VITERBI DECODER WITH REDUCED NUMBER OF DATA MOVE OPERATIONS

VITERBI DECODER WITH REDUCED NUMBER OF DATA MOVE OPERATIONS FIELD OF THE INVENTION The present invention relates to Viterbi decoders for determining a maximum likelihood path through a trellis created by a convolutional code.

BRIEF DESCRIPTION OF THE DRAWINGS Objects and advantages of the present invention will- become apparent as the following description proceeds, taken in conjunction with the accompanying drawings in which:

FIGURE 1A is a diagram for a trellis for a convolutionally coded signal, showing all of the branches extending from the states at time unit j = m to j = m + 1, prior to the selection of the surviving path.

FIGURE IB shows the trellis after the surviving paths have been selected.

FIGURE 1C shows the surviving path extended from the time unit j = m + ltoj = m + 2.

FIGURE 2A shows the prior art scheme used for storing a plurality of surviving paths.

s .

FIGURE 2B shows the paths shifted in the registers in a manner indicated by the arrows in FIGURE 2A.

FIGURE 3 is a simplified block diagram of the environment in which the present invention is implemented.

FIGURE 4A shows a four state convolutional encoder which is used in combination with a first embodiment of the present invention.

FIGURE 4B illustrates a trellis for the four state convolutional encoder of FIGURE 4A.

FIGURE 5 shows a look-up table wherein the new state in conjunction with the old state allows for the access of subset-defining bit pairs.

FIGURE 6 is a look-up table of a first embodiment of the present invention implemented in computer memory wherein the surviving paths are stored.

FIGURE 7 shows a block diagram of a transmitter of a modem for another implementation of the present invention.

FIGURE 8A (sheet 5) shows the upper two quadrants of the complex plane of the signal constellation used with the present invention.

FIGURE 8B (sheet 6) shows the lower two quadrants of the complex plane of the signal constellation of the present invention.

FIGURE 9 shows a fragmented portion of the upper, right hand quadrant of the signal constellation and illustrates the distance properties in the constellation.

FIGURE 10 (sheet 7) shows a data packet utilized in the present invention.

FIGURE 11 is a schematic diagram of a portion of the convolutional encoder which generates the subset-selecting bits.

FIGURE 12 shows a frame of the subset-selecting bits generated by the convolutional encoder.

FIGURE 13 shows the structure of the convolutional encoder.

FIGURE 14 shows two states of the trellis with the branches shown between the present state zero and the next state zero.

FIGURE 15 is a block diagram of the receiver of the present invention.

FIGURE 16A shows the memory arrangement of a second embodiment of the present invention.

FIGURE 16B shows the memory arrangement one group interval after that of FIGURE 16A.

FIGURE 16C shows the memory arrangement on group interval after that of FIGURE 16B.

FIGURE 17 is a flow chart for the write operation for the memory shown in FIGURES 16A-C.

2 0 FIGURE 18 is a flow chart of the retrieval operation for the memory shown in FIGURES 16A-C.

FIGURE 19 shows the modulo characteristic of the write and retrieve operations.

DESCR1PTION OF THE PRIOR ART A Viterbi decoder is used to decode convolutional codes, such codes being used to reduce error rates and increase the signal-to-noise ratio in a transmission system over band limited transmission channels.

The Viterbi decoder is described in detail in an article entitled "The Viterbi Algorithm" in the Proceedings of the IEEE, Vol. 61, No. 3, March, 1973.

Depending on the encoder design, 2-diinensional convolutional encoding is used as shown in O.S. Patent 4,601,044 to Kromer, in. et al or multidimensional coding (where the numbers of dimensions is greater than 2) is used as shown in European patent application 85300803.5, publication No.

0,154,415 to Gallager, published August 9, 1985. The Viterbi decoder in the receiver uses the received 2-diinensional or multidimensional signals to estimate the original path the encoder through the trellis. The maximum likelihood path is determined by finding the possible sequence of 2-dimensional symbol subsets (which trellis path) which is closest to the sequence of received signals. The receiver computes the minimum cost path and from the path history, determines an estimated received symbol.

As to the specific way in which the Viterbi decoder was implemented in the computer programs of the prior art modems, the following description is provided. Referring to FIGURES 1A through 1C, the prior art implementation of the Viterbi decoder will be illustrated in a simple example. In this example, the Viterbi algorithm is used in conjunction with a simple convolutional encoder having four states identified as 0, 1, 2, and 3 in the treUis of FIGURE 1A. A convolutional encoder is a finite state device which passes through a succession of the states, with each state being separated in time as shown by time unit j.

The trellis is used to diagram all permissible transitions between the present state to the next state of the encoder, with each of the branches between states being identified by a unique encoder binary output, referred to as the subset-selecting bits and designated in FIGURE 1A as a.

b, e, op d. Each of these subset-selecting bits (a pair of bits in this case) map upon a given subset of symbols.

Beginning at time unit j = m+1, the branch metric is calculated for the paths, i.e., branches, entering each state 0, 1, 2 and 3, as shown in FIGURE 1A. For each state at time unit m + 1, the survivor path with the smallest metric w' is determined, as shown in FIGURE IB.

The paths are stored, either by storing the subset-identifying bits or the states, and the metrics w' for the paths are stored. Referring to FIGURE 1C, the unit time j is increased by one. Metrics w" for all of the paths entering the states at j = m + 2 are computed by adding calculated branch metrics of the branches from j = m + ltoj = m + 2to the surviving metrics w' of the connecting survivor paths at the preceding unit. At this point, each surviving metric w' includes only the previously calculated branch metric, but with the passing of more time units, each surviving metric w' for a given path will be the summation of all the branch metrics of the path up to that time unit. Again for each state, the path with the smallest metric (the survivor) is selected, eliminating all other paths and reducing the trellis to that shown in FIGURE 1C. For each state, the surviving path is stored, together with its metric.

The path histories are extended each time unit as described above. Once the stored path histories are extended so that j = m + L - 1 (where L is the truncation length), the decoder finds the minimum cost path (smallest metric) among the surviving paths. From the surviving path, the decoder provides the best estimate of the subset or state at j = m. This is repeated each time unit, so that, e.g., at j = m + L, the decision is made for the state or subset at j + m + 1.

j.

There are 2 states in the trellis (k = 2 in this example).

t.

Consequently, for each time unit, 2 words of storage are needed for the survivors and to store the entire path histories, an information sequence of length 2 x L must be stored. As shown in FIGURE 2A, the prior art technique for storing these information sequences in memory comprised a memory array as shown. For each time unit, the subset-selecting bits (or states) of a given surviving path are stored in a row. Referring to FIGURE 2A, if we assume at time interval j - 1 that the subset-selecting bits are ordered in the trellis as a, b, c, d and that the extension of the path will result in the subset-selecting bits being reorganized in the order of a, c, b, and d at time unit j, each of the rows representing the path histories must be shifted as shown by the arrows in FIGURE 2A to obtain a rearranged memory of FIGURE 2B. To accomplish this, it is assumed that the states are sequentially updated. At state 0, register 3 is written in FIGURE 2A to the first row in FIGURE 2B. As a result, the register 0 is destroyed by register 3. So when state 1 is updated, the survivor in register 0 is lost. One way of solving problem is to store register 3 in a buffer (not shown) and then copy register 2 in FIGURE 2A to row 3 in FIGURE 2B, register 1 in FIGURE 2A to row 2 in FIGURE 2B, register 0 in FIGURE 2A to row 1 in FIGURE 2B, and store the row stored in the buffer back to row 0. For a eonvolutional code with many states, the algorithm is very complex.

Consequently, from an implementation point of view, each time j is updated, each state could come from multiple states of time j - 1. Therefore, all the paths need to be reshuffled according to the new order of the subset-selecting bits at time j. In addition, in order to rearrange the path of a particular state when the survivors of all the other states have not been determined, it is necessary to have temporary buffers to prevent overwriting of the path's history. An alternative would be to maintain two tables.

As a result of this prior art design, at each time instant j, the operation of reading the past paths and writing to the new locations is required. For a eonvolutional code with many states, it creates a great number of data moves. For example, a 64 states eonvolutional code with a truncation length of 20 needs 64 x 20 read and write operations.

In prior art schemes the states have been stored instead of the subset-selecting bits and a look-up table has been used to generate the subset-selecting bits, when needed. In other words, in the above example, with the knowledge of two sequential inputs to the encoder, i.e., the old and new states, an output of the encoder, i.e., the subset-selecting bits, can be determined or vice versa. Since the encoder adds one bit of redundancy per time unit, the prior art scheme took advantage of this fact by storing the input of the encoder instead of the required output, so as to reduce memory requirements at the cost of having to use the look- up table to recreate the subset-selecting bits.

The above example is for 2-dimensional coded modulation wherein the output of the encoder consists of subset-defining bits that map onto two-dimensional subsets of signal points. The unit time is one baud, i.e., symbol interval or modulation period. The above example is also applicable to an N-dimensional coded modulation system (N > 2), wherein the subset-defining bits map onto a multidimensional subset. In this case, the unit time is a group interval comprising a plurality of bauds and the trellis has parallel branches between states.

SUMMARY OF THE INVENTION The present invention relates to a Viterbi decoder wherein a maximum likelihood path through a trellis is selected from a plurality of surviving paths. The improvement of the present invention comprises a memory array divided into a plurality of groups of addressable locations.

Each group is associated with, i.e., indexed to, a unique one of a plurality of states of the trellis. Each addressable location within each group is associated with, i.e., indexed to, a given time unit of a sequence of time units. From each current state, to which each memory location is also associated with, a unique branch extends backward in time to a previous state. At this point in time, each of these branches form a part of a unique surviving path terminating at the current state, although such branches may cease to be a part of a surviving path in subsequent time periods. The storage of each surviving path is accomplished by storing a previous state of the surviving path in the addressable location associated with the current state of the surviving path. Knowing this previous state determines the addressable location associated with a new current state, that contains the next previous state and so on until the entire surviving path is extracted from memory. The need for extracting the surviving path from memory occurs each time unit when the surviving path with the lowest metric, i.e., the maximum likelihood path, must be extracted to determine transition bits after a delay of a number of time units. The oldest two adjacent states from the maximum likelihood path are converted by use of a look-up table to the desired transition bits.

Depending upon the embodiment, the transition bits comprise the subset- defining bits or are used to obtain the subset-defining bits.

The invention makes use of the recognition of a property of the trellis that at a given time unit, each new current state in the trellis is connected by the surviving path to only one previous state, even though no path, one path, or a plurality of paths may extend to the next successor state.

The invention is implemented by using a wraparound memory for storing the plurality of groups of addressable locations wherein the newest states for the newest time unit are stored in the group of addressable locations previously containing the oldest states for the oldest time unit. The position of the group of addressable location is kept track of by using an offset pointer which cycles in a modulo manner so that no shifting in or out of the groups of addressable locations is required.

The present invention avoids the prior art schemes which require a tremendous amount of data moves to shift the data around. For example, the invention only requires L indirect reads of the memory (L is the truncation length of the Viterbi decoder). Consequently, for a 64 state convolutional code with L = 20, the invention can save up to 90% of the processing time of a TMS 32020 processor. Additionally, there are other advantages in being able to vary the truncation length and to have a piggy back arrangement for parallel transition branches.

Description of Preferred Embodiments Referring to FIGURE 3, for the purposes of illustrating the invention in a simplified and clear manner, the improved Viterbi decoder of the present invention first is shown implemented with a 2-dimensional convolutional code in a transmitter 1 of a modem 2. The 2-dimensional convolutional code is created by an illustrative convolutional encoder 3 with four states. In FIGURE 4A the convolutional encoder 3a is shown and includes two delay elements 3b and 3c and one Exclusive Or 3d. This convolutional encoder 3a is shown in FIGURE 6 of and described in detail in the mentioned . U.S. Patent 4,601,044 to Kromer et al. After aa description of this first embodiment, the improved Viterbi decoder is described in detail in conjunction with its implementation in a second embodiment for a N-dimensional coded modulated systems, where N is .

greater than 2.

FIGURE 4B represents an illustrative trellis 4 of the convolutional encoder 3a of FIGURE 4A. In the trellis, for each time unit j - i (where 0 £ i < 7), the four states are repeated and are identified as 0, 1, 2 and 3. In the first illustrative embodiment, the time unit is one baud, i.e., one symbol interval or modulation period. Each branch 5 of the trellis 1 represents a permissible transition of the encoder 3a from one state to another. Consequently, the trellis is used to diagram all permissible transitions between the present state and the next state of the encoder. Since the encoder passes through a succession of states, a plurality of paths 8 are defined by the interconnection of the branches 5.

As shown at time unit j - 2, although a plurality of paths may extend forward in time from a given state, there is only one path extending backward in time from any given state. For example at j - 2, the state 1 has two paths extending forward therefrom, but extending backward the two paths have merged to a single path.

For each permissible transition from one state to another, the encoder 3a provides a two bit output (identified as subset-defining bit group), which is mapped onto one of four possible subsets of signai symbols (points) in a 2-dimensional signal constellation, as shown in mentioned U.S. Patent 4,601,044. Referring to FIGURE 5, for each permissible transition from an old state to a new state, the subset- defining bits of the encoder 3a are shown in decimal form, thereby defining the interval logic of the convolutional encoder 3a. The "x's" represent an impermissible transition.

In a conventional manner, as shown in U.S. Patent 4,501,044, the convolutionally encoded signal is modulated in the transmitter 1 and is transmitted over a band limited transmission medium 7, e.g., telephone lines, as shown in FIGURE 3. In a receiver 8 of a modem 9, the received signal is demodulated to provide received 2-dimensional, subset defining bit groups. In a well-known manner in a Viterbi decoder 9a, by means of the maximum likelihood sequence estimation technique using the Viterbi algorithm, the maximum likelihood path is determined.

As is well-known, the Viterbi decoder 9a, during each symbol interval, i.e., modulation period or baud, extends each of the surviving paths to their next successor state. The metric of the extended path is obtained by adding the metric of the surviving path to the branch metric of the corresponding branch which extends the path. The survivor at the next frame and 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. Thus, the path history of the surviving path is extended. Once the path history is extended the Viterbi decoder finds the minimum cost path among the survivors, and from the path history outputs the best estimate of the subset-defining bit group for the symbol interval that is a -n- predetermined number of symbol intervals in the past. As an example, in FIGURE 4B, the path with the minimum cumulative metric is shown by asterisks extending from time unit j back in time to time unit j - 7. In time unit j, the decoder would determine the best estimate of the state of the encoder at j - 7 to be state 1.

Up to this point in the description, the structure described is of a well-known conventional design. The novelty of the invention resides m the manner in which the path histories of the trellis are stored in memory for use in calculating the best estimates of the subset-defining bit groups. The invention makes use of following properties of the trellis:

1. At a given time unit j - i, each new state in the trellis is connected by one of the surviving paths to only one previous state. In a trellis for a 2-dimensional coded modulated system, as shown in FIGURE 48, each new state is connected to only one old state by a single branch As wUl be discussed hereinafter, in an N-dimensional modulated system (N > 2), there will be a plurality of parallel branches extending backward from the new state to the single old state.

2. By knowing the states at j - i and j - i - l, the subset-defining bit groups, which are indexed to the transitions from j - i to j - i - i, can be determined, by using, for example, the look-up table of FIGURE 5 In the preferred embodiment, this table is stored in the memory of a nmcrocomputer in the receiver 8 of FIGURE 3. For the purposes of the claims, in this embodiment, transition bits are the subset defining bits with there being one subset-defining bit group for each baud.

As a result of these two properties, the most likely path is reconstructed and retrieved by use of a look-up table, i.e., the predecessor states table, of FIGURE 6. Such a look-up table is stored in memory. The columns of memory of FIGURE 6 are associated with (indexed to) a time unit j - i, where 0 < i < L - 1 (L being the truncation length of the Viterbi decoder). Since a decision at time unit j is made to determine the best estimate of the state at J - L - 1, for each of the surviving paths, L states must be stored and the cumulative metric of the path. For this simple embodiment, each stored state comprises a 2-bit word. Although for the purposes of simplicity, L is shown to be 7 time umts are shown in FIGURE 6, typically L is 16 to 20 time units.

In addition to the indexing of each time unit, each row feroup) of the memory of FIGURE 6 is associated with (indexed to) a unique one of the four states of the trellis. Consequently, at a given time umt i - i, each addressable location is associated with a unique branch As previously explained, a unique one of the branches extend backward to the previous state from each current state, even if no branch, one branch or a plurality of branches extend forward from each current state Consequently, each addressable location is associated with not only a umque state, but also a unique branch extending backward from that state. At that point in time (time unit J), each unique branch (four in number) extending back from the current states forms a part of one of four unique surviving paths, although for later time units that branch may cease to be part of one of the surviving branches, as illustrated by branch 5a in FIGURE 4A.

The memory locations preferably, but not necessarily, are indexed from the top to the bottom to the states shown adjacent to the nght-most column. Each of the four surviving paths ends with its current state at j in one of the addressable locations in the memory column for j There ,» only one surviving path ending in each of these locations. In summary, for each path history (and therefore each row), there must be L addressable memory locations, with each location having an n bit word, where n=2 in the illustrative first embodiment (L and n are integers) There * only one row associated with each state and each addressable location therein is associated with a given time unit from a sequence of consecutive time units. In FIGURE 6, the columns are numbered 1 through 7 in the table.

In the column of time unit j - i, in each addressable location representing the termination of a branch at a state indexed to that location, there is stored the location at time unit j - i - 1 of the starting state of that branch. For example, assuming that the path ending in at state 2 (row 3) has the smallest metric, its history can be retrieved back to time unit j-6, as shown by the asterisk values. The history of this maximum likelihood path is extracted one branch at a time. More specifically, at j, the stored value in row 3 indicates that the previous state at j-1 is 2. This defines a branch extending from state 2 at time unit j to state 2 at time unit j - 1. At j-1, going to state 2 (row 3) the table indicates that the previous state at j-3 is 3. At the time unit j-3, going to state 3 (row 4), the table indicates the previous state at j-4 is 0, and so on to j-6. With each look-up, the path is extended back by another branch. It should be noted that the pattern of the asterisk values in FIGURE 6 is roughly that of the minimum metric path shown in FIGURE 4.

To retrieve the path history of the minimum cost path (i.e., maximum likelihood path) indicated in FIGURES 4 and 6, the addressabl memory of FIGURE 6 would be accessible as follows. A memory array L(n) is defined wherein n is an index of the time units of the path history. Consequently, once the minimum metric surviving path is determined, it is known that its first location is state 2 at time unit j, which allows for the table look-ups to proceed as follows:

e Time Unit t=j t = j-1 t = j-2 t = j-3 t=j-4 t = j-5 t = j-6 Previous State 2 = h(j-l) = table (j, 1) 1 = h(j-2) = table (j-1, 2) 3 = h(j-3) = table (j-2, 1) 2 = h(j-4) = table (j-3, 3) 1 = h(j-5) = table (3-4, 2) 2 = h(j-6) = table (j-5, 1) 1 = h(j-7) = table (j-6, 2) Note that each determined previous state defines in the look-up operation, the location to find the next previous state. The path recovered is as follows: 1, 2, 1, 2, 3, 1, and 2. The oldest two states at time units j - 5 and j - 6 define the new state and old state, respectively, in the conversion table of FIGURE 5, thereby providing the required subset defining bit group for j - 5, which is the bit pair represented in decimal form as 0.

In summary, the look-up table is formed in memory wherein a group of at least L addressable memory locations (one each time unit) is indexed to each of the states of the trellis. At a given time unit, a surviving path having a given carrent state will have stored at the addressable location indexed to that state, the previous state of the surviving path. For the previous time unit, knowing the previous state defines the addressable location associated with the next current state from which the next previous state can be found, and so on, until the entire path history has been retrieved from a (L) x (M) memory array, wherein M is the number of states. For the purpose of this description and the claims, each addressable location is considered indexed to a "current» state, while the contents of that location designates a "previous" state for the path having that particular current state.

Referring to FIGURE 6, the invention is implemented in a wrap around computer memory in a manner that makes it unnecessary to shift the columns of states each time a new group of current states is received and the oldest group of states is no longer required. As well be described in more detail hereinafter in the discussion of the 19.2K modem implementation, an offset pointer 9b is used to mark the most current column for time unit j. For the purposes of explanation, the columns of the look-up table are labeled 1 through 7. Assuming the initial time units associated with the columns are as shown in FIGURE 6, the pointer is initially pointing to column 1 as having the most current group of states.

After the next time unit, the contents of column 7 (and columns before it, if any, which are not shown in the drawing) are no longer needed and the most current states are stored in the column associated with the oldest time unit, with the pointer 9b designating this time unit. For the purposes of simplicity, as shown in FIGURE 6, it is assumed that the number of columns in memory equals the truncation length L. The oldest group of states stiU needed would then be in column 6, with the time units associated with columns 7 through 1 being reordered across the top of the table to be, respectively, j, j - 6, j - 5, j - 4, j - 3, j - 2, j - 1, After the next group of states are received, they would be stored in column 6 (with the pointer designating the same) and the time units would be reordered to j - 1, j, j - 6, j - 5, j - 4, j - 3, and j - 2. This advancement of the pointer would continue for each time unit, cycling in a direction from column 7 to column 1 and then starting over at column 7 and so on. In the case where the truncation length is less than the number of columns available (L < 7 in the simplified example), then the current states are still stored in the column associated with the oldest time unit.

Consequently, this offset pointer indicates the current time frame with the current states. The pointer is updated by modulo arithmetic so that the wrap around nature of the data storage prevents the need for shifting the data from one column to the next.

As to the advantages of the invention, in this state history method, there is no need within the same register to connect the present path to the correct old path. At time j - i, it is only necessary to store the state that the path connects to at time j - i - 1. When the minimum metric is determined, the subset-defining bits are derived by searching the state history table of FIGURE 6 and then the conversion table of FIGURE 5. This only requires L indirect reads {L is the truncation length). For example, as will be described in another embodiment hereinafter, for a 64 states convolutional encoder with L=20, it can save up to 9096 of TMS32020 processing time.

As will become apparent from the detailed description of the embodiment for the multidimensional coded system, the invention provides other advantages. If the look-up table is large enough, the truncation length can be varied by only changing the loop count in the above method. This offers programming flexibility and no extra buffers and rearrangements are needed. If there are parallel transition branches from the old state to the new state, as there is with the multidimensional coded systems, an index can be piggy backed to indicate which parallel branch is the one entering the state. To accomplish this, an extra table read can be done or the transition branch can be derived by using the piggy back index.

Hereinafter, the improved Viterbi decoder of the present invention will be illustrated implemented in a 19.2K bit per second modem using N-dimensional convolutional coding. Referring to FIGURE 7, there is shown a block diagram of the generalized structure of a transmitter 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.

The 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 7 is shown in FIGURES 8A and 8B, with upper two quadrants and lower two quadrants of the complex plane of the constellation 36 being shown, respectively, in FIGURES 8A and 8B. 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 up 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 A09 Al, B0, Bl, 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, Al, B0, and Bl, 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. The units of the graph of FIGURES 2A and 2B are shown in volts 0.

Each subset is represented by a two bit number shown in Table 1.

Subset Subset Index MSB LSB A0 0 0 Bl 0 1 BO 1 0 Al 1 1 Table 1 Thus, the subset-selecting bits 26 form an output of the (m, m-1) convolutional encoder 18 of FIGURE 7 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.

To 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 8A is enlarged and shown in FIGURE 9. The symbols of the constellation 36 are on a square grid of the complex plane. From FIGURE 9 it can be seen that the maximum between BO and Bl. The minimum distance of d is between AO and BO, AO and Bl, Al and BO and Al and Bl. The maximum Hamming distance also occurs between AO and Al, as well as between BO and Bl.

FIGURE 10 shows a packet of data generated by the serial to parallel converter 16 of FIGURE 7 which comprises 28 bits which go to the convolutionai encoder 18. The bits x8-x28 are the symbol-selecting bits 20. As shown in FIGURE 11, bits x x2, x3, x4, x5, x6, and x7 form the input bits 24 to convolutionai encoder 18. As shown in FIGURES 11 and 12, the output bits y7, y6, y5, y4, y3, y2, ylf and y0 of the encoder 18 are grouped into four bit groups (y7 y6), (y5 y4), (y3 y2) and (yi yo) that defines a frame 48 of bits. As will be explained hereinafter, bits x1-x7 form only a portion of the expanded bit sequence output of the encoder 18. FIGURES 11 and 12 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 corresponds to one of the four 2-dimensional subsets A0, Al, B0 and Bl.

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 eight bit output of the encoder 18 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 Al B0 A0 B0 (11100010) corresponds to 2-dimensional symbol subsets (11), (10), (00) and (10).

From 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 12, bit x7 occurring in the nth frame will be referred to as xn . Table 2 shows the assignment of points from inner and outer points of the constellation 36 shown in FIGURES 8A-8B for different values of o Q 10 ft xn , xn and x n. Bit xn indicates whether the frame 48 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 x1£ 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.

w , x baud 1 n baud 2 baud 3 baud 4 Inner Inner Inner Inner Outer Inner Inner Inner Inner Outer Inner Inner Inner Inner Inner Inner Outer Inner Inner Outer Table 2 Table 3 shows how the symbol-selecting bits 20 of FIGURES 7 and 10 are assigned with each group of 8 9 _, x,» x„, and x .

n' n' n n "n " n uauu i 0 X X x9-x13n n 1 0 0 n n 1 0 1 11 15x ~-xn n 1 1 1 x11-x15n n baud 1 baud 2 baud 3 x1l»-x18 n n x-x18 n n n n x16-x20 n n x -x -> n n x1 n n x n"55 n x21-x25 n n baud 4 x24-x28 n n x2iU28 n n x -x n n xn-xn x26-x28 n n Table 3 FIGURE 13 shows the convolutional encoder 18 of FIGURES 7 and 11 in detail. The encoder 18 is a 64 state (8, 7) encoder. As previously described, the input bits to the encoder are x1 , where i = 1 through 28 and the output bits are y , 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 through x2 are passed through unchanged and unused to become output bits y J through y2. Inputs are used but pass through the encoder 18 without being changed jme OUtnilt hi»-o v _,.' tu- : .. i-!*- 1 3 X -X n n and become output bits /V. The input bits .,3 are expanded to create the output bits y-yj. The output bits of y" through y28 are identified as the expanded bit sequence, which includes the subset- selecting bits y0-y7 and the symbol-seiecting bits 28 The convolutional encoder 18 includes four modulo 2 adders 52, 54, 56, and 58, each connected to receive at least some of the bits x3, x2, and x1. ' The encoder 18 includes six delay elements SO, 62, 64, Be" 68, and 70, with each element providing a delay of one group interval (four bauds).

With reference to FIGURE 13, the input-output relationships of the subset-selecting bits 26 of the encoder 18 are as follows:

yn = xn y5 = x5 •'n n "n " Xn D2©©*n (D © © xn © *n © n = xn D2©xn <1 © D © d2) © *„ © xn © x!! Vn = xn (1©D©D2)©x2 ( 1 © D) © @ ,6 @ x5 @ £ (Equations 1) where D is a delay element and © means an exclusive OR operation.

The convolutional encoder 18 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 14, the encoder 18 has 84 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. Which next state occurs is a function of the present state and on the combination of the x\ x2, and x3 bits received each group interval. Each transition from one Jtate to another state is identified as a branch, 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 branches 74 between present state 0 and next state 0 are shown in FIGURE 14. 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 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.

From FIGURE 13 and the Equations 1 for encoder 18, it can be shown that xj, xjj, x and xj 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 14. This set of sixteen parallel branches will be 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 14, 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 the multidimensional symbol subsets of the coset associated with the branches going from present state 0 to next state 0 are shown in FIGURE 14. The multidimensional symbol subsets associated with each branch can be indexed by integers ranging from 0 to 255. The index can be expressed as:

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) + 3° index of subset for baud 1.

As to the multidimensional symbol subsets associated with each branch, the 2-dimensional subsets therein preferably, but not necessarily, are ranged 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.

12S8519 A receiver 80 of the modem 10 is generally shown in the block diagram of FIGURE 15 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 to produce a baseband signal consisting of a stream of received 2-dimensional symbol signals. As described hereinafter in detail, the rest of the circuitry processes the stream of received 2-dimensional symbol signals and decides which of the multidimensional symbols were sent by means of the maximum likelihood sequence estimation technique using the Viterbi algorithm. The maximum likelihood path is determined by which permissible multidimensional symbol sequence is closest in Euclidean distance squared (as will be described hereinafter) to that of received multidimensional symbol sequence.

A branch metric 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 is the arithmetic sum of the metrics of the individual branches which form the particular path. Hence, it is necessary to first obtain the branch metric for each of the previously described 256 branches. As mentioned earlier, the indices of the ideal multidimensional symbols are in one to one correspondence with the sequence of baud based 2-dimensional symbol sets. Hence, in order to find the distance between a received branch and an ideal branch with a certain index value, it is necessary to know when the receiver frame starts.

Referring to FIGURE 15, 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(l), R(2), R(3), and R(4), where:

R(J) = [RX(J), RY(J)], 1 < J < 4 (Equation 2) A symbol (ideal point) in the signal constellation can be represented as:

IDL(L, N) = [IDLX (L, N), IDLY(L, N)] (Equation 3) where L = 1 means that the point belongs to subset AO L = 2 means that the point belongs to subset Bl 5 L = 3 means that the point belongs to subset BO L = 4 means that the point belongs to subset Al and N = index of the point in the subset.

For each L and J, the slicer 102 finds the N = N j , such that |R(J) - IDL(Lf N . )| < JR(J) - IDL(L, N) | where |R(J) - IDL(L, N)|2 = |RX(J) - IDLX(L, N) |2+ |RY(J) - IDLY(L, N)j2 (Equation 4) The slicer 102 thus finds four nearest points for the received signal of each baud. For each baud of the four baud frames, we can denote the distance (metric) between each of the ideal point and the received point as d[J, L(J), Nm.n(J)]. Thus [d [J, L(J), N .(J)] = |R(J) - IDLIUJ), N .(J)]| nun min ' (Equation 5) 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, i.e., AO AO AO AO through Aj A1 A1 Al. The branch metric computer 100 computes the branch metric term, which can be either equal to or proportional to dBRANCH(I) where dBRANCH (I) is defined by:

dBRANCH «) = [d2(l, L<1>, Nmin(1)) + d2(2, L<2>, Nmin(2)) d2{3, L(3), Nm.n(3)) + d2(4, L(4), tf) (Equation 6) The index I identifies the branch and extends from 0 to 255, showing that this calculation is repeated for each branch. Consequently, at this point we have 256 d 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.

Next, the branch metric computer 100 selects for each coset (group of 16 branches each) that branch which has the minimum branch metric, which results in reducing the number of contending branches (multidimensional symbol subsets) from 256 to 16. Only one branch of each coset can form a part of the maximum likelihood path. 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.

Generally, the Viterbi decoder 104 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 14. During every frame, the receiver computes the minimum cost path and from the path history, determines the estimated multidimensional symbol. As 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 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:

c(i, n) = Minimum {c(j, n-1) + r(i, j)} (Equation 7) where c(j, n-1) is the surviving metric of the jth state at time t = (n-l)Tframe, r(i> j) is the branch metric of the output branch which connects the jth state at time t = (n-l)Tf to the ith state at t = nTf. c{i, n), the cost at the ith state at time t = nTf, 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 conventional 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.

From the retained knowledge of the received multidimensional 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.

The 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 7, 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 15, all of the receiver modem functions of FIGURE 15 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.

Referring to FIGURE 16A, a memory array 130 is formed in RAM and consists of a plurality of memory blocks 132 (only part shown), with each memory block being composed of a plurality of addressable locations 134 (only part shown). The number of memory blocks 132 is equal to the maximum possible depth (max depth) of the Viterbi algorithm. The truncation length (L) of the Viterbi algorithm, which can be selectively varied by the user, cannot exceed the max depth. In the illustrative embodiment max depth is equal to 32 group intervals, i.e., 128 bauds. Typically, the truncation length would be selected to be 20 or more group intervals, depending upon the condition of the transmission lines. For example, when the truncation length is 20 group intervals, this means the memory blocks 132 numbered 1 through 20 are used. Each block of this sequence of blocks are associated with (indexed to) one of a plurality of time units (group intervals) j - i, where 0 < i < 31, but only those blocks associated with time units j - i, where 0< i < L-l, contain sets of states that are currently used in the path retrieval. As is well- known, the truncation length can be varied as desired. The number of addressable locations in each memory block 132 is equal to the number of states (P) of the trellis, which is 64 in the illustrative embodiment.

Consequently, the size of memory array 130 is equal to the max depth times the number of states P and the memory mapping is accomplished by dividing the memory into groups of addressable locations, each of which is associated with (indexed to) a given state, with each of the addressable locations 134 in a given group being associated with a unique time unit and a unique block 132.

The information contained in the memory array 130 can be formatted in a predecessor state, look-up table, such as that shown in FIGURE 6. Each memory block 134 correlates with a column and the addressable addresses for the same state of the columns correlates with a row. However, as a practical matter, when the look-up table is implemented in memory, the addressable locations form one long sequence.

Each addressable location 134, which is associated (indexed to) a present state of one of the surviving paths (as described in detail heretofore), contains a previous state of that surviving path. With the passing of each time unit, the states contained within a given block 132 are associated with a progressively older time unit, until the contents of that block is disposed of by using that block for the storage of the most current (newest) states, in a manner to be described hereinafter.

The memory array 130 is updated as follows. Referring to FIGURE 16A, when L = 20, the illustrative memory block numbers 1 through 20 represent blocks of addressable locations having the states of the surviving paths occurring in time units that initially become progressable older with the block number. For the purposes of explanation, it is initially assumed in FIGURE 16A that block 1 is indexed to newest time unit j and that block 20 is indexed to the time unit j - L - 1. Each group interval, when the states for a new time unit are received fop current time unit j, the blocks of states previously indexed to the time units j - i, are now indexed to the time units j - i - 1. Although the contents of each block could be shifted downward to make room for the newest states (each block always indexed to the same time unit), instead, according to the present invention the newest states are put in the block previously associated with the oldest time unit as shown in FIGURE 16B. Initially, this is block 32. Each block 132 is identified in the drawings by its block number, followed by the time unit that it is associated with, relative to the current time unit j. Continuing, as shown in FIGURE 16C, the position indexed time units rotate on a modulo basis. As shown in FIGURES 16A-16C, to keep track of this rotation, a program stored in ROM uses a data address pointer 136 (labeled offset pointer) to indicate in which memory block 132 the location of the states indexed to the most current (newest) time unit, and, once knowing this, the position of the states for the other time units are known. The pointer 136 jumps from block 1 to block 32, and then each time unit progressively moves to blocks with decreasing block numbers until it again reaches block 1 when it again jumps to block 32.

This offset pointer 136 is updated by modulo arithmetic after each data entry is completed. Also, because of the wrap around nature of modulo arithmetic, no shift-in and shift-out operations are needed.

Referring to FIGURES 16A-C, the path update is reduced down to P writes. In the illustrative embodiment, modulo Q arithmetic is used, with the base Q being equal to the number of blocks (and therefore the number of indexed time units), which is 32 in this case (identified as depth max).

In the case where L = Q, then the block indexed to j - 31, when previously containing the oldest states used during the previous time unit, is used to store the newest states during the subsequent time unit. Where L = 20, the newest states replace states that have not been used in path retrieval for several time units.

Referring to FIGURE 17, as shown by the flow chart starting at block 138, at block 140 the starting address for the write operation is calculated to be equal to the table origin plus the offset (number of blocks from the table origin to, but not including, the current memory block designated by the offset pointer) times the number of states <P) in the block. The counter at block 142 is set to 1 and at block 144 each of the 64 addressable locations are written to by cycling through the loop of 144, 146 and 148, and the program exits from the loop at block 150 when KK = n state (64 states in the illustrative embodiment) at block 146.

Referring to the flow chart in FIGURE 18 and the following discussion of a computer program, the data is retrieved from the memory array in a manner to be described hereinafter. In the program, the following symbols are defined:

mod (n, m) min node table (a, b) pstate tptr newstat trans bits Content of the address Modulo base m of n Minimum node (first and most recent addressable location of maximum likelihood path) table look-up for the transition bits where a & b are row & column past (previous) state table pointer present (current) state transition bits First, as previously described, the path with the least cost is selected and the current addressable location (min - node) of that path is therefore known. The program will have the following steps. The blocks of the flow diagram of FIGURE 18 are correlated to the program steps by showing the blocks in brackets adjacent to the appropriate steps in the program.

tptr = table origin + Q x mod (offset, depth max) tmin = tptr + min node ! Table entry [see block 152] pstate = < tmin > > state enters min node C loop to retrieve data L frames ago C notice that the depth could change accordingly. It is C dependent of depth max. Notice that there are L - 1 C indirect-read do for kk = 1, L - 1 [see blocks 154 and 156] newstat = pstate ! Store pstate tptr = table origin + Q x mod (offset + kk, depth max) + pstate pstate = < tptr > ! state enters kk frames ago '5 end do C table look-up for transition bits (trans bits) [see block 158] C update offset trans bits = table (pstate, newstat) i.e., trans bits is the recover data offset = mod (offset-1, depth max) C end of routine, wait for next update Referring to FIGURE 19, the arrow on line A-A the points in the direction of rotation of the offset pointer, which wraps around every depth max and the arrow on line B-B shows the direction of the tptr pointer, which wraps around every depth max in the opposite direction from the offset pointer.

Referring back to FIGURES 14 and 15, for each coset, in the branch metric computer 100, one of sixteen parallel branches is selected from each coset based upon a cost. This branch is defined by the subset- defining bits y° to y of FIGURE 13. As previously explained, these selected branches form part of the surviving paths. In its simplest implementation, the subset-defining bits forming a given, selected branch could be stored in the addressable location for the current state for that transition. However, because of practical memory size locations existing in actual implementation of the present invention, it has been found desirable to use a more elaborate arrangement. More specifically, there is stored in the addressable location for a given current state a branch index number from 1 to 16 which identifies the branch (subset-defining bits) within a given coset that was selected based on cost for the transition from a previous state to that particular current state.

This requires only 4 bits instead of the 8 bits required to store the branch required to store the subset-defining bits. Additionally, the current state and the previous state are used to access the above-described look-up table discussed with respect to the computer program to obtain the transition bits. In this case, with reference to FIGURE 13, the transition bits are a portion of the subset-defining bits, bits y° - y, when output bits yn to yn are all zero. In other words, when these latter bits are zero, they do not change the output of the bits y0 to y". However, unlike the 2-dimensional modulation embodiment, for the other possibilities in the coset, the transitional bits are not the same as the subset defining bits. However, a portion of the transition bits, which are unique to each coset, are used to identify the coset. With the coset as identified by the transition bits and the branch index number, a second look-up table operation is performed. The contents, which are addressable as a function of the coset number and the index number is shown below in Table 4 and each of the accessed values are the subset- defining bits (as shown by their subset designations) for a given branch within a given subset. It should be noted that the last two subset designations in the first branch of each coset (where y4 - y7 are zero) is the binary equivalent of the decimal coset number, e.g." for Coset 7, BIA1 equals 7.

12885X9 Branch Index Number Coset 7 Coset 4 Coset 9 Coset 1 A0AOB1A1 A0A0B1A0 AOAOB0B1 AOAOBOBO 2 A0B1A0A0 A0B1A0A1 A0B1A1B0 A0B1A1B1 3 A0B0A1A0 A0B0A1A1 AOBOAOBO A0B0A0B1 4 A0A1B0A1 A0A1B0A0 A0A1B1B1 A0A1B1B0 B1A0BOBO B1A0B0B1 B1A0B1A0 B1A0B1A1 6 B1B1A1B1 B1B1A1B0 B1B1A0A1 B1B1A0A0 7 B1B0A0B1 B1B0A0B0 B1B0A1A1 B1B0A1A0 8 B1A1B1B0 B1A1B1B1 B1A1B0A0 B1A1B0A1 9 B0A0B0B1 BOAOBOBO B0A0B1A1 BOAOB1A0 B0B1A1B0 B0B1A1B1 B0B1A0A0 B0B1A0A1 11 BOBOAOBO B0B0A0B1 B0B0A1A0 B0B0A1A1 12 B0A1B1B1 B0A1B1B0 B0A1B0A1 BOA1B0A0 13 A1A0B1A0 A1A0B1A1 A1A0B0B0 A1A0B0B1 14 A1B1A0A1 A1B1A0A0 A1B1A1B1 A1B1A1B0 A1B0A1A1 A1B0A1A0 A1B0A0B1 A1B0A0B0 16 A1A1BOA0 A1A1B0A1 A1A1B1B0 A1A1B1B1 Coset 5 Coset 6 Coset 11 Coset 8 1 A0A0B1B1 A0A0B1B0 A0AOB0A1 AOAOBOAO 2 A0B1A0B0 AOB1A0B1 A0B1A1A0 A0B1A1A1 3 A0B0A1BO A0B0A1B1 AOBOAOAO A0B0A0A1 4 A0A1B0B1 AOA1BOB0 A0A1B1A1 A0A1B1AO B1A0B0A0 B1A0B0A1 BÎA0B1B0 B1A0B1B1 6 B1B1A1A1 B1B1A1A0 B1B1A0B1 B1B1A0BO 7 B1BOA0A1 B1B0A0A0 B1B0A1B1 B1B0A1B0 8 B1A1B1A0 B1A1B1A1 B1A1B0B0 B1A1B0B1 9 B0A0B0A1 BOAOBOAO B0A0B1B1 B0A0B1B0 B0B1A1A0 B0BÎA1A1 B0B1A0BÛ B0B1A0B1 11 BOBOAOAO BOBOA0A1 B0B0A1B0 B1B0A1B1 12 B0A1B1A1 B0A1B1A0 B0A1B0B1 B1A1B0B0 13 A1A0B1B0 A1A0B1B1 A1A0B0A0 A1A0B0A1 14 A1B1A0B1 A1B1A0B0 A1B1A1A1 A1B1A1A0 A1B0A1B1 A1B0A1B0 A1B0A0A1 A1B0A0A0 16 A1A1B0B0 A1A1B0B1 A1A1B1A0 A1A1B1A1 Coset 15 Coset 12 Coset 1 Coset 2 1 A0A0A1A1 A0A0A1A0 A0A0A0B1 AOAOAOBO 2 AOB1B0A0 A0B1B0A1 A0B1B1B0 A0B1B1B1 3 AOBOB1A0 A0B0B1A1 AOBOBOBO A0B0B0B1 4 A0A1A0A1 A0A1A0A0 A0A1A1B1 A0A1A1B0 B1AOAOB0 B1AOA0B1 B1A0A1A0 B1A0A1A1 6 B1B1B1B1 B1B1B1B0 B1B1B0A1 B1B1B0A0 7 B1B0B0B1 B1B0B0B0 B1B0B1A1 B1B0B1AO 8 B1A1A1B0 B1A1A1B1 B1A1A0A0 B1A1A0A1 9 B0A0A0B1 BOAOAOBO B0A0A1A1 B0A0A1A0 B0B1B1B0 B0B1B1B1 BOB1B0A0 BOB1B0A1 11 BOBOBOBO B0B0B0B1 BOB0B1A0 B0B0B1A1 12 B0A1A1B1 B0A1A1B0 BOA1A0A1 BOA1A0A0 13 A1AOA1A0 A1A0A1A1 A1A0A0B0 A1A0A0B1 14 A1B1B0A1 A1B1B0A0 A1B1B1B1 A1B1B1B0 A1B0B1A1 A1B0B1A0 A1B0B0B1 AlBOBOBO 16 A1A1A0A0 A1A1A0A1 A1A1A1B0 A1A1A1B1 Coset 13 Coset 14 Coset 3 Coset 0 1 A0AOA1B1 A0A0A1B0 A0A0A0A1 AOAOAOAO 2 A0B1B0B0 A0B1B0B1 A0B1B1A0 A0B1B1A1 3 AOB0B1BO A0B0B1B1 AOBOBOAO A0B0B0A1 4 A0A1A0B1 A0A1A0B0 A0A1A1A1 A0A1A1A0 B1A0A0A0 B1A0A0A1 B1A0A1B0 B1A0A1B1 6 B1B1B1A1 B1B1B1A0 B1B1B0B1 B1B1B0B0 7 B1B0B0A1 B1B0B0A0 B1B0B1B1 B1B0B1B0 8 B1A1A1A0 B1A1A1A1 B1A1A0B0 B1A1A0B1 9 B0A0A0A1 BOAOAOAO B0A0A1B1 BOA0A1B0 B0B1B1A0 B0B1B1A1 BOBIBOBO B0B1B0B1 11 BOBOBOAO B0B0B0A1 BOBOBIBO B0B0B1B1 188519 12 B0A1A1A1 BOA1A1AO 13 A1AOA1BO A1AOA1B1 14 A1B1BOB1 A1B1BOBO A1BOB1B1 A1BOB1BO 16 A1A1AOBO A1A1AOB1 BOA1A0B1 BOA1AOBO A1AOAOAO A1AOAOA1 A1B1B1A1 A1B1B1AO A1BOBOA1 A1BOBOAO A1A1A1AO A1A1A1A1 Table 4 Instead of storing the previous state in the addressable location associated with each current state as in the preferred implementation, an alternative would be to store the subset-defining bit group, (i.e., selected branch, as shown in Table 4) that correlates with the transition from the previous state to the particular current state in the addressable location associated with this current state. In retrieving the path, the current state, along with its associated stored subset-defining bit group can be used to access a look-up table to obtain the previous state, which, as with the preferred embodiment, is used to locate the next addressable location in the previous time unit to be read. Consequently, such an implementation requires an added step for obtaining the previous state. It should be understood that in the claims where it is recited that the addressable location contains the "previous state» of the path, that this term, when used in the claims, not only is intended to cover stored information directly specifying the previous state, but is also intended to cover and read upon the storage of the corresponding subset-defining bit group, which in fact leads to the determination of the previous state. In this latter implementation, after the previous state is looked up, no further look-up operations are required as in the preferred implementation, to obtain the subset-defining bit group, since that information has already been obtained. For convolutional encoders with a large number of states as in this 19.2K modem implementation, this latter technique is not preferred, due to the great size of the look-up table.

Although particular embodiments of the invention have been shown and described here, there is no intention to thereby limit the invention to the details of such embodiments. On the contrary, the intention is to cover all modifications, alternatives, embodiments, usages and equivalents of the subject invention as fall within the spirit and scope of the invention, specification and the appended claims.