Abstract:
A multiple description TCQ arrangement employs a trellis graph that is the tensor product, such as T 1 {circle around (X )} T 2 , of two trellis graphs, such as T 1  and T 2 . The codevectors of the tensor-product trellis, c i , are incorporated within trellis graph T 1  and also within trellis graph T 2 . The incorporation within trellis graph T 1  is effected by assigning the c i  codevectors to sets, and by deriving therefrom codevectors for trellis graphs T 1  and T 2 . The actual values that these codevectors take on are arranged to insure certain distortion results. Specifically, the distortion measure of the tensor-product trellis is minimized, subject to the condition that the distortion measures of approximations made by decoders operating with the T 1  and T 2  trellises do not exceed a predetermined value. Consequently, an improved arrangement is realized in which two encoders are cooperatively generating separate trellis-coded descriptions of the input sequence. Three different fidelity levels can thus be achieved: a first when only a first description is employed, a second when the second description is employed, and a third (and highest level of fidelity) when both descriptions are employed. This allows for use of receivers that are responsive to different rates, or the use of receivers that have adaptable rates.

Description:
RELATED APPLICATION 
     This application is related to, and claims priority from, a provisional application filed on Jan. 16, 1998 which carries the Ser. No. 60/071,749. This application is also related to an application filed on even date herewith, which carries the Ser. No. 09/072,782, and is titled “Successively Refinable Trellis Coded Quantization”. 
    
    
     BACKGROUND OF THE INVENTION 
     This invention relates to quantizers and, more particularly, to trellis-coded quantizers. 
     In recent years rate-scalable source coders have received growing attention. By selecting different sub-streams of the output of such coders, various levels of encoding rate and distortion can be achieved. 
     The ability to select different sub-streams is important in various applications. One such application, for example, may relate to receivers that are adapted to operate at one data rate under normal conditions, and adapted to accept a lower data rate when transmission conditions are adverse, while still producing a bona fide output, albeit, of lower fidelity. 
     A powerful source coding scheme for memoryless sources is trellis coded quantization. See, for example, M. W. Marcellin and T. R. Fischer, “Trellis coded quantization of memoryless and Gauss-Markov sources,”  IEEE Trans. Comm. , vol. 38, pp. 82-93, January 1990. It has been shown that for a memoryless uniform source, trellis coded quantizers (TCQs) provide mean squared errors (MSEs) within 0.21 dB of the theoretical distortion bounds (for given rates). The performance of a trellis-coded quantization (TCQ) arrangement is much better than that of the best scalar quantizer (Lloyd-Max quantizer) at the same rate. 
     Rate scalability can be achieved with successive refinement, as well as with multiple descriptions. Successive refinement refers to the notion of a transmitter sending one stream of data which can decode the desired signal, albeit with lower fidelity, and one or more additional streams of data that refined the decoded output. Although until now, it has been thought that trellis coding does not lend itself to successive refinability, the aforementioned co-pending application discloses a successively refinable trellis quantizer. 
     Multiple description refers to the notion of a transmitter sending more than one description of a given sequence. A receiver accepting one of the descriptions can reproduce the signal with a certain fidelity, and a receiver accepting both descriptions can reproduce the signal with a higher fidelity. Unlike with successive refinement, either one of the multiple descriptions can be used to decode the signal. 
     Multiple description scenarios have a well-established history, but no present publications exist that disclose the use of multiple description coding in the context of trellis quantization. The challenge is to realize simple and effective multiple description arrangements for trellis coding. 
     SUMMARY 
     A multiple description TCQ arrangement is achieved by employing a trellis graph that is the tensor product, such as T 1  {circle around (X)} T 2 , of two trellis graphs, such as T 1  and T 2 . The codevectors of the tensor-product trellis, c i , are incorporated within trellis graph T 1  and also within trellis graph T 2 . The incorporation within trellis graph T 1  is effected by assigning the c i  codevectors to sets, and by deriving therefrom codevectors for trellis graphs T 1  and T 2 . The actual values that these codevectors take on are arranged to insure certain distortion results. Specifically, the distortion measure of the tensor-product trellis is minimized, subject to the condition that the distortion measures of approximations made by decoders operating with the T 1  and T 2  trellises do not exceed a predetermined value. Consequently, an improved arrangement is realized in which two encoders are cooperatively generating separate trellis-coded descriptions of the input sequence. Three different fidelity levels can thus be achieved: a first when only a first description is employed, a second when the second description is employed, and a third (and highest level of fidelity) when both descriptions are employed. This allows for use of receivers that are responsive to different rates, or the use of receivers that have adaptable rates. 
    
    
     BRIEF DESCRIPTION OF THE DRAWING 
     FIG. 1 shows a one-bit trellis coding graph; 
     FIG. 2 presents a two-bit, four-state, trellis coding graph arrangement with parallel transitions; 
     FIG. 3 illustrates the structure of a tensor-product trellis graph created from trellis graphs having the FIG. 1 structure; 
     FIG. 4 depicts a block diagram of an arrangement where an encoder that produces multiple (two) descriptions of an input sequence feeds three receivers, and where one receiver is responsive to one description, a second receiver is responsive to the second description, and one receiver is responsive to both descriptions; 
     FIG. 5 shows a heuristic approach for making index assignments; and 
     FIG. 6 presents an algorithm for ascertaining optimized encoder threshold levels. 
    
    
     DETAILED DESCRIPTION 
     For sake of simplicity and ease of understanding, the trellis graph that is employed in the following disclosure is the relatively simple, four-state, trellis graph shown in FIG.  1  and described in the aforementioned Marcellin et al publication. It should be understood, however, that use of the FIG. 1 trellis graph is not a requirement of this invention. It is merely illustrative. Other one-bit-per sample trellis graphs can be used, as well as trellis graphs employing a larger number of bits per sample. 
     Multiple Transitions Trellis 
     The four-state trellis of FIG. 1 has two transitions from each state, and a signal level that is associated with each transition. For example, a level c 0  for transitions V 0  to V 0  and V 2  to V 1 , c 1  for transitions V 1  to V 2  and V 3  to V 3 , c 2  for transitions V 0  to V 1  and V 2  to V 0 , and c 3  for transitions V 1  to V 3  and V 3  to V 2 , where c 0 &lt;c 1 &lt;c 2 &lt;c 3 . When encoding, each input sample causes a transition in the graph from a current state to a next state, based on the signal level of the input sample and the signal levels that are associated with the transitions emanating from the current state. More specifically, in determining which transition to take, a distortion cost measure is evaluated by ascertaining the distance between the input sample and the two levels that are relevant in the current state of the trellis. The transition selected dictates the value of an output bit of the encoder. For example in state V 1  only levels c 1  and c 3  are relevant, and a transition from state V 1 to V 3  might indicate that the input sample is closer to level c 3  than to level c 1 . 
     Trellises that have a larger number of signal levels, and a corresponding larger number of transitions from any one state, produce a larger number of bits per sample. Such trellises can have, but do not have to have, more than four states. Indeed, a four-state trellis of the type shown in FIG. 1 can be employed for any number of bits per sample, simply by employing parallel transitions. Accordingly, even if one were to restrict all embodiments to four-state trellis graphs, such as the one depicted in FIG. 1, the generality of the present disclosure would not be diminished. The following briefly reviews the concept of multiple transitions. 
     When it is desired to quantize a sequence of signal samples with R bits per sample,  2   R+1  signal levels c i , i=0,1, . . . , ( 2   R+1 −1) are used in the trellis encoding process. In the decoding process, the received bits select from among signal levels e i , i=0,1, . . . , ( 2   R+1 −1), thereby approximating the sequence of input samples. Typically, the e i  levels are equal to the c i  levels. If levels c i  are enumerated in ascending order, and if the four-transition trellis graph of FIG. 1 is to be employed for encoding, then the set of levels is partitioned into four subsets in such a manner that every fourth level belongs to the same subset, i.e. each c 4i+m  for i=0,1, . . . goes into subset A m , where m=0,1,2,3. The levels in each subset A m  are then assigned to parallel transitions from one particular state to another particular state, yielding a four-state trellis graph with multiple transitions and  2   R+1  levels. Typically, the c i &#39;s and the e i &#39;s levels are scalar. Generally, however, they can be multi-dimensional vectors and, therefore, some practitioners refer to these levels as “codevectors”. 
     To illustrate, FIG. 2 depicts a trellis graph for R=3, where there are four transitions from any one state to another state. Thus, when an encoder is residing in state  31  in FIG. 2, a label D0, D2, D4, D6, D8, D10, D12, or D14 is selected based on whether the sample to be quantized is closest to either c 0 , c 2 , c 4 , c 6 , c 6 , C 8 , c 10 , c 12 , or c 14 , respectively. Labels D0, D4, D8, and D12 correspond to a transition from state  31  to state  41  in FIG. 2, and labels D2, D6, D10 and D14 correspond to a transition from state  31  to state  42  in FIG.  2 . The output bits that are delivered by the quantizer are selected based on the label attached, and can be, for example, as shown in the table below: 
     
       
         
               
               
             
               
               
               
             
           
               
                   
                   
               
               
                   
                 Output 
               
               
                   
                 bits 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Transition 
                   
               
               
                   
                 from 31 to 
                   
               
               
                   
                 41 &amp; 42 
                   
               
               
                   
                 D0 
                 000 
               
               
                   
                 D2 
                 100 
               
               
                   
                 D4 
                 001 
               
               
                   
                 D6 
                 101 
               
               
                   
                 D8 
                 010 
               
               
                   
                  D10 
                 110 
               
               
                   
                  D12 
                 011 
               
               
                   
                  D14 
                 111 
               
               
                   
                 Transition 
                   
               
               
                   
                 from 32 to 
                   
               
               
                   
                 43 &amp; 44 
                   
               
               
                   
                 D1 
                 000 
               
               
                   
                 D3 
                 100 
               
               
                   
                 D5 
                 001 
               
               
                   
                 D7 
                 101 
               
               
                   
                 D9 
                 010 
               
               
                   
                  D11 
                 110 
               
               
                   
                  D13 
                 011 
               
               
                   
                  D15 
                 111 
               
               
                   
                   
               
             
          
         
       
     
     Tensor-Product Trellis 
     If T 1  and T 2  denote trellises with states v 1 ,v 2 , . . . ,v 2 b 1 , and w 1 , w 2 , . . . , w 2 , . . . ,w 2 b 2 , respectively, then the tensor-product trellis T 1  {circle around (X)} T 2  is a trellis with  2   b1+b2  states v p ×w q , where p=1,2, . . . ,2 b1  and q=1,2, . . . , 2 b2 . This is illustrated in FIG. 3 for b 1 =b 2 = 2 , where two four-state trellis graphs are combined to form a tensor-product trellis. As can be seen from FIG. 3, a transition between states v p ×w q  and v r ×w s  in T 1  {circle around (X)} T 2  exists if, and only if, there exist transitions between v p  and v r  in T 1  and between w q  and w s  in T 2 . For example, there exists a transition between state v 1 ×w 1  of the T 1  {circle around (X)} T 2  graph and state v 2 ×w 1  of the T 1  {circle around (X)} T 2  graph in FIG. 3, because there exists a transition between states v 1  and v 2  in the T 1  trellis graph of FIG. 3, and there is also a transition between states w 0  and w 0  in the T 2  trellis graph of FIG.  3 . This attribute allows the tensor-product trellis to be used as a TCQ that develops a multiply-descriptive output code. 
     In other words, it is possible to construct a trellis-coded encoder that produces a coded sequence of bits corresponding to trellis T 1  and a coded sequence of bits corresponding to trellis T 2 , and a decoder that is responsive to the two-coded sequences with codevectors that are associated with the tensor-product trellis T 1  {circle around (X)} T 2 . 
     Illustrative Embodiment 
     An illustrative physical embodiment of such an arrangement is presented in FIG. 4, where encoder  100  receiving the aforementioned input samples, includes an encoder  110  which belongs to a first TCQ (TCQ 1 ), characterized by trellis graph T 1 , and an encoder  120  which belongs to a second TCQ (TCQ 2 ) characterized by trellis graph T 2 . Encoder  100  thus outputs two distinct signals. One of the signals, passing through channel  130  and having the rate R1 bits per sample, arrives at receiver  140 . That same signal passes through channel  130  as it arrives at receiver  150 . The other of the signals, passing through channel  135  and having the rate R2 bits per sample, arrives at receives  160 , and that same signal passes through channel  135  as it arrives at receiver  150 . Receiver  140  includes a decoder for TCQ 1  (employing trellis graph T 1 ), and receiver  160  includes a decoder for TCQ 2  (employing trellis graph T 2 ). Receiver  150 , realizing a combined, or central, TCQ (TCQ 0 ), includes a decoder employing the tensor-product trellis T 1  {circle around (X)} T 2 . As described more fully below, receiver  150  develops an approximation of the encoded signal with a certain, minimized, level of distortion, while receivers  140  and  160  develop an approximation of the encoded signal with a larger level of distortion than that of receiver  150 . The distortion produced by receivers  140  and  160  need not be equal, however. 
     Although the FIG. 4 arrangement depicts three receivers, it should be understood that this depiction is primarily for the purpose of describing the encoding and decoding operations, and the design process involved. In practice, it is expected that a transmitter will be at least capable of outputting multiple descriptions, but in some applications it might not actually output those descriptions automatically. In some applications the multiple descriptions will be transmitted automatically, perhaps over disparate channels. In other applications, the multiple descriptions might be transmitted only in response to certain conditions. The control over what is transmitted by the encoders within the transmitter resides in controller  115 . Controller  115  can, for example, direct the encoders to output their sequences seriatim. FIG. 4 shows two TCQ encoders on the encoding side, but it should be understood that a larger number of TCQ encoders are possible, with each providing its own description of the input sequence of samples. Obviously, receiver  150  that includes a decoder for TCQ 0  that employs tensor-product trellis T 1  {circle around (X)} T 2  can be easily converted to a receiver that decodes TCQ 1  or TCQ 2 . 
     On the decoding side, in some applications one may find three kinds of receivers that are used concurrently by different users. A receiver may be adapted to decode the sequence that arrives at the rate R1bits per sample (one description), another receiver may be adapted to decode the sequence that arrives at the rate R2 bits per sample (another description), and still another receiver may be adapted to decode the sequence that arrives at the rate R1+R2 bits per sample. In other applications, receivers may be adapted to accept any of the above three rates, based, perhaps, on channel conditions or wishes of the user. More particularly, a receiver may be adapted to accept any subset of the multiple descriptions and create an output that is based on the received (i.e., received without errors that are not correctable) descriptions. 
     Trellis Codevectors 
     Returning to the design issues of the three TCQs, the variables that need to be ascertained are 
     a) codevectors employed in encoder  110  and receiver  140  (codevectors b 1   k  for TCQ 1 ); 
     b) codevectors employed in encoder  120  and receiver  160  (codevectors b 2   j  for TCQ 2 ); and 
     c) codevectors employed in receiver  150  (codevectors c i  for TCQ 0 ). 
     It is clear that codevectors c i  need to be selected to produce a good output at receiver  150 , and codevectors b 1   k  and b 2   j  need to be selected to produce a good output at receivers  140  and  160 , respectively. Of course, one would rightly expect that receiver  150 , which is responsive to more information, would produce an output with less distortion than the distortion at the outputs of receivers  140  and  160 , which are responsive to less information. In deciding on the proper values of c i , b 1   k , and b 2   j , one could select the b 1   k  and b 2   j . codevectors that independently produce the lowest distortion out of receivers  140  and  160 , respectively, and then decide on the c 1  codevectors that would give the lowest distortion at the output of receiver  150 , given the fact that the signal was encoded with the previously selected b 1   k  and b 2   j  codevectors. Thereafter, one would undertake a refinement process to modify the selected codevectors so as to improve the distortion measure at the output of receiver  150 , at the cost of degrading the performance at the output of receivers  140  and  160 , to insure that receiver  150  provides the least-distorted output. 
     Conversely, one could select the c i  codevectors that would give the lowest distortion at the output of receiver  150 , then select the b 1   k  and b 2   j  codevectors and ascertain whether some preselected maximum level of distortion has been exceeded. If so, one would undertake a refinement process to modify the selected c i  codevectors, and corresponding b 1   k  and b 2   j  codevectors, to improve performance at the output of receivers  140  and  160 , at the expense of performance at the output of receiver  150 . 
     We have ascertained that a relationship exists between the c i  codevectors and the b 1   k , and b 2   j  codevectors, and in accordance with the principles disclosed herein, the c i , b 1   k , and b 2   j  codevectors are determined concurrently. 
     Proceeding with the relationship between b 1   k , b 2   j , and c i , codevectors c i  are partitioned into two groups. The first group, set aside for TCQ 1 , contains  2   R     1     +1  subsets B 1   k , k=0,1, . . . , ( 2   R     1     +1 −1 ), where each subset consists of  2   R     1     +1  c i &#39;s. Similarly, the second group, set aside for TCQ 2 , contains  2   R     2     +1  subsets B 2   j  where each subset consists of  2   R     1     +1  c i &#39;s. To illustrate, when R1=R2=1, the partitioning results in four subsets B 1   0 , B 1   1 , B 1   2 , and B 1   3  for TCQ 1 , and four subsets B 2   0 , B 2   1 , B 2   2 , and B 2   3  for TCQ 2 . 
     When R1 or R2 is greater than 1, the number of subsets in the corresponding group is a higher power of two, and is divisible by four. If, as indicated above, the designer wishes to employ a four-state trellis with parallel transitions, such as in FIG. 2, then a subset having more than four members is itself partitioned into four subsets A 1   m  and/or A 2   m , to create a TCQ 1 , or a TCQ 2 , as the case may be, with multiple transitions. 
     Every pair of paths in T 1  and T 2  identifies a path in T 1  {circle around (X)} T 2 , and it corresponds to a path of length N in T 1  and a path of length N in T 2 . The transitions in T 1  correspond to codevectors selected from b 1   0 , b 1   1 , b 1   2 , and b 1   3  which are derived from subsets B 1   0 , B 1   1 , B 1   2  , and B 1   3  , respectively. Likewise, the transitions in T 2  correspond to codevectors selected from b 2   0 , b 2   1 , b 2   2 , and b 2   3 , which are derived from subsets B 2   0 , B 2   1 , B 2   2 , and B 2   3 , respectively. More specifically, the b 1   k  levels correspond to the centroids of B 1   k , and the b 2   j  levels correspond to the centroids of B 2   j . 
     In the context of this disclosure, the centroid of a subset corresponds to the sum of all members of the set, c i , each multiplied by a fraction that corresponds to the area under the probability distribution of the input signals, in the region where input signals are closer to member c i  than to any other member of the set. For example, a simple illustrative probability distribution for input signals might be triangular in shape, beginning with 0 at −1.0, increasing linearly to a maximum of 1.0 at 0, and then decreasing linearly to 0 at +1.0. Given, for example, a subset containing members {−0.8, −0.7, −0.6, −0.5}, the four resulting intervals are −1.0 to −0.75, −0.75 to −0.65, −0.65 to −0.55, and −0.55 to +1.0. The centroid would then be            -     (     0.8   ·       0.25   2     2       )       -       (   0.7   )                     (   0.1   )                     (   0.3   )       -       (   0.6   )                     (   0.1   )                     (   0.4   )       -       (   0.5   )     ·     (         1.45   ·   0.55     2     +   0.5     )         =     -   0.522775                            
     The n th  transition in the TCQ 0  trellis is defined by the codevector that remains from the intersection of the B subsets (i.e., B 1   k ∩ B 2   j ) that are involved in the n th  transition of TCQ 1  and TCQ 2 . Clearly, then, the assignment of codevectors c i  to sets B 1   k  and B 2   j  should be such that B 1   k  ∩ B 2   j  is non-zero, and unique for all i and j. This translates to a requirement that precisely one c i  should be shared between any pair of B 1   k  and B 2   j . This, in turn, means that members of any particular subset B 1   k  must be members in different ones of subsets B 2   j . A simple way to achieve such an assignment is to use a two-dimensional table, such as the one shown in FIG. 5, where each row specifies the c i &#39;s that are assigned for a particular b 1   k  of TCQ 1 , and each column specifies the c i &#39;s that are assigned for a particular b 2   j  of TCQ 2 . 
     Heuristically, it has been shown that a reasonable starting assignment is achieved by ordering the c i &#39;s in ascending order and by assigning the ordered c i &#39;s in accordance with a path such as the one outlined in FIG.  5 . With reference to FIG. 3, the assignments in FIG. 5 correspond to transition assignments expressed in the following Table 1. 
     
       
         
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 b 0   1   
                 b 1   1   
                 b 2   1   
                 b 3   1   
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 c 0   
                 c 1   
                 c 5   
                 c 6   
               
               
                 b 0   2   
                 v 0  × w 0  → v 0  × w 0   
                 v 0  × w 1  → v 0  × w 2   
                 v 0  × w 0  → v 0  × w 1   
                 v 0  × w 1  → v 0  × w 3   
               
               
                   
                 v 0  × w 2  → v 0  × w 1   
                 v 0  × w 3  → v 0  × w 3   
                 v 0  × w 2  → v 0  × w 0   
                 v 0  × w 3  → v 0  × w 2   
               
               
                   
                 v 2  × w 0  → v 1  × w 0   
                 v 2  × w 1  → v 1  × w 2   
                 v 2  × w 0  → v 1  × w 1   
                 v 2  × w 1  → v 1  × w 3   
               
               
                   
                 v 2  × w 2  → v 1  × w 0   
                 v 2  × w 3  → v 1  × w 3   
                 v 2  × w 2  → v 1  × w 0   
                 v 2  × w 3  → v 1  × w 2   
               
               
                   
                 c 2   
                 c 4   
                 c 7   
                 c 12   
               
               
                 b 1   2   
                 v 1  × w 0  → v 2  × w 0   
                 v 1  × w 1  → v 2  × w 2   
                 v 1  × w 0  → v 2  × w 1   
                 v 1  × w 1  → v 2  × w 3   
               
               
                   
                 v 1  × w 2  → v 2  × w 1   
                 v 1  × w 3  → v 2  × w 3   
                 v 1  × w 2  → v 2  × w 0   
                 v 1  × w 3  → v 2  × w 2   
               
               
                   
                 v 3  × w 0  → v 3  × w 0   
                 v 3  × w 1  → v 3  × w 2   
                 v 3  × w 0  → v 3  × w 1   
                 v 3  × w 1  → v 3  × w 3   
               
               
                   
                 v 3  × w 2  → v 3  × w 1   
                 v 3  × w 3  → v 3  × w 3   
                 v 3  × w 2  → v 3  × w 0   
                 v 3  × w 3  → v 3  × w 2   
               
               
                   
                 c 3   
                 c 8   
                 c 11   
                 c 13   
               
               
                 b 2   2   
                 v 0  × w 0  → v 1  × w 0   
                 v 0  × w 1  → v 1  × w 2    
                 v 0  × w 0  → v 1  × w 1   
                 v 0  × w 1  → v 1  × w 3   
               
               
                   
                 v 0  × w 2  → v 1  × w 1   
                 v 0  × w 3  → v 1  × w 3    
                 v 0  × w 2  → v 1  × w 0   
                 v 0  × w 3  → v 1  × w 2   
               
               
                   
                 v 2  × w 0  → v 0  × w 0   
                 v 2  × w 1  → v 0  × w 2    
                 v 2  × w 0  → v 0  × w 1   
                 v 2  × w 1  → v 0  × w 3   
               
               
                   
                 v 2  × w 2  → v 0  × w 1   
                 v 2  × w 3  → v 0  × w 3    
                 v 2  × w 2  → v 0  × w 0   
                 v 2  × w 3  → v 0  × w 2   
               
               
                   
                 c 0   
                 c 10   
                 c 14   
                 c 15   
               
               
                 b 3   2   
                 v 1  × w 0  → v 3  × w 0   
                 v 1  × w 1  → v 3  × w 2    
                 v 1  × w 0  → v 3  × w 1   
                 v 1  × w 1  → v 3  × w 3   
               
               
                   
                 v 1  × w 2  → v 3  × w 1   
                 v 1  × w 3  → v 3  × w 3    
                 v 1  × w 2  → v 3  × w 0   
                 v 1  × w 3  → v 3  × w 2   
               
               
                   
                 v 3  × w 0  → v 2  × w 0   
                 v 3  × w 1  → v 2  × w 2    
                 v 3  × w 0  → v 2  × w 1   
                 v 3  × w 1  → v 2  × w 3   
               
               
                   
                 v 3  × w 2  → v 2  × w 1   
                 v 3  × w 3  → v 2  × w 3    
                 v 3  × w 2  → v 2  × w 0   
                 v 3  × w 3  → v 2  × w 2   
               
               
                   
               
             
          
         
       
     
     Table 2 below shows the correspondence between the codevectors of TCQ 0  and the codevectors for TCQ 1  and TCQ 2  for the assignments made in FIG.  5 . 
     
       
         
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                 TCQ0 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
               
                   
               
             
             
               
                 TCQ1 
                 0 
                 0 
                 1 
                 2 
                 1 
                 0 
                 0 
                 1 
                 2 
                 3 
                 3 
                 2 
                 1 
                 2 
                 3 
                 3 
               
               
                 TCQ2 
                 0 
                 1 
                 0 
                 0 
                 1 
                 2 
                 3 
                 2 
                 1 
                 0 
                 1 
                 2 
                 3 
                 3 
                 2 
                 3 
               
               
                   
               
             
          
         
       
     
     Consequently, the c i &#39;s that are assigned to subsets B 1    k  and B 2   j  are: 
     
       
         B 1   0 ={c 0 ,c 1 ,c 5 c 6 } B 2   0 ={c 0 ,c 2 ,c 3 ,c 9 } 
       
     
     
       
         B 1   1 ={c 2 ,c 4 ,c 7 ,c 12 } B 2   1 ={c 1 ,c 4 ,c 8 , c 10 } 
       
     
     
       
         B 1   2 ={c 3 ,c 8 ,c 11 ,c 13 } B 2   2 ={c 5 ,c 7 ,c 11 ,c 14 } 
       
     
     
       
         B 1   3 ={c 9 ,c 10 ,c 14 ,c 15 } B 2   3 ={c 6 ,c 12 ,c 13 ,c 15 } 
       
     
     To complete the design, the actual values of the c 1 , b 1   k , and b 2   j  codevectors need to be selected and, in accordance with the principle disclosed above, concurrently optimized for signals having a particular signal probability distribution. Thus, the codevectors need to be optimized in conformance with the objective to minimize the distortion measure D of TCQ 0 , subject to the condition that the resulting distortion measure D1 of TCQ 1  is not greater than d1 and the resulting distortion measure D2 of TCQ 2  is not greater than d2. 
     Using Lagrange multipliers λ 1  and λ 2 , the above-expressed constrained minimization problem can be converted to a non-constrained minimization problem of the form: 
     
       
         min {D+λ 1 D 1 +λ 2 D 2 }  (1)  
       
     
     For a given pair of multipliers λ 1  and λ 2 , and a training sequence x(n) with N samples, an illustrative distortion measure cost function can be expressed by                J   =         1   N            ∑     n   =   1     N            [       x        (   n   )       -       x   ^          (   n   )         ]     2         +         λ   1          [       x        (   n   )       -         x   ^     1          (   n   )         ]       2     +         λ   2          [       x        (   n   )       -         x   ^     2          (   n   )         ]       2         ,           (   2   )                                
     where {circumflex over (x)}(n) is the decoded approximation of sample x(n) in TCQ 0 , {circumflex over (x)} 1  (n) is the decoded approximation of sample x(n) in TCQ 1 , and {circumflex over (x)} 2 (n) is the decoded approximation of sample x(n) in TCQ 2 . The objective, then, is to minimize the combined distortion measure, J, by iteratively modifying the values of codevectors b 1   k , b 2   j , and c i  until the distortion constraints are met, and further modifications fail to sufficiently improve the value of J to merit continued modifications. 
     In the process of optimizing the values of codevectors c i , b 1   k , and b 2   j , if Y i  is the set of all training samples which are encoded as c i  in TCQ 0 , and |Y i | is the number of elements in set Y i , then replacing each codevector c i  with a new codevector {tilde over (c)} i  defined by                    c   ~     i     =       1          Y   i                   ∑       x        (   n   )       ∈     Y   i              x        (   n   )             ,           (   3   )                                
     results in a lower distortion D, when the same path and codewords which have been used for the old codevectors are utilized. Note that since we are dealing with a trellis coder, the fact that some x(n) ∈ Y i  does not necessarily means that c i  is the closest codevector to x(n). As the c i  values are modified in each iteration, the corresponding b 1   k , and b 2   j  values are also modified, as disclosed above. 
     FIG. 6 presents a flow chart of the process for ascertaining the actual c i , b 1   k , and b 2   j  values. Block  200  is the initialization block. This block identifies the probability distribution for the input signals and, in conformance with that distribution, selects a training sequence. It then chooses an initial set of C i  codevectors, and computes an initial set of b i   k , and b 2   j  codevectors. It selects λ 1  and λ 2 , sets the 0 th  iteration of the distortion measure, J (O) , to a large number, and sets iteration index r to 1. Control then passes to block  210  where the training sequence is encoded, using the Viterbi algorithm, using equation (2) as the distortion measure. Block  220  decodes the encoded signals using codevectors c i , b 1   j , and b 2   j  to obtain sequences {circumflex over (x)}(n), {circumflex over (x)} 1 (n), and {circumflex over (x)} 2 (n) , respectively, and block  230  computes J (r) . Decision block  240  evaluates              J     (     r   -   1     )       -     J     (   r   )           J     (   r   )         ,                          
     which is a measure of improvement in distortion measure, J, and compares it to a preselected threshold, ε. If the measure of improvement is greater than ε, control passes to block  250  which updates codevectors c i  in accordance with equation (3), and correspondingly updates codevectors b 1   k , and b 2   j . Thereafter, the index r is incremented in block  260 , and control returns to block  210 . If the measure of improvement is not greater than ε, then the iterative process for the selected Lagrange multipliers terminates. Block  270  then computes distortions D1 and D2 by evaluating                D1   =       1   N            ∑     n   =   1     N            [       x        (   n   )       -         x   ^     1          (   n   )         ]     2                         and           (   4   )               D2   =       1   N            ∑     n   =   1     N              [       x        (   n   )       -         x   ^     2          (   n   )         ]     2     .                 (   5   )                                
     Block  280  determines whether either one of the distortions is greater than the maximum specified distortion (e.g., the condition is TRUE if D1&gt;d1). If so, control passes to block  290 , where the corresponding Lagrange multiplier is reduced (in this example, λ 1 ) to allow that component to have greater effect in the calculations of equation (2). Block  290  also resets the iteration index, r, to 1, and returns control to block  210 . The process is repeated until both D1 and D1 are not greater than d1 and d2, respectively. 
     Since the FIG. 6 process alters the c i  levels, it is possible that the order of the c i  thresholds might change. For example, it is possible that the value of c 5  has grown to be larger than c 7  and that the value of c 4  has become less than the value of c 3 . Repositioning the TCQ 1  and TCQ 2  codevectors, the altered table would be 
     
       
         
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
               
                   
               
               
                 TCQ 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
               
                   
               
             
             
               
                 TCQ1 
                 0 
                 0 
                 1 
                 1 
                 2 
                 0 
                 1 
                 0 
                 2 
                 3 
                 3 
                 2 
                 1 
                 2 
                 3 
                 3 
               
               
                 TCQ2 
                 0 
                 1 
                 0 
                 1 
                 0 
                 3 
                 2 
                 2 
                 1 
                 0 
                 1 
                 2 
                 3 
                 3 
                 2 
                 3 
               
               
                   
               
             
          
         
       
     
     And the new set of assignments for B 1   k  and B 2   j  becomes: 
     
       
         B 1   0 ={c 0 ,c 1 ,c 5 ,c 7 } B 2   0 ={c 0 ,c 2 ,c 4 ,c 9 } 
       
     
     
       
         B 1   1 ={c 2 ,c 3 ,c 6 ,c 12 } B 2   1 ={c 1 ,c 3 ,c 8 ,c 10 } 
       
     
     
       
         B 1   2 ={c 4 ,c 8 ,c 11 ,c 13 } B 2   2 ={c 5 , c 6 ,c 11 ,c 14 } 
       
     
     
       
         B 1   3 ={c 9 ,c 10 ,c 14 ,c 15 } B 2   3 ={c 7 ,c 12 ,c 13 ,c 15 } 
       
     
     The above disclosure illustrates the principles of this invention; however, various modifications and enhancements can be introduced by artisans without departing from the spirit and scope of this invention, which is defined by the following claims. For example, the trellises that form the “building-blocks” of the tensor trellis can be replaced with other trellises, and do not have to be the same as the trellises. Also, although the above discloses the principles of this invention in the context of trellis coded quantization (TCQ) using scalar levels, it should be understood that these principles also apply to vectors (TCVQ), to entropy-coded TCQ, and to entropy-coded TCVQ. The generalized class that includes TCQ, TCVQ, entropy-coded TCQ, and entropy-coded TCVQ is termed herein GTCQ.