, will achieve the same effective d min for essentially the same decoding complexity as the same raster encoding in a system that does not work with partial response.

An illustrative embodiment of the present invention is based on a known eight-dimensional two-state lattice code, similar to that described by Ungerbroeck in the above-mentioned article, which uses a two-dimensional set of 128 points to provide 6 bits per signal ( two-dimensional). (It is also similar to the code used in CCITT Recommendation V.33 for a 14.4 k bits per second data modem.) Figure 2 shows the encoder 20 for this code. For each 40 6 bit symbol 21, supplied by a data source 23 to the encoder, δ £ ö 0 4 7 4 11 20, two of the six input bits go to an 8 state convolutional encoder with 2/3 input / output bit ratio. The 3 output bits of this encoder are used in a subset selector 24 to select one of the 8 subsets from a 128 point signal set illustrated in Figure 3; there are 16 points in each subset (the points in the 8 subsets are indicated by A to H respectively). The remaining four "uncoded bits" 26 (FIG. 2) are used in a signal point selector 28 to select the (two-dimensional-10) signal point to be transmitted from the selected subset. The code achieves an improvement in d ^ nin by a factor of 5 (7 dB) over an uncoded system, but loses about 3 dB by using a 128 point set instead of a 64 point set, so that the net co- gain is approximately 4 dB.

15

One-dimensional implementation of the two-dimensional Ungerboeck code

The sequence of symbols x ^, broadcast across the channel, is one-dimensional in a 1-D baseband system with partial response. It is therefore useful (although not essential) to translate known grating codes into one-dimensional form. There are two aspects to this transformation: first, characterizing the two-dimensional subsets as combinations of constituent one-dimensional subsets, and second, characterizing the finite two-dimensional set as a combination of constituent one-dimensional sets. We will now demonstrate how this decomposition is performed for the exemplary two-dimensional Ungerboeck code, and then indicate how it can be performed in the general case of an N-dimensional grating code.

30 As a first step, it is noted that each of the 8 two-dimensional subsets A, B, ..., can be considered as the union of two smaller two-dimensional subsets, eg Aq and Αχ, Bo and Βχ, etc., where each of the 16 smaller subsets can be characterized as follows. The possible values of each coordinate of a signal point are divided into four classes a, b, c, d; each of the smaller two-dimensional subsets then consists of points whose coordinates are in a specified pair of classes. A suitable mathematical expression for this decomposition is obtained if we scale Figure 3 such that 40 signal points are one unit apart in each dimension (and the .sec r · ·. · 12 coordinates of each point are half integers); in that case the classes a, b, c, d are equivalent classes (modulo 4), and each of the 16 groups Ag, Αχ, Bq, ... are the points whose two coordinates are congruent to a given pair (x , y) modulo 4, wherein x and y can each take one of the four values [a, b, c, d], for example [± 1/2, ± 3/2]. These four values are referred to as the (one-dimensional) "group gathering representatives". The points from the constellation of figure 3 are marked with zeros and ones to show a possible configuration of the 16 subsets. For example, the Gg point 29 has the coordinates x = 5/2, y = 9/2, and its jointly established representatives are (5/2, 9/2) modulo 4 or (-3/2, 1/2 ).

We can now modify Figure 2 as follows. With reference to Figure 4, the three output bits of the encoder 22 plus one of the unencrypted bits 30 have been used as input signals to a sub-15 set selector 32, which selects one of the 16 subsets based on the four input bits, the unencoded bit 30 chooses between Ag and Αχ, or Bg and Βχ etc., depending on which of the original 8 subsets is selected by the three convolutional coded bits produced by the encoder 22. In fact, the encoder 22 and the bit 30 represent an 8-state encoder with a 3/4 input / output bit ratio, the output selecting one of the 16 subsets, although the group of possible signal point sequences has not been changed. Then, each of these 16 smaller subsets is designated by a pair of one-dimensional group set representatives 34, one for each coordinate, each group set representative q <taking one of four values. The group of group collection representatives is denoted by (c ^, cgfc)

One aspect of the invention is the observation that all of the good codes mentioned above, i.e. those of Ungerboeck, Gallager, Wei, and Calderbank and Sloane, can be transformed in the same manner. That is, any one of these N-dimensional grating codes can be generated by an encoder selecting one of the 4 ^ subsets, the subsets being specified by N4-valued one-dimensional groupset representatives, corresponding to the congruent classes for any coordinate (modulo 4). In some cases it is only necessary to use the 2N subsets specified by N 2-valued one-dimensional group set representatives (e.g., [± 1/2]) corresponding to the congruent classes of each coordinate (modulo 2); for example .8600474 13 image for the 4 states 2D code from Ungerboeck, the 8 states 4D code from Galleger (and the similar code from Calderbank and Sloane), the 16 states 4D code from Wei and the 64 states 8D code, etc. We have also observed that many good grating codes can be transformed in this way; i.e., the Schlafli grating code D4 and the Gosset grating code Eg can be represented by arrays of 4 or 8 bivalent dimensional group set representatives (modulo 2); the grids Αχβ and A32 from Barnes-Wall and the grille A24 from Leech can be represented by quadrilateral one-10 dimensional group collection representatives (modulo 4).

A general form for all of these codes is shown in Figure 5. The encoder is N-dimensional and operates once for every N signals sent over the channel. With each operation, p bits enter the binary encoder C33 and are encoded in p + r encoded bits. These coded bits select (in selector 35) one of the 2P + r subsets of an N-dimensional constellation (the subsets correspond to the 2P + r groupings of a subgrid A 'of an N-dimensional lattice Λ so that each subset 2n "P points). A further number of np 20 uncoded bits select (in selector 37) a signal point from the selected subset. Thus, the code sends n bits for each N-dimensional symbol, using a constellation of 2n + r N-dimensional signal points N. The encoder C and the grid division Λ / Λ * ensure a certain minimum squared distance

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25 d π, ίπ between any two signal point series belonging to a possible subset series.

The above observation (regarding the transformability of all good codes) is the result of the mathematical observation that for each of the good grating and grid codes mentioned, the grid 30 4Z of the N-folds of whole multiples of 4 is a part grid of the grid A '(and in some cases equals 2Z ^).

For certain integers q, A 'is then the set of 2 ^ group sets of 4zN in A'. The practical effect of this observation is that, provided nq + p, we can take p + r encoded 35 bits with q uncoded bits in a subset selector that is one of 2 ^ + Ρ + Γ subgroups of 4Z ^ in Λ further, these groupings can be identified by a set of N four-valued one-dimensional groupset representatives (οχ ^, cgk, ···, qjk), in which Cjk is separated by 40 integers equivalence classes (modulo 4). represents. Thus, Figure 5 can already be modified as shown in Figure 6. In this modification, we can assume that the signal constellation of 2n + r points is divided equally into 2q + p + r subsets, each with the same number of sig-5 sew points (2n-cl "P).

The illustrative Ungerboeck code implementation is an example where N = 2, Λ = Z2, Λ '= 2RZ2, p = 2, p + r = 3, q = 1 and n = 6.

The second step is to decompose the constellation into composite one-dimensional constellations. For the constellation in Figure 3, each coordinate can take one of 12 values that can be grouped as 8 “internal points” (eg [± 1/2, ± 3/2, ± 5/2, ± 7/2]) and 4 "external points" (eg [± 9/2, ± 11/2]), as suggested by the boundary 31 in Figure 3. There are two internal points and an external point in each of the four one-dimensional equivalence classes ( for example, the class whose group set representative equals +1/2 contains the two internal points +1/2 and -7/2 and the external point 9/2, because these three points are congruent with +1/2 (modulo 4) ). Starting from a particular group set representative, it is therefore only necessary to indicate whether a point is an internal or an external point and, if it is an internal point, which of the two internal points it is. This can be done with two bits, for example bik (= inside or outside) and b2k (= which of the innermost) (or with a trivalent parameter a ^) 25 We can say that the pair (b ^ k »^ k) a range identification parameter a ^, which takes one of three values, indicating the following three ranges (a) from 0 to 4 (internal point, positive); (b) from -4 to 0 (internal point, negative); 30 (c) from -6 to -4 and from 4 to 6 (external point).

The fact that each trajectory spans a part of the real axis line with a total width of 4, which part contains exactly a point congruent with any real number (modulo 4), means that the trajectory identification parameter a ^ plus the group set representative 35 c | <specify a unique signal point for any value of Ck

The signal point selector 36 of Figure 4 can then be resolved as follows. With reference to Figure 4, three unencoded bits 40 enter a range identifying parameter selection element 42 for each 40 pairs of coordinates. An uncoded bit determines whether there is an external one. 8 8 0 U k i 4 15 point has been sent. If so, a second bit determines which coordinate will contain the outer point, and the third bit selects which inner point is in the other coordinate. If not, both coordinates are internal points and the 5 second and third bits select which internal point is in each coordinate. As a total, the element 42 thus maps the three uncoded input bits 40 into two pairs of output bits 44 aj = (bn, bi2), and a2 = (b2i, b22) »where each pair of bits used to determine a coordinate in conjunction with the corresponding -10 the group set representative c ^ or C £ is generated by the group set representative pair selector 46. Thus, the entire encoder is reduced to a form in which each coordinate X | <(48) is selected (in a coordinate selector 50) by four bits. , two representing and two representing a ^ = (b ^, 15 b2k).

All constellations generally used with the above codes can be decomposed in this way. The principles are similar to those discussed in my U.S. Patent 4,597,090 and in the aforementioned article by Forney et al., "Effi-cient Modulation ...", in which N-dimensional constellations were constructed from constituent two-dimensional constellations; a similar structure of two-dimensional constituent constellations was used by Wei in conjunction with grating codes in its above-mentioned United States patent application.

The general form of an encoder for N-dimensional codes is shown in Figure 8. For every N coordinates, p bits 51 enter an encoder 52 and p + r encoded bits 54 are produced; these bits are entered with q uncoded bits 56 into a selector 58 which selects a set of N group set representatives% (60) -30; the remaining npq uncoded bits 62 are transformed (in a selector 64) into a series of trajectory identification parameters aj <(66) which together with the q <determine a series of N signal point values (70) (in a signal point selector 68) by a signal point selection function f (c ^, a ^) operating on a one-dimensional basis. In general, the trajectory identification parameter a ^ defines a subset of the real axis (one-dimensional constellation) with a width (measure) 4 that contains exactly an element that is congruent with every possible value (modulo 4), and the function f ^, a ^) selects that element. For all the codes mentioned, the alphabet of 40 group set representatives can be taken as 4 on whole numbers. 11 C C 4 7 li 16 * numerous values apart from each other (modulo 4); for some codes, the alphabet of group set representatives can be taken as two fully spaced values (modulo 2) (in which case the trajectories have a width of 2). The size of a 5 a | <alphabet is as large as needed to transmit the n bits per N coordinates. The signal point strings generated by this embodiment of the encoder are generally the same as those in the original code, and in particular they are separated by the same minimum squared distance d 1 m-yn as the original code.

Subgroup precoder nq N-dimensional signal point series, generated by good grating codes known per se, if serially converted to one-dimensional signal points, generally cannot be used as input values for a channel with partial response according to figure 1 p without degradation of ddmjn ( due to the interference between the symbols). However, a technique that we will denote with group collection precoding allows for adaptation of these known codes to partial response systems without increasing Sx or the d2m-jn gradation. The general technique is illustrated in Figure 9.

We use the same convolutional encoder 52 as was used for the known grating code, preferably in the form of figure 8. The p + r encoded output bits 54 do not directly select a subset, but are converted (as in figure 8) into a group collection selector / series former 70 in a sequence c of N one-dimensional group collection representatives Cfj, corresponding to the subset which would be selected in a non-partial response system. These group set representatives are then "pre-coded" (in a pre-coder 72) into an alternative (or "pre-coded") group set representative set c'k (74) for which c'k = c'k-i + c | <(modulo 4) (in those cases where it is possible to use modulo 2 group set-35 representatives, this pre-coding modulo 2 can be performed). Thus, the precoded group set representative set 74 is the running digital sum modulo 4 (or 2) of the ordinary set of set group representatives. The pre-encoded group set representatives C can then be grouped N at a time 40 in a group 75 to specify (in a signal point selector / serial generator 76) an N-dimensional subset; a signal point can then be selected in the usual manner (based on the unencoded bits 78); and the resulting signal point can be transmitted as a sequence x (D) of N one-dimensional signals xk over the partial response channel (in the same order in which they are precoded).

It should be noted that if the c ^ are half integers, then c'fc alternates between two groups of four values, one group being displaced 1/2 from the other. This has only a minor impact; we can, for example, "ditter" alternate coordinates X [<with +1/4 and -1/4 as an adjustment to this periodicity. Alternatively, integer values can be assumed for the Ck alphabet, e.g. [0, 1, 2, 3]; in that case c'k is always from the same alphabet, e.g. [± 1/2, ± 3/2]. This offset of c'k or Ck 15 does not affect the d ^ ml-n of the code.

If the encoder is in the form of Figure 8, then Figure 9 can be brought in the form of Figure 10 in which the same blocks perform the same things. In particular, since we have characterized the function f (ck »ak) as a function that selects the unique 20 element congruent of Ck in a trajectory identified by ak, it does not matter whether the precoding changes the c'k alphabet with respect to from the Ck alphabet; the (modulo 4) in the pre-coding unit is indeed not necessary in principle, although it may be useful in practice.

With figure 9 or figure 10 it can be shown that the PRC

series yk = Xk - Xk-1 has elements that are congruent to Ck (modulo 4) and thus fall into the subsets of the original grating code, and therefore have at least the same d2mfn * The RDS series Xk has the same average energy Sx 30 as in the original grating code if the c'k alphabet is the same as the Ck alphabet; even if it is not, equality still applies approximately. (Deη the illustrative embodiment, the mean energy per coordinate is 10.25, with signals separated by integer values). If the Ck are integers 35 then the Xk are independent, identically distributed random variables, and thus (a) $ y = 2SX; (b) the spectrum of the RDS series [xkl is flat (white) in its Nyquist band; 40 (c) the spectrum of the PRC series [yk] is the same as that of the partial response channel 1800474 s.

Even if the ck are not integers, these statements are still approximately true.

The group set precoding can be modified for other types of systems with partial response as follows. For a 1 + D (one-dimensional) system with partial response, the same system is used except that c'k_i is subtracted instead of added in precoder 72, resulting in c'k = C | <- c'k_i ( modulo 4). For a 1-DL system, the delay element D is replaced by a delay element DL, so that c \ = c '^ - L + c ^. For a 1 + D two-dimensional system, two 1 + D precoding units are used in parallel with pairs of output signals from the subset selector / series generator serving as input signals and the two output signals determining the 15 real and imaginary (in phase and quadrature) parts of the two-dimensional signal point being transformed.

RPS feedback

Depending on the application, it may be desirable to reduce the average energy Sy of the PRC series at the expense of an increase in the average energy Sx of the RDS series. This also tends to flatten the PRC spectrum, while the low frequency content of the RDS spectrum will increase. In "Information Rates and Power Spectra of Digital Codes", Justesen has IEEE Trans. Inform. Theory, Vol. IT-28, pp. 457-472, 1982 introduced the term "cutoff frequency" fg 25 below which the PRC spectrum is small and above which it tends to run flat, and has shown that fg is approximated by fg · «. (Sy / 2Sx) ffl, where f ^ is the Nyquist band edge frequency.

A general method of applying this compromise and still maintaining 30 d2min of the grating code in the PRC arrays is to extend the encoder of Figures 9 and 10 in the following manner.

The PRC series can be calculated from the RDS series; for the 1-D channel, each PRC signal is exactly equal to y ^ = xk - xk-1 · With reference to Figure 11, we can have the signal point selector 80 each X | <based on X [<_ i (by x | <back via a delay element 82) as well as on the current pre-coded group set representatives c'k and on the trajectory identification parameter aj <such that large PRC values y ^ (calculated in sum 40 unit 84) are avoided. As long as the signals x ^ are still chosen .8800474 e 19 as congruent to c'k (modulo 4), the signals y ^ congru-

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be grafted to c | <(modulo 4) and therefore the grille code will be maintained. (It is noted that although the idea is to calculate the PRC value y ^ so that it is kept small 5, in fact the previous RDS value Χ | <_χ is returned so that we can speak of RDS feedback).

For the illustrative embodiment, this can work as follows.

As already noted, the normal selector function f (c | <, a ^) of the voter can be characterized by saying that the 8 internal 10 points are the 8 half integer values ranging from -4 to +4, while the 4 external points are the 4 half integer values that lie in the range of -6 to -4 and +4 to +6. We can vary the internal point range and the external point range as a function of as long as the internal point range 8 spans 15 signal points, 2 from each equivalence class, while the external point range spans 4 signal points, one from each equivalence class.

A general way to do this is to translate all trajectories by a translation variable R (xk-1) which is a function of 20 Xfc.p. That is to say, in the illustrated embodiment, that the internal point trajectory is modified to run from -4 + R (xfc-i) to 4 + R (xfc_i), and the external point range runs from -6 + R (xk-l) to -4 + R (xk-l) and from 4 + R (xk -l) to 6 + R (xk-1) * 25 The function R (xk-i) must generally increase by x ^ -i in order to reduce the y ^. We have been able to show that the optimal choice is R (xk-l) = Bxk-1 * where β is a parameter in the range o ^ β <1. If β = 0 then the RDS feedback disappears via the element 82 and in figure 11 the group set precoding is reduced as in figure 10. If with this choice Sq is the value of Sx in the normal case (β = 0), then it follows that (a) Sx = S0 / (X - (b) Sy = 2S0 / (1 + 6); 35 (c) The spectrum Sx (f) of the RDS series is proportional to 1 / (1 - 2fcos θ + β2), where Θ = irf / f ^; (d) The spectrum Sy (f) of the PRC series is proportional to 2 (1 - cos0) / (1 - 2β cosO + β2), the "cutoff frequency" fg is equal to (1-β) ίΝ · 40 (e) The X | <are limited to the range from -Μ / 2 (1 + β) to. & E C (i 7 h 20 M / 2 (1-3), and the are limited by the range from -M to + M if the range of the coordinates in the original code is from + M / 2 to + M / 2.

If 3 approaches 1, Sy approaches Sq, and Sy (f) approaches a flat spectrum with a sharp zero at zero frequency. Meanwhile, Sx becomes large and Sx (f) approaches a 1 / (1-D) spectrum, except that it remains finite near the zero frequency. We have been able to demonstrate that this is the best possible compromise between Sx, Sy, and Sq10. Figures 12, 13, 14 show three equivalent ways to generate Xk and / or y ^ based on C | <, ak, and X | <-1 · Figure 12 corresponds very closely to figure 11.

In Figure 12, the feedback variable c'k-i in the group collection precoder 72 is replaced by X | <-is because c'k-i 15 = X | <_i (modulo 4), and only the value of c'k (modulo 4) is used in the selector 80. R (a | <) denotes the trajectory identified by a | <, and R (x | <-i) represents the trajectory translation variable introduced by the RDS feedback. Because y ^ = xk - xk-1 = c'k - x'k-1 (modulo 4) and c'k = Ck + 20 Xk_i (modulo 4), yk 5 Ck (modulo 4).

Figures 13 and 14 are mathematically equivalent to Figure 12 in that if they have the same initial value Xk_i and the same series of input values (ck, ak) they will produce the same group of output values (xk * yk). In Figure 13, yk is chosen as the unique element which is congruent to Ck in the range R (ak) + R (xk-i) - Xk-1> and xk is determined from yk using xk = yk + xk -l »such that xk = c'k = ck + xk-i (modulo 4) and the unique element in the series R (ak) + R (xk-1) is congruent to c'k (modulo 4). In Figure 14, an innovation variable 30 ik is chosen as the unique element congruent with c "k ξ Ck +

Xk-l - R (xk-l) (modulo 4) in the range R (ak), and Xk is determined from I as xk - I + R (xk-l) »so that Xk ξ c" k + R (Xk -1) = c'k (modulo 4), and the unique element R (ak) + R (xk-1) is congruent to c'k (modulo 4) Figure 12 35 combines the delay element in the pre-encoder with the delay element required for the RDS feedback, and is very useful if Xk is the desired output value and c'k is always from the same alphabet, eg [± 1/2, ± 3/2]. Figure 13 completely eliminates the pre-encoder and is very useful if yk is the desired 40 starting value and Ck is always from the same alphabet, .680 04 74% 21 * eg [± 1/2, ± 3/2] Figure 14 takes the trajectory translation variable R (xk-1 ) outside the selector, so that the I is always chosen from the same trajectory (the union of all R (ak)}; the innovation sequence i (D) is approximately a series of independently identically distributed 5 random variables i ^ (where the very small variations, introduced by the c'k congruent! condition are neglected) and this auxiliary sequence may be useful if a white (spectrally flat) sequence is desired which is deterministic related to x (D) or y (D).

Figures 15, 16, 17 illustrate three equivalent filter configurations for use with the x (D), y (D), and i (D) series from Figures 12, 13, 14. In Figure 15, the RDS series x (D) filtered in a transmit filter Hy (f) before it is transmitted (as signal s (t)) over the actual channel (not shown). In Figure 16, the PRC series y (D) is filtered in a transmit filter H'j (f) whose response is equivalent to that of a cascade of 1 / (1-D) on sampled data working filters and Hj ( f); since y (D) has a zero point at the zero frequency, it doesn't matter that the response of 1 / (1-D) is infinite at the zero frequency (especially if Hy (f) also has the zero at the zero frequency ). In Figure 17, the innovation sequence i (D) is filtered in a transmit filter H '^ f) whose response is equivalent to that of a cascade of 1 / (1-eD) on sampled data working filters and Hj (f) ; this is equivalent to Figures 15, 16 if R (xk_i) = gxf <-i; in other cases, the equivalent filter acting on sampled data, the filter corresponding to Xk = ik + 25 R (xk-i) s is generally not linear. Any of these equivalent embodiments may be preferred depending on Hy (f), R (xk-i), and the implementation technology employed.

Certain modifications of the RDS feedback systems discussed above may be desirable in practice. For example, it may be desirable to change the shape of the trajectories R (afc) from those used if R (xk-i) = 0. In the illustrative embodiment, a simple RDS feedback is implemented as follows: if xk- l is positive, y ^ is chosen as usual in the range from -4 to +4 if aj <indicates the internal point, but if ak indicates an external point then yk is chosen as a number congruent to ck in the range from - 4 to -8; if xk-χ is negative then the range from 4 to 8 is used for the external points. Then applies

(a) the range of the PRC series yk is limited from -7 1/2 40 to +7 1/2, instead of -11 to +11 if there is no RDS

. 8 e o o - “4 22 feedback occurs; (b) the PRC variance Sy has been reduced from 20.5 to 13.25, a reduction of 1.9 dB, and about 1.1 dB above Sg = 10.25; (C) the mean of yk is -3/2 if Xk_i is positive and +3/2 if negative, so that the RDS sequence tends to remain close to zero. Although it is difficult to calculate Sx exactly, it follows from the facts E [ykxk-l3 = Sy / 2 and 10 E £ ykXk-l3 = “(3/2) EUxk-ll3 that the mean of the absolute value of Xk is equal is on Sy / 3 = 4.42, so that the RDS series Xk is very well defined. (Without RDS feedback, the average of the absolute value of Xk is 2.75); (D) the variance of the yk is at a given Xk_i equal to So = 11, about 0.3 dB higher than the So = 10.25 possible without RDS feedback. The minimum possible Sx for Sy = 13.25, Sg = 11, is Sx ~ 19.5 corresponding to honor 0.66. Because Sx = 20 Sjx | + E [| x |] 2, Sx must be greater than (4.42) 2 i9s5 so that with this simple method we achieve less than the optimal spectral compromise; (e) every possible yk is associated with a unique pair (ck, ak). As we will discuss in more detail below, this means that a decoder must keep an estimate of the running digital sum of the estimated PRC sequence and that no error propagation will occur in the decoder.

In summary, this simple method does not achieve the best power compromise between Sx and Sy, on the other hand, not only Sy is effectively limited, but also the peak values of yk, delineating the RDS series xk fairly well, and error propagation to the receiver appearance.

With these methods, therefore, a compromise is possible between Sx 35 and Sy (i) for the unbound case, in which the xk sequence is uncorrelated, Sx has the same energy So as is necessary for transmitting n bits per symbol in the non-partial response case, and Sy = 2SX, (ii) up to the case where the yk sequence is uncorrelated, Sy = Sq »and Sx becomes very large. These compromises 40 are possible for all mentioned grating and grating codes.

, 8 8 e 0 "v m β 23

Decoder! Ng

The above methods result in the generation of PRC

strings that belong to a known good code, and therefore have a 2 d min that is at least as large as that of the code.

Thus, with reference to Figure 18, a suitable detector for the noise-received PRC series z (D) = y (D) + n (D) is a maximum likelihood series estimation unit (according to the Viterbi algorithm) for the known good code, which is modified as follows: 10 (a) a first step in the decoding may consist of, for each noise received PRC value z ^ = yj <+ n ^, for each of the four classes of the real numbers congruent with the four one-dimensional group set representatives c · ^ (modulo 4), j = 1, 2, 3, 4, finding (in block 92) the nearest element in each class at z ^, and its "metric" mjk = ( yjk - zk) ^ (the squared distance from zfe); (b) in a code based on an N-dimensional lattice distribution Λ / Λ ', a second decoding step may be, for example, finding for each of the 2P + r subgroups of Λ' in \ the best (lowest metric) of the 2Z group sets of 4Z, the association of which is the group sets of A 'by summing the respective mechanics of the constituent one-dimensional metric metrics mjk and comparing these sums with each other (block 94); (c) the decoding can then proceed in the usual manner (block 96) using the best metric determined in step (b) as metric for each group sets of A '. The decoder will eventually produce an estimate of a set of group sets of A ', which can be mapped to a set of estimated group set representatives that can be mapped to the corresponding y ^, from which the original and% can be recovered, if desired (block 98). These last steps require the decoder to keep the running digital sum Xk-1 of the estimates y <.

40 Because the PRC strings are in the known code, the error will be '. 88 0 0 "24 appearances of this decoder are at least as good as those of the known code, in that at least the same effective dim is maintained. However, since the PRC sequences are really only a subset of the known code sequences, such a decoder is not a true maximum probability sequence estimator for the PRC sequences. As a result, she may occasionally decode to a string that is not a legitimate PRC string. Legitimate PRC strings must meet the following two additional conditions: 10 (a) A legitimate finite PRC streak y (D) must be divisible by 1-D; i.e. the sum of its coordinates must be zero; (b) the trajectory restrictions imposed by the signal point selector must be satisfied by all yk (or equivalent 15 by χ | <or ik) based on the reconstructed

a

first values of the RDS Xk-i *

If this decoder makes a normal decoding error, corresponding to a short period of erroneous group collection estimates, followed by correct group collection estimates, then it is possible that the corresponding finite PRC error sequence will have a running digital sum different from zero. This will lead to a persistent error in the estimated running digital sum χ | <_ι of the decoder which may lead to a possible back-mapping of errors on the? K »and eventually Xk, even though the group 25 sets Ck are correct, for as long as the error persists in the RDS estimate.

The decoder must therefore continuously observe (block 99) whether the path conditions in the reconstructed y ^ and ^ k are satisfied. If not met then the estimated RDS 30 xk-1 is known to be incorrect; the xk_i must be corrected by the minimum amount required to meet the range constraint, assuming the group set string Ck is correct. With probability 1, this may result in a resynchronization of the estimated RDS to the correct value, and normal decoding may continue. However, a significant period of error propagation can occur.

Avoidance of error propagation

We will now indicate a general method to avoid error propagation to the receiver. The method works best if the signal-40 constellation consists of all points in Λ within an N cube, but .6600474 25 is not limited to that case. It can be considered a generalization of the principles of previous precoding forms (modulo M) for use with coded strings.

The basic idea is that every possible PRC value must correspond to a unique (c ^, a ^) value if the code can be formulated in one-dimensional form as in figure 7; or more specifically, that each group of N yk values must correspond not only to a unique set of NC | <values but also to a unique group of uncoded bits, if a general N-dimensional signal point selector is used as in Figure 6 Then, the inverse A, mapping the decoded yk to the encoded and unencoded bits is independent of the estimate of the running digital sum by the decoder so that (a) the decoder does not have to follow the RDS; 15 (b) no error propagation occurs.

Thus, in Figure 18, block 99 can be eliminated.

Figure 19 shows how this can be done if the code can be formulated in one-dimensional form as in the illustrated embodiment. From ck and ak, a signal point selector selects a value of Sk = f (ck »ak) zoaï in Figure 8. In the illustrative embodiment, $ k assumes one of 12 values, namely, the half integer values in the range of -6 to +6. In general, Sk will take one of the values from an alphabet of integer spacings in a range of width M; we indicate this range 25 with Rq. As in Figure 13, yk is then selected as the unique number which is congruent with Sk (modulo M) in the range Rg + R (xk_i) - Xk-1 ^ t width M, where R (xk-i) has an RDS feedback trans ! is variable, and xk_i is the preceding RDS signal point. The current RDS Xk is calculated as yk + Xk-1 · 30 Figures 20 and 21 show equivalent methods for generating Xk and / or yk from the series sk such that yk = Sk (modulo M), analogous to the figures 12 and 14. In Figure 21, an innovation variable I is generated which is more or less white and uniformly distributed over the range Rq, such that its variance Sg is approximately M / 12; thus, So 12 for the illustrative embodiment, a deterioration of about 0.7 dB above the value of Sq = 10.25 achievable without RDS feedback. As in Figures 12, 13, 14, all three series Xk> yk and I carry the same information, and as in Figures 15, 16, 17, arbitrary 40 one can be used as input string to a filter which, 88, ent 26 spectrum for transformation is transformed.

The deterioration in the innovation variance is eliminated if the original code coordinates are uniformly distributed over a range Rol, i.e. if the original constellation is bounded by an N cube with side Rg.

As an illustrative example with a rectangular constellation, we use the same two-dimensional 8-state Ungerboeck encoder as in Figure 2, but now with the 128 point constellation of Figure 22 instead of that of Figure 3. The constellation consists of 10 alternating points from the conventional 256 point 16 x 16 constellation; the coordinates therefore have the 16 half integer values [± 1/2, ± 3/2, ..., ± 15/2], but with the restriction that the sum of the two coordinates is an even integer (0, modulo 2 ) must be. So the minimum squared distance between the signal points is 2 instead of 1; and the d2m-jn of the code is 10 instead of 5. The variance of each coordinate is now 21.25 instead of 10.25, which after scaling by a factor of 2 gives a loss of 0.156 dB compared to the constellation from figure 3, because the intersection looks more like a circle than at the square. (In grid terminology, we now use an 8-way grid partition RZ2 / 4Z3U2 instead of Z2 / 2RZ2).

It can be observed that now each of the 8 subsets corresponds to a unique pair of group set representatives (cj, C2) modulo 4 such that c ^ + C2 = 0 (modulo 2). The three encoded bits of Figure 2 therefore directly determine a pair of group collection representatives in the subcollection selector 24 instead of using the 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 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 is suitably expressed by the two bit trajectory identification parameters (a ^, ag) each representing one of the four values [± 2, ± 6]; the coordinate selection function is then simply 35 S | <= f (q <, a | <) = C | <+ ak · It is noted that the possible values for Sk are the 16 half integer values in the range Rg from -8 to 8 with a width M = 16.

The conventional precoding can then be performed modulo 16. The entire encoder is illustrated in Figure 23. The RDS value 40 Xk is the sum Sk + Xk_i (modulo 16). In this case, the Xk, 8800474t 27 * values are essentially independent identically distributed (white) random variables, and yj <= xk - xk-i = sj <(modulo 16).

To achieve a spectral compromise via RDS feedback such as -5 as in Figures 12, 13, 14, we assume that continuously represents the desired congruence class of (modulo 16) and that R (xk-1) is an RDS feedback variable such as in Figures 12, 13, 14, which is ideally equal to βχ | <-ΐ · Figures 24, 25, 26 then show three equivalent methods to calculate the series x ^ and / or 10 yk = xk “xk- 1 obtain such that yk = S | <(modulo 16) and $ x and Sy reach the desired compromise, given Sq = 21.25. Here Rq is the range from -8 to +8.

In this case, the innovation variable i ^ has a variance Sg λ; 16.2 / 12 = 21.33, substantially equal to the variance of each 15 coordinate in Figure 22, so that no deterioration is greater than 0.16 dB if Figure 22 is used instead of Figure 3.

As already noted, the decoder must follow the RDS because, based on the estimated PRC sequence y ^, the c ^, <i and u ^ “, the original input bit sequence is finally uniquely determined. However, if the decoder maintains the estimated RDS as well as the corresponding ranges in which the may fall, it can detect that an error has occurred when a decoded yk is outside the estimated range. Even if not used for error correction, such a trajectory violation monitor can lead to an estimate of the decoding error amount.

Improved decoding units

A true maximum probability sequence estimator would take into account the entire state of the encoder and the channel, which will generally include the value of RDS xk_i (the channel state) as well as the state of the coder C Such a decoder will yield the actual d2min of the PRC strings and will be free from error propagation. However, since Xk_i generally assumes a large number of values, in principle an infinite number is possible with RDS 35 feedback, such a decoder is not very practical.

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In addition, to achieve the actual dcml * n it may be necessary to enter a substantially infinite decoding delay because the code / channel combination becomes quasi-catastrophic if n is large, as we will explain in more detail below.

40 It may be worth considering an enlargement of the decoding unit in order to obtain at least the actual part of the code. Since all finite PRC strings are divisible by 1-D, all finite weight strings must be of equal weight. The actual d ^ m-jn is therefore always momentary. In an illu-

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5, the real d π, -, - η is in fact equal to 6 and not equal to 5.

A general method of obtaining the real d2ml-n in such cases where only the effective number of states in the decoder is doubled is the following. In the decoder 10, each state of the encoder C is divided into two states, one corresponding to an even RDS and one corresponding to an odd RDS. During the decoding, two series will then only converge in the same state if their estimated RDS has the same value (modulo 2). Thus, it becomes impossible for two series, differing by an odd weight error sequence, to converge, so that the effective d min is the weight of the minimum even weight error sequence in the original code. Furthermore, if a decoding error occurs that results in a persistent estimated RDS error, as discussed above, then this error must be at least 2, and will therefore be detected more quickly.

The decoder of Figure 18 can be used in the modified manner shown in Figure 27. For most codes, each of the subsets of the signal constellation (groupings of A 'in Λ) will contain points, all of which have an even or odd sum of 25 coordinates. For example, in Figure 3, four of the eight subsets have points whose coordinate sum equals 0 (modulo 2), and four contain points whose coordinate sum equals 1 (modulo 2). The metric of each subset (subset of Λ 'in A) can be determined as before in blocks 92 and 94; the maximum probability sequence estimator unit 196 is then modified to find the best set of subgroups (a) contained in the code and (b) having a running digital sum that is congruent with zero (modulo 2). The decoded subgroup sequence is mapped back to ^ and \ by the block 99 if necessary (post-35 statements will now be performed in multiples of 2).

However, there is a drawback to this technique besides doubling the decoding state space. Two strings can differ with an odd weight error string and are followed by a long string of zeros (no differences). The decoder can then store parallel state pairs in the decoder for a very long time. 8 8 P i 'ί * 29 lie without resolving the ambiguity. Ultimately, this "quasi-catastrophic" behavior can only be resolved by the maximum likelihood sequence unit by a trajectory violation due to different RDS parity on the two roads. Thus, the decoding delay required to determine the real .d2min can be very large.

For this reason, it will generally be preferable to choose an encoder C with twice the number of states, and to use an unenlarged decoder for C. For example, there is a 16-state two-dimensional Ungerboeck code with d2m ^ n = 6; even though it has a slightly greater error coefficient than the 8-state code with an enlarged 16-state decoding unit, we assume it will be preferable in practice.

It is worth noting that PRC strings, derived from the four states two-dimensional Ungerboeck code, always have a true d2ml-n of 6 because that code has d2min = 4, where the only error strings with a weight of 4 are equal to simple coordinate errors with a major 2, which are also not divisible by 1-D. A 16-state decoder that maintains the RDS modulo 4 can determine this d2m-jn. In this case, however, the code is not only quasi-catastrophic, but the error coefficient is also large, so that again the normal 16-state 2D Ungerboeck code seems to be preferable.

25 Quadrature systems

As previously noted, a complex (or quadrature) system operating with partial response (a QPRS system) can be modeled as a 1 + D sampled data filter operating on an RDS series x (D) with complex values to produce a PRC series y (D) = (1 + D) x (D) 30 with complex values, ie ^ =) ^ + When used with double sideband quadrature amplitude modulation over a band-pass channel, such a system results in zeros on both band edges, fc ± f ^, where fc is the carrier frequency and f ^ = 1 / 2T is the width of the single Nyquist band.

If N is even and 4Z ^ is a partial grid of A ', as in the case of all the good codes mentioned above, then we can adapt a known good code for use in a QPRS system by essentially the same principles as in the apply the above. A group set of 4Z ^ can be specified by N / 2 40 complex group set representatives c ^, in which the group set, ê & c (; - 30 sharing representatives take one of 16 possible values, corresponding to four mutually worthy values (modulo 4) for the respective real and imaginary parts of c | <. The general picture of figure 8 then applies with the exception of 5 the fact that the group collection selector 58 and the path identification parameter selector 64 are the N / 2 complex valued group collection representatives C | < as well as the range identification parameters a | <, and that the signal point selector functions once per quadrature signal and outputs complex valued output signals X | <. The group-10 set pre-coding is performed as in Figure 9 by forming the complex pre-coded group set c'k = c ^. -c'k-i (modulo 4) once per quadrature symbol The RDS feedback is briefly as in Figures 11, 12, 13 performed by using a function R (a | <) that identifies an area in complex space 16 iden-15 that contains exactly one element of each group set of p4Z, and a complex trans! atation variable R (x | <-i) »which is ideally equal to βχ | <-ΐ · In the cases where 2Z ^ or 2RZ ^ form part raster of A ', precoding can be performed modulo 2 resp. 2 + 2i, and R (a ^) can identify a part of a region 4 or 8 which contains exactly one element for each group set of 2Z ^ resp. 2RZ ^.

Higher dimensional systems

We have shown embodiments in which coordinates of N-dimensional symbols are formed on a signal-for-signal (one- or two-dimensional) basis, with signal-for-signal feedback from the previous RDS value X | <-i results can be obtained with systems that select signals on a higher dimensional basis. In such systems, the precoded group set of reprepresentatives must be grouped as in Figure 9 in order to select subsets in the appropriate dimension, then select signal points in that dimension and then re-serialize the coordinates for transmission over the channel. If the order of the group sets is maintained, such a scheme retains the property that the PRC strings of the given code are 35 and have the specified d / ml-n. In such a system, it may be more natural to perform the (RDS) feedback on a higher dimensional basis rather than on each signal. N-dimensional codes

Although one-dimensional representation of codes is desirable, it is not essential. In this section we will show, 8 ε o u <:: v β 31 »how codes can 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 embodiments can be obtained.

5 We will again use the 8 states two-dimensional Ungerboeck type code from figure 2, with the two-dimensional 128 point constellation of figure 3. We remind you that every coordinate in this constellation takes values from the alphabet of the 12 half integer values in the range of -6 10 to +6; the two-dimensional constellation uses 128 of the 144 possible pairwise combinations of elements from this alphabet.

As a first step, we expand the signal constellation to an infinite number of values in the following manner. The expanded constellation consists of all pairs of numbers that are congruent with a given point in the original (Figure 3) constellation (modulo 12). Thus, the points in the expanded constellation consist of pairs of half integer values. If we consider the original constellation as a cell bounded by a 12 x 12 square 98, then the expanded constellation consists of an infinite repetition of this cell in two-dimensional space, as shown schematically in Figure 28.

It should be noted that each cell contains only 128 of the 144 possible points; there are "holes" 99 of 4 x 4 in the expanded constellation.

The key feature of this expanded constellation 101 is that if we place a 12 x 12 square randomly anywhere in the plane (with the sides horizontally and vertically oriented), this square will enclose exactly 128 points, a congruent with each of the points in the original constellation. Even an even more general theorem holds: if we place a diamond 102 with horizontal width 12 and vertical height 12 (see figure 29) arbitrarily somewhere on the plane 30, then this will also enclose 128 points, a congruent with each point in the original constellation.

With reference to Figure 30, we can now implement the RDS feedback on a two-dimensional basis as follows. Suppose x | <_i represents the running digital sum of all yk preceding the current (two-dimensional) symbol. Suppose that R (xk-i) now designates an area of the plane corresponding to a 12 x 12 diamond as shown in Figure 29, where both the shape and the position of the diamond may depend on Χ [<_ ι · Suppose that (yo , k »yo, k + 1) indicates the point in the original constellation that would be selected 40 (in voters 104, 105) by the three encoded bits and the four not. 8 £ 0 0; 32 coded bits in accordance with the unrestricted code (Figure 2). Then (in selector 106) (yk, yk + i) is selected as a unique point in the two-dimensional expanded constellation that lies within the region R (xk-i) and that is congruent to 5 (yo, k> yo, k + l ) (modulo 12); these will be the two coordinates yk. We can obtain (x ^ Xk + l) from Xk = ^ k + xk-l> xk + l = yk + 1 + xk ° P in the manner shown.

We will now show that this two-dimensional scheme can produce the same output values as the one-dimensional RDS feedback-10 scheme (modulo 12) shown earlier, with the optimal one-dimensional RDS feedback variable R (xk-1) = pxk-1 to figure 31, in a dimension, given x ^ y ^, is chosen as the unique value in the range Rq + 6Xk-i - Xk-1 congruent to Sk (modulo 12), in which we now recognize that $ k congru-15 ent is with yo, k · A coordinate of the rhombus, which is used in the two-dimensional system, can therefore be taken to lie within the range of the same width 12. Given x ^ .j and yk, and thus also Xk = yk + xk-l », yk + χ is then chosen as the unique value in the range Ro - (1 - $) xk = Ro - (1 20 - B) yk - (1 - B) xk-l which is congruent to Sk + i = yo, k + l (modulo 12). yk + i is therefore in the range Rq - (1 - &) xk-l (same as yk) shifted by - (1 - p) yk ·

Thus, by a suitable choice of the pane, we can emulate the behavior of a one-dimensional (modulo 12) RDS feedback system with a two-dimensional system. Thus, it will have the same advantages, including avoiding error propagation and including a near-optimal compromise between Sx, Sy and So and with the same drawbacks, notably the increase in Sq to 12 over the otherwise possible 10.25.

30 We can choose other two-dimensional RDS feedback variable and (offer) to further simplify implementation, and achieve other benefits at the cost of a sub-optimal power compromise. A system nearly identical to the previously described simplified one-dimensional system results if we make 35 R (xk-i) equal to a square 120 with side 12, centered on (-2, -2) if Xk_i is positive, and the square 122 centered on (+2, +2) if Xk_i is negative. So we use one of the two constellations 124, 126 shown schematically in Figure 32.

As in the previous one-dimensional system, the internal points are always selected from the same set regardless of x ^ -i * but the external points are varied to bias in a positive or negative direction. The ranges of y ^ are strictly limited from -7 1/2 to 7 1/2. In fact, this system 5 is identical to the earlier simplified system, except that y ^ + i was chosen based on rather than x ^. In practice, all performance and spectrum values will be nearly equal.

Another variant results in a system related to the type of Calderbank, Lee and Mazo. A CLM type system uses an expanded signal constellation with twice the original number of signal points, divided into two separate constellations, one to be used if x ^ i is positive and the other if x | <_i is negative. For example, Figure 33 shows a 15 square 16 x 16 constellation divided into two separate constellations 110, 112 of 128 points each such that each of these constellations is equally divided into 8 subsets of 16 points each. A constellation consists of points whose sum of the coordinates is positive or zero and is used when x ^ _i nega-20 is active; the other consists of points whose sum of coordinates is negative or zero and is used if it is positive. In two dimensions, doubling the constellation size doubles Sy and therefore does not result in a favorable power compromise; in higher dimensions, however, the drawback that results from the use of two separate constellations is less.

These ideas can be generalized to N dimensions in the following manner. If there is a one-dimensional formulation of the code as in Figure 8, using a modulus M, then an N cube 30 with side M will fully comprise the N-dimensional constellation, and the resulting cell can be replicated to the N-space to be filled without compromising the minimum squared distance between the code strings congruent to the original code strings modulo M. We can then use an N-dimensional RDS feedback function R (xk-l) 35 where for each Xk-i »R (xk-1) there is an area from the N-space with a volume Mws containing exactly one point in each equivalent class of N-vectors modulo M in an N-dimensional analog of Figure 30.

Other embodiments are within the scope of the following claims.

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Claims (33)

1. Device for generating a series of digital signals xk and / or a series of digital signals k = 1, 2, ... such that the relationship between the xk signals and yk signals is equal to yk = yk ± xk With L an integer, which signals yk are a series with a given modulation code, the device comprising an encoder for selecting J from said signals yk, where J ^ 1, (yk, yk + i. ••• yk + Jl) »which are congru-10ent to a set of J group set representatives ck (modulo M), with M an integer, which group set representatives are specified in accordance with said modulation code given, which J symbols are selected from one of a number of J-dimensional constellations, which choice is based on a previous xk ', with k' <k, wherein at least one of said constellations contains both a point with a positive sum of the coordinates and another point with a negative sum of the coordini which encoder is arranged such that said signals xk have a finite variant? e Sx. 2. Device for generating a series of digital signals xk and / or a series of digital signals yk, k = 1, 2, ... »such that the relationship between the xk signals and yk signals is equal to yk = xk ± xk-L »not L an integer, which signals yk are a sequence with a given modulation code, which device comprises an encoder for selecting said signals xk congruent to a series of alternative group set representatives c ' k (modulo M) where c'k = ck - c'k_L (modulo M) in case yk = xk 30 + xk_ | _ 'c'k = ck + c'k_L (modulo M) in case yk = xk - Xk + L 'and where ck is a group collection representative specified in accordance with said modulation code. 3. Device for generating a series of digital signals xk and / or a series of digital signals yk, k = 1, 2 ..., capable of representing n bits per signal, such that the relationship between xk and yk equals yk = xk ± xk_L> with L an integer, where the signals xk and yk have variances sx and 40 Sy, which yk signals fall within an alphabet of possible sec! 474 35, ke yk signals having an equal distance within said alphabet equal to the distance Δ, which device is provided with an encoder which ensures that the series has a variance 5 Sy less than 2 $ o and the said series xjc has a variance Sx not greater than S2y / 4 (5y-Sg), where Sg is approximately the minimum signal power required to represent n bits per signal in an alphabet spaced Δ. The device of claim 3, wherein said sequence y ^ is a sequence in a given modulation code. 5. Device for generating a series of digital signals xj (and / or a series of digital signals y ^, k = 1, 2, such that the relationship between the Χ | ς signals and y ^ signals is equal to y ^ = x ^ ± Χ | <-1_ »L has a 9th integer, which x ^ and y ^ series have variances Sx and Sy, which symbols yj <are a series in a given modulation code, which device comprises an encoder said x ^ and y ^ signals have a randomly selectable variance of sx and sy within predetermined ranges. 6. Device as claimed in claim 5, wherein said signal series are able to represent n bits per signal, said yk signals fall within an alphabet with possible y ^ signals with equal distances equal to the value Δ, the tra objects are controlled by a parameter 3, and Sx is approximately equal to S0 / (l-32), and Sy is approximately equal to 2 $ o / (l + 3) s where Sq is approximately the nominal signal power required for representing n bits per symbol within a spaced alphabet equals Δ in accordance with said 30 code. 7. Apparatus for generating a sequence in a given N-dimensional modulation code by generating a sequence of one-dimensional signals, which modulation code is based on an N-dimensional constellation divided into subsets associated with said code which subsets each comprise N-dimensional signal points, and the choice of each subset is based on coded bits and unencrypted bits of said signal points, the device comprising an encoder for deriving for each said N-dimensional symbol from said coded and uncoded bits of ..8 8 0 0 "7 a set of N, M-valued one-dimensional group set representatives C | <corresponding to the congruent classes of each of the N coordinates (modulo M) group collection representative indicates a subset of one-dimensional values in a one-dimensional constellation of possible coordinate values for each of said N dimensions, and each of said one-dimensional signals in said series is selected from said possible coordinate values based on the uncoded bits. The device of claim 1, 2, 3 or 5, further comprising an output to which said sequence is supplied. The device of claim 1, 2, 3 or 5, further comprising an output to which said series X | <is fed. 10. Device according to claim 1, 2, 3 or 5, in which L equals 15. The device of claim 1, 2, 3 or 5, wherein the relationship between the X | <signals and y | <signals is equal to y ^ = x | <- xx_L, with L an integer. 12. Device as claimed in claim 1, 2, 3, 5 or 7, wherein the modulation code is a grating code. The device of claim 1, 2, 3, 5 or 7, wherein the modulation code is a schedule code. The device of claim 1, 2, 3, 5 or 7, wherein M is equal to 2. The device of claim 1, 2, 3, 5 or 7, wherein M = 4. The device of claim 1, 2, 3, 5 or 7, wherein M is a multiple of 4. The device of claim 1, wherein J is equal to 1. 18. The apparatus of claim 1, wherein J is equal to the number N of the dimensions in said modulation code. The device of claim 1, wherein k 'is equal to k-1. The device of claim 1, wherein J is equal to 1 and each of said constellations is a one-dimensional range of values centered on β | <_ι, 0 <β <1. The device of claim 20, wherein β> 0. The device of claim 1, wherein there is a finite group of said J-dimensional constellations. The device of claim 22, wherein there are two of said J-dimensional constellations. 24. Device according to claim 1, 2, 3 or 5, wherein y ^ and \ .88 (ιv, have real values. The device of claim 1, 2, 3 or 5, wherein yk and xk have complex values. 26. Device according to claim 25, in which M equals 2 + 5 2i. The device of claim 1, 2, 3 or 5, wherein at least two of said J-dimensional constellations are not separated from each other. 28. Decoder for decoding a series Zk = yk + 10 nk, k = 1, 2, ..., into a decoded series yk, wherein the series of signals y ^ is such that (a) said series is a given has modulation code; (b) the running digital sum Xk = yk-i + yk-1 + Yk-2 + ... has a finite variance Sx; (C) said signals yj fall in a predetermined allowable range depending on x ^, k '<k; and represents the% noise series, provided with a trajectory exceedance monitor which reconstructs the estimated running digital sum Xk = Pk + Pk-1 + ·· *>, compares said decoded series ^ with said predetermined allowable trajectory based on said estimated running digital sum xk ', k' <k, and generates an indication if said y [<is outside said allowable range. The decoder of claim 28, wherein said Λ. Estimated running digital sum xk is adjusted based on said indication that yk will be within said allowable range. 30. The decoder of claim 29, wherein said readjustment is performed with a minimum possible value such that y k falls within said allowable range. 31. Decoder for decoding a sequence Xk = yk + nk, k = 1, 2, ..., wherein the sequence of signals yk is such that (a) said sequence originates from a given modulation code, which code is capable of being generated by an encoder with a finite number of states equal to Q; (b) yk = xk ± Xk-L »with L an integer, wherein said series xk has a finite variance Sx, and the series nk represents noise, comprising 40 a modified maximum probability series -8800474 unit of estimate adapted for finding MQ partially decoded strings, up to a certain point in time K, one of said strings for any combination of said finite number Q states and each of a finite number M of values modulo M mutually spaced those integer values are such that each sequence (a) in said code is up to time K; (b) corresponds to said encoder that is in a certain state at said time K; (c) corresponds to a value of at said time-10 dot K which is congruent with one of the given values modulo M. The decoder of claim 31 wherein M is equal to 2. The decoder of claim 31 wherein M equals 4. ********. 8 t C 01.7 4