USRE33041E - Multi-dimensional coding for error reduction - Google Patents
Multi-dimensional coding for error reduction Download PDFInfo
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- USRE33041E USRE33041E US07/085,689 US8568987A USRE33041E US RE33041 E USRE33041 E US RE33041E US 8568987 A US8568987 A US 8568987A US RE33041 E USRE33041 E US RE33041E
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L1/00—Arrangements for detecting or preventing errors in the information received
- H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
- H04L1/0056—Systems characterized by the type of code used
- H04L1/0059—Convolutional codes
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
- H03M13/25—Error detection or forward error correction by signal space coding, i.e. adding redundancy in the signal constellation, e.g. Trellis Coded Modulation [TCM]
Definitions
- This invention relates to coding information so as to reduce errors caused by transmission from being included in the received signal and, in particular, to multidimensional coding.
- the information signals are often distorted by noise and other causes. Sometimes the information signals are distorted to such an extent that the received signals do not duplicate the information sent.
- the information is coded at the transmitter.
- block coding introduces n-k redundant bits to a block of k bits of information to derive a coded block of n bits which is transmitted to a receiver.
- code words there are 2 k code words, each of length n, and the set is called (n, k) block code.
- Convolutional coding introduces m bits from previous blocks of information, each having k bits, to derive a coded signal having n bits.
- the encoder is said to have a memory order of m.
- the set of codes is called an (n, k, m) convolutional code.
- the code rate is k/n.
- U.S. Pat. No. 4,077,021 teaches a technique called set partitioning that assigns signal points to successive blocks of input data. More particularly, a code rate of 4 bits/2-dimensional symbol is shown. Also, a coding gain of 4 db over standard uncoded transmission is obtained. That is, noise immunity is obtained without increasing the power required for transmission. As will be described fully in the detailed description of the present invention, it is desirable to obtain a more efficient coding scheme.
- a multidimensional coder which (1) reduces the power consumed, (2) achieves a high code rate, and (3) provides an efficient scheme by low power use and low error in the received signal.
- an information block comprising eight bits of input to the coder is converted into any one of five hundred and twelve four-dimensional code words.
- the eight bits are change into .[.an output having.]. nine bits by .[.retaining.]. .Iadd.a convolutional coder which retains .Iaddend.two input bits from the immediately preceding block and one input bit from .[.two preceding blocks.]. .Iadd.the block before that one.Iaddend..
- the .[.output is.]. .Iadd.nine bits are .Iaddend.divided into two signal groups: a first group comprises five bits and a second comprises four bits.
- the numbers appearing in the thirty-two code words include 1, -1, 3, -3, 5, and -5.
- the second group of four bits make up a vector which determines the sign of the four-dimensional code word read from the read only memory.
- a trellis code with rate of eight bits per four-dimensional symbol with a gain of 4.7 decibels over standard uncoded transmission corresponds to improving the block error rate by a factor of 10.
- the signal constellation comprises five hundred twelve four-dimensional signal points, that is, each symbol has a set of four numbers which defines a signal point in four-dimensional space.
- a trellis code with rate twelve bits per four-dimensional symbol provides a gain of 4.9 db over standard uncoded transmission. This method is suggested for data transmission rates of 14.4 kbits/sec.
- FIG. 1 is a prior art system for transmission of encoded signals
- FIG. 2 illustrates the concept of convolutional encoders
- FIG. 3 shows possible transitions of states between two instants in time
- FIG. 4 illustrates error events and the term Euclidean distance
- FIG. 5 shows a convolutional encoder of memory order three
- FIG. 6 shows a signal constellation for uncoded transmission at the rate four bits per two dimensional symbol
- FIG. 7 shows a convolutional encoder embodying the present invention
- FIG. 8 shows a rectangular constellation for uncoded transmission at six bits per two dimensional symbol
- FIG. 9 shows a signal constellation for trellis codes at the rate four bits per two dimensional symbol.
- FIG. 1 there is shown a prior art system comprising a digital source 10 supplying a signal to encoder 12.
- the coded signal from encoder 12 is modulated by device 14 for transmission to a distance location where the signal is demodulated at device 18.
- the demodulated signal is thereafter decoded at device 20 and sent on to a digital sink 22.
- the use of encoder 12 reduces the errors caused by noise in the transmission facility 16.
- the present invention relates to an improvement in the aforesaid encoder 12.
- the constraint length v is given by ##EQU1##
- the output of x i of the encoder is a fixed function x of the (v+k) variables a 1 i , . . . , a 1 i-v1 ; a 2 i , . . . , a 2 i-v2 ; a 3 i , . . . , a 3 i-v3 ; . . . ; a k i , . . . , a k i-vk . That is,
- the v-tuple (a 1 1 a 1 i-1 . . . a 1 i-v1 ; . . . ; a k i a k 1-l . . . a k i-v1 ) is the state of the encoder and there are 2 k states.
- the three bits shown at time i are the bits stored for a 2 at time (i-1), that is, a 2 i-1 ; for a 3 at time (i-1), that is, a 3 i-1 ; and a 3 at time (i-2), that is, a 3 i- 2.
- An edge joins two states at time i and i+1. Each edge is double in this case, as will be explained below.
- the state transition diagram can be understood by examining, for example, the transition from the state ⁇ 010 ⁇ at time i to a state at time i+1.
- the 1 in state ⁇ 010 ⁇ is a 3 i-1
- this bit 1 will now be a 3 i-2 and can be any one of the four codes which end in a 1:001, 101, 011, or 111 depending on the next value of a 2 i-i and a 3 i-1 .
- noise in the transmission channes distorts the sequence of signals from the output of the coder:
- r ij x ij +z ij , the z ij being the noise component.
- the noise samples z ij are independent zero-mean Gaussion variables of variance, ⁇ 2 .
- the Viterbi algorithm disclosed at 61 Proceedings of the IEEE 268-278 (No. 3, March, 1973), is used to find the most likely path through the trellis given the observed sequence ⁇ r i ⁇ .
- the path chosen will not always coincide with the correct path, but will occasionally diverge from it and remerge at a later time. This is called an error event.
- An error event E of the length l lasts from j to (j+1), the sequence x j , . . . , x.sup.(j+l-L) instead of the correct sequence x j , . . . , x.sup.(j+1-L).
- the squared Euclidean distance d 2 that is d 2 (E), between the two paths of E is given by ##EQU2## where ⁇ ⁇ denotes the usual Euclidean norm.
- the bits a 2 i-1 , a 3 i-1 , and a 3 i-2 describe the state of the encoder.
- the possible transitions between states of such an encoder is shown hereinabove in FIG. 3.
- the binary notation 0,1 is then changed to 1, -1, respectively.
- the minimum squared distance of this trellis code is simply four times the free distance of the original binary convolutional code, namely 16. This is so because 0 opposite 1 contributes 1 to the free distance while 1 opposite -1 contributes 4 to the squared minimum distance.
- the average power or energy is ##EQU7## that is, 20. Because the minimum squared distance between distinct signals is four, the figure of merit is given by the formula: ##EQU8##
- the first column of Table 1 shows the code words representative of signal points in four dimensional space.
- the number of permutations for each representative code word is shown in column three, giving a total of thirty-two code words stored in ROM 74.
- the list of thirty-two code words is obtained by permuting the coordinates of the code words in column one.
- the second set of leads 51 from serial to parallel converter 70 carries three of the binary input bits to a device 50 which stores three bits from prior blocks as disclosed in detail earlier herein with reference to FIG. 5, and delivers four bits on leads 53 to a device 80, At device 80 the binary bit 0 is converted to 1 and a binary 1 is converted to a -1.
- the vector W of four bits w 1 , w 2 , w 3 , and w 4 is then delivered to multipliers 82 . . . 88.
- the number y 1 of the accessed code word is then multiplied by w 1 from lead 81 at multiplier 82 to generate output x 1 . That is, y 1 is negated if w 1 is -1 but remains unchanged if w 1 is +1.
- x 2 , x 3 , and x 4 are generated and sent over leads 89.
- the thirty-two code words correspond to the entries in ROM 74. The entire list of thirty-two code words in ROM 74 is shown in Table 2 hereinbelow.
- the set of coordinates (x 1 , x 2 , x 3 , x 4 ) representing the input block of 8 bits on leads 89 is then sent on to a modulator.
- the present invention relates to an encoder only and therefore other equipment cooperating with the encoder is not disclosed in any detail, beyond what was disclosed earlier with reference to FIG. 1.
- the distance d(A, B) between two sets of vectors A and B is given by the expression: ##EQU9##
- the partition into sets S(w) satisfies the following metric properties: (M1): if x, y ⁇ S(w), then ⁇ x-y ⁇ 2 ⁇ 16; and
- five uncoded bits were added, obtaining an input of eight parallel sequences to the encoder in FIG. 7: ⁇ a 1 i ⁇ , ⁇ a 2 i ⁇ , ⁇ a 3 i ⁇ , . . . , ⁇ a 8 i ⁇ .
- the sequences ⁇ a 2 i ⁇ , ⁇ a 3 i ⁇ determine the state a 2 i-1 a 3 i-1 a 3 i-2 of the encoder in FIG. 7.
- FIG. 8 there is shown a signal constellation comprising sixty-four signal points which is used for standard uncoded transmission at the rate of 6 bits/2-dimensional symbol.
- a constellation comprising two such copies are needed.
- the average signal power P is ##EQU15## Thus, ##EQU16##
- Coded transmission requires 2.sup.(k+1) signal points.
- the points of the lattice (2Z+1) 4 where (2Z+1) represents the set of odd numbers, lie in shells around the origin consisting of sixteen vectors of energy four, sixty-four vectors of energy twelve, . . . , shown summarized in Table 3.
- the 2 k+1 signal points are obtained by taking all points of energy four, twelve, twenty, . . . and just enough points of a final shell to bring the total number up to 2 k+1 .
- the signal constellation is partitioned into sixteen sets S(w) according to congruence of the entries modulo four Each set contains 2 k-3 signal points.
- Edges in the eight state trellis originally labeled ⁇ w are now labelled with the 2 k-2 vectors in S(w) U S(-w).
- the metric properties (M1) and (M2) guarantee that the minimum squared distance of this trellis is sixteen.
- the scheme disclosed by the present invention is an efficient method of coding.
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Abstract
Description
x.sup.i =x(a.sub.1.sup.i a.sub.1.sup.i-1 . . . a.sub.1.sup.i-v1 ; . . . ; a.sub.k.sup.i a.sub.k.sup.i-1 . . . a.sub.k.sup.i-v1).
x.sup.i =(x.sup.i1, x.sup.i2, x.sup.i3, x.sup.i4).
r.sup.i =(r.sup.i1, r.sup.i2, r.sup.i3, r.sup.i4)
v.sub.1.sup.i =a.sub.1.sup.i,
v.sub.2.sup.i =a.sub.1.sup.i +a.sub.2.sup.i +a.sub.2.sup.i-1 +a.sub.3.sup.i-1,
v.sub.3.sup.i =a.sub.1.sup.i +a.sub.2.sup.i-1 +a.sub.3.sup.i +a.sub.3.sup.i-2, and
v.sub.4.sup.i =a.sub.1.sup.i +a.sub.2.sup.i +a.sub.3.sup.i +a.sub.3.sup.i-2.
v.sub.1.sup.i =a.sub.1.sup.i,
v.sub.2.sup.i =a.sub.1.sup.i +a.sub.2.sup.i-1 +a.sub.2.sup.i +a.sub.3.sup.i-1,
v.sub.3 =a.sub.1.sup.i +a.sub.2.sup.i-1 +a.sub.3.sup.i +a.sub.3.sup.i-2, and
v.sub.4 =a.sub.1.sup.i +a.sub.2.sup.i +a.sub.3.sup.i +a.sub.3.sup.i-2.
TABLE 1 ______________________________________ Representative Signal Point Energy Level or Average Power ##STR1## ______________________________________ (1111) 4 1 (3111) 12 4 (3311) 20 6 (5111) 28 4 (3331) 28 4 (5311) 36 12 (3333) 36 1 Total number of code 32 words ______________________________________
TABLE 2 ______________________________________ Input to ROM 74 Code Output From ROM 74 a.sub.4 a.sub.5 a.sub.6 a.sub.7 a.sub.8 y.sub.1 y.sub.2 y.sub.3 y.sub.4 ______________________________________ 1 1 1 1 1 1 1 1 -1 1 1 1 1 3 1 1 1 1 -1 1 1 1 1 3 1 1 -1 -1 1 1 1 1 1 3 1 1 1 -1 1 1 1 1 1 3 -1 1 -1 1 1 3 3 1 1 1 -1 -1 1 1 3 1 3 1 -1 -1 -1 1 1 3 1 1 3 1 1 1 -1 1 1 3 3 1 -1 1 1 -1 1 1 3 1 3 1 -1 1 -1 1 1 1 3 3 -1 -1 1 -1 1 5 1 1 1 1 1 -1 -1 1 1 5 1 1 -1 1 -1 -1 1 1 1 5 1 1 -1 -1 -1 1 1 1 1 5 -1 -1 -1 -1 1 3 3 3 1 1 1 1 1 -1 3 3 1 3 -1 1 1 1 -1 3 1 3 3 1 -1 1 1 -1 1 3 3 3 -1 -1 1 1 -1 5 3 1 1 1 1 -1 1 -1 5 1 3 1 -1 1 -1 1 -1 5 1 1 3 1 -1 -1 1 -1 3 5 1 1 -1 -1 -1 1 -1 1 5 3 1 1 1 1 -1 -1 1 5 1 3 -1 1 1 -1 -1 3 1 5 1 1 -1 1 -1 -1 1 3 5 1 -1 -1 1 -1 -1 1 1 5 3 1 1 -1 -1 -1 3 1 1 5 -1 1 -1 -1 -1 1 3 1 5 1 -1 -1 -1 -1 1 1 3 5 -1 -1 -1 -1 -1 3 3 3 3 ______________________________________
TABLE 3 ______________________________________ Representative Energy ##STR2## ______________________________________ (1111) 4 1 (3111) 12 4 (3311) 20 6 (5111). (3331) 28 4 + 4 = 8 (5311). (3333) 36 12 + 1 = 13 (5331) 44 12 (7111). (5333). (5511) 52 4 + 4 + 6 = 14 (7311). (5531) 60 12 + 12 = 24 (7331). (5533) 68 12 + 6 = 18 (7333). (5551). (7511) 76 4 + 4 + 4 + 12 = 20 (9111). (7531). (5553) 84 4 + 24 + 4 = 32 (9311). (7533) 92 12 + 12 = 24 (9331). (7711). (7551). 100 12 + 6 + 12 + 1 = 31 (5555) (7553). (9333). (9511). 108 12 + 4 + 12 + 12 = 40 (7731) (9531). (7733) 116 24 + 6 = 30 (9533). (11111). (7751). 124 12 + 4 + 12 + 4 = 32 (7555) (11311). (9711). (9551). 132 12 + 12 + 12 + 12 = 48 (7753) (11331). (9731). (9553) 140 12 + 24 + 12 = 48 (11333). (11511). (9733). 148 4 + 12 + 12 + 6 + 4 = 38 (7755). (7771) (11531). (9555). (9751) 156 24 + 4 + 24 + 4 = 56 (7773) (11533). (9911). (9753) 164 only 13 ______________________________________
Claims (23)
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Cited By (3)
Publication number | Priority date | Publication date | Assignee | Title |
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US5058135A (en) * | 1988-07-22 | 1991-10-15 | Telecommunications Radioelectriques Et Telephoniques T.R.T. | Modulating arrangement for phase and amplitude modulation of a wave |
US5515400A (en) * | 1989-02-17 | 1996-05-07 | Fujitsu Limited | Method for arranging signal points in a quadrature amplitude modulation/demodulation system |
WO2001065735A1 (en) * | 2000-02-28 | 2001-09-07 | University Of Maryland Baltimore County | Error mitigation system using line coding for optical wdm communications |
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Cited By (3)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US5058135A (en) * | 1988-07-22 | 1991-10-15 | Telecommunications Radioelectriques Et Telephoniques T.R.T. | Modulating arrangement for phase and amplitude modulation of a wave |
US5515400A (en) * | 1989-02-17 | 1996-05-07 | Fujitsu Limited | Method for arranging signal points in a quadrature amplitude modulation/demodulation system |
WO2001065735A1 (en) * | 2000-02-28 | 2001-09-07 | University Of Maryland Baltimore County | Error mitigation system using line coding for optical wdm communications |
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