Modem and method using multidimensional coded modulation.

1.

Field of the Invention

0001The present invention relates to transmitting digital data over a band limited channel by using convolutional coding and multidimensional coded modulation.

2.

Background of the Invention

0002U.S. patent 4,077,021 to Csajka et al. used 2-dimensional coding to improve the performance of the modem against Gaussian noise. Although this coding scheme improved the performance of the system against noise, it also doubled the number of points in the signal constellation making the system more susceptible to other impairments. This is mitigated by the use of a multidimensional code, as described in European patent application EP-A-0 154 415 to Gallager and U.S. patent 4,581,601.

0003In a modem using multidimensional coded modulation, to obtain fractional bits per baud, the signal constellation can be divided into inner points and outer points, with one of the uncoded bits indicating for the n 2-dimensional (symbols) whether there are any outer points and at least one uncoded bit to indicate in which of the n bauds the outer point occurs, as shown in the article entitled "Efficient Modulation for Band Limited Channels", G. David Forney et al., IEEE Journal On Selected Areas in Communication, Vol. SAC, 2 September, 1984, pp. 632-645.

0004European patent application EP-A-0 200 505 teaches, for a multidimensional coded modulation system, a method wherein the most likely sequence of multidimensional signal points to have been sent is determined from the received signal. In this method, the distance is computed between each received constituent signal (with dimensions fewer than that of the multidimensional constellation) and the nearest constituent signal points in the corresponding constituent subsets. The distance between each received multidimensional signal and the nearest multidimensional signal point in each multidimensional subset is determined based on a combination of distances with respect to constituent subsets corresponding to the multidimensional subset. The most likely sequence is determined based on the distances between each received multidimensional signal and the nearest multidimensional signal point in each multidimensional subset.

0005European patent application EP-A-0 200 505, on page 20, second column, first paragraph suggests that an improper number of outer points can be determined from the detected bits and that "Then a modification is made on those four groups to render them together valid". However, no method of accomplishing this "modification" is taught.

0006U.S. Patent 4,597,090 to Forney discloses that in a block coded system two code words which have the same group bit but different subset bits must differ in at least two of their subset bits.

SUMMARY OF THE INVENTION

0007A first aspect of the present invention is directed to a modulation-demodulation apparatus for transmitting a plurality of information bits over a band limited channel, said apparatus including a transmitter having convolutional encoder means, said convolutional encoder means providing one of a plurality of members of a coset during each one of a plurality of group intervals, said coset being one of a plurality of cosets with each of said cosets being associated with a unique permissible transition of said convolutional encoder means from a given present state to one of a plurality of next states, each said group interval having a plurality of bauds, each of said members having a plurality of components with each of said components being associated with a unique one of said bauds, said apparatus further including a receiver having branch cost calculator means for selecting one of said members from each of said cosets for said group interval based upon said selected member having associated therewith a minimum member cost relative to a plurality of member costs of the other said members of said coset, each of said member costs being a sum of a plurality of component costs for said components of one of said members, characterized by: <ul id="ul0001" list-style="dash"><li>said convolutional encoder means being designed to provide at least one group of at least two of said members of one of said cosets with each of said members of said group having a non-common portion and a common portion, each of said non-common portions including at least one non-common component which is different between said at least two members and each of said common portions including at least one common component which is the same between said at least two members;</li><li>said branch cost calculator means having comparing means for comparing at least two non-common portion costs associated with said non-common portions and for selecting a surviving one of said non-common portions with a minimum one of said non-common portion costs, whereby said selection of said surviving non-common portion based upon said minimum non-common portion cost determines which of said at least two members has said minimum member cost.</li></ul>

0008In a second aspect the invention resides in a modulation-demodulation method for transmitting a plurality of information bits over a band limited channel, said method including convolutionally encoding said plurality of information bits using a convolutional encoder so as to provide one of a plurality of members of a coset during each one of a plurality of group intervals, said coset being one of a plurality of cosets with each of said cosets being associated with a unique permissible transition of said convolutional encoder from a given present state to one of a plurality of next states, each said group interval having a plurality of bauds, each of said members having a plurality of components with each of said components being associated with a unique one of said bauds, said method further including at a receiver using a branch cost calculator for selecting one of said members from each of said cosets for said group interval based upon said selected member having associated therewith a minimum member cost relative to a plurality of member costs of the other said members of said coset, each of said member costs being a sum of a plurality of component costs for said components of one of said members, characterized by the steps of: <ul id="ul0002" list-style="dash"><li>providing at said transmitter at least one group of at least two of said members of one of said cosets so that said members of said group each have a non-common portion and a common portion, each of said non-common portions including at least one non-common component which is different between said at least two members and each of said common portions including at least one common component which is the same between said at least two members;</li><li>comparing at said receiver at least two non-common portion costs associated with said non-common portions and for selecting a surviving said non-common portion with a minimum said non-common portion cost, whereby said selection of said surviving non-common portion based upon said minimum non-common portion cost determines which of said at least two members has said minimum member cost.</li></ul>

BRIEF DESCRIPTION OF THE DRAWINGS

0009Further objects and advantages of the present invention will become apparent as the following description proceeds, taken in conjunction with the accompanying drawings in which:

0010FIGURE 1 shows a block diagram of a transmitter of a modem of the present invention and the channel.

0011FIGURE 2A shows the upper two quadrants of the complex plane of the signal constellation used with the present invention.

0012FIGURE 2B shows the lower two quadrants of the complex plane of the signal constellation of the present invention.

0013FIGURE 3 shows a fragmented portion of the upper, right hand quadrant of the signal constellation and illustrates the distance properties in the constellation.

0014FIGURE 4 shows a data packet utilized in the present invention.

0015FIGURE 5 is a schematic diagram of a portion of the convolutional encoder which generates the subset-selecting bits.

0016FIGURE 6 shows a frame of the subset-selecting bits generated by the convolutional encoder.

0017FIGURE 7 shows the structure of the convolutional encoder.

0018FIGURE 8 shows two states of the trellis with the branches shown between the present state zero and the next state zero.

0019FIGURE 9 is a block diagram of the receiver of the present invention.

0020FIGURE 10 is a flow chart of the branch metric calculator of the present invention.

0021FIGURE 11 is a flow chart of the branch metric calculator of the present invention.

0022FIGURE 12 is a flow chart of the branch metric calculator of the present invention.

0023FIGURE 13 is a flow chart of the branch metric calculator of the present invention.

0024FIGURE 14 is a flow chart of the Viterbi decoder of the present invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENT

0025Referring to FIGURE 1, there is shown a block diagram of the generalized structure of a transmitter 10 of a modem 12 in which the present invention is implemented. In the transmitter 10 an information bit stream 14 is made up of serial digital data from a data source (not shown). From the information bit stream 14, bit sequences of fixed length are taken by a serial to parallel converter 16 to form packets of bits which are passed on to a convolutional encoder 18. A portion of the bits of each packet, symbol-selecting bits 20, are passed through the convolutional encoder 18 to a symbol generator 22. The remaining portion 24 of the bits of the packet are passed on to the convolutional encoder 18. The convolutional encoder 18 preferably, but not necessarily, is an encoder which allows for 2n dimensional coded modulation. In the preferred embodiment, for each packet, a portion of the output of the convolutional encoder 18, i.e., subset-selecting bits 26, form a frame of an even number of bits, i.e., the subset-selection bits in a frame consist of 2n bits. In other implementations, the frame can have an odd number of bits. The subset-selecting bits 26 are passed to the symbol generator 22. For each received packet, the subset-selecting bits 26 and symbol-selecting bits 20 form an output of the encoder 18 that will be identified as the expanded bit sequence. The symbol generator 22 typically uses a pair (i.e., bit group) of the 2n subset-selecting bits each baud (i.e., symbol interval/modulation period) to specify one of four subsets into which a symbol constellation is partitioned. The symbol-selecting bits 20 are used by the symbol generator 22 to select a symbol (i.e., signal point in constellation) from each subset. Consequently, each packet from the converter 16 is mapped onto a group of n 2-dimensional symbols in n bauds (one group interval), where n is > 1 and defines a multidimensional point (symbol) 28. The multidimensional symbols 28, in a conventional manner, are filtered by bandwidth limiting filters and used to modulate a carrier in modulator and filter means 30 to provide a modulated carrier signal 32 to a band limited channel 34. The symbol generator 22 and modulator and filtering means 30 define a multidimensional coded modulation means 35.

0026The specific structure of the transmitter 10 will now be described. The modem 12 in which the present invention is preferably, but not necessarily, implemented transmits data at a rate of 19,200 bits per second. The symbol constellation 36 used by the symbol generator 22 of FIGURE 1 is shown in FIGURES 2A and 2B, with upper two quadrants and lower two quadrants of the complex plane of the constellation 36 being shown, respectively, in FIGURES 2A and 2B. For purposes of illustration, the transmitter 10 uses eight dimensional coded modulation. The baud rate is 2742.86 symbols per second with 7 bits per baud. The constellation 36 has 160 complex 2-dimensional symbols, i.e., signal points, with each symbol having a specific alphanumerical identifier. The symbols are divided into outer and inner points, with a circle around each signal point that is an outside point. The symbol constellation 36 also is divided into four subsets A0, A1, B0, B1, which are the first portion of the symbol identifier. After the subset identifier, separately for both inner or outer symbol groups, the symbols within each subset are designated in the symbol identifier by the number after the hyphen. For example, the symbol identifier "A0-5" for a circled point indicates an outer point from the subset A0, with that particular point being designated by number 5 within the outer points of the subset A0. In summary, the constellation is not only partitioned into four subsets of A0, A1, B0, and B1, but the constellation is also divided into a portion having inner points and a portion having outer points. Although the preferred embodiment has four subsets, different numbers and arrangements of subsets are possible.

0027Each subset is represented by a two bit number shown in Table 1. <tables id="tabl0001" num="0001"><img file="EP0341256B1_D0001.tif" /></tables>

0028Thus, the subset-selecting bits 26 of FIGURE 1 form an output of the (m, m-1) convolutional encoder 18 of FIGURE 1 which can be represented by n 2-dimensional subsets of the form shown in Table 1. The integer m is equal to 2n plus the number of symbol selecting bits. MSB and LSB stand for most significant bit and least significant bit, respectively.

0029To better illustrate the subset to symbol mapping provided by the symbol generator 22, a portion of the first quadrant of the symbol constellation 36 shown in FIGURE 2A is enlarged and shown in FIGURE 3. The symbols of the constellation 36 are on a square grid of the complex plane. From FIGURE 3 it can be seen that the maximum Euclidean distance of 2^1/2(d) occurs between A0 and A1 as well as between B0 and B1. The minimum distance of d is between A0 and B0, A0 and B1, A1 and B0 and A1 and B1. The maximum Hamming distance also occurs between A0 and A1, as well as between B0 and B1.

0030FIGURE 4 shows a packet of data generated by the serial to parallel converter 16 of FIGURE 1 which comprises 28 bits which go to the convolutional encoder 18. The bits x₈-x₂₈ are the symbol-selecting bits 20. As shown in FIGURE 5, bits x₁, x₂, x₃, x₄, x₅, x₆, and x₇ form the input bits 24 to convolutional encoder 18 of FIGURE 1. As shown in FIGURES 5 and 6, the output bits y₇, y₆, y₅, y₄, y₃, y₂, y₁, and y₀ of the encoder 18 are grouped into four bit groups (y₇ y₆), (y₅ y₄), (y₃ y₂) and (y₁ y₀) that defines a frame 48 of bits of FIGURE 6. As will be explained hereinafter, bits x₁-x₇ form only a portion of the expanded bit sequence output of the encoder 18 of FIGURE 1. FIGURES 5 and 6 shows the assignment of bit groups for each baud in the frame 48, with each bit group 50 defining a bit pair in the preferred embodiment. Each frame 48 has the duration of a group interval (time interval) that is four bauds in duration. Via Table 1, each bit group 50 of FIGURE 5 corresponds to one of the four 2-dimensional subsets A0, A1, B0 and B1. A symbol is transmitted from one of the 2-dimensional subsets in each of the four bauds. When viewed on a multidimensional level, each frame 48 of FIGURE 6 of eight bit output of the encoder 18 of FIGURE 1 corresponds to one of 256 8-dimensional symbol subsets, with each multidimensional symbol subset corresponding to a different combination of four 2-dimensional symbol subsets. For example, the multidimensional symbol subset A1 B0 A0 B0 (11100010) corresponds to 2-dimensional symbol subsets (11), (10), (00) and (10).

0031From this point on we will use superscripts to indicate the order of the bits inside the frame and subscripts to refer to the frame number in which the bits occurs. For example, referring to frame 48 in FIGURE 6, bit x₇ occurring in the nth frame will be referred to as x<maths id="math0001" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>7</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0002.tif" /></maths>. Table 2 shows the assignment of points from inner and outer points of the constellation 36 shown in FIGURES 2A-2B for different values of x<maths id="math0002" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>8</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0003.tif" /></maths> , x<maths id="math0003" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>9</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0004.tif" /></maths> and x<maths id="math0004" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>10</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0005.tif" /></maths>. Bit x<maths id="math0005" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>8</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0006.tif" /></maths> indicates whether the frame 48 of FIGURE 6 has any outer points or not. If the frame 48 (i.e., one of the bauds of the frame) were to have an outer point, then the dibit x<maths id="math0006" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>9</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0007.tif" /></maths> x<maths id="math0007" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>10</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0008.tif" /></maths> would indicate in which baud (and therefore bit group) the outer point would occur. In Table 2, "Inner" means inner point and "Outer" means outer point. <tables id="tabl0002" num="0002"><img file="EP0341256B1_D0009.tif" /></tables>

0032Table 3 shows how the symbol-selecting bits 20 of FIGURES 1 and 4 are assigned with each group of x<maths id="math0008" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>8</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0010.tif" /></maths>, x<maths id="math0009" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>9</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0011.tif" /></maths>, and x<maths id="math0010" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>10</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0012.tif" /></maths>. <tables id="tabl0003" num="0003"><img file="EP0341256B1_D0013.tif" /></tables>

0033FIGURE 7 shows the convolutional encoder 18 of FIGURES 1 and 5 in detail. The encoder 18 is a 64 state (8, 7) encoder. As previously described, the input bits to the encoder are x<maths id="math0011" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>i</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0014.tif" /></maths>, where i = 1 through 28 and the output bits are y<maths id="math0012" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>i</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0015.tif" /></maths>, where i = 0 through 28. As is well-known, the encoder adds one redundancy bit; however, in the preferred embodiment the encoder adds less than one bit per baud of redundancy into the system. Encoders with redundancies greater than one bit (and therefore redundancy of one or greater per baud) can be used with the present invention. Input bits x<maths id="math0013" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>8</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0016.tif" /></maths> through x<maths id="math0014" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>28</mtext></mrow><mrow><mtext> n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0017.tif" /></maths> are passed through unchanged and unused to become output bits y<maths id="math0015" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>8</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0018.tif" /></maths> through y<maths id="math0016" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>28</mtext></mrow><mrow><mtext> n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0019.tif" /></maths>. Inputs x<maths id="math0017" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>4</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0020.tif" /></maths>-x<maths id="math0018" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>7</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0021.tif" /></maths> are used but pass through the encoder 18 without being changed and become output bits y<maths id="math0019" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>4</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0022.tif" /></maths>-y<maths id="math0020" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>7</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0023.tif" /></maths>. The input bits x<maths id="math0021" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>1</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0024.tif" /></maths>-x<maths id="math0022" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>3</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0025.tif" /></maths> are expanded to create the output bits y<maths id="math0023" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>0</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0026.tif" /></maths>-y<maths id="math0024" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>3</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0027.tif" /></maths>. The output bits of y<maths id="math0025" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>0</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0028.tif" /></maths> through y<maths id="math0026" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>28</mtext></mrow><mrow><mtext>n </mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0029.tif" /></maths> are identified as the expanded bit sequence, which includes the subset-selecting bits y<maths id="math0027" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>0</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0030.tif" /></maths>-y<maths id="math0028" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>7</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0031.tif" /></maths> and the symbol-selecting bits y<maths id="math0029" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>8</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0032.tif" /></maths>-y<maths id="math0030" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>28</mtext></mrow><mrow><mtext> n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0033.tif" /></maths>. The convolutional encoder 18 includes four modulo 2 adders 52, 54, 56, and 58, each connected to receive at least some of the bits x<maths id="math0031" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>3</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0034.tif" /></maths>, x<maths id="math0032" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>2</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0035.tif" /></maths>, and x<maths id="math0033" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>1</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0036.tif" /></maths>. The encoder 18 includes six delay elements 60, 62, 64, 66, 68, and 70, with each element providing a delay of one group interval (four bauds). The input bit x<maths id="math0034" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>3</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0037.tif" /></maths> is fed to the modulo adders 52 and 58 via line 70a, the input bit x<maths id="math0035" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>2</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0038.tif" /></maths> is fed to the modulo adders 54 and 58 via line 70b, and the input bit x<maths id="math0036" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>1</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0039.tif" /></maths> is fed to the modulo adders 56 and 58 via line 70c. With a one group interval delay, the bit x<maths id="math0037" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>3  </mtext></mrow><mrow><mtext>n-1</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0040.tif" /></maths> is fed to the modulo adders 52 and 58 via line 70d, the bit x<maths id="math0038" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>2  </mtext></mrow><mrow><mtext>n-1</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0041.tif" /></maths> is fed to the modulo adders 54 and 58 via line 70e, and the bit x<maths id="math0039" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>1  </mtext></mrow><mrow><mtext>n-1</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0042.tif" /></maths> is fed to the modulo adders 52, 54 and 56 via line 70f. With a two group interval delay, the bit x<maths id="math0040" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>3  </mtext></mrow><mrow><mtext>n-2</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0043.tif" /></maths> is fed to the modulo adders 54 and 58 via line 70g, the bit x<maths id="math0041" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>2  </mtext></mrow><mrow><mtext>n-2</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0044.tif" /></maths> is fed to the modulo adders 52 and 56 via line 70h, and the bit x<maths id="math0042" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>1  </mtext></mrow><mrow><mtext>n-2</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0045.tif" /></maths> is fed to the modulo adders 54 and 56 via line 70i. The bit x<maths id="math0043" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>7</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0046.tif" /></maths> is fed to the modulo adders 52, 54, and 56 via line 70j. The bit x<maths id="math0044" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>6</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0047.tif" /></maths> is fed to the modulo adders 52, 54, and 58 via line 70k. The 4 bit x<maths id="math0045" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>5</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0048.tif" /></maths> is fed to the modulo adders 52, 56, and 58 via line 70l. The bit x<maths id="math0046" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>4</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0049.tif" /></maths> is fed to the modulo adders 54, 56 and 58 via line 70m.

0034With reference to FIGURE 7, the input-output relationships of the subset-selecting bits 26 of the encoder 18 are as follows:<maths id="math0047" num=""><math display="block"><mrow><msubsup><mrow><mtext>y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>7</mtext></mrow></msubsup><msubsup><mrow><mtext> = x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>7</mtext></mrow></msubsup><mspace linebreak="newline" /><msubsup><mrow><mtext> y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>6</mtext></mrow></msubsup><msubsup><mrow><mtext> = x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>6</mtext></mrow></msubsup><mspace linebreak="newline" /><msubsup><mrow><mtext> y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>5</mtext></mrow></msubsup><msubsup><mrow><mtext> = x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>5</mtext></mrow></msubsup><mspace linebreak="newline" /><msubsup><mrow><mtext> y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>4</mtext></mrow></msubsup><msubsup><mrow><mtext> = x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>4</mtext></mrow></msubsup><mspace linebreak="newline" /><msubsup><mrow><mtext> y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>3</mtext></mrow></msubsup><msubsup><mrow><mtext> = x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>3</mtext></mrow></msubsup><msubsup><mrow><mtext> (1 ⊕ D) ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>2</mtext></mrow></msubsup><msubsup><mrow><mtext> (D ⊕ D²) ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>1</mtext></mrow></msubsup><msubsup><mrow><mtext> D ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>7</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>6</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>5</mtext></mrow></msubsup><mspace linebreak="newline" /><msubsup><mrow><mtext> y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>2</mtext></mrow></msubsup><msubsup><mrow><mtext> = x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>3</mtext></mrow></msubsup><msubsup><mrow><mtext> D² ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>2</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>1</mtext></mrow></msubsup><msubsup><mrow><mtext> (D ⊕ D²) ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>7</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>6</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>4</mtext></mrow></msubsup><mspace linebreak="newline" /><msubsup><mrow><mtext> y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>1</mtext></mrow></msubsup><msubsup><mrow><mtext> = x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>2</mtext></mrow></msubsup><msubsup><mrow><mtext> D² ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>1</mtext></mrow></msubsup><msubsup><mrow><mtext> (1 ⊕ D ⊕ D²) ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>7</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>5</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>4</mtext></mrow></msubsup><mspace linebreak="newline" /><msubsup><mrow><mtext> y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>0</mtext></mrow></msubsup><msubsup><mrow><mtext> = x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>3</mtext></mrow></msubsup><msubsup><mrow><mtext> (1 ⊕ D ⊕ D²) ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>2</mtext></mrow></msubsup><msubsup><mrow><mtext> (1 ⊕ D) ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>1</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>6</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>5</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ x</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>4</mtext></mrow></msubsup><mtext>   (Equations 1)</mtext></mrow></math><img file="EP0341256B1_D0050.tif" /></maths> where D is a delay element and ⊕ means an exclusive OR operation.

0035The convolutional encoder 18 of FIGURE 1 is a finite state device which passes through a succession of states, with each state being separated in time by one group interval. Referring to the trellis diagrams in FIGURE 8, the encoder 18 of FIGURE 1 has 64 possible states (not all shown). Due to the fact that the encoder 18 has three binary inputs, for a given present state, only 8 of the next states are permissible, i.e., can be reached from the present state. Which next state occurs is a function of the present state and on the combination of the x<maths id="math0048" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>1</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0051.tif" /></maths>, x<maths id="math0049" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>2</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0052.tif" /></maths>, and x<maths id="math0050" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>3</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0053.tif" /></maths> bits received each group interval. Each permissible transition from one state to another state is identified by a coset of parallel branches. In general, the trellis can be used to diagram all permissible transitions between the present state to the next state of the encoder, although only the parallel branches 74 between present state 0 and next state 0 are shown in FIGURE 8. There are 256 possible branches which exist in the trellis. These 256 branches are subdivided into two groups of 128 branches each. The branches from a present state to a next state can be either from one or the other group. Extending from the present state 0, 128 branches exist in the trellis 72, with 16 branches (one coset) going to each of 8 permissible next states (only one set of 16 is shown). Likewise, 128 branches can extend from each present state in the same manner and this is extended continuously between adjacent states separated by group intervals. In the preferred embodiment, the group used of the two groups of 128 branches alternates with the states, e.g., state 0 using one group and state 1 using the other group and so on.

0036From FIGURE 7 and the Equations 1 for encoder 18 of FIGURE 1, it can be shown that x<maths id="math0051" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>7</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0054.tif" /></maths>, x<maths id="math0052" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>6</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0055.tif" /></maths>, x<maths id="math0053" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>5</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0056.tif" /></maths> and x<maths id="math0054" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>4</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0057.tif" /></maths> do not affect the states. In other words, changing any one or all of these bits does not change the state. Hence, between any two given states occurring at successive group intervals, there will be the previously described sixteen parallel branches, with each of the sixteen branches having a different multidimensional symbol subset associated with it, as shown in FIGURE 8. As previously mentioned, this set of sixteen parallel branches is referred to as a coset. The 256 branches can be decomposed into 16 cosets. Hence, between a present state and each of 8 next permissible states, there will be 16 branches with each branch being labeled by one of the 16 multidimensional symbol subsets of a given coset. With respect to FIGURE 8, in the preferred embodiment, 8 of the cosets are assigned to even number states and 8 are assigned to odd number states. For the purposes of illustration, only those multidimensional symbol subsets of the coset associated with the branches going from present state 0 to next state 0 are shown in FIGURE 8. The multidimensional symbol subsets associated with each branch can be indexed by integers ranging from 0 to 255. The index can be expressed as:<maths id="math0055" num=""><math display="block"><mrow><mtext>Index of 8 dimensional subset = 64(index of subset for baud 4) + 16(index of subset for baud 3) + 4(index of subset for baud 2) + index of subset for baud 1.</mtext></mrow></math><img file="EP0341256B1_D0058.tif" /></maths> As to the multidimensional symbol subsets associated with each branch, the 2-dimensional subsets therein preferably, but not necessarily, are arranged as a frame of subsets for bauds 4, 3, 2, 1, in that order. The encoder 18 transmits 29 bits in 4 bauds or 7.25 bits per baud.

0037A receiver 80 of the modem is generally shown in the block diagram of FIGURE 9 and includes a demodulator 82. The noise affected modulated carrier signal is received over the channel 34 and in a conventional manner that will not be detailed herein is passed through the demodulator 82 (which is normally preceded by a Hilbert filter and passband equalizer) to produce a baseband signal consisting of a stream of received 2-dimensional symbol signals.

0038The decoding process is repeated for each frame of received subset-defining bits (e.g., bits defining a multidimensional symbol subset), which is the received version of the frame transmitted and shown in FIGURE 6. To accomplish this, in the receiver 80 of FIGURE 10, the receiver operation is synchronized to the frames of the incoming signal i.e., the receiver must know the beginning of each frame. In other words, the receiver must determine which bit group in the received frame of subset-defining bits corresponds to the bit group (y₁ y₀) in the transmitted subset-defining bits. Having found this starting bit group, the receiver frame of received subset-defining bits is synchronized with the transmitter frame of subset-defining bits.

0039Referring to FIGURE 9, the operation of slicer 102 will be described. Each received multidimensional signal provides information in the form of the multidimensional subsets of the multidimensional points sent. As previously mentioned, the received 8-dimensional signal can be decomposed into four 2-dimensional signals which occur consecutively. More specifically, each frame of the received multidimensional signal can be represented as R(1), R(2), R(3), and R(4), where:<maths id="math0056" num=""><math display="block"><mrow><mtext>R(J) = [RX(J), RY(J)], 1 &lt; J &lt; 4   (Equation) 2</mtext></mrow></math><img file="EP0341256B1_D0059.tif" /></maths>

0040In a first embodiment of the branch metric computer 100 of the present invention (to be described hereinafter) the inner and outer points are taken into account in the cost calculations. Consequently a symbol (ideal point) in the signal constellation can be represented as:<maths id="math0057" num=""><math display="block"><mrow><mtext>IDL(K, L, N) = [IDLX (K, L, N), IDLY(K, L, N)]   (Equation 3)</mtext></mrow></math><img file="EP0341256B1_D0060.tif" /></maths> where    K = 0 means that the point is inner    K = 1 means that the point is outer    L = 1 means that the point belongs to subset A0    L = 2 means that the point belongs to subset B1    L = 3 means that the point belongs to subset B0    L = 4 means that the point belongs to subset A1 and    N = index of the point in the subset.

0041In the first embodiment, N can take on values from 0 to 31, if the point belongs to the inner part of the constellation (that part comprised of inner points) and N = 0 to 7 for points on the outer part of the constellation (that part comprised of outer points). For each K, L and J, the slicer 102 finds the N = N_min, such that<maths id="math0058" num=""><math display="block"><mrow><msub><mrow><mtext>|R(J) - IDL(K, L, N</mtext></mrow><mrow><mtext>min</mtext></mrow></msub><mtext> )| &lt; |R(J) - IDL(K, L, N)| where</mtext><mtext> |R(J) - IDL(K, L, N)|² = |RX(J) - IDLX(K, L, N)|² + |RY(J) - IDLY(K, L, N)|²   (Equation 4)</mtext></mrow></math><img file="EP0341256B1_D0061.tif" /></maths>

0042The slicer 102 thus finds eight nearest points for the received signal of each baud. Of these eight ideal points, four are inner points and four are outer points. A pair of inner and outer points belong to each of the four subset A0, B0, B1 and A1. For each baud of the four baud frames, we can denote the Euclidean distance (metric) between each of the ideal point and the received point as d[J, K(J), L(J), N _min(J)]. Thus<maths id="math0059" num=""><math display="block"><mrow><msub><mrow><mtext>d [J, K(J), L(J), N </mtext></mrow><mrow><mtext>min</mtext></mrow></msub><msub><mrow><mtext>(J)] = |R(J) - IDL[K(J), L(J), N </mtext></mrow><mrow><mtext>min</mtext></mrow></msub><mtext> (J)]|   (Equation 5)</mtext></mrow></math><img file="EP0341256B1_D0062.tif" /></maths>

0043In a second embodiment of the branch metric calculator 100 of the present invention wherein no distinction is made between inner and outer points, the "K" terms are removed from the above equations, and the slicer 102 thus finds the four nearest points to the received signal for each baud, regardless of whether the nearest point is an inner point or an outer point. Only the costs (metrics) of the four nearest points to the received signal are needed for the branch metric (cost) calculator 100 of the second embodiment and there is no need for keeping track of the points themselves. However, as will be described hereinafter, for the Viterbi decoder 104 of the second embodiment, it is necessary to calculate Equations 4 and 5 for both inner and outer points (K = 0 and 1), but only for the subsets of the estimated multidimensional symbol subset. For each baud there needs to be only one inner and one outer point determined (L has a single value each baud as defined by the estimated multidimensional symbol subset.)

0044A branch metric (cost) is the distance between a received multidimensional signal and possible multidimensional symbol from the multidimensional symbol subset corresponding to the branch of the trellis. A Viterbi decoder (to be described hereinafter) computes a cumulative path metric for each of the survivor paths during each frame. This cumulative path metric (cost) is the arithmetic sum of the metrics of the individual branches which form the particular path.

0045The branch metric calculator 100 selects for each coset (group of 16 parallel branches each) that branch, i.e., multidimensional symbol subset, which has the minimum branch metric, which results in reducing the number of contending branches from 256 to 16. As will be discussed hereinafter, only one branch of each coset can be a candidate to be part of one of the surviving paths of the trellis during the decoding process. Thus, the described reduction by a factor of 16 of the contending branches is based upon which branch of said coset represents the multidimensional symbol subset which is closest to the received multidimensional signal.

0046The present invention concerns the computations performed by the branch cost calculator 100. In a brute force implementation of a branch metric computer, the branch metric calculator 100 would compute the branch metric term, which can be either equal to or proportional to d_BRANCH(I) where d_BRANCH (I) is defined by:<maths id="math0060" num=""><math display="block"><mrow><msub><mrow><mtext>d</mtext></mrow><mrow><mtext>BRANCH</mtext></mrow></msub><msub><mrow><mtext> (I) = [d²(1, K(1), L(1), N</mtext></mrow><mrow><mtext>min(1)</mtext></mrow></msub><msub><mrow><mtext>) + d²(2, K(2), L(2), N</mtext></mrow><mrow><mtext>min(2)</mtext></mrow></msub><msub><mrow><mtext>) + d²(3, K(3), L(3), N</mtext></mrow><mrow><mtext>min(3)</mtext></mrow></msub><msub><mrow><mtext>) + d²(4, K(4), L(4), N</mtext></mrow><mrow><mtext>min(4)</mtext></mrow></msub><mtext>)]   (Equation 6)</mtext></mrow></math><img file="EP0341256B1_D0063.tif" /></maths> As previously explained, each branch of the trellis corresponds to a possible multidimensional symbol subset (therefore a four baud signal in the frame) and in this embodiment there are 256 branches, as identified by multidimensional symbol subsets A0 A0 A0 A0 through A1 A1 A1 A1. For each multidimensional symbol subset, five possible signal combinations of 2-dimensional symbols can be constructed having inner (In) and outer (Out) points (symbols) as shown in Table 4. <tables id="tabl0004" num="0004"><img file="EP0341256B1_D0064.tif" /></tables> In Table 4 we assume that the right most signal of the frame belongs to the first baud and the left most signal is that of the fourth baud. In the case where inner and outer points are to be taken into account, the brute force computation of Equation 6 would be done for each of the five combinations shown in Table 4 and the combination with the least value is chosen as the metric for the branch. After this reduction, we have 256, d² values, one for each of the 256 branches, with each squared distance d² value representing the squared distances between the received 2-dimensional signal and the contending 2-dimensional symbol subsets. As previously mentioned, the contending branches are next reduced from 256 to 16, based upon selecting the lowest cost branch for each coset.

0047The present invention is directed toward making it unnecessary to determine the branch metric for each branch and to directly compare all of the branch metrics (cost) during each group interval. As will be discussed in detail, in looking for commonality in certain components, i.e., 2-dimensional symbol subsets, the brute force approach can be avoided. In order to avoid confusion with similar terms, in the following discussion, the term "component" will be used in place of "2-dimensional symbol subset".

0048As previously described, a branch is associated with a unique multidimensional symbol subset, which is a sequence of components and has n components. Each multidimensional symbol subset of a coset is defined as being a "member" of that coset. A submember of a member consists of part of the components of the member. The submember does not have all of the components of the member and only has components that form some part of the member in the same bauds. If two or more members (or submembers) have the same components for the same bauds, they are said to have common components, if not they are said to have non-common components in those bauds. The present invention makes use of the fact that several members have a common submember and the cost of this common submember (once calculated) can be used for all the members of any coset which has this common submember without having to be calculated repeatedly. With respect to FIGURE 8, it is important to note that the parallel branches 74 between states are merely added to conceptually illustrate that when the convolutional encoder passes from one state to another, as shown by the lines connecting the states, one of a plurality of multidimensional symbol subsets have been generated. Of course, the lines do not physically exist and what in fact exists is only a number representative of the multidimensional symbol subset.

0049The 256 multidimensional symbol subsets (members) associated with 256 branches of the trellis for the encoder of FIGURE 6 are provided in Table 5. <tables id="tabl0005" num="0005"><img file="EP0341256B1_D0065.tif" /></tables><tables id="tabl0006" num="0006"><img file="EP0341256B1_D0066.tif" /></tables><tables id="tabl0007" num="0007"><img file="EP0341256B1_D0067.tif" /></tables><tables id="tabl0008" num="0008"><img file="EP0341256B1_D0068.tif" /></tables>

0050Referring to Table 5, the 256 multidimensional symbol subsets are tabulated into sixteen cosets, with there being 16 members for each coset. The left most component is that of the fourth baud, the next is that of third baud, the next is that of second baud and the right most is that of first baud. The order of the components within a given member are predetermined; however, it should be understood that this order can be reversed, i.e., the first component can be for the first baud and the last component for the fourth baud.

0051According to the present invention, the arrangement of cosets and the members in each coset brings out the commonality of the components. This commonality reduces the number of calculations needed to compute the value of the branch cost as given by Equation 6. Hereafter, it will be seen that use of this commonality reduces computations in two ways. First, it reduces the computations required to eliminate members within a coset. Second, costs calculated for one coset can be used in the other cosets. Hereafter, the preferred arrangement of the first embodiment of the present invention will be described, with the properties generated by such arrangements being also described.

0052In Table 5, the cosets are grouped into four sets. The set of four cosets is identified as a supercoset. Similarly, each coset is divided into four sets, with each set having four members. This set is identified as a subcoset. According to the present invention, the members within a coset are arranged so that the four members of a subcoset have a property that the left most component is common for all. As an example, subcoset 1 belonging to coset 15 is as follows: <tables id="tabl0009" num="0009"><img file="EP0341256B1_D0069.tif" /></tables> For the purposes of identification for each coset, the top most subcoset in each coset is identified as subcoset 1, with each one thereafter being progressively numbered until the bottom one, which is subcoset four. The four cosets which belong to the same supercoset have what is defined as a common three member property. This property can be stated as follows. Any subcoset in a coset can be transformed to one of the subcoset in each of the other cosets belonging to the same supercoset by just changing the common components, i.e., the components within the same baud that are the same for all four members of the subcoset. Although the four components of four members in baud four consist of the same component, for this description and in the claims, they will be referred to in the plural, i.e., components. Likewise, wherever there is a repetition of a common component in the same baud, when referring to the group, the term "components" is used. These common components will be referred to as the common-subcoset-components. As an example, the subcoset <tables id="tabl0010" num="0010"><img file="EP0341256B1_D0070.tif" /></tables> belonging to coset 15 can be transformed to subcoset of coset 12. <tables id="tabl0011" num="0011"><img file="EP0341256B1_D0071.tif" /></tables> In other words, by changing the first column of common subset components from A0′s of subcoset 1 of coset 15 to A1′s, B0′s, or B1′s one obtains, respectively, subcoset 4 of coset 12, subcoset 3 of coset 1, or subcoset 2 of coset 2, all of which are in the same supercoset. This type of relationship holds true for all cosets within all supercosets.

0053In addition to the commonality in one column of four components, for each subcoset there is commonality of two pairs of components in a second column. More particularly, for example, in coset 15, in the column for baud one, there are two common components of A1′s and two common components of A0′s. It is important to note that commonality is determined with reference to comparisons made between components of different members, not with components of the same member. Because each member has its components in a row, the commonality appears in the columns, although this could be reversed.

0054Referring to Table 5, there are sixteen submembers containing the components for baud three and baud two, which can be split into eight sets of two submembers each. In each subcoset, the component of each submember (consisting of two components) are the non-common components. In other words, with respect to each set, the components of each submember are non-common components relative to the components of the other submember when a comparison is made between components of the two submembers of the same baud. These sets are shown in Table 6. <tables id="tabl0012" num="0012"><img file="EP0341256B1_D0072.tif" /></tables><tables id="tabl0013" num="0013"><img file="EP0341256B1_D0073.tif" /></tables> Submember sets (1), (2), (3) and (4) of the first group are common to the supercoset 3 containing cosets 15, 12, 1 and 2, as well as to the supercoset 4 containing cosets 13, 14, 3 and 0. Similarly, submember sets (5), (6), (7), and (8) of Group Two are common to supercoset 1 containing cosets 7, 4, 9, and 10 as well as supercoset 2 containing cosets 5, 6, 11 and 8. The subcosets in cosets 7, 4, 9, and 10 illustrate this point as follows in Table 7. <tables id="tabl0014" num="0014"><img file="EP0341256B1_D0074.tif" /></tables>

0055It can be seen that the submembers <tables id="tabl0015" num="0015"><img file="EP0341256B1_D0075.tif" /></tables> (two of the sets of Table 6) are common to the subcosets shown in Table 7. Additionally, the members of each subcoset are categorized into two pairs, with each pair having one of the sets. With respect only to the members of each pair of members, each submember forming part of the set is defined as the non-common portion of the member and the balance of each member that does not form part of the set is defined as the common portion of the member. In determining the member of each pair forming one of the sets with the minimum metric (cost), it can be determined by comparing only the summed costs of the two different components, thereby eliminating the need to sum the cost of all four components prior to comparing the cumulative metrics. Consequently, within each coset there are eight pairs of members having commonality, when compared with each other, in two common components. When further grouped into subcosets, each subcoset has common components for all members in one baud.

0056As a result of the commonality in two of the columns previously described, there is no commonality in the other two columns, i.e., the two columns for bauds three and two, as dictated by the encoder used in embodiment shown. More specifically, each of the four components A0, A1, B0, and B1 can be found, in differing members in both bauds three and two. The lack of commonality is contained within these two bauds. With this particular encoder, there will be commonality in two bauds, one with all components being in common and the other with two pairs being common. By selecting convolutional encoders having the above described commonality in two columns, with the four submembers of three bauds each being repeated in four cosets, the embodiments of the present invention have maximized the use of such commonality. However, other encoders have different patterns of commonality wherein the present invention would find use, even though such commonality may not be as useful. Moreover, with respect to the "common portion and the "non-common" portion of each branch, for the purposes of defining the invention in the claims, the components of each portion do not have to be adjacent to each other. For example, although the submembers of Table 6 comprise two "non-common components" which are adjacent to each other and two "common components" which are separated, other encoder schemes might lead to the non-common components being separated and so on.

0057As previously discussed in the cost calculations, it is necessary to determine which of the 16 members for each coset has the smallest cost. There is a cost associated with each member and that member having the lowest summed cost (minimum metric) of its components is to be determined. In doing the comparison within a given subcoset, the components for the fourth baud do not affect the elimination of a member. Likewise, when comparing the first pair of members and the second pair of members, the commonality in the first baud eliminates the cost of the components of the first baud from being a factor. For this step of the comparison, only the components of bauds two and three, i.e., the sets defined in Table 6, need to be considered. Moreover, this set reappears in eight cosets. Hence, the cost computations between pairs of submembers of Table 6 need to be done only once, even though these submembers occur in eight different subcosets. By doing the cost calculations for each set, there is a reduction of eight to one.

0058Moreover, as previously described, the components for bauds three, two and one, which define four submembers, are repeated four times. Therefore, the cost computations done for four submembers in one coset can be used for the submembers in three other cosets. Hence, the cost computation is reduced by a factor of four.

0059With reference to the first embodiment, another factor for reduction can be seen by looking at Table 2. As previously discussed, the outer point occurs only in one baud of a multidimensional-symbol-subset. In the first embodiment of the branch cost metric calculator 100 of FIGURE 9 wherein the cost calculations are to be performed for each combination of inner and outer pointers, an algorithm to minimize the number of computations can be based on the following principle which we will denote as principle of algorithm one, which is used in the first embodiment. With algorithm one, as previously described, the slicer 102 of FIGURE 9 computes the cost (metric or distance) of the nearest inner and outer point of each component (subset) to the received signal in each baud. According to the invention, if for any component in any baud the outer point is closer than the inner point, then for all subsequent bauds for computing the submember or member cost containing that component only inner points will be considered. Otherwise, the minimum of the costs of inner and outer point are again considered. In computing the branch cost, the cost of baud three is added to baud two and the resulting cost is added to the cost of baud one and the second resulting cost is added to the cost of baud four. With respect to the first embodiment, referring to the flow chart of FIGURE 10, steps involved in the process of branch computation are as follows: <ul id="ul0003" list-style="none"><li>1. At block 110 slicer 102 of FIGURE 9 computes the distance of the received point from the closest inner and outer points for each component in each baud, as specified in Equations 3, 4 and 5.</li><li>2. After a counter is incremented at block 111, at block 112, for the component for baud two, the determination is made as to whether the cost is smaller for the inner or outer point. The inner or outer point is chosen based upon whether inner cost is smaller or outer cost is smaller.</li><li>3. With the members arranged according to Table 5, the submember cost are formed for submembers of baud three and baud two. There are sixteen submembers which are shown in Table 6:<img file="EP0341256B1_D0076.tif" /> The submember costs are calculated as follows. Referring to blocks 114, 116 and 118, if for any component in baud two, the minimum is for an outer point as determined at the block 114, then baud three is assumed to be an inner point at the block 118 and then for a submember containing that baud two component the cost of the inner point of the corresponding subset for baud three is added at the block 120. Otherwise, it is determined at the block 116 whether the inner or outer point of baud three has the minimum cost and at block 120 this minimum cost is added to the cost of the component at baud two. For example, for submember B0A0, component B0 represents baud three and component A0 represents baud two. If for component A0 the outer point is chosen, then for component B0 an inner point is chosen at the block 118. Otherwise, the minimum of the inner cost and outer cost for component B0 is chosen in baud three at the block 116. This is repeated through a loop including blocks 111 and 122 until these calculations have been done for all 16 of the above described submembers.</li><li>4. The sixteen submembers (two components), as previously described, are separated into 8 sets (pairs) of Table 6, at block 124. The cost for each has already been determined. The costs of each pair are compared and the surviving submember with the lowest cost is selected at block 126. Since these pairs of submembers are in every subcoset, comparing their cost to eliminate the submember of higher cost thus reduces the total number of comparisons to be made.</li><li>5. There will be eight survivor members in the comparison process in the block 126. By individually adding the cost of the four components of baud one to each of these survivors, 32 costs are formed for three-component submembers, each of which will be a sum of three costs, as illustrated in Table 7.<img file="EP0341256B1_D0077.tif" /><img file="EP0341256B1_D0078.tif" /> From Table 7, it is seen that four three-component submembers result when each surviving two-component submember has the components of baud one added thereto, which are all of the components A0, B1, B0 and A0. This addition of baud one components, one at a time, generates half of the three-component submembers that exist in Table 5. Utilizing this relationship, the flow chart of FIGURE 11 demonstrates how the three-component submembers are generated. Again the principle of algorithm one is applied. Counters for the submembers, components, and other variables used later are initialized at block 128 and the submember counter is incremented at block 130. As mandated by loop 131, for each of the two-component submembers for bauds three and two, at block 132 a determination is made as to whether each two-component submember has an outer point in either component. If there is an outer point therein, then the component for baud one (as determined by a component counter) is assumed to be an inner point and one of four three-component members formed by use of the current two-component member is defined at block 136. If there is no outer point, then one of the four three-component members derived from a given two-component member is defined at block 138. Based upon the component values of 1, 2, 3, and 4, the baud one component added is A0, B1, B0 and A1, respectively, at the blocks 136 and 138. At block 140, the costs of the inner and outer points for baud one are compared and the minimum cost component of the two is selected. Once the appropriate cost has been determined at the block 140 or the block 136, then the cost of the three-component submember is calculated at block 143 by adding the two-component member cost to the appropriate cost of the component for baud one. The component counter is incremented at block 144. Via decision blocks 141 and 142, each of the four components are used. Each of the four three-component submembers having as part thereof one of the eight two-component submembers is formed via loop 145. After all eight two-component submembers are processed through the loop 131, the program exits via decision block 146.</li><li>6. The three-component submembers from which the subcosets can be constructed are shown in Table 8.<img file="EP0341256B1_D0079.tif" /> Referring to Table 8, as previously mentioned, the four three-component submembers of bauds three, two, and one found in each coset are repeated four times within a given supercoset. The four three-component submembers will be identified as a submember group. Consequently, within each supercoset, there are only four unique submember groups and among all the supercosets, there are 16 unique submember groups. In Table 8, the columns of submember groups are arranged in columns, going from left to right, from supercosets three, four, one, and two, respectively. Referring to Table 8, all of the 64 three-component submembers for bauds three, two and one are shown. Out of these 64 submembers, 32 have been eliminated in block 126 by taking the commonality of the two-component submembers. Referring to the flow chart of FIGURE 12, for each of the submember groups, the cost of the two surviving three-component submembers are compared, with the one having the minimum cost being selected and retained. There will be sixteen of these in total. Referring to the flow chart, after the submember group counter is incremented at block 148 and the one surviving three-component submember is selected at block 150 for each submember group as dictated by block 152.</li><li>7. Next, the member of each subcoset with the least cost is formed by adding to the cost of the one surviving three-component submember to the cost of the corresponding component of baud four. At this point there is left four members and their associated cost for each coset, i.e., one for each subcoset. Three comparisons are made for each coset to obtain the minimum cost member for each coset with an allowable sequence.</li></ul>

0060Referring to the flow chart of FIGURE 12, again the rule is that, if the prior three-component submember contained an outer point as determined at block 154, then to the submember defined at block 156, the cost of the inner point for baud four is added at block 162. If there is no outer point, and for the member defined at block 156, the smaller cost of the two costs, one for the inner point and one for the outer point, is determined at block 150. As to the four-component members being defined in blocks 156 and 160, this is merely a matter of sequentially associating the baud four component (subset) to the appropriate submember. Each surviving three-component branch has the four components of baud four added thereto to generate the four members in four cosets of the same supercoset, as defined by the loop having blocks 164, 168 and 166. This is repeated for each surviving three-component submember as dictated by the loop having blocks 170 and 153. The cost of baud four is added to the three-component cost at block 162 and this is repeated for all 64 members.

0061At this point, there is left the costs of four members for each coset. At block 174 three comparisons are made for each coset to obtain the minimum cost member. Due to blocks 172, 176, 178, and 180, the comparisons are made three times for each coset, so that a single member and member cost is arrived at for each of the 16 cosets with an allowable sequence. There is a tenfold reduction in the number of comparison and the number of additions is reduced by a factor of 34. While this is a suboptimum decoder, its performance is close to the ideal maximum likelihood decoder for high signal to noise ratio.

0062Contrary to the first embodiment, in the second embodiment of the present invention, the branch metric calculator 100 of FIGURE 9 does not take into account whether a point is an inner point or an outer point. A second algorithm is used which requires a lesser number of operations than algorithm one of the first embodiment and will be referred to as algorithm two. In algorithm two no distinction is made between the cost of inner and outer points. The algorithm will now be described in detail and basically results in a simplified version of the first and has the following steps, with Tables 7 and 8 being again applicable. 1) For each baud, the cost of the closest point in each component (subset) to the received point in slicer 102 as previously described. 2) Next, the cost of the component in baud two is added to the cost of the component in baud three to form sixteen costs for submembers with two-components. 3) Next, eight comparisons of the sixteen submembers are made, which leaves eight submembers. Each comparison involves a pair of submembers having the same common component in baud one. 4) The cost of component of baud one is added to the cost of the surviving submembers. There are 32 additions and 32 new costs will be formed. 5) Sixteen comparisons are made between pairs of three-component submembers of the same subcoset to come up with sixteen three-component member costs. 6) The cost of the appropriate component in baud four is added to the three-component submember to form 64 costs of 64 branches. 7) The four costs in each coset are compared to find the lowest cost of each coset.

0063The number of additions for algorithm two is the same as the number of additions for algorithm one. The number of comparisons for algorithm 2 is less than the number of comparisons for algorithm one by sixteen. However the overall complexity of algorithm two is smaller than that of algorithm one. For example, the slicer 102 of FIGURE 9 for algorithm two has to do half the number of computations as the slicer for algorithm one, since the former does not have to search for inner and outer points separately. In algorithm one, for the case of all inner points the members are needed during each addition. These members are tested to determine if the previous submember contains any outer point or not. Depending upon whether the previous submembers contained any outer point or not the subsequent instructions will have to be branched to the right location. This increases the number of instruction cycles needed to code the algorithm.

0064Generally, the Viterbi decoder 104 for the first embodiment of the present invention in the receiver 80 uses the received multidimensional signals to estimate the original path of the encoder 18 through the trellis, such trellis being discussed with respect to FIGURE 8. The maximum likelihood path is determined by finding the possible sequence of multidimensional symbol subsets (which trellis path) which is closest to the sequence of received multidimensional signals. During every frame, the receiver computes the minimum cost path and from the path history, determines the estimated multidimensional symbol.

0065As is well-known, the Viterbi decoder 104, during each frame (group interval) extends each of the surviving paths to their successor state at the next frame. The metric (cost) of the extended path is obtained by adding the metric of the survivor to the branch metric of the corresponding branch which extends the path. The survivor at the next frame at a given state is found by comparing the metrics of all the extended paths which connect to a given state and choosing the minimum. More specifically, extending the paths' histories can be mathematically characterized by computing:<maths id="math0061" num=""><img file="EP0341256B1_D0080.tif" /></maths> where c(j, n-1) is the surviving metric of the jth state at time t = (n-1)T_frame, r_(i,_j) is the branch metric of the output branch which connects the jth state at time t = (n-1)T_f to the ith state at t = nT_f. c(i, n), the cost at the ith state at time t = nT_f, is obtained by minimizing over all j which are connected to state i. Thus, the path history of the surviving paths is extended. Once the path history is extended the decoder 104 finds the minimum cost path among the survivors, and from the path history outputs the best estimate of the multidimensional symbol subset for the group interval to a convolutional decoder 106. The delay between the present frame and the present estimate (of the past symbol) can be 3v frames, where v is the constraint length of the convolutional encoder 18 and in this case is 6.

0066From the retained knowledge of the received multidimensional signal sequence and from the knowledge of the closest multidimensional symbol subset of the maximum likelihood path, the best estimate of the transmitted multidimensional symbol (point) is obtained. From the estimated, transmitted multidimensional signal the convolutional decoder 106 produces the packet data. Finally a serial to parallel converter 108 converts this packet to serial data.

0067The Viterbi decoder 104 of FIGURE 9 for the second embodiment of the present invention is exactly the same conventional arrangement, except as follows. Once the minimum cost sequence is found and the corresponding member has been estimated at the receiver, it is necessary to determine (with knowledge of the corresponding received sequence, i.e., received multidimensional signal) which sequence of inner and outer points is closest to the received sequence which has just been estimated. As previously described and shown in Table 4, each member can be composed of five possible signal combinations of inner and outer points (2-dimensional symbols). The inner and outer points of the signal constellation are defined in Tables 2 and 3. It has been discovered by the inventors that in the second embodiment the component costs of nearest ideal points can be used in the branch cost calculations (regardless of whether the nearest point is an inner or outer point) without significant loss in performance, if any resultant impermissible sequence of inner and outer signal points is corrected in the Viterbi decoder. After the determination of the best estimate of a member (which forms a part of the minimum cost path), there is known 1) the received multidimensional signal, i.e., multidimensional point (symbol) received, and 2) the best estimate of the member, i.e., a multidimensional symbol subset. Instead of merely selecting the ideal multidimensional symbol (four 2-dimensional symbols) from the multidimensional symbol subset that is closest to the received multidimensional signal, the Viterbi decoder 104 calculates d_BRANCH(I) of Equation 6 five times, one for each of the five combinations of Table 4. In Equation 6, the "L" term is defined by the known member and the "K" term is defined by the particular sequence of inner and outer points. Consequently, there are five summed costs associated with the five combinations, and the combination with the minimum (smallest) summed cost is selected to be the multidimensional symbol provided to the convolutional decoder 106.

0068The slicer output of the cost of the closest (in Euclidean distance) 2-dimensional symbol for each 2-dimensional symbol subset for each baud is used by the branch cost calculator. However, for the Viterbi decoder 104 of FIGURE 9, the slicer must provide for each best estimated member (multidimensional-symbol-subset) the cost of the closest inner point and outer point, but only for those 2-dimensional symbol subsets of the best estimated member. Referring to the flow chart of FIGURE 16, this is done at block 182 for each best estimated member determined by the Viterbi decoder. Consequently, there are two 2-dimensional costs for each baud for a given best estimated member. At block 184, five multidimensional symbols are selected from the best estimated multidimensional-symbol-subset (member) that have the sequences of inner and outer 2-dimensional symbols (components) shown in Table 4. At block 186, the 2-dimensional cost provided above are summed to define a member (multidimensional cost), i.e. d_BRANCH(I) of Equation 6, for each of the five multidimensional symbols. At block 188, the multidimensional symbol with the minimum multidimensional cost (metric) is selected by the Viterbi decoder to be outputted to the convolutional decoder 106 and is defined as being the best estimated multidimensional symbol selected from the best estimated multidimensional-symbol-subset based upon the knowledge of the received multidimensional signal and the requirement of meeting one of the permissible sequences of Table 4.

0069For the purpose of this disclosure including the claims "multidimensions" shall mean more than 2-dimensions.

0070The modem in which the present invention is implemented is a microprocessor based modem. As will be appreciated by those in the art, the microprocessor-based modem control and data processing circuits also typically include the usual data storage elements (e.g., ROM for program control storage and the like, and RAM for variable input/output/intermediate result data, etc.) conventionally associated with a microprocessor CPU for performing desired manipulation of digital signals in accordance with a stored program. In the presently preferred exemplary embodiment, these already present microprocessor CPU, ROM and RAM elements are also utilized to perform the functions of the present invention. With the present invention, the modem functions are implemented in Texas Instrument's TMS 32020 processors. With respect to FIGURE 1, all of the transmitter modem functions therein are preferable, but not necessarily, performed by the digital signal processor. As is well-known, after filtering in block 30, a D/A converter and antialasing filters are used. With respect to FIGURE 9, all of the receiver modem functions of FIGURE 9 are preferably, but not necessarily, performed by the digital signal processor. As is well-known, prior to demodulation, the received signal is processed by an A/D converter and filters.

0071Referring back to the discussion of the convolutional encoder of FIGURE 7, the dependence of y<maths id="math0062" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>3</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0081.tif" /></maths>, y<maths id="math0063" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>2</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0082.tif" /></maths>, y<maths id="math0064" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>1</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0083.tif" /></maths> and y<maths id="math0065" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>0</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0084.tif" /></maths> on x<maths id="math0066" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>7</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0085.tif" /></maths>, x<maths id="math0067" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>6</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0086.tif" /></maths>, x<maths id="math0068" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>5</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0087.tif" /></maths> and x<maths id="math0069" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>4</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0088.tif" /></maths> in Equation 1 can be expressed by the vector:<maths id="math0070" num=""><img file="EP0341256B1_D0089.tif" /></maths> where the first row corresponds to y<maths id="math0071" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>3</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0090.tif" /></maths>, the second row to y<maths id="math0072" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>2</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0091.tif" /></maths>, the third row to y<maths id="math0073" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>1</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0092.tif" /></maths> and the fourth row to y<maths id="math0074" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>0</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0093.tif" /></maths>. By interchanging the rows, a total of 24 different vectors can be obtained. Since for each arrangement of the vector there is a convolutional encoder, it then follows that there are 24 different convolutional encoders which can be realized and for each of the encoders a receiver with reduced number of computations - using the principles of this invention - can be designed.

0072With respect to FIGURE 7, as merely a matter of convenience the unchanged bits y<maths id="math0075" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>4</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0094.tif" /></maths> through y<maths id="math0076" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>28</mtext></mrow><mrow><mtext>n </mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0095.tif" /></maths> are shown as being part of the inputs and outputs of the convolutional encoder, although they pass through the convolutional encoder unchanged and of that group only bits y<maths id="math0077" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>4</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0096.tif" /></maths> through y<maths id="math0078" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>7</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0097.tif" /></maths> are used by the encoder. However, with respect to the claims, it is clear that the unchanged bits do not have to be considered as part of the convolutional encoder. The convolutional encoder outputs (y<maths id="math0079" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>7</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0098.tif" /></maths> y<maths id="math0080" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>6</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0099.tif" /></maths> y<maths id="math0081" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>5</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0100.tif" /></maths> y<maths id="math0082" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>4</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0101.tif" /></maths> y<maths id="math0083" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>3</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0102.tif" /></maths> y<maths id="math0084" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>2</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0103.tif" /></maths> y<maths id="math0085" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>1</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0104.tif" /></maths> y<maths id="math0086" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>0</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0105.tif" /></maths>) satisfy the parity check equation:<maths id="math0087" num=""><math display="block"><mrow><munder accentunder="true"><mrow><mtext>y</mtext></mrow><mo>̲</mo></munder><mtext></mtext><munder accentunder="true"><mrow><mtext>H</mtext></mrow><mo>̲</mo></munder><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>T</mtext></mrow></msup><mtext> = </mtext><munder accentunder="true"><mrow><mtext>0</mtext></mrow><mo>̲</mo></munder><mtext>   (Equation 9)</mtext></mrow></math><img file="EP0341256B1_D0106.tif" /></maths> where <u style="single">y</u> = (y<maths id="math0088" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>7</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0107.tif" /></maths> y<maths id="math0089" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>6</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0108.tif" /></maths> y<maths id="math0090" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>5</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0109.tif" /></maths> y<maths id="math0091" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>4</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0110.tif" /></maths> y<maths id="math0092" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>3</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0111.tif" /></maths> y<maths id="math0093" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>2</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0112.tif" /></maths> y<maths id="math0094" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>1</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0113.tif" /></maths> y<maths id="math0095" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>0</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0114.tif" /></maths>) <u style="single">H</u> = (H⁸(D) H⁷(D) H⁶(D) H⁵(D) H⁴(D) H³(D) H²(D) H¹(D) H⁰(D) where D is the delay operator and T means transpose. Here Hⁱ(D) = a_v D^v ⊕ a_v-1 D^v-1 ⊕ .... a₀ where i = 0 .... 7 a₁ = 0 or 1 and 1 = 0 ... v where v is the constraint length of the encoder. In the present embodiment, the constraint length of the encoder is six, since there are six shift registers. The concept of the parity check equation and constraint length are well-known in the literature of convolutional codes as shown by U.S. patent 4,601,044 to Kromer et al. For the present encoder the parity check equation is given by<maths id="math0096" num=""><math display="block"><mrow><msubsup><mrow><mtext>P = y</mtext></mrow><mrow><mtext>n-6</mtext></mrow><mrow><mtext>7  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-2</mtext></mrow><mrow><mtext>7  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-1</mtext></mrow><mrow><mtext>7  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>7</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-6</mtext></mrow><mrow><mtext>6  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-5</mtext></mrow><mrow><mtext>6  </mtext></mrow></msubsup><mspace linebreak="newline" /><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-4</mtext></mrow><mrow><mtext>6  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>6</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-6</mtext></mrow><mrow><mtext>5  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-4</mtext></mrow><mrow><mtext>5  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-3</mtext></mrow><mrow><mtext>5  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-2</mtext></mrow><mrow><mtext>5  </mtext></mrow></msubsup><mspace linebreak="newline" /><msubsup><mrow><mtext> ⊕y</mtext></mrow><mrow><mtext>n-1</mtext></mrow><mrow><mtext>5  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>5</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-6</mtext></mrow><mrow><mtext>4  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-5</mtext></mrow><mrow><mtext>4  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-2</mtext></mrow><mrow><mtext>4  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-1</mtext></mrow><mrow><mtext>4  </mtext></mrow></msubsup><mspace linebreak="newline" /><msubsup><mrow><mtext> ⊕y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>4</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-6</mtext></mrow><mrow><mtext>3  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-5</mtext></mrow><mrow><mtext>3</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-3</mtext></mrow><mrow><mtext>3  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>3</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-6</mtext></mrow><mrow><mtext>2  </mtext></mrow></msubsup><mspace linebreak="newline" /><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-4</mtext></mrow><mrow><mtext>2  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>2</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-6</mtext></mrow><mrow><mtext>1  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-5</mtext></mrow><mrow><mtext>1  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-4</mtext></mrow><mrow><mtext>1  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-3</mtext></mrow><mrow><mtext>1  </mtext></mrow></msubsup><mspace linebreak="newline" /><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-2</mtext></mrow><mrow><mtext>1  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-1</mtext></mrow><mrow><mtext>1  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>1</mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-6</mtext></mrow><mrow><mtext>0  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n-3</mtext></mrow><mrow><mtext>0  </mtext></mrow></msubsup><msubsup><mrow><mtext> ⊕ y</mtext></mrow><mrow><mtext>n</mtext></mrow><mrow><mtext>0</mtext></mrow></msubsup><mtext>= 0   (Equation 10)</mtext></mrow></math><img file="EP0341256B1_D0115.tif" /></maths>

0073The total number of terms in the parity check equation is an even number. As a result, if each of the output of the encoder is inverted, i.e., y<maths id="math0097" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>i</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0116.tif" /></maths> goes to<img file="EP0341256B1_D0117.tif" /> then P is still zero, because of the even number of terms in the parity check equation. Whenever there is a phase hit of 180 degrees, the received points are 180 degrees from the transmitted points. Under this condition, the received 2-dimensional subsets are changed as follows:<maths id="math0098" num=""><img file="EP0341256B1_D0118.tif" /></maths> As we pointed out earlier, the parity check equation P remains zero even though y<maths id="math0099" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>i</mtext></mrow><mrow><mtext>n</mtext></mrow></mfrac></mrow></math><img file="EP0341256B1_D0119.tif" /></maths> goes to<img file="EP0341256B1_D0120.tif" />. In other words, for all the various encoders, even with 180 degrees phase shift, the output sequence satisfies the parity check equation.