Differentially nonlinear convolutional channel coding with expanded set of signalling alphabets

• US 4,520,490 A
• Filed: 08/05/1983
• Issued: 05/28/1985
• Est. Priority Date: 08/05/1983
• Status: Expired due to Term

• Current Assignee: Lucent Technologies, Inc. (Nokia Corporation)
• Original Assignee: AT&T Information Systems Inc. (AT&T, Inc.)
• Inventors: Wei, Lee-Fang
• Primary Examiner(s): Safourek, Benedict V.
• Application Number:
  US06/520,691
• Time in Patent Office: 662 Days
• Field of Search: 375/25, 375/27, 375/39, 375/42, 455/60, 371/43, 371/44, 371/45, 332/9 R, 370/19, 370/20, 370/21
• US Class Current: 375/246
• CPC Class Codes: H04L 1/0054 Maximum-likelihood or sequential decoding, e.g. Viterbi, Fano, ZJ algorithms
• CPC Class Codes: H04L 27/3433 using an underlying square constellation

Abstract

A technique for converting a sequence of binary digits into a sequence of signalling alphabets of a modulated carrier signal for data transmission. By differentially encoding incoming data, non-linearly encoding the differentially encoded output in a state-dependent sequential manner and mapping the non-linearly encoded output into an expanded set of signalling alphabets having 90, 180 and 270 degree phase ambiguities, enlarged minimum Euclidean distance between possible sequences of signalling alphabets is achieved while the effect of the phase ambiguities of the expanded set of signalling alphabets can be removed in the receiver. This results in a reduced error probability when maximum likelihood decoding and differential decoding are applied in the receiver.

Claims

1. A method for transmitting a bit sequence by modulating a carrier signal to sequentially assume one of a plurality of pairs of discrete signal values, comprising the steps of:

(a) differentially encoding bits in each group of r bits in said sequence by rotating the values of bits in the previous differentially encoded r bit group in accordance with the values of bits in the current r bit group,

(b) expanding each differentially encoded r bit group into a group of r+1 bits by a non-linear convolutional coding process utilizing m internal states, the current state of said convolutional code being determined by p stored bits, where 2^p =m, said p stored bits being functions of the previous bits input to said convolutional encoder and affecting the output of said convolutional encoder,

(c) selecting one of 2^r+1 signalling alphabets from a signal constellation, each alphabet corresponding to a pair of discrete signal values, said selection being made as a function of the r+1 bit group so that 2^r signalling alphabets correspond to all permitted transitions from each current internal state of said convolutional coding process to m possible next states of said convolutional coding process, and so that the 2^r signalling alphabets have a larger minimum distance from each other than the minimum distance within the complete set of 2^r+1 signalling alphabets, and

(d) transmitting the pairs of discrete signal values corresponding to the selected signalling alphabets in sequence,

wherein said signal constellation is arranged so that the same signal constellation is obtained when the original constellation is rotated by 90, 180 or 270 degrees, and wherein

rotation of said constellation by 90, 180 or 270 degrees produces the same effect on said transmitted discrete signal values as rotation of said differentially encoded bits in each r bit group.

2. A technique for modulating in-phase and quadrature phase carriers in accordance with an applied sequence of binary digits, including the steps of:

(a) differentially encoding bits in each current group of r bits in said applied sequence by rotating the values of previous differentially encoded bits by a position differential determined by said bits in said current group,

(b) encoding each differentially encoded r bit group into an associated r+1 bit group in accordance with a multi-state non-linear convolutional code, the current state of said convolutional code being determined in accordance with bits in previous differentially encoded r bit groups,

(c) associating each expanded r+1 bit group produced by said convolutional code with a signalling alphabet in a two-dimensional constellation, and

(d) modulating said in-phase and quadrature phase carriers in accordance with the coordinates of said signalling alphabets,

wherein said signalling alphabets are arranged in said constellation such that

(1) the 2^r alphabets which correspond to all permitted transitions from each state of said convolutional code to m possible following states of said convolutional code, have a larger minimum distance than the complete set of 2^r+1 signalling alphabets,

(2) the 2^r alphabets which correspond to all permitted transitions from m possible previous states of said convolutional code to each current state of said convolutional code have a larger minimum distance than the complete set of 2^r+1 signalling alphabets,

(3) the coordinates of all signalling alphabets in said constellation are preserved when said constellation is rotated by 90, 180 or 270 degrees, and

(4) rotation of said differentially encoded bits in each r bit group results in association of the produced r+1 bit group with a signalling alphabet in said rotated constellation which corresponds to the alphabet in the original constellation associated with the r+1 bit group produced without rotation.

3. A technique for modulating in-phase and quadrature phase carriers in accordance with an applied sequence of binary digits, including the steps of:

(a) differentially encoding at least some of the binary digits in each current group of r bits in said applied sequence by rotating the values of previous differentially encoded binary digits by a position differential determined by said binary digits in said current group,

(b) expanding c bits in each group of r differentially encoded bits into a group of c+1 bits by a non-linear convolutional encoding process utilizing m internal states, the current state of said convolutional code being determined by p stored bits, where 2^p =m, said p stored bits being determined by bits in previous differentially encoded r bit groups, said c bits including only bits in each r bit group which are differentially encoded,

(c) selecting one of 2^r+1 signalling alphabets from a signal constellation as a function of the expanded r+1 bit group which includes said c+1 convolutionally encoded bits and the remaining r-c bits produced by said differential encoding step, the selection being made so that 2^r signalling alphabets correspond to all permitted transitions from each current internal state of the convolutional coding process to m possible next states of the convolutional coding process, and that the 2^r signalling alphabets have a larger minimum distance from each other than the minimum distance within the complete set of 2^r+1 signalling alphabets, and

(d) modulating said in-phase and quadrature-phase carriers in accordance with the sequence of selected signalling alphabets,

wherein said signalling alphabets form a constellation having 90, 180 and 270 degree symmetry, the location of each signalling alphabet in said constellation being uniquely defined by its associated coordinates, the mapping between expanded r+1 bit groups and said alphabets in said signal constellation being arranged so that rotation of said signal constellation by 90, 180 or 270 degrees results in selection of the same signalling alphabet as rotation of those differentially encoded bits in each r bit group included among the c convolutionally encoded bits.

4. A method for transmitting a bit sequence by modulating in-phase and quadrature phase carriers to sequentially assume one of a plurality of discrete values in accordance with the coordinates associated with a sequence of signalling alphabets, comprising the steps of:

(a) differentially encoding at least two bits in each group of r sequential bits by rotating the values of the corresponding bits in previous differentially encoded r bit group in accordance with the values of said at least two bits,

(b) expanding each differentially encoded r bit group into a group of r+1 bits by a non-linear convolutional coding process utilizing multiple internal states, the current state of said convolutional code being determined by bits in previous differentially encoded r bit groups,

(c) selecting one out of 2^r+1 signalling alphabets from a signal constellation as a function of each expanded r+1 bit group, the selection being made so that 2^r signalling alphabets correspond to all permitted transitions from each current internal state of the coding process to possible follower states and that the 2^r signalling alphabets have a larger minimum distance from each other than the minimum distance within the complete set of 2^r+1 signalling alphabets, and

(d) modulating said in-phase and quadrature phase carriers in accordance with the sequence of selected signalling alphabets,

wherein said signalling alphabets form a constellation having 90, 180 and 270 degree symmetry, the location of each signalling alphabet in said constellation being uniquely defined by its associated coordinates, the mapping between each expanded r+1 bit group and said alphabet in said constellation being arranged so that rotation of said signal constellation by 90, 180 or 270 degrees results in selection of the same signalling alphabet as rotation of said at least two differentially encoded bits in each r bit group.

5. A technique for modulating in-phase and quadrature phase carriers to assume a sequence of discrete values u_n and v_n, respectively, in accordance with an applied sequence of bits, including:

(1) expanding each current group of r bits in said applied sequence into a sequence of r+1 bits in accordance with the state of an m-state convolutional code, each of said m-states being defined by p stored bits, where 2^p =m, said p stored bits being determined by bits in previous r bit groups,

(2) selecting said discrete values in accordance with the coordinates of the one of 2^r+1 signalling alphabets associated with the values of the bits in each expanded r+1 bit group, said selection being made so that 2^r signalling alphabets correspond to all permitted transitions from each current one of said m states to m possible next states, and so that the minimum distance between said 2^r signalling alphabets is larger than the minimum distance between the complete set of 2^r+1 signalling alphabets, and

(3) modulating said in-phase and quadrature-phase carriers in accordance with said selected discrete values,

characterized in that

(4) said technique further includes the step of differentially encoding at least one pair of bits in each group of r bits before said expanding step by rotating the values of a pair of previous differentially encoded bits by a position differential determined by the values of said bits in said at least one pair of bits,

(5) said expanding step includes determining said p bits which define said next state as a non-linear function of said p bits which define said current state and bits in said current r bit group, and

(6) said technique further includes forming said signalling alphabets in a constellation with only 90, 180 and 270 degree symmetry such that rotation of those of said differentially encoded bits processed in said expanding step has the same effect as rotation of said signalling alphabets by 90, 180 or 270 degrees.

6. A technique for modulating in-phase and quadrature phase carriers in accordance with a sequence of binary digits, by associating each group of r binary digits with a signalling alphabet in a constellation, the coordinates of each alphabet determining the amplitudes of said in-phase and quadrature phase modulations, respectively, including the steps of:

(a) rotating at least some of the binary digits in each of the r bit groups of binary digits output from a differential encoder in accordance with the values of the corresponding bits in each current r bit group,

(b) expanding each differentially encoded r bit group into an associated group of r+1 bits by a non-linear convolutional code utilizing m states, the current state of said convolutional code being determined by p bits in previous differentially encoded r bit groups, where p is an integer such that 2^p =m, and

(c) selecting for each of said expanded r+1 bit groups a particular signalling alphabet in said constellation, said selection being from a set of 2^r alphabets which correspond to all permitted transitions from said current state of said convolutional code to m possible next states of said convolutional code, alphabets in said 2^r set having a larger minimum distance from each other than alphabets in the complete set of 2^r+1 alphabets,

wherein said constellation is chosen such that the same constellation is obtained when the original constellation is rotated by 90, 180 or 270 degrees, and a different constellation is obtained for other amounts of rotation, and

wherein the mapping between said signalling alphabets and said groups of r+1 bits is arranged such that rotation of the sequence of mapped signalling alphabets by 90, 180 or 270 degrees has the same effect on said modulated carriers as is obtained by rotating consistently at least some of said differentially encoded bits in each r bit group before said expansion.

7. A technique for modulating in-phase and quadrature phase carriers to assume a sequence of discrete values u_n and v_n, respectively, in accordance with an applied sequence of bits, including:

(1) expanding each current group of r bits in said applied sequence into a group of r+1 bits in accordance with the state of an m-state convolutional code,

(2) selecting said discrete values in accordance with the coordinates of the one of 2^r+1 signalling alphabets associated with the values of the bits in each expanded r+1 bit group, said selecting being made so that 2^r signalling alphabets correspond to all permitted transitions from each current one of said m states to m possible next states, and so that the minimum distance between said 2^r signalling alphabets is larger than the minimum distance between the complete set of 2^r+1 signalling alphabets, and

(3) modulating said in-phase and quadrature-phase carriers in accordance with said selected discrete values,

characterized in that

(4) said technique further includes the step of differentially encoding at least one pair of bits in each group of r bits before said expanding step by rotating the values of a pair of previous differentially encoded bits by a position differential determined by the values of said bits in said at least one pair of bits,

(5) said expanding step includes determining bits in said r+1 bit group as a non-linear function of bits in said current and previous r bit groups, and

(6) said technique further includes forming said signalling alphabets in a constellation with only 90, 180 and 270 degree symmetry such that rotation of said differentially encoded bits processed in said expanding step has the same effect as rotation of said signalling alphabets by 90, 180 or 270 degrees.

8. Apparatus for transmitting a bit sequence by modulating a carrier signal to sequentially assume one of a plurality of pairs of discrete signal values, comprising

(a) means for differentially encoding bits in each group of r bits in said sequence by rotating the values of bits in the previous differentially encoded r bit group in accordance with the values of bits in the current r bit group,

(b) means for expanding each differentially encoded r bit group into a group of r+1 bits by a non-linear convolutional coding process utilizing m internal states, the current state of said convolutional code being determined by p stored bits, where 2^p =m, said p stored bits being functions of the previous bits input to said convolutional encoder and affecting the output of said convolutional encoder,

(c) means for selecting one of 2^r+1 signalling alphabets from a signal constellation, each alphabet corresponding to a pair of discrete carrier signal values, said selection means being arranged so that 2^r signalling alphabets correspond to all permitted transitions from each current internal state of said convolutional coding process to m possible next states of said convolutional coding process, and so that the 2^r signalling alphabets have a larger minimum distance from each other than the minimum distance within the complete set of 2^r+1 signalling alphabets, and

(d) means for transmitting the pairs of discrete carrier signal values corresponding to the selected signalling alphabets in sequence,

wherein said signal constellation is arranged so that the same signal constellation is obtained when the original constellation is rotated by 90, 180 or 270 degrees, and wherein

rotation of said constellation by 90, 180 or 270 degrees produces the same effect on said transmitted discrete signal values as rotation of said differentially encoded bits in each r bit group.

9. Apparatus for modulating in-phase and quadrature phase carriers in accordance with an applied sequence of binary digits, including

(a) means for differentially encoding bits in each current group of r bits in said applied sequence by rotating the values of previous differentially encoded bits by a position differential determined by said bits in said current group,

(b) means for encoding each differentially encoded r bit group into an associated r+1 bit group in accordance with a multi-state non-linear convolutional code, the current state of said convolutional code being determined in accordance with bits in previous differentially encoded r bit groups,

(c) means for associating each expanded r+1 bit group produced by said convolutional code with a signalling alphabet in a two-dimensional constellation,

and

(d) means for modulating said in-phase and quadrature phase carriers in accordance with the coordinates of said signalling alphabets,

wherein said signalling alphabets are arranged in said constellation such that

(1) the 2^r alphabets which correspond to all permitted transitions from each state of said convolutional code to m possible following states of said convolutional code, have a larger minimum distance than the complete set of 2^r+1 signalling alphabets,

(2) the 2^r alphabets which correspond to all permitted transitions from m possible previous states of said convolutional code to each current state of said convolutional code have a larger minimum distance than the complete set of 2^r+1 signalling alphabets,

(3) the coordinates of all signalling alphabets in said constellation are preserved when said constellation is rotated by 90, 180 or 270 degrees, and

(4) rotation of said differentially encoded bits in each r bit group results in association of the produced r+1 bit group with a signalling alphabet in said rotated constellation which corresponds to the alphabet in the original constellation associated with the r+1 bit group produced without rotation.

10. Apparatus for modulating in-phase and quadrature phase carriers in accordance with an applied sequence of binary digits, including

(a) means for differentially encoding at least some of the binary digits in each current group of r bits in said applied sequence by rotating the values of previous differentially encoded binary digits by a position differential determined by said binary digits in said current group,

(b) means for expanding c bits in each group of r differentially encoded bits into a group of c+1 bits by a non-linear convolutional encoding process utilizing m internal states, the current state of said convolutional code being determined by p stored bits, where 2^p =m, said p stored bits being determined by bits in previous differentially encoded r bit groups, said c bits including only bits in each r bit group which are differentially encoded,

(c) means for selecting one of 2^r+1 signalling alphabets from a signal constellation as a function of the expanded r+1 bit group which includes said c+1 convolutionally encoded bits and the remaining r-c bits produced by said differential encoding step, said selecting means being arranged so that 2^r signalling alphabets correspond to all permitted transitions from each current internal state of the convolutional coding process to m possible next states of the convolutional coding process, and that the 2^r signalling alphabets have a larger minimum distance from each other than the minimum distance within the complete set of 2^r+1 signalling alphabets, and

(d) means for modulating said in-phase and quadrature-phase carriers in accordance with the sequence of selected signalling alphabets,

wherein said signalling alphabets form a constellation having 90, 180 and 270 degree symmetry, the location of each signalling alphabet in said constellation being uniquely defined by its associated coordinates, the mapping between expanded r+1 bit groups and said alphabets in said signal constellation being arranged so that rotation of said signal constellation by 90, 180 or 270 degrees results in selection of the same signalling alphabet as rotation of those differentially encoded bits in each r bit group included among the c convolutionally encoded bits.

11. Apparatus for transmitting a bit sequence by modulating in-phase and quadrature phase carriers to sequentially assume one of a plurality of discrete values in accordance with the coordinates associated with a sequence of signalling alphabets, comprising

(a) means for differentially encoding at least two bits in each group of r sequential bits by rotating the values of the corresponding bits in previous differentially encoded r bit group in accordance with the values of said at least two bits,

(b) means for expanding each differentially encoded r bit group into a group of r+1 bits by a non-linear convolutional coding process utilizing multiple internal states, the current state of said convolutional code being determined by bits in previous differentially encoded r bit groups,

(c) means for selecting one out of 2^r+1 signalling alphabets from a signal constellation as a function of each expanded r+1 bit group, the selection being made so that 2^r signalling alphabets correspond to all permitted transitions from each current internal state of the coding process to possible follower states and that the 2^r signalling alphabets have a larger minimum distance from each other than the minimum distance within the complete set of 2^r+1 signalling alphabets, and

(d) means for modulating said in-phase and quadrature phase carriers in accordance with the sequence of selected signalling alphabets,

wherein said signalling alphabets form a constellation having 90, 180 and 270 degree symmetry, the location of each signalling alphabet in said constellation being uniquely defined by its associated coordinates, the mapping between each expanded r+1 bit group and said alphabet in said constellation being arranged so that rotation of said signal constellation by 90, 180 or 270 degrees results in selection of the same signalling alphabet as rotation of said at least two differentially encoded bits in each r bit group.

12. Apparatus for modulating in-phase and quadrature phase carriers to assume a sequence of discrete values u_n and v_n, respectively, in accordance with an applied sequence of bits, including:

(1) means for expanding each current group of r bits in said applied sequence into a sequence of r+1 bits in accordance with the state of an m-state convolutional code, each of said m-states being defined by p stored bits, where s^p =m, said p stored bits being determined by bits in previous r bit groups,

(2) means for selecting said discrete values in accordance with the coordinates of the one of 2^r+1 signalling alphabets associated with the values of the bits in each expanded r+1 bit group, said selection being made so that 2^r signalling alphabets correspond to all permitted transitions from each current one of said m states to m possible next states, and so that the minimum distance between said 2^r signalling alphabets is larger than the minimum distance between the complete set of 2^r+1 signalling alphabets, and

(3) means for modulating said in-phase and quadrature-phase carriers in accordance with said selected discrete values,

characterized in that

(4) said apparatus further includes means for differentially encoding at least one pair of bits in each group of r bits applied to said expanding means by rotating the values of a pair of previous differentially encoded bits by a position differential determined by the values of said bits in said at least one pair of bits,

(5) said expanding means includes means for determining said p bits which define said next state as a non-linear function of said p bits which define said current state and bits in said current r bit group, and

(6) said selecting means is arranged to form said signalling alphabets in a constellation with only 90, 180 and 270 degree symmetry such that rotation of those of said differentially encoding bits processed by said expanding means has the same effect as rotation of said signalling alphabets by 90, 180 or 270 degrees.

13. Apparatus for modulating in-phase and quadrature phase carriers in accordance with a sequence of binary digits, by associating each group of r binary digits with a signalling alphabet in a constellation, the coordinates of each alphabet determining the amplitudes of said in-phase and quadrature phase modulations, respectively, including

(a) means for rotating at least some of the binary digits in each of the r bit groups of binary digits output from a differential encoder in accordance with the values of the corresponding bits in each current r bit group,

(b) means for expanding each differentially encoded r bit group into an associated group of r+1 bits by a non-linear convolutional code utilizing m states, the current state of said convolutional code being determined by p bits in previous differentially encoded r bit groups, where p is an integer such that 2^p =m, and

(c) means for selecting for each of said expanded r+1 bit groups a particular signalling alphabet in said constellation, said selection being from a set of 2^r alphabets which correspond to all permitted transitions from said current state of said convolutional code to m possible next states of said convolutional code, alphabets in said 2^r set having a larger minimum distance from each other than alphabets in the complete set of 2^r+1 alphabets,

wherein said constellation is chosen such that the same constellation is obtained when the original constellation is rotated by 90, 180 or 270 degrees, and a different constellation is obtained for other amounts of rotation, and

wherein the mapping between said signalling alphabets and said groups of r+1 bits is arranged such that rotation of the sequence of mapped signalling alphabets by 90, 180 or 270 degrees has the same effect on said modulated carriers as is obtained by rotating consistently at least some of said differentially encoded bits in each r bit group before said expansion.

14. Apparatus for modulating in-phase and quadrature phase carriers to assume a sequence of discrete values u_n and v_n, respectively, in accordance with an applied sequence of bits, including:

(1) means for expanding each current group of r bits in said applied sequence into a group of r+1 bits in accordance with the state of an m-state convolutional code,

(2) means for selecting said discrete values in accordance with the coordinates of the one of 2^r+1 signalling alphabets associated with the values of the bits in each expanded r+1 bit group, said selecting means being arranged so that 2^r signalling alphabets correspond to all permitted transitions from each current one of said m states to m possible next states, and so that the minimum distance between said 2^r signalling alphabets is larger than the minimum distance between the complete set of 2^r+1 signalling alphabets, and

(3) means for modulating said in-phase and quadrature-phase carriers in accordance with said selected discrete values,

characterized in that

(4) said apparatus further includes means for differentially encoding at least one pair of bits in each group of r bits applied to said expanding means by rotating the values of a pair of previous differentially encoded bits by a position differential determined by the values of said bits in said at least one pair of bits,

(5) said expanding means is arranged to determine bits in said r+1 bit group as a non-linear function of bits in said current and previous r bit groups, and

(6) said selecting means is arranged to form said signalling alphabets in a constellation with only 90, 180 and 270 degree symmetry such that rotation of said differentially encoded bits processed in said expanding step has the same effect as rotation of said signalling alphabets by 90, 180 or 270 degrees.

15. A method for transmitting a bit sequence by modulating a carrier signal to sequentially assume one of a plurality of pairs of discrete signal values, comprising the steps of:

(a) expanding each current bit group into a group of r+1 bits by a finite state non-linear convolutional coding process utilizing p stored bits, the current values of said p stored bits affecting the current output of said convolutional encoder and being determined by both the previous r bit group input to said convolutional encoder and previous values of said p stored bits,

(b) selecting one of 2^r+1 signalling alphabets from a signal constellation, each alphabet corresponding to a pair of discrete signal values, said selection being made as a function of the r+1 bit group so that 2^r signalling alphabets correspond to all permitted transitions from each current internal state of said convolutional coding process to all possible next states of said convolutional coding process, and so that the 2^r signalling alphabets have a larger minimum distance from each other than the minimum distance within the complete set of 2^r+1 signalling alphabets, and

(c) transmitting the pairs of discrete signal values corresponding to the selected signalling alphabets in sequence,

wherein said signal constellation is arranged so that the same signal constellation is obtained when the original constellation is rotated by 90, 180 or 270 degrees, and

wherein rotation of said constellation by 90, 180 or 270 degrees produces the same effect on said transmitted discrete signal values as modification of only said current values of said p stored bits without modification of said values of said current r bit group.

16. A method for transmitting a bit sequence by modulating a carrier signal to sequentially assume one of a plurality of pairs of discrete signal values, comprising the steps of:

(a) expanding each current bit group into a group of r+1 bits by finite state non-linear convolutional coding process utilizing p stored bits, the current values of said p stored bits affecting the current output of said convolutional encoder and being determined by both the previous r bit group input to said convolutional encoder and previous values of said p stored bits,

(b) selecting one of 2^r+1 signalling alphabets from a signal constellation, each alphabet corresponding to a pair of discrete signal values, said selection being made as a function of the r+1 bit group so that 2^r signalling alphabets correspond to all permitted transitions from each current internal state of said convolutional coding process to all possible next states of said convolutional coding process, and so that the 2^r signalling alphabets have a larger minimum distance from each other than the minimum distance within the complete set of 2^r+1 signalling alphabets, and

(c) transmitting the pairs of discrete signal values corresponding to the selected signalling alphabets in sequence,

wherein said signal constellation is arranged so that the same signal constellation is obtained when the original constellation is rotated by 90, 180 or 270 degrees, and

wherein in said expanding step there are, for each encoder input bit sequence and each set of initial values of said p stored bits, 3 other sets of initial values of said p stored bits such that the 4 corresponding sequences of pairs of discrete signal values are related to each other by rotation of said signal constellation by 90, 180 or 270 degrees.

17. A method for encoding a sequence of bits for transmission to a receiver via a modulated carrier including the steps of:

(a) representing each group of r bits in said bit sequence by an associated r+1 bit group in accordance with the values of said r bits and the current internal state of and m-state non-linear convolutional encoder, where said states are designated by an index i=1,2, . . . m,

(b) mapping each r+1 bit group to an associated signalling alphabet in a constellation having 90, 180 or 270 degree symmetry, and

(c) modulating in phase and quadrature phase components of said carrier in accordance with the coordinates of said associated signalling alphabet,

wherein said convolutional encoder is arranged to allow transitions from each of said current states to only selected next states, each of said allowed transitions having an associated set of signalling alphabets, such that (1) three one-to-one functions f₁, f₂, and f₃, map each encoder state to another encoder state, and (2) for each pair of current encoder state i and next encoder state j, the three sets of signalling alphabets associated with transitions from current encoder state f (i) to next encoder state f (j)=1,2,3 can be obtained from the set of signalling alphabets associated with the transition from current encoder state i to next state j by rotating the last mentioned set of signalling alphabets by 90, 180 or 270 degrees, respectively,

whereby the values of said bits in said r bit group are recoverable at said receiver despite rotations of the signalling constellation by 90, 180 or 270 degrees.

18. Apparatus for transmitting a bit sequence by modulating a carrier signal to sequentially assume one of a plurality of pairs of discrete signal values, comprising:

means for expanding each current bit group into a group of r+1 bits by a finite state non-linear convolutional coding process utilizing p stored bits, the current values of said p stored bits affecting the current output of said convolutional encoder and being determined by both the previous r bit group input to said convolutional encoder and previous values of said p stored bits,

(b) means for selecting one of 2^r+1 signalling alphabets from a signal constellation, each alphabet corresponding to a pair of discrete signal values, said selection being made as a function of the r+1 bit group so that 2^r signalling alphabets correspond to all permitted transitions from each current internal state of said convolutional coding process to all possible next states of said convolutional coding process, and so that the 2^r signalling alphabets have a larger minimum distance from each other than the minimum distance within the complete set of 2^r+1 signalling alphabets, and

means for transmitting the pairs of discrete signal values corresponding to the selected signalling alphabets in sequence,

wherein said signal constellation is arranged so that the same signal constellation is obtained when the original constellation is rotated by 90, 180 or 270 degrees, and

wherein rotation of said constellation by 90, 180 or 270 degrees produces the same effect on said transmitted discrete signal values as modification of only said current values of said p stored bits without modification of said values of said current r bit group.