Abstract:
In digital data transmission and storage, binary data strings are encoded into symbols from a quaternary alphabet by a specific inventive use of finite-state machines. The novel, high-rate quaternary codes offer spectral shaping properties together with a significant increase in noise margin when used on channels which lend themselves to partial-response shaping. In principle, the encoding occurs in three steps: a u-state transition diagram generates quaternary symbols whose running digital sum assumes values from a given set; the u-state transition diagram is converted into a v-state machine, each of the v states being associated with a number of transitions sufficient to encode the input data bytes into output code words; finally, the v-state machine is switched into a next state depending on its current state and the last encoded input data byte.

Description:
BACKGROUND OF THE INVENTION 
     The present invention relates generally to the encoding of data strings, here the encoding of sequential binary data, for digital transmission or recording systems. In particular, the novel coding technique employs high-rate quaternary codes that offer advantageous properties for transmission over metallic cables or optical fibers and for recording in storage media. Further, the present invention is directed to an encoder utilizing the novel coding technique and to new codes, and to a transmission or recording system employing the above method of encoding and the new codes. 
     In the prior art, many digital transmission or recording systems require the use of codes which impose restrictions on the channel input sequences. Such codes, often termed constrained, modulation, or channel codes, are mainly used to improve timing and gain control in the receiver (or decoder in a recording system), to reduce intersymbol interference, and to shape the spectrum of the transmitted or recorded sequences so that the spectrum matches the frequency characteristics of the channel. In particular, many line codes were designed to suppress the spectral components of encoded sequences near the zero frequency. Such codes are instrumental in reducing the effects of baseline wander in a receiver. 
     Codes of that kind are DC-free, i.e., have a first order spectral null at zero frequency, if, and only if, the running digital sum (RDS) is bounded. The number of values N that the RDS can assume determines the low frequency content of the code spectrum. For example, the modified version of the ternary MS43 line code described by P. A. Franaszek in &#34;Sequence-State Coding for Digital Transmission&#34;, Bell System Technical Journal, Vol. 47, pp. 143-157, January 1968, which was adopted by the German Bundespost as a standard for an ISDN transceiver, is based on a bounded RDS constraint with N=6. 
     Binary and multilevel codes with a spectral null at zero frequency possess desirable distance properties in addition to their spectrum shaping properties and can therefore be used to improve the reliability of transmission over noisy channels. In particular, they can be used to increase the Euclidian distance at the output of partial-response channels which are often encountered in digital recording and wire transmissions. 
     Binary Codes for partial-response channels that expand bandwidth by using convolutional codes with good Hamming Distance properties have been considered, e.g., by J. K. Wolf and G. Ungerboeck in &#34;Trellis Codes for Partial Response Channels&#34;, IEEE Trans. Commun., Vol. COM-34, pp. 765-773, August 1986. Binary codes that expand bandwidth by enforcing a spectral null have been given, e.g., in U.S. Pat. No. 4,888,775 by R. Karabed and P. H. Siegel, issued in 1989. All these coding schemes are binary, and are well suited for applications such as saturation recording. 
     Multilevel coding schemes, as discussed, e.g., by G. D. Forney and A. R. Calderbank in &#34;Coset Codes for Partial Response Channels; or Coset Codes with Spectral Nulls&#34;, IEEE Trans. Inform. Theory, Vol. IT-35, pp. 925-943, September 1989, achieve higher immunity against noise by expanding the signal alphabet and using trellis or known coset codes adapted for partial-response channels. Low rate (R less than or equal to 1 bit/symbol) quaternary trellis codes for partial response channels have also been constructed using concatenated coding schemes and set partitioning of the channel output signal set. Multilevel coding schemes are well suited for applications such as bandwidth efficient digital transmission and AC-bias recording. 
     OBJECTS OF THE INVENTION 
     A primary object of the present invention is to provide an optimal and advantageous method and apparatus for storing or transmitting data using optimized quaternary codes. 
     A further object of the present invention is to provide a method and means for constructing and applying optimized quaternary codes in the above apparatus. These optimized codes offer the spectral shaping properties of line codes, in particular for baseband wire/cable transmission. 
     Another object of the present invention is to increase/improve the noise margin during transmission or storage, especially in partial response class IV channels. 
     A still further object of the present invention is to reduce the average transmitted power at the channel input. 
     SUMMARY OF THE INVENTION 
     A method of encoding and decoding data according to the principles of the present invention meeting these and other objects comprises forming a v-state machine according to a v-state transition diagram and generating a first set of quaternary symbol sequences that is part of a second set of quaternary symbol sequences derived from a u-state transition diagram wherein the running digital sum of the sequences of the second set assumes values taken from a finite set of integers, such that each of v states is associated with 2K transitions, each transition corresponding to an m-symbol code word; encoding in said v-state machine a k-bit data byte into an m-symbol quaternary code word and generating a next state of said v-state machine as a function of its current state and the last encoded k-bit data byte. Sequences of m-symbol quaternary code words are decoded by estimating the encoded k-bit data bytes from output samples received from a partial response channel by generating the most likely sequence of m-symbol words that can be derived from the u-state transistion diagram, and mapping said generated m-symbol codes words into k-bit data bytes. The present invention also describes encoder/decoder apparatus for performing the disclosed encode/decode methods. 
     The invention meets these and other objects by the methods stated in claims 1 and 2 and the apparatus defined in claims 6 and 7. preferred embodiments and details of the methods and apparatus according to the invention are defined in the respective subclaims. 
     In brief, the quaternary coding technique disclosed here differs from conventional multilevel coding schemes for partial-response channels in that it optimizes the data transfer by applying a novel encoding method, i.e., a novel selection process for the code in which the data are to be transmitted or stored. The resulting transmitted (or stored) code exhibits spectral nulls at all frequencies where the channel transfer function vanishes. 
     Advantages are, first, less errors since the Euclidian distance at the channel output is increased; second, lower power consumption; and third, a relatively simple decoder design, including, e.g., a Viterbi detector. 
    
    
     BRIEF DESCRIPTION OF THE DRAWING 
     In the following, the preferred embodiment of the invention shall be described in detail with reference to the appended drawing, starting with the mathematical background, discussion of simulation results, and implementations, and in which: 
     FIG. 1 is an infinite-state transition diagram illustrating quaternary sequences having a spectral null at DC; 
     FIG. 2 illustrates a first u-state transition diagram according to the present invention; 
     FIG. 3 illustrates a first v-state transition diagram according to the present invention; 
     FIG. 4 illustrates a second u-state transition diagram according to the present invention; 
     FIG. 5 illustrates a second v-state transition diagram according to the present invention; 
     FIG. 6 illustrates a third u-state transition diagram according to the present invention; 
     FIG. 7 illustrates a third v-state transition diagram according to the present invention; 
     FIG. 8 is a graph illustrating the event error probability as a function of the energy-per-symbol to noise-power-density ratio; 2E s  /RN n , scaled by the rate loss; 
     FIG. 9 is a graph illustrating the power spectral density of various codes generated according to the present invention; 
     FIG. 10 is a block diagram illustrating a first encoder implementation according to the present invention; 
     FIG. 11 is a block diagram illustrating a first decoder implementation according to the present invention; 
     FIG. 12 is a block diagram illustrating a second encoder implementation according to the present invention; and 
     FIG. 13 is a block diagram illustrating a second decoder implementation according to the present invention. 
    
    
     THE TABLES 
     The tables form part of the description and arc, for the sake of convenience, appended to the following text that describes the invention in detail, and in which: 
     Table 1 lists the capacity C (bit/symbol) for finite-state subdiagrams of the infinite-state transition diagram shown in FIG. 1; 
     Table 2 specifies the encoder mapping for the encoder of FIG. 3; 
     Table 3 lists code words in hexadecimal presentation for the encoder of FIG. 3; 
     Table 4 describes a 10-state trellis structure for Viterbi-detecting the code of Table 3; 
     Table 5 specifies the encoder mapping for the encoder of FIG. 5; 
     Table 6 lists code words in hexadecimal representation for the encoder of FIG. 5; 
     Table 7 describes an 8-state trellis structure for Viterbi-detecting the code of Table 6; 
     Table 8 specifies the encoder mapping for the encoder of FIG. 7; 
     Tables 9A/9B list code words in hexadecimal presentation for the encoder ofFIG. 7; 
     Table 10 describes a 12-state trellis structure for Viterbi-detecting the code of Table 9; and 
     Table 11 provides a code comparison. 
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     Coding Scheme 
     The codes given in this section are based on Finite-State Transition Diagrams (FSTDs) whose edges are labeled by symbols from the quaternary alphabet {-3,-1,+1,+3}. Referring now to FIG. 1 an infinite-state transition diagram whose subdiagrams generate quaternary sequences with bounded RDS, i.e., with a spectral null at DC is illustrated. Let G N  be a finite-state subdiagram of this infinite-state transition diagram obtained by restricting the state set to N consecutive states. In this case, the number of states in G N  and the number of distinct values assumed by the RDS are the same. Table 1 contains the capacity C, i.e., the maximum achievable code rate, associated with G N  for different values of N. 
     The communication system model consists of a transmitter, a partial-response channel and a receiver. The transmitter employs finite-state machine encoders which map binary data streams into quaternary sequences. The set of all possible encoded sequences is a subset of the set of all sequences generated by the FSTD on which the encoder is based. 
     The receiver utilizes a soft-decision Viterbi detector followed by a block decoder which inverts the encoder mapping. The Viterbi detector performs sequence estimation on the trellis that combines the FSTD and the partial-response channel trellis. In other words, it chooses the most likely sequence generated by the FSTD in the sense that when it is transmitted over a noise-free channel, it results in a string of symbols which has a minimum Euclidean distance to the noisy received sequence. Thecomplexity of the resulting Viterbi detectors can be kept within manageablelimits without sacrificing performance by exploiting the periodic nature ofquaternary FSTDs. 
     The construction of efficient codes is usually based on a technique known as state splitting. In the most common form of state splitting, referred to as output state splitting, the outgoing transitions (or edges) of a split state are distributed among the two resulting offsprings, whereas the incoming transitions of the split state are duplicated. A sequence of such splittings results in an equivalent FSTD where the number of outgoingtransitions from each state is now sufficient to perform rate k/m bit/symbol coding. This increase in outgoing transitions has been achievedat the expense of an increase in decoder look-ahead. In other words, the sliding window of the sliding block decoder has become larger. In the construction of the disclosed quaternary codes, state splitting has been applied in a reverse manner. The incoming transitions of a split state aredistributed among the two resulting offsprings whereas the outgoing transitions of the split state are duplicated. Therefore, rather than increasing the number of outgoing transitions, this technique, referred toas input state splitting, allows redirection of transitions without incurring any increase in decoder look-ahead. As a consequence, quasi-catastrophic error propagation can be avoided without climinating a large number of low-power code words and/or increasing decoder look-ahead.Keeping as many as possible low-power code words has the advantage of reducing the average transmitted power. 
     In general, the p-th power of G N  is an N-state transition diagram where a transition from state i to state j corresponds to a p-step path from state i to state j in G N  and vice versa The codes disclosed hereare based on 3-state transition diagrams derived from the second power (p=2) of G 7  and G 11 . In this case, each transition is labeled with two quaternary symbols. 
     Codes based on G 7   
     The maximum rate that encoders derived from G 7  N=7, can achieve, is 1.566 bit/symbol (see Table 1). In this case, the number of identical symbols at the input and output of the dicode channel is restricted to 6 and 5, respectively. 
     FIG. 2 shows a u-state transition diagram, i.e, the 3-state irreducible component of the second power of G 7  from which one of the 8B/6Q codes(rate R=8/6 bit/symbol) is derived. The third power of the FSTD in FIG. 2 gives rise to a new 3-state transition diagram where each state has at least 256 outgoing 6-symbol transitions allowing the realisation of a 3-state 8B/6Q encoder. 
     One observes that it is possible to start at different encoder states and generate exactly the same sequences. Such sequences give rise to the phenomenon of quasi-catastrophic error propagation and should be eliminated. This can be achieved by eliminating transitions that start andend at the same state and cause error propagation, and rearranging transitions via input state splitting of the center state. In other words,transitions are rearranged such that edges arriving at the center state arepartitioned and redirected to its two offsprings and the outgoing edges areduplicated. This leads to a 4-state transition diagram that does not generate sequences which could give rise to quasi-catastrophic error propagation. Since there are more than 256 transitions emanating from eachof the 4 states, one can discard the transitions that contribute most to the average transmitted power. The reduced average transmitted power associated with sequences generated by this encoder is thus P s  =2.93 which is considerably smaller than P s  =5, i.e., the average transmitted power of a reference 2B/1Q scheme. 
     The structure of a resulting v-state encoder (v=4), often referred to as a v-state machine, is shown in FIG. 3. The encoder mapping is specified in Table 2 and, finally, the list of code words in hexadecimal representationis given in Table 3. The two quaternary symbols corresponding to a hexadecimal number are obtained by grouping the binary representation of ahexadecimal number into 2 dibits and converting every dibit 00, 01, 10, 11 into -3, -1, +1, +3, respectively. As an example, the code word D99 has the binary representation 110110011001 and corresponds to the 6 symbols +3-1+1-1+1-1 at the input of the channel. The bar on a list of i code words such as L(i) indicates that the signs of code word symbols in the original list of code words L(i) have been inverted. 
     For example, +1-1+1-1+1-1 (D99) is in A(43) (see Table 3) and represents the label of one of the 43 transitions in A(43) from state 1 to state 2 orfrom state 2 to state 4 (see FIG. 3). The inverse of D99 is -3+1-1+1-1+1 and belongs to A(43). Table 2 consists of two subtables specifying the mapping of 8-bit data words into 6-symbol code words. Subtable 2.1 provides data word to code word assignment when the encoder is in state 1 and state 4. Subtable 2.2 does the same for the encoder states 2 and 3. The entries of the first column in both subtables indicate lists of data words d in decimal form. For example, the entry d=0-42 in the first columnof Subtable 2.1 indicates the data words 00000000=0, 00000001=1, . . . , 00101010=42. The entries of the second and third columns in both tables are of the form L(i)/s where L(i) is a list of i code words and s=1,2,3,4 indicates the next encoder state. For example, Δ(43)/2 in the first column of Subtable 2.1 indicates a list of 43 code words that can be generated when the encoder changes its state from state s=1 (present state) to state s=2 (next state). 
     Decoding is accomplished in two steps. First, a Viterbi detector with a path memory of 24 symbols operating every two symbol intervals on the 10-state combined FSTD and channel trellis releases two quaternary symbolsafter a delay equal to the path memory. The 10-state trellis is described in Table 4 where the (i,j)th location--i vertically and j horizontally indicated--provides the channel input and output symbol pairs associated with the transition from trellis state i to trellis state j. For example, the transition from trellis state 3 to trellis state 5 is associated with the channel input symbol pair +1 -1 and the channel output symbol pair 2 -2. In a second step, a block decoder without look-ahead operating on groups of 6 quaternary symbols inverts the encoder mapping in Table 2 and releases 8 information bits. 
     An 8B/6Q code that requires only 8 trellis states for Viterbi detection hasalso been designed. The construction of this code is based on the u-state transition diagram shown in FIG. 4. This FSTD is a subdiagram of the FSTD shown in FIG. 2, obtained by eliminating four edges arriving at the centerstate labeled by (-1, +3), (+1, -3), (+3, -3), (-3, +3). The RDS of sequences generated by this FSTD takes still 7 different values as before.However, the capacity has now decreased to 1.431 bit/symbol. Applying inputstate splitting to the center state in the third power of the FSTD in FIG. 4, one can design an 8B/6Q encoder with reduced average transmitted power,no quasi-catastrophic error propagation and a corresponding block decoder without look-ahead. The structure of the resulting 4-state encoder is shown in FIG. 5, the encoder mapping is specified in Table 5 and, finally,the list of code words in hexadecimal representation is given in Table 6. The average transmitted power now is P s  =3.29 which is slightly higher than in the previous case. Decoding is performed on an 8-state trellis obtained by combining the FSTD in FIG. 4 and the trellis of the dicode channel. This trellis is described in Table 7. Finally, the path memory of the Viterbi detector is limited to only 24 symbols. 
     Codes based on G 11   
     A higher rate code that requires 12 trellis states for Viterbi detection has also been designed. The construction of this code is based on a subdiagram of the 5-state irreducible component of the second power of G 11 , N=11. Elimination of the two corner states at both ends of this 5-state component leads to a u-state transition diagram shown in FIG. 6, wherein u-3. The RDS of sequences generated by this FSTD takes 11 different values and the capacity is 1.613 bit/symbol. Applying input state splitting to the center state in the third power of the FSTD in FIG.6, one can design an 9B/6Q encoder with reduced average transmitted power, no quasi-catastrophic error propagation, and a corresponding block decoderwithout look-ahead. The structure of the resulting 4-state encoder is shownin FIG. 7 and the encoder mapping is specified in Table 8. Subtable 8.1 provides data byte to code word assignment when the encoder is in state 1 or state 4 (present state), whereas Subtable 8.2 shows the same for encoder states 2 and 3. Finally, the list of code words in hexadecimal representation is given in Tables 9A and 9B. The average transmitted powernow is P s  =4.02 which is slightly higher than in the previous two cases, but still 20% lower than the average transmitted power of the uncoded baseline system. Decoding is performed on a 12-state trellis obtained by combining the FSTD in FIG. 6 and the trellis of the dicode channel. This trellis is described in Table 10. Finally, the path memory of the Viterbi detector is again limited to only 24 symbols. 
     For example, the 6-symbol code word +1 -3 +1 +3 -1 -1 has the hexadecimal representation 8B5 and belongs to the list of code words F(18) (see Table 9). The code word 8B5 in F(18) can be generated when the encoder is in state s=2 or s=3 (see Subtable 8.2 and FIG. 7). In both cases, the next state is state s=2 or s=3 as it can be seen from Subtable 8.2 and FIG. 7. 
     Simulation Results 
     Simulation results for the proposed codes have verified the expected codinggains. FIG. 8 shows event error probabilities, P E  as a function of theenergy-per-symbol to noise-power-density ratio scaled by the rate loss, 2E s/RND  in dB. Curve 1 in FIG. 8 gives the theoretical performance ofan MLSE (Maximum Likelihood Sequence Estimation) receiver for uncoded quaternary transmission over the dicode channel (R=2 bit/symbol) estimatedby ##EQU1## 
     Curves 2-4 in FIG. 8 correspond to the performance of receivers for coded quaternary transmission at rates R=8/6 bit/symbol and R=9/6 bit/symbol over the dicode channel. 
     Curve 2 shows the case of the rate R=9/6 bit/symbol code where the Viterbi detector operates on the 12-state combined trellis described in Table 10. Compared to curve 1, a gain in noise margin of 2.9 dB is obtained. Curve 3corresponds to a rate R=8/6 bit/symbol code where the Viterbi detector operates on the 8-state combined trellis in Table 7. Compared to curve 1, a gain of approximately 3.4 dB is obtained. Finally, curve 4 shows the case of a rate R=8/6 bit/symbol code where the receiver utilizes a 10-state Viterbi detector with a trellis described in Table 4. A comparison of curve 4 with curve 1 indicates a gain of 3.8 dB in noise margin. 
     It is also of interest to calculate the spectrum of the particular codes proposed here. This will shed some light on spectral characteristics of encoded sequences such as width of spectral notch at f=0 and average transmitted power. FIG. 9 shows the power spectral densities of the various codes described in the previous section versus the normalized frequency, f/f s , where f stands for frequency and f s  represents the symbol frequency, i.e. the inverse of the duration of a quaternary symbol. The area under the curves in FIG. 9 corresponds to the average transmitted power P s . It can be seen that all three codes have reduced average transmitted power, i.e. their average transmitted power isconsiderably less than the average transmitted power in the case of uncodedquaternary transmission. Finally, Table 11 summarizes the parameters associated with the three codes described in the previous section and compares them against the baseline system of uncoded quaternary transmission. 
     Implementation 
     The finite-state machine encoders and block decoders disclosed here can be implemented either by random logic or by Read Only Memory blocks (ROMs). 
     Referring now to FIGS. 10 and 11, a ROM-based realization of the two 8B/6Q finite-state machine encoders described in Tables 2 and 5 is shown. It consists of an 8-bit input register 1, a 2-bit state register 2, a 1k×14 ROM 3, and a converter 4. Registers 1 and 2 are clocked at a rate of f s  /6, where f s  is the symbol rate. 
     At the end of an encoder cycle, a 12-bit code word is generated which is serialized and mapped into 6 quaternary symbols in converter 4. Decoding of the 8B/6Q codes is accomplished by a Viterbi detector 5 in cascade witha block decoder 6 shown in FIG. 11. 
     Channel output samples are quantized to q bits. A serial-to-parallel converter, consisting of two q-bit registers 7 and 8 and a 2q-bit register9, supplies a Viterbi detector 5 with two q-bit channel output samples. Registers 7 and 8 are clocked at a rate f s , whereas register 9 is clocked at a rate f s  /2. Viterbi detector 5 accepts the two q-bit samples and delivers four bits corresponding to delayed estimates of two quaternary symbols. The decision delay is equal to the path memory of the Viterbi detector. Three consecutive 4-bit outputs of Viterbi detector 5 stored in registers 10, 11, and 12, respectively, are passed on to a 12-bit register 13 whose content is interpreted as the 12-bit address of a4k×8 ROM 6. This ROM 6 constitutes the block decoder. Registers 10, 11, and 12 are clocked at a rate f s  /2, whereas register 13 is clocked at a rate f s  /6. The content of an 8-bit output register 14 represents the transmitted data byte. The output register 14 is clocked ata rate f s  /6. 
     Referring now to FIGS. 12 and 13, a ROM-based realization of the 9B/6Q finite-state machine encoder described in Table 8 is shown. It consists ofa 9-bit input register 15, a 2-bit state register 16, a 2k×14 ROM 17,and a converter 18. 
     At the end of an encoder cycle, a 12-bit code word is generated which is serialized and mapped into 6 quaternary symbols by converter 18. Decoding of the 9B/6Q code is accomplished by a 12-state Viterbi detector 19 in cascade with a block decoder 20, here a 4k×9 ROM, shown in FIG. 13. 
     Channel output samples are quantized to q bits. A serial to parallel converter, consisting of two q-bit registers 21 and 22 and a 2q-bit register 23, supplies Viterbi detector 19 with two q-bit channel output samples. Registers 21 and 22 are clocked at a rate f s , whereas register 23 is clocked at a rate f s  /2. Viterbi detector 19 accepts the two q-bit samples and delivers four bits corresponding to delayed estimates of two quaternary symbols. The decision delay is equal to the path memory of the Viterbi detector. Three consecutive 4-bit outputs of Viterbi detector 19 stored in registers 24, 25, and 26, respectively, are passed on to a 12-bit register 27 whose content is interpreted as the 12-bit address of a 4k×9 ROM 20. This ROM 20 constitutes the block decoder. Registers 24, 25, and 26 are clocked at a rate f s  /2, whereas register 27 is clocked at a rate f s  /6. The content of a 9-bit output register 28 represents the transmitted data byte. The output register 28 is clocked at a rate f s  /6. 
     While the present invention has been particularly shown and described with reference to preferred embodiments thereof, it will be understood by thoseskilled in the art that various changes in form and detail may be made therein without departing from the spirit, scope and teaching of the invention. Accordingly, the invention herein disclosed is to be consideredmerely as illustrative and limited in scope only as specified in the appended claims. 
     
                       TABLE 1______________________________________N       C               N     C______________________________________4       1.0000          10    1.75735       1.2924          11    1.79336       1.4499          12    1.82187       1.5664          13    1.84478       1.6508          14    1.86359       1.7111          15    1.8790______________________________________ 
    
     
                       TABLE 2______________________________________Subtable 2.1          1            4______________________________________ 0-42          A(43)/2      .sup.-- A(43)/3 43-66         B(24)/2      .sup.-- B(24)/3 67-131        C(65)/3      .sup.-- C(65)/2132-153        D(22)/1      .sup.-- D(22)/4154-255        E(102)/4     .sup.-- E(102)/1______________________________________Subtable 2.2          2            3______________________________________ 0-42          A(43)/4      .sup.-- A(43)/1 43-54         F(12)/2      F(12)/2 55-66         .sup.-- F(12)/3                       .sup.-- F(12)/3 67-131        G(65)/1      G(65)/1132-196        .sup.-- G(65)/4                       .sup.-- G(65)/4197-255        H(59)/3      .sup.-- H(59)/2______________________________________ 
    
     
                                           TABLE 3__________________________________________________________________________Encoder tables: 8B6Q 10-state VD__________________________________________________________________________List A(43)  799     697        A57           6D6              A5A                 A5D                    A66 A69                           6D9                              676  A96     A99        69A           A8A              69D                 679                    A2A 66D                           6A6                              6A9  967     96A        96D           75A              976                 979                    D5A 667                           D66                              D69  997     766        99A           99D              D96                 D99                    769 9A6                           9A9                              9D6  9D9     66A        796List B(24)  A2D     A27        C6A           C9A              CA6                 CA9                    A93 A63                           78A                              A87  A8D     A36        A39           72A              6CA                 757                    75D D57                           D5D                              6A3  D8A     9A3        D2A           9CAList C(65)  E95     E98        E92           E86              E89                 E1A                    E26 E29                           7A5                              A6C  7A8     A75        A78           7A2              A72                 A9C                    AA5 AA8                           AA2                              7D5  6E5     AD5        AD8           AD2              AC6                 AC9                    6E8 6E2                           B56                              B59  B65     B68        B62           B4A              B95                 B98                    B92 B86                           B89                              B1A  B26     B29        6AC           D75              6B5                 DA5                    DA8 DA2                           9AC                              DD5  9B5     9B8        775           9B2              6B8                 E56                    E59 9E5                           E65                              E68  9E8     E62        6B2           9E2              E4AList D(22)  AA4     AA1        B55           B49              B94                 B91                    B85 B16                           B19                              B25  DA4     DA1        E55           E58              E52                 E64                    E61 E46                           E49                              E94  E91     E85List E(102)  D76     D79        AD9           9B9              67A                 D97                    D9A D9D                           797                              ACA  6DA     DA6        DA9           9D7              79A                 9DA                    9DD 69E                           A2B                              79D  A2E     DD6        6DD           DD9              9E6                 66B                    9E9 A3A                           67D                              75B  B57     7A6        B5A           6B6              B5D                 7A9                    75E B66                           677                              B69  6E6     E57        E5A           96B              E5D                 A5B                    6B9 A5E                           E66                              96E  E69     767        6E9           A67              B96                 76A                    B99 A6A                           977                              E96  A6D     E99        97A           6A7              B8A                 E8A                    97D A76                           A79                              76D  66E     7D6        6AA           7D9              E2A                 A97                    A9A B2A                           99B                              A9D  99E     776        AA6           69B              AA9                 9A7                    779 9AA                           AA3                              D5B  D5E     9AD        6AD           D67              A8B                 D6A                    CAA 6D7                           D6D                              A8E  9B6     AD6List F(12)  A53     A47        A4A           A4D              A17                 A1A                    A1D D1A                           D4A                              71A  74A     C5AList G(65)  5B4     5B1        5E4           5E1              674                 671                    6A4 6A1                           6D4                              6D1  6C5     755        758           752              764                 761                    746 749                           794                              791  785     716        719           725              974                 971                    9A4 9A1                           9D4                              9D1  9C5     A55        A58           A52              A64                 A61                    A46 A49                           A94                              A91  A85     A88        A82           A16              A19                 A25                    A28 A22                           D55                              D58  D52     D64        D61           D46              D49                 D94                    D91 D85                           D16                              D19  D25     C56        C59           C65              C95List H(59)  A95     A98        A92           A86              A89                 756                    759 A26                           A29                              765  795     675        D56           D59              D65                 D62                    678 726                           D95                              729  D98     672        D92           D86              D89                 D26                    D29 C66                           C69                              6A5  96C     C96        C39           975              CA5                 6A8                    978 972                           6A2                              99C  9A5     9A8        9A2           6D5              9D5                 9D8                    6D8 9D2                           9C6                              9C3  6D2     6C6        6C3           A56              A59                 A5C                    A65 A68                           A62__________________________________________________________________________ 
    
     
                                           TABLE 4__________________________________________________________________________0        1     2     3     4     5     6     7     8     9__________________________________________________________________________0  -1 +1 +1 -1 +3 -3 -1 +3 +1 +1 +3 -1       +1 +3 +3 +1   -2 +2 0  -2 +2 -6 -2 +4 0  0  +2 -4       0  +2 +2 -21  -1 +1 +1 -1 +3 -3 -1 +3 +1 +1 +3 -1       +1 +3 +3 +1   0  +2 +2 -2 +4 -6 0  +4 +2 0  +4 -4       +2 +2 +4 -22  -1 +1 +1 -1 +3 -3 -1 +3 +1 + 1                            +3 -1       +1 +3 +3 +1   +2 +2 +4 -2 +6 -6 +2 +4 +4 0  +6 -4       +4 +2 +6 -23  -3 +1 -1 -1 +1 -3 -3 +3 -1 +1 +1 -1 +3 -3 -1 +3 +1 +1     +3                                                 -1   -6 +4 -4 0  -2 -4 -6 +6 -4 +2 -2 -2 0  -6 -4 +4 -2 0      0                                                 -44  -3 +1 -1 -1 +1 -3 -3 +3 -1 +1 +1 -1 +3 -3 -1 +3 +1 +1     +3                                                 -1   -4 +4 -2 0  0  -4 -4 +6 -2 +2 0  -2 +2 -6 -2 +4 0  0      +2                                                 -45  -3 +1 -1 -1 +1 -3 -3 +3 -1 +1 +1 -1 +3 -3 -1 +3 +1 +1     +3                                                 -1   -2 +4 0  0  +2 -4 -2 +6 0  +2 +2 -2 +4 -6 0  +4 +2 0      +4                                                 -46  -3 +1 -1 -1 +1 -3 -3 +3 -1 +1 +1 -1 +3 -3 -1 +3 +1 +1     +3                                                 -1   0  +4 +2 0  +4 -4 0  +6 +2 +2 +4 -2 +6 -6 +2 +4 +4 0      +6                                                 -47        -3 -1 -1 -3       -3 +1 -1 -1 +1 -3 -3 +3 -1 +1     +1                                                 -1    - 6       +2 -4 -2       -6 +4 -4 0  -2 -4 -6 +6 -4 +2     -2                                                 -28        -3 -1 -1 -3       -3 +1 -1 -1 +1 -3 -3 +3 -1 +1     +1                                                 -1    -4 +2 -2 -2       -4 +4 -2 0  0  -4 -4 +6 -2 +2     0                                                 -29        -3 -1 -1 -3       -3 +1 -1 -1 +1 -3 -3 +3 -1 +1     +1                                                 -1    -2 +2 0  -2       -2 +4 0  0  +2 -4 -2 +6 0  +2     +2                                                 -2__________________________________________________________________________ 
    
     
                       TABLE 5______________________________________Subtable 5.2          1            4______________________________________ 0-31          A(32)/2      .sup.-- A(32)/3 32-47         B(16)/2      .sup.-- B(16)/3 48-94         C(47)/3      .sup.-- C(47)/2 95-140        D(46)/1      .sup.-- D(46)/4141-255        E(115)/4     .sup.-- E(115)/1______________________________________Subtable 5.1          2            3______________________________________ 0-31          A(32)/4      .sup.-- A(32)/1 32-71         F(40)/4      .sup.-- F(40)/1 72-81         G(10)/2      G(10)/2 82-91         .sup.-- G(10)/3                       .sup.-- G(10)/3 92-153        H(62)/1      H(62)/1154-215        .sup.-- H(62)/4                       .sup.-- H(62)/4216-255        I(40)/3      .sup.-- I(40)/2______________________________________ 
    
     
                                           TABLE 6__________________________________________________________________________Encoder tables: 8B6Q 8-state VD__________________________________________________________________________List A(32)  A96     66A        A69           A66 66D 96D                      99D                         D5A D5D                                99A  D2A     D2D        9A9           6A9 6A6 D99                      D96                         D69 D66                                9A6  6D9     6D6        96A           A5A A5D 69A                      A2A                         A2D 69D                                9D9  9D6     A99List B(16)  D8A     D8D        A8A           A8D 9CA 9CD                      6CA                         6CD C9A                                C9D  C6A     C6D        CA9           CA6 CD9 CD6List C(47)  E4A     E4D        E1A           E1D E59 E56                      AA5                         AA2 E29                                E26  AD5     E95        E92           AD2 E65 E62                      9B5                         A75 E35                                A72  9B2     9E5        9E2           CB5 DA5 DA2                      DD5                         CE5 D75                                D72  B4A     B4D        6B5           B1A B1D 6B2                      B59                         B56 6E5                                B29  B26     6E2        B95           B92 B65 B62                      B35List D(46)  AA4     AA1        AD4           AD1 A74 A71                      DA4                         DA1 D74                                D71  B49     B46        B19           B16 B55 B52                      B58                         B25 B22                                B28  B94     B91        B64           B61 E49 E46                      E19                         E16 E55                                E52  E58     E25        E22           E28 E94 E91                      E64                         E61 9B4                                9B1  9E4     9E1        6B4           6B1 6E4 6E1List E(115)  CA7     6DD        B69           B66 B63 AD9                      AD6                         6D7 B36                                9AA  9AD     9A7        A79           A76 E4B E4E                      A73                         CDA 9B9                                E1B  E1E     6B9        D5B           D5E 6B6 6B3                      E5A                         E5D E57                                CB6  D2B     D2E        E2A           E27 9B6 66B                      66E                         9B3 E99                                E96  D8B     D8E        CE6           E69 E66 E63                      6E9                         6E6 6E3                                E36  D9A     D9D        D97           9CE 99B A5B                      A5E                         96B D6A                                D6D  D67     96E        6CB           C9B C9E DA9                      DA6                         DA3 A2B                                A2E  6CE     9E9        DD9           DD6 9E6 9E3                      99E                         A8B D79                                D76  A8E     C6B        C6E           9DA 6AA B4B                      B4E                         6AD A9A                                A9D  A97     6A7        B1B           B1E 9DD 9D7                      A6A                         B5A B5D                                B57  A6D     A67        9CB           69B 69E B2A                      B27                         6DA AA9                                AA6  AA3     B99        B96           CAA CADList F(40)  757     75A        75D           72A 72D 727                      799                         796 793                                769  766     763        739           736 979 976                      973                         679 676                                697  667     A57        A27           D57 D27 967                      997                         A39 A36                                D39  D36     9A3        A93           A63 9D3 D93                      D63                         6A3 6D3                                673List G(10)  A4A     A4D        A1A           A1D 8B5 8B2                      8E5                         8E2 2B5                                2E5List H(62)  8B4     8B1        8E4           8E1 9A4 9A1                      9D4                         9D1 974                                971  6A4     6A1        6D4           6D1 674 671                      A49                         A46 A4C                                A19  A16     A1C        A55           A52 A58 A25                      A22                         A28 A94                                A91  A64     A61        A34           A31 D49 D46                      D19                         D16 D55                                D52  D58     D25        D94           D91 D64 D61                      749                         746 719                                716  755     752        758           725 794 791                      764                         761 5B4                                5B1  5E4     5E1List I(40)  A59     A56        A29           A26 A95 A92                      A65                         A62 A35                                A32  D59     D56        9A5           9A2 D29 D26                      9D5                         D95 D92                                9D2  D65     D62        975           D35 972 759                      756                         729 726                                795  792     765        762           735 6A5 6A2                      6D5                         6D2 675                                672__________________________________________________________________________ 
    
     
                                           TABLE 7__________________________________________________________________________0       1     2     3     4     5     6     7__________________________________________________________________________0 -1 +1 +1 -1 +3 -3 +1 +1 +3 -1 +1 +3 +3 +1  -2 +2 0  -2 +2 -6 0  0  +2 -4 0  +2 +2 -21 -1 +1 +1 -1 +3 -3 +1 +1 +3 -1 +1 +3 +3 +1  0  +2 +2 -2 +4 -6 +2 0  +4 -4 +2 +2 +4 -22 -1 +1 +1 -1 +3 -3 +1 +1 +3 -1 +1 +3 +3 +1  +2 +2 +4 -2 +6 -6 +4 0  +6 -4 +4 + 2                                 +6 -23 -3 +1 -1 -1 +1 -3 -1 +1 +1 -1 -1 +3 +1 +1 +3 -1  -4 +4 -2 0  0  -4 -2 +2 0  -2 -2 +4 0  0  +2 -44 -3 +1 -1 -1 +1 -3 -1 +1 +1 -1 -1 +3 +1 +1 +3 -1  -2 +4 0  0  +2 -4 0  +2 +2 -2 0  +4 +2 0  +4 -45       -3 -1 -1 -3 -3 +1 -1 -1 -3 +3 -1 +1 +1 -1   -6 +2 -4 -2 -6 +4 -4 0  -6 +6 -4 +2 -2 -26       -3 -1 -1 -3 -3 +1 -1 -1 -3 +3 -1 +1 +1 -1   -4 +2 -2 -2 -4 +4 -2 0  -4 +6 -2 +2 0  -27       -3 -1 -1 -3 -3 +1 -1 -1 -3 +3 -1 +1 +1 -1   -2 +2 0  -2 -2 +4 0  0  -2 +6 0  +2 +2 -2__________________________________________________________________________ 
    
     
                       TABLE 8______________________________________Subtable 8.1          1            4______________________________________ 0-138         A(139)/2     A(139)/3139-229        B(91)/3      B(91)/2230-301        C(72)/1      C(72)/4302-511        D(210)/4     D(210)/1______________________________________Subtable 8.2          2            3______________________________________ 0-138         A(139)/4     .sup.-- A(139)/1139-145        E(7)/4       .sup.-- E(7)/1146-163        F(18)/2      F(18)/2164-181        .sup.-- F(18)/3                       .sup.-- F(18)/3182-292        G(111)/1     G(111)/1293-403        .sup.-- G(111)/4                       .sup.-- G(111)/4404-511        H(108)/3     H(108)/2______________________________________ 
    
     
                                           TABLE 9A__________________________________________________________________________Encoder tables: 9B6Q 12-state VD__________________________________________________________________________List A(139)  63D     637        979           976              97C                 75A                    75D                       757                          6A9                             39A  39D     397        72A           72D              727                 6A6                    6AC                       6A3                          973                             78A  78D     787        36A           36D              367                 799                    796                       79C                          793                             93A  93D     6D9        769           766              76C                 763                    6D6                       7C9                          7C6                             6DC  6D3     739        736           679              676                 3A9                    3A6                       67C                          673                             937  3D9     3D6        99A           379              376                 96A                    A5A                       A5D                          A57                             9CA  9A9     C9A        C9D           A2A              A2D                 A27                    C97                       9A6                          69A                             69D  A8A     A8D        A87           697              C6A                 C6D                    A99                       A96                          A9C                             A93  C67     9AC        9A3           A69              A66                 A6C                    A63                       9CD                          9C7                             CCA  AC9     AC6        66A           A39              A36                 A33                    66D                       667                          96D                             9D9  9D6     9DC        CA9           D5A              D5D                 D57                    CA6                       CAC                          CA3                             6CA  D2A     D2D        D27           6CD              6C7                 CD9                    CD6                       D8A                          D8D                             D87  9D3     D99        D96           D9C              D93                 967                    99D                       C79                          D69                             D66  D6C     D63        C76           DC9              DC6                 997                    63A                       D39                          D36List B(91)  DA5     DA2        DA8           DD5              DD2                 DD8                    CE5                       CE2                          D75                             D72  D78     9B5        9B2           7A5              7A2                 7A8                    7D5                       7D2                          7D8                             9B8  775     772        778           B4A              B4D                 B47                    B1A                       B1D                          B17                             B59  B56     B5C        B53           B29              B26                 6B5                    6B2                       B89                          B86                             B95  B92     B98        6B8           B65              B62                 B68                    9E5                       3B5                          B35                             B32  BC5     9E2        635           3E5              E4A                 E4D                    E47                       6E2                          6E8                             9E8  E1A     E1D        E17           E59              E56                 E5C                    E53                       E29                          E26                             E89  E86     E95        E92           E98              E65                 E62                    E68                       E35                          E32                             EC5  AA5     AA2        AA8           AD5              AD2                 AD8                    A75                       A72                          A78                             CB5  CB2List C(72)  AA4     AA1        AD4           AD1              A74                 A71                    DA4                       DA1                          DD4                             DD1  D74     D71        7A4           7A1              7D4                 7D1                    774                       771                          B49                             B46  B4C     B19        B16           B1C              B55                 B52                    B58                       B25                          B22                             B28  B85     B82        B88           B94              B91                 B64                    B61                       B34                          B31                             E49  E46     E4C        E19           E16              E1C                 E55                    E52                       E58                          E25                             E22  E28     E85        E82           E88              E94                 E91                    E64                       E61                          E34                             E31  9B4     9B1        9E4           9E1              6B4                 6B1                    6E4                       6E1                          CB4                             CB1  CE4     CE1__________________________________________________________________________ 
    
     
                                           TABLE 9B__________________________________________________________________________Encoder tables: 9B6Q 12-state VD__________________________________________________________________________List D(210)  6BC 9E3 B1B             B1E                CDA CDD CD7                           9EC D2B D2E  9A7 B5A B5D             B57                37A 93E 99E                           6E9 D8B B2A  B2D B27 B8E             6E6                6E3 3B9 3B6                           B8A B8D B87  C7A B9A D9D             D97                B99 B96 B93                           B9C C77 6EC  63B 3E9 3E6             B69                B66 B63 B6C                           D6A D6D D67  B39 B36 63E             99B                97A CB9 CB6                           BC3 BC6 DCA  DCD DC7 A5B             A5E                E4B E4E 97D                           977 69B C9B  D3A D37 E1B             E1E                C9E CE9 CE6                           A2B A2E DA9  DA6 E5A E5D             E57                DA3 DAC 69E                           96B 6AA E2A  E2D E27 DD9             DD6                DD3 6AD 6A7                           A8B E8A E8D  E87 A8E 9CB             D79                D76 D73 E99                           E96 E93 E9C  9CE 39B 39E             C6B                E69 E66 E63                           E6C C6E 96E  A9A 75B 75E             E39                E36 A9D A97                           D9A 9B9 9B6  EC9 EC6 66B             36B                72B 72E 36E                           6DA A6A A6D  A67 6DD 78B             78E                6D7 66E 9B3                           ACA ACD 79A  79D 797 AC7             9BC                9DD 9D7 93B                           76A 76D 767  A3A A3D A37             9AA                7CA 67A 67D                           AA9 AA6 AA3  AAC 677 73A             737                9AD 6CB 6CE                           CAA AD9 7A9  7A6 7A3 7AC             AD6                AD3 ADC 3AA                           3AD 7D9 7D6  3A7 CAD CA7             A79                A76 779 776                           773 A73 A7C  9E9 9E6 6B9             6B6                6B3 B4B B4E                           3DA D5B D5EList E(7)  A3C AC3 ACC             C3A                3A3 3CA 33AList F(18)  8B5 8B2 8B8             8E5                8E2 8E8 A4A                           A4D A47 A1A  A1D A17 2B5             2B2                2B8 2E5 2E2                           2E8List G(111)  5B4 5B1 5E4             5E1                2B4 2B1 2E4                           2E1 8B4 8B1  8E4 8E1 9A4             9A1                9D4 9D1 974                           971 6A4 6A1  6D4 6D1 674             671                CA4 CA1 CD4                           CD1 C74 C71  3A4 3A1 3D4             3D1                374 371 A49                           A46 A4C A43  A19 A16 A1C             A13                A55 A52 A58                           A25 A22 A28  A85 A82 A88             A94                A91 A64 A61                           A34 A31 AC4  AC1 D49 D46             D4C                D43 D19 D16                           D1C D13 D55  D52 D58 D25             D22                D28 D85 D82                           D88 D94 D91  D64 D61 D34             D31                DC4 DC1 749                           746 74C 743  719 716 71C             713                755 752 758                           725 722 728  785 782 788             794                791 764 761                           734 731 7C4  7C1List H(108)  6A5 6A2 6A8             6D5                6D2 6D8 675                           672 678 CA5  CA2 CA8 CD5             CD2                CD8 C75 C72                           C78 3A5 3A2  3A8 3D5 3D2             3D8                375 372 378                           A59 A56 A5C  A53 A29 A26             A2C                A23 A89 A86                           A8C A83 A95  A92 A98 A65             A62                A68 9A5 9A2                           A35 A32 A38  9A8 AC5 AC2             AC8                9D5 9D2 9D8                           975 972 978  D59 D56 D5C             D53                D29 D26 D2C                           D23 D89 D86  D8C D83 D95             D92                D98 D65 D62                           D68 D35 D32  D38 DC5 DC2             DC8                759 756 75C                           753 729 726  72C 723 789             786                78C 783 795                           792 798 765  762 768 735             732                738 7C5 7C2                           7C8__________________________________________________________________________ 
    
     
                                           TABLE 10__________________________________________________________________________0      1   2     3     4   5     6     7   8   9     10  11__________________________________________________________________________0  -3 +3  -1 +1      +1 -1 +3 -3 -1 +3                      +1 +1 +3 -1     +1 +3                                          +3 +1   -6 +6  -4 +2      -2 -2 0  -6 -4 +4                      -2 0  0  -4     -2 +2                                          0  -21  -3 +3  -1 +1      +1 -1 +3 -3 -1 +3                      +1 +1 +3 -1     +1 +3                                          +3 +1   -4 +6  -2 +2      0  -2 +2 -6 -2 +4                      0  0  +2 -4     0 +2                                          +2 -22  -3 +3  -1 +1      +1 -1 +3 -3 -1 +3                      +1 +1 +3 -1     +1 +3                                          +3 +1   -2 +6  0 +2      +2 -2 +4 -6 0 +4                      +2 0  +4 -4     +2 +2                                          +4 -23  -3 +3  -1 +1      +1 -1 +3 -3 -1 +3                      +1 +1 +3 -1     +1 +3                                          +3 +1   0 +6  +2 +2      +4 -2 +6 -6 +2 +4                      +4 0  +6 -4     +4 +2                                          +6 -24      -3 +1      -1 -1 +1 -3 -3 +3                      -1 +1 +1 -1 +3 -3                                      -1 +3                                          +1 +1 +3 -1  -6 +4      -4 0  -2 -4 -6 +6                      - 4                         +2 -2 -2 0 -6                                      -4 +4                                          -2 0  0 -45      -3 +1      -1 -1 +1 -3 -3 +3                      -1 +1 +1 -1 +3 -3                                      -1 +3                                          +1 +1 +3 -1  -4 +4      -2 0  0  -4 -4 +6                      -2 +2 0  -2 +2 -6                                      -2 +4                                          0  0  +2 -46      -3 +1      -1 -1 +1 -3 -3 +3                      -1 +1 +1 -1 +3 -3                                      -1 +3                                          +1 +1 +3 -1  -2 +4      0  0  +2 -4 -2 +6                      0  +2 +2 -2 +4 -6                                      0 +4                                          +2 0  +4 -47      -3 +1      -1 -1 +1 -3 -3 +3                      -1 +1 +1 -1 +3 -3                                      -1 +3                                          +1 +1 +3 -1  0 +4      +2 0  +4 -4 0 +6                      +2 +2 +4 -2 +6 -6                                      +2 +4                                          +4 0  +6 -48          -3 -1 -1 -3     -3 +1 -1 -1 +1 -3                                      -3 +3                                          -1 +1 +1                                                    +3 -3      -6 +2 -4 -2     -6 +4 -4 0  -2 -4                                      -6 +6                                          -4 +2 -2                                                    02 -69          -3 -1 -1 -3     -3 +1 -1 -1 +1 -3                                      -3 +3                                          -1 +1 +1                                                    +3 -3      -4 +2 -2 -2     -4 +4 -2 0  0 -4                                      -4 +6                                          -2 +2 0 -2                                                    +2 -610         -3 -1 -1 -3     -3 +1 -1 -1 +1 -3                                      -3 +3                                          -1 +1 +1                                                    +3 -3      -2 +2 0  -2     -2 +4 0  0  +2 -4                                      -2 +6                                          0  +2 +2                                                    +4 -611         -3 -1 -1 -3     -3 +1 -1 -1 +1 -3                                      -3 +3                                          -1 +1 +1                                                    +3 -3      0  +2 +2 -2     0  +4 +2 0  +4 -4                                      0 +6                                          +2 +2 +4                                                    +6 -6__________________________________________________________________________ 
    
     
                                           TABLE 11__________________________________________________________________________Rate (R)   Net increase in            Average Power                     Maximum #Viterbi                                   Path Memory                                          #Encoder                                                 Decoder Window(bit/symbol)   noise margin (dB)            (P.)     Zero Runlength                             States                                   (symbols)                                          States (symbols)__________________________________________________________________________2/1     0        500      ∞  4    ∞                                          1      18/6     34       329      5        8    24     4      68/6     38       293      5       10    24     4      69/6     29       402      5       12    24     4      6__________________________________________________________________________