Symbol state trellis maximum likelihood detection method

Claim 2

In a continuous phase modulation (CPM) system having M symbols and an intersymbol interference of L, a symbol state trellis maximum likelihood detection method for limiting the number of interfering symbols which are required for examination, said symbol state trellis maximum likelihood detection method comprising the steps of: determining Euclidean branch distances for a plurality of M symbols from a present time period T N to said plurality of M symbols at a previous time period T N-1 ; determining an error of total Euclidean path distances corresponding to each of said Euclidean branch distances for said plurality of M symbols from said present time period T N to said previous time period T N-1 ; saving a particular path to each of said plurality of M symbols at time period T N from time period T N-1 , each of said particular paths corresponding to a minimum error of said total Euclidean path distances; and detecting symbols received by retracing said Euclidean path distances corresponding to said minimum error for M time periods from said symbol at time period T N having said minimum error.

Claim 3

A symbol state trellis maximum likelihood detection method as claimed in claim 1, wherein said step of determining Euclidean branch distances includes the step of setting symbol 1, a symbol at time period T N-1 , with an initial symbol value of said plurality of M symbols.

Claim 4

A symbol state trellis maximum likelihood detection method as claimed in claim 2, wherein said step of determining Euclidean branch distances further includes the steps of: looking-up a value of previously stored symbols prior to symbol 1; setting symbol N (for N greater than or equal to 2) equal to said value; and iterating said steps of looking-up and setting for L time periods.

Claim 5

A symbol state trellis maximum likelihood detection method as claimed in claim 3, wherein said step of determining Euclidean branch distances further includes the step of setting symbol 0 with said initial symbol value of said plurality of M symbols.

Claim 6

A symbol state trellis maximum likelihood detection method as claimed in claim 4, wherein said step of determining Euclidean branch distances further includes the step of: clearing an accumulated phase indicator; and setting an indication of a current sample to be that of a first sample.

Claim 7

A symbol state trellis maximum likelihood detection method as claimed in claim 5, wherein said step of determining Euclidean branch distances further includes the steps of: finding a received phase for said current sample; looking-up an expected phase for said current sample; subtracting said expected phase from said phase of said current sample to produce a difference; determining the absolute value of said difference of said expected phase from said phase of said sample; and adding said absolute value to said accumulated phase indicator to produce an accumulated phase error.

Claim 8

A symbol state trellis maximum likelihood detection method as claimed in claim 6, wherein said step of determining Euclidean branch distances further includes the steps of: incrementing a current sample count indicator; determining whether said current sample count indicator is less than or equal to a predetermined maximum sample counts per period; and iterating said steps of claims 6 and 7, where said current sample count indicator is less than or equal to said predetermined maximum.

Claim 9

A symbol state trellis maximum likelihood detection method as claimed in claim 7, wherein said step of determining Euclidean branch distances further includes the steps of: saving said accumulated phase error for each of said branch distances to symbol 0; incrementing or decrementing said value of symbol 0 by a predefined number, so that the value of symbol 0 corresponds to a next sequential symbol of said plurality of M symbols; determining whether said value of symbol 0 is less than or equal to M-1 (greater than or equal to -(M-1); and iterating said steps of claims 5 through 8, when said value of symbol 0 is less than or equal to M-1.

Claim 10

A symbol state trellis maximum likelihood detection method as claimed in claim 8, wherein said step of determining Euclidean branch distances further includes the steps of: incrementing or decrementing said value of symbol 1 by said predefined number, so that the value of symbol 1 corresponds to said next sequential symbol of said plurality of M symbols; determining whether said value of symbol 1 is less than or equal to M-1 (greater than or equal to -(M-1); and iterating said steps of claims 4 through 9, when said value of symbol 1 is less than or equal to M-1.

Claim 11

A symbol state trellis maximum likelihood detection method as claimed in claim 9, wherein said step of determining an error of total Euclidean path distances includes the steps of: setting a lowest path distance indicator for all symbols with a predetermined largest possible value; and setting said value of symbol 0 with said value of the largest negative symbol of said plurality of M symbols.

Claim 12

A symbol state trellis maximum likelihood detection method as claimed in claim 10, wherein said step of determining an error of total Euclidean path distances further includes the steps of: setting a lowest path distance indicator for symbol 0 only with said largest possible value; and setting the value of symbol 1 with said value of the largest negative symbol of said plurality of M symbols.

Claim 13

A symbol state trellis maximum likelihood detection method as claimed in claim 11, wherein said step of determining an error of total Euclidean path distances further includes the steps of: adding a total accumulated phase error for a path distance to symbol 1 to the accumulated phase error for a branch distance from symbol 1 at time period T N-1 to symbol 0 at time period T N ; comparing said accumulated phase error to the lowest path distance indicator for symbol 0 only; determining whether said accumulated phase error is less than said lowest path distance indicator for symbol 0 only.

Claim 14

A symbol state trellis maximum likelihood detection method as claimed in claim 12, wherein said step of determining an error of total Euclidean path distances further includes the steps of: incrementing or decrementing said value of symbol 1 by said predetermined number, so that the value of symbol 1 corresponds to the next sequential symbol of said plurality of M symbols; determining whether said value of symbol 1 is less than or equal to M-1 (greater than or equal to -(M-1); and iterating said steps of claims 12 through 14, when said value of symbol 1 is less than or equal to M-1.

Claim 15

A symbol state trellis maximum likelihood detection method as claimed in claim 13, where said step of determining an error of total Euclidean path distances further includes the steps of: normalizing each of said path distances to the value of symbol 0 based upon said lowest path distance indicator for symbol 0 only; adding the path distance from symbol 1 to symbol 0 to the previous path to symbol 1; saving the added path distance; and comparing the lowest path distance indicator for symbol 0 only to the lowest path distance indicator for all symbols.

Claim 16

A symbol state trellis maximum likelihood detection method as claimed in claim 14, said step of determining an error of total Euclidean path distances further including the step of determining whether said lowest path distance indicator for symbol 0 is less than the lowest path distance indicator for all symbols.

Claim 17

A symbol state trellis maximum likelihood detection method as claimed in claim 15, wherein further included the steps of: incrementing or decrementing said value of symbol 0 by said predefined number, so that the value of symbol 0 corresponds to next sequential symbol of said plurality of M symbols; determining whether said value of symbol 0 is less than or equal to M-1 (greater than or equal to -(M-1); and iterating said steps of claims 11 through 18, when said value of symbol 0 is less than or equal to M-1.

Claim 18

In a continuous phase modulation (CPM) system having M symbols and an intersymbol interference of L, a symbol state trellis maximum likelihood detection method for limiting the number of interfering symbols which are required for examination, said symbol state trellis maximum likelihood detection method comprising the steps of: determining Euclidean branch distances for plurality of M symbols from a present time period T N to said plurality of M symbols at previous time periods T N-1 ; determining an error of total Euclidean path distances corresponding to each of said Euclidean branch distances for said plurality of M symbols from said present time period T N to said previous time period T N-1 ; and detecting symbols received by retracing said Euclidean path distances corresponding to said minimum error for M time periods from said symbol at said time period T N having a minimal phase error.

Claim 19

In a continuous phase modulation (CPM) system having M symbols and an intersymbol interference of L, a symbol state trellis maximum likelihood detection method for limiting the number of interfering symbols which are required for examination, said symbol state trellis maximum likelihood detection method comprising the steps of: determining Euclidean branch distances for plurality of M symbols from a present time period T N to said plurality of M symbols at previous time periods T N-1 ; determining an error of total Euclidean path distances corresponding to each of said Euclidean branch distances for said plurality of M symbols from said present time period T N to said previous time period T N-1 ; and detecting symbols received by retracing said Euclidean path distances corresponding to said minimum error for M time periods from said symbol at said time period T N having a minimal phase error.

Claim 20

In a continuous phase modulation (CPM) system having M symbols and an intersymbol interference of L, a symbol state trellis maximum likelihood detection method for limiting the number of interfering symbols which are required for examination, said symbol state trellis maximum likelihood detection method comprising the steps of: determining Euclidean branch distances for a plurality of M symbols from a present time period T N to said plurality of M symbols at a previous time period T N-1 ; determining an error of total Euclidean path distances corresponding to each of said Euclidean branch distances for said plurality of M symbols from said present time period T N to said previous time period T N-1 ; saving a particular path to each of said plurality of M symbols at time period T N from time period T N-1 , each of said particular paths corresponding to a minimum error of said total Euclidean path distances; detecting symbols received by retracting said Euclidean path distances corresponding to said minimum error for M time periods from said symbol at time period T N having said minimum error; said step of determining Euclidean branch distances includes the step of setting symbol 1, a symbol at time period T N-1 , with an initial symbol value of said plurality of M symbols; said step of determining Euclidean branch distances further includes the steps of: looking-up a value of previously stored symbols prior to symbol 1; setting symbol N (for N greater than or equal to 2) equal to said value; and iterating said steps of looking-up and setting for L time periods; said step of determining Euclidean branch distances further includes the step of setting symbol 0 said initial symbol value of said plurality of M symbols; said step of determining Euclidean branch distances further includes the step of: clearing an accumulated phase indicator; and setting an indicator of a current sample to be that of a first sample; said step of determining Euclidean branch distances further includes the steps of: finding a received phase for said current sample; looking-up an expected phase for said current sample; subtracting said expected phase from said phase of said current sample to produce a difference; determining the absolute value of said difference of said expected phase from said phase of said simple; and adding said absolute value to said accumulated phase indicator to produce an accumulated phase error; said step of determining Euclidean branch distances further includes the steps of: incrementing a current sample count indicator; determining whether said current sample count indicator is less than or equal to a predetermined maximum sample counts per period; and iterating said steps of finding, looking-up, subtracting, determining, adding, incrementing, and determining, when said current sample count indicator is less than or equal to said predetermined maximum; said step of determining Euclidean branch distances further includes the steps of: saving said accumulated phase error for each of said branch distances to symbol 0; incrementing or decrementing said value of symbol 0 by a predefined number, so that the value of symbol 0 corresponds to a next sequential symbol of said plurality of M symbols; determining whether said value of symbol 0 is less than or equal to M-1 (greater than or equal to -(M-1); and iterating said steps of clearing, setting, finding looking-up, subtracting, determining, adding, incrementing, iterating, saving, incrementing, determining, and iterating, when said value of symbol 0 is less than or equal to M-1; said step of determining Euclidean branch distances further includes the steps of: incrementing or decrementing said value of symbol 1 by said predefined number, so that the value of symbol 1 corresponds to said next sequential symbol of said plurality of M symbols; determining whether said value of symbol 1 is less than or equal to M-1 (greater than or equal to -(M-1); and iterating said steps of clearing, setting, finding looking-up, subtracting, determining, adding, incrementing, iterating, saving, incrementing, determining, iterating, incrementing, and determining, when said value of symbol 1 is less than or equal to M-1; said step of determining an error of total Euclidean path distances includes the steps of: setting a lowest path distance indicator for all symbols with a predetermined largest possible value; and setting said value of symbol 0 with said value of the largest negative symbol of said plurality of M symbols; said step of determining an error of total Euclidean path distances further includes the steps of: setting a lowest path distance indicator for symbol 0 only with said largest possible value; and setting the value of symbol 1 with said value of the largest negative symbol of said plurality of M symbols; said step of determining an error of total Euclidean path distances further includes the steps of: adding a total accumulated phase error for a path distance to symbol 1 to the accumulated phase error for a branch distance from symbol 1 at time period T N-1 to symbol 0 at time period T N ; comparing said accumulated phase error to the lowest path distance indicator for symbol 0 only; determining whether said accumulated phase error is less than said lowest path distance indicator for symbol 0 only; and said step of determining an error of total Euclidean path distances further includes the steps of: setting said lowest path distance indicator for symbol 0 only with the smallest determined accumulated phase error for the path corresponding to symbol 0; saving said path distance from said previous symbol to symbol 0 as a new value for symbol 1; and omitting said steps of setting said lowest path distance and saving said path distance, when said accumulated phase error is greater than or equal to said lowest path distance indicator for symbol 0 only.

Claim 21

In a continuous phase modulation (CPM) system having M symbols and an intersymbol interference of L, a symbol state trellis maximum likelihood detection method for limiting the number of interfering symbols which are required for examination, said symbol state trellis maximum likelihood detection method comprising the steps of: determining Euclidean branch distances for a plurality of M symbols from a present time period T N to said plurality of M symbols at a previous time period T N-1 ; determining an error of total Euclidean path distances corresponding to each of said Euclidean branch distances for said plurality of M symbols from said present time period T N to said previous time period T N-1 ; saving a particular path to each of said plurality of M symbols at time period T N from time period T N-1 , each of said particular paths corresponding to a minimum error of said total Euclidean path distances; detecting symbols received by retracting said Euclidean path distances corresponding to said minimum error for M time periods from said symbol at time period T N having said minimum error; said step of determining Euclidean branch distances includes the step of setting symbol 1, a symbol at time period T N-1 , with an initial symbol value of said plurality of M symbols; said step of determining Euclidean branch distances further includes the steps of: looking-up a value of previously stored symbols prior to symbol 1; setting symbol N (for N greater than or equal to 2) equal to said value; and iterating said steps of looking-up and setting for L time periods; said step of determining Euclidean branch distances further includes the step of setting symbol 0 said initial symbol value of said plurality of M symbols; said step of determining Euclidean branch distances further includes the step of: clearing an accumulated phase indicator; and setting an indicator of a current sample to be that of a first sample; said step of determining Euclidean branch distances further includes the steps of: finding a received phase for said current sample; looking-up an expected phase for said current sample; subtracting said expected phase from said phase of said current sample to produce a difference; determining the absolute value of said difference of said expected phase from said phase of said simple; and adding said absolute value to said accumulated phase indicator to produce an accumulated phase error; said step of determining Euclidean branch distances further includes the steps of: incrementing a current sample count indicator; determining whether said current sample count indicator is less than or equal to a predetermined maximum sample counts per period; and iterating said steps of finding, looking-up, subtracting, determining, adding, incrementing, and determining, when said current sample count indicator is less than or equal to said predetermined maximum; said step of determining Euclidean branch distances further includes the steps of: saving said accumulated phase error for each of said branch distances to symbol 0; incrementing or decrementing said value of symbol 0 by a predefined number, so that the value of symbol 0 corresponds to a next sequential symbol of said plurality of M symbols; determining whether said value of symbol 0 is less than or equal to M-1 (greater than or equal to -(M-1); and iterating said steps of clearing, setting, finding looking-up, subtracting, determining, adding, incrementing, iterating, saving, incrementing, determining, and iterating, when said value of symbol 0 is less than or equal to M-1; said step of determining Euclidean branch distances further includes the steps of: incrementing or decrementing said value of symbol 1 by said predefined number, so that the value of symbol 1 corresponds to said next sequential symbol of said plurality of M symbols; determining whether said value of symbol 1 is less than or equal to M-1 (greater than or equal to -(M-1); and iterating said steps of clearing, setting, finding looking-up, subtracting, determining, adding, incrementing, iterating, saving, incrementing, determining, iterating, incrementing, and determining, when said value of symbol 1 is less than or equal to M-1; said step of determining an error of total Euclidean path distances includes the steps of: setting a lowest path distance indicator for all symbols with a predetermined largest possible value; and setting said value of symbol 0 with said value of the largest negative symbol of said plurality of M symbols; said step of determining an error of total Euclidean path distances further includes the steps of: setting a lowest path distance indicator for symbol 0 only with said largest possible value; and setting the value of symbol 1 with said value of the largest negative symbol of said plurality of M symbols; said step of determining an error of total Euclidean path distances further includes the steps of: adding a total accumulated phase error for a path distance to symbol 1 to the accumulated phase error for a branch distance from symbol 1 at time period T N-1 to symbol 0 at time period T N ; comparing said accumulated phase error to the lowest path distance indicator for symbol 0 only; determining whether said accumulated phase error is less than said lowest path distance indicator for symbol 0 only; and said step of determining an error of total Euclidean path distances further includes the steps of: setting said lowest path distance indicator for symbol 0 only with the smallest determined accumulated phase error for the path corresponding to symbol 0; saving said path distance from said previous symbol to symbol 0 as a new value for symbol 1; and omitting said steps of setting said lowest path distance and saving said path distance, when said accumulated phase error is greater than or equal to said lowest path distance indicator for symbol 0 only.

Claim 22

In a continuous phase modulation (CPM) system having M symbols and an intersymbol interference of L, a symbol state trellis maximum likelihood detection method for limiting the number of interfering symbols which are required for examination, said symbol state trellis maximum likelihood detection method comprising the steps of: determining Euclidean branch distances for a plurality of M symbols from a present time period T N to said plurality of M symbols at a previous time period T N-1 ; determining an error of total Euclidean path distances corresponding to each of said Euclidean branch distances for said plurality of M symbols from said present time period T N to said previous time period T N-1 ; saving a particular path to each of said plurality of M symbols at time period T N from time period T N-1 , each of said particular paths corresponding to a minimum error of said total Euclidean path distances; detecting symbols received by retracting said Euclidean path distances corresponding to said minimum error for M time periods from said symbol at time period T N having said minimum error; said step of determining Euclidean branch distances includes the step of setting symbol 1, a symbol at time period T N-1 , with an initial symbol value of said plurality of M symbols; said step of determining Euclidean branch distances further includes the steps of: looking-up a value of previously stored symbols prior to symbol 1; setting symbol N (for N greater than or equal to 2) equal to said value; and iterating said steps of looking-up and setting for L time periods; said step of determining Euclidean branch distances further includes the step of setting symbol 0 with said initial symbol value of said plurality of M symbols; said step of determining Eucliden branch distances further includes the step of: clearing an accumulated phase indicator; and setting an indication of a current sample to be that of a first sample; said step of determining Euclidean branch distances further includes the steps of: finding a received phase for said current sample; looking-up an expected phase for said current sample; subtracting said expected phase from said phase of said current sample to produce a difference; determining the absolute value of said difference of said expected phase from said phase of said simple; and adding said absolute value to said accumulated phase indicator to produce an accumulated phase error; said step of determining Euclidean branch distances further includes the steps of: incrementing a current sample count indicator; determining whether said current sample count indicator is less than or equal to a predetermined maximum sample counts per period; and iterating said steps of finding, looking-up, subtracting, determining, adding, incrementing, and determining, when said current sample count indicator is less than or equal to said predetermined maximum; said step of determining Euclidean branch distances further includes the steps of: saving said accumulated phase error for each of said branch distances to symbol 0; incrementing or decrementing said value of symbol 0 by a predetermined number, so that the value of symbol 0 corresponds to a next sequential symbol of said plurality of M symbols, determining whether said value of symbol 0 is less than or equal to M-1 (greater than or equal to -(M-1); and iterating said steps of clearing, setting, finding looking-up, subtracting, determining, adding, incrementing, iterating, saving, incrementing, determining, and iterating, when said value of symbol 0 is less than or equal to M-1; said step of determining Euclidean branch distances further includes the steps of: incrementing or decrementing said value of symbol 1 by said predetermined number, so that the value of symbol 1 corresponds to said next sequential symbol of said plurality of M symbols; determining whether said value of symbol 1 is less than or equal to M-1 (greater than or equal to -(M-1); and iterating said steps of clearing, setting, finding looking-up, subtracting, determining, adding, incrementing, iterating, saving, incrementing, determining, iterating, incrementing, and determining, when said value of symbol 1 is less than or equal to M-1; said step of determining an error of total Euclidean path distances includes the steps of: setting a lowest path distance indicator for all symbols with a predetermined largest possible value; and setting said value of symbol 0 with said value of the largest negative symbol of said plurality of M symbols; said step of determining an error of total Euclidean path distances further includes the steps of: setting a lowest path distance indicator for symbol 0 only with said largest possible value; and setting the value of symbol 1 with said value of the largest negative symbol of said plurality of M symbols; said step of determining an error of total Euclidean path distances further includes the steps of: adding a total accumulated phase error for a path distances to symbol 1 to the accumulated phase error for a branch distance from symbol 1 at time period T N-1 to symbol 0 at time period T N ; comparing said accumulated phase error to the lowest path distance indicator for symbol 0 only; determining whether said accumulated phase error is less than said lowest path distance indicator for symbol 0 only; and said step of determining an error of total Euclidean path distances further includes the steps of: setting said lowest path distance indicator for symbol 0 only with the smallest determined accumulated phase error for the path corresponding to symbol 0; saving said path distance from said previous symbol to symbol 0 as a new value for symbol 1; and omitting said steps of setting said lowest path distance and saving said path distance, when said accumulated phase error is greater than or equal to said lowest path distance indicator for symbol 0 only; said step of determining an error of total Euclidean path distances further includes the steps of: incrementing or decrementing said value of symbol 1 by said predetermined number, so that the value of symbol 1 corresponds to the next sequential symbol of said plurality of M symbols; determining whether said value of symbol 1 is less than or equal to M-1 (greater than or equal to -(M-1); and iterating said steps of adding, comparing, determining, incrementing, and determining, when said value of symbol 1 is less than or equal to M-1; said step of determining an error of total Euclidean path distances further includes the steps of: normalizing each of said path distances to the value of symbol 0 based upon said lowest path distance indicator for symbol 0 only; adding the path distance from symbol 1 to symbol 0 to the previous path to symbol 1; saving the added path distance; and comparing the lowest path distance indicator for symbol 0 only to the lowest path distance indicator for all symbols; said step of determining an error of total Euclidean path distances further including the step of determining whether said lowest path distance indicator for symbol 0 is less than the lowest path distance indicator for all symbols; said step of saving a particular path includes the steps of: setting said lowest path distance indicator for all symbols equal to said lowest path distance indicator for symbol 0 only; and saving symbol 0 as the current symbol; and omitting said steps of setting said lowest path distance indicator and saving symbol 0, when said lowest path distance indicator for symbol 0 only is greater than or equal to the lowest path distance indicator for all symbols.

Claim 23

In a continuous phase modulation (CPM) system having M symbols and an intersymbol interference of L, a symbol state trellis maximum likelihood detection method for limiting the number of interfering symbols which are required for examination, said symbol state trellis maximum likelihood detection method comprising the steps of: determining Euclidean branch distances for a plurality of M symbols from a present time period T N to said plurality of M symbols at a previous time period T N-1 ; determining an error of total Euclidean path distances corresponding to each of said Euclidean branch distances for said plurality of M symbols from said present time period T N to said previous time period T N-1 ; saving a particular path to each of said plurality of M symbols at time period T N from time period T N-1 , each of said particular paths corresponding to a minimum error of said total Euclidean path distances; detecting symbols received by retracting said Euclidean path distances corresponding to said minimum error for M time periods from said symbol at time period T N having said minimum error; said step of determining Euclidean branch distances includes the step of setting symbol 1, a symbol at time period T N-1 , with an initial symbol value of said plurality of M symbols; said step of determining Euclidean branch distances further includes the steps of: looking-up a value of previously stored symbols prior to symbol 1; setting symbol N (for N greater than or equal to 2) equal to said value; and iterating said steps of looking-up and setting for L time periods; said step of determining Euclidean branch distances further includes the step of setting symbol 0 with said initial symbol value of said plurality of M symbols; said step of determining Eucliden branch distances further includes the step of: clearing an accumulated phase indicator; and setting an indication of a current sample to be that of a first sample; said step of determining Euclidean branch distances further includes the steps of: finding a received phase for said current sample; looking-up an expected phase for said current sample; subtracting said expected phase from said phase of said current sample to produce a difference; determining the absolute value of said difference of said expected phase from said phase of said simple; and adding said absolute value to said accumulated phase indicator to produce an accumulated phase error; said step of determining Euclidean branch distances further includes the steps of: incrementing a current sample count indicator; determining whether said current sample count indicator is less than or equal to a predetermined maximum sample counts per period; and iterating said steps of finding, looking-up, subtracting, determining, adding, incrementing, and determining, when said current sample count indicator is less than or equal to said predetermined maximum; said step of determining Euclidean branch distances further includes the steps of: saving said accumulated phase error for each of said branch distances to symbol 0; incrementing or decrementing said value of symbol 0 by a predetermined number, so that the value of symbol 0 corresponds to a next sequential symbol of said plurality of M symbols, determining whether said value of symbol 0 is less than or equal to M-1 (greater than or equal to -(M-1); and iterating said steps of clearing, setting, finding looking-up, subtracting, determining, adding, incrementing, iterating, saving, incrementing, determining, and iterating, when said value of symbol 0 is less than or equal to M-1; said step of determining Euclidean branch distances further includes the steps of: incrementing or decrementing said value of symbol 1 by said predetermined number, so that the value of symbol 1 corresponds to said next sequential symbol of said plurality of M symbols; determining whether said value of symbol 1 is less than or equal to M-1 (greater than or equal to -(M-1); and iterating said steps of clearing, setting, finding looking-up, subtracting, determining, adding, incrementing, iterating, saving, incrementing, determining, iterating, incrementing, and determining, when said value of symbol 1 is less than or equal to M-1; said step of determining an error of total Euclidean path distances includes the steps of: setting a lowest path distance indicator for all symbols with a predetermined largest possible value; and setting said value of symbol 0 with said value of the largest negative symbol of said plurality of M symbols; said step of determining an error of total Euclidean path distances further includes the steps of: setting a lowest path distance indicator for symbol 0 only with said largest possible value; and setting the value of symbol 1 with said value of the largest negative symbol of said plurality of M symbols; said step of determining an error of total Euclidean path distances further includes the steps of: adding a total accumulated phase error for a path distances to symbol 1 to the accumulated phase error for a branch distance from symbol 1 at time period T N-1 to symbol 0 at time period T N ; comparing said accumulated phase error to the lowest path distance indicator for symbol 0 only; determining whether said accumulated phase error is less than said lowest path distance indicator for symbol 0 only; and said step of determining an error of total Euclidean path distances further includes the steps of: setting said lowest path distance indicator for symbol 0 only with the smallest determined accumulated phase error for the path corresponding to symbol 0; saving said path distance from said previous symbol to symbol 0 as a new value for symbol 1; and omitting said steps of setting said lowest path distance and saving said path distance, when said accumulated phase error is greater than or equal to said lowest path distance indicator for symbol 0 only; said step of determining an error of total Euclidean path distances further includes the steps of: incrementing or decrementing said value of symbol 1 by said predetermined number, so that the value of symbol 1 corresponds to the next sequential symbol of said plurality of M symbols; determining whether said value of symbol 1 is less than or equal to M-1 (greater than or equal to -(M-1); and iterating said steps of adding, comparing, determining, incrementing, and determining, when said value of symbol 1 is less than or equal to M-1; said step of determining an error of total Euclidean path distances further includes the steps of: normalizing each of said path distances to the value of symbol 0 based upon said lowest path distance indicator for symbol 0 only; adding the path distance from symbol 1 to symbol 0 to the previous path to symbol 1; saving the added path distance; and comparing the lowest path distance indicator for symbol 0 only to the lowest path distance indicator for all symbols; said step of determining an error of total Euclidean path distances further including the step of determining whether said lowest path distance indicator for symbol 0 is less than the lowest path distance indicator for all symbols; said step of saving a particular path includes the steps of: setting said lowest path distance indicator for all symbols equal to said lowest path distance indicator for symbol 0 only; and saving symbol 0 as the current symbol; and omitting said steps of setting said lowest path distance indicator and saving symbol 0, when said lowest path distance indicator for symbol 0 only is greater than or equal to the lowest path distance indicator for all symbols.