Abstract:
A method of transmitting data in a cable modem system includes the steps of encoding the data using forward error correction. The data is then encoded with Turbo encoding. The data is then sent to a modulation scheme. The data is then transmitted over a cable channel. The data is then demodulated. The data is then decoded using a Turbo decoder. An inverse of the forward error correction is then applied to the data.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
   This application is related to commonly assigned application Ser. No. 10/208,045, filed on Jul. 31, 2002, entitled TURBO-CODING DOCSIS INFORMATION FOR SATELLITE COMMUNICATION, which is incorporated by reference herein. 
   This application claims priority to U.S. Provisional Patent Application No. 60/436,470, filed on Dec. 27, 2002, entitled TURBO CODING FOR UPSTREAM AND DOWNSTREAM TRANSMISSION IN CABLE SYSTEMS, which are both incorporated by reference herein. 

   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates to encoding of data transmissions and cable modem systems. 
   2. Background Art 
   Forward error correction (FEC) is required in cable modem systems to provide high quality communication over the RF propagation channel, which induces signal waveform and spectrum distortions. These impairments drive the design of the transmission and receiver equipment, the design objective which is to select modulation formats, error control schemes, demodulation and decoding techniques and hardware components that together provide an efficient balance between system performance and implementation complexity. 
   Traditional forward error correction (FEC) schemes for communication systems include use of convolutional codes, block codes such as Reed-Solomon or BCH codes, and/or concatenated coding schemes. Turbo Codes are a relatively new class of codes that have been demonstrated to yield bit error rate (BER) performance close to theoretical limits on important classes of channels by means of an iterative soft-decision decoding method. A Turbo encoder consists of a parallel or serial concatenation of typically two systematic, recursive convolutional codes (“constituent codes”) separated by an interleaver that randomizes the order of presentation of information bits to a second constituent encoder with respect to a first constituent encoder. The performance of a Turbo Code depends on the choice of constituent codes, interleaver block size (which generally increases with higher block length), and number of decoder iterations. For a particular Turbo Code, in which the constituent codes are fixed, one can ideally adjust the block size and number of decoder iterations to trade-off performance, latency, and implementation complexity requirements. As the block size changes, however, a new interleaver matched to that block size is required. 
   Accordingly, there is a continued need for coding schemes that provide higher performance under noise conditions prevailing in cable modem systems. 
   BRIEF SUMMARY OF THE INVENTION 
   There is provided a method of transmitting data in a cable modem system including the steps of encoding the data using forward error correction. The data is then encoded with Turbo encoding. The data is then sent to a modulation scheme. The data is then transmitted over a cable channel. The data is then demodulated. The data is then decoded using a Turbo decoder. An inverse of the forward error correction is then applied to the data. 
   Additional features and advantages of the invention will be set forth in the description that follows, and in part will be apparent from the description, or may be learned by practice of the invention. The advantages of the invention will be realized and attained by the structure particularly pointed out in the written description and claims hereof as well as the appended drawings. 
   It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are intended to provide further explanation of the invention as claimed. 

   
     BRIEF DESCRIPTION OF THE FIGURES 
     The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention. 
     In the drawings: 
       FIG. 1  illustrates a cable modem transmitter and receiver system; 
       FIG. 2  illustrates a Turbo Encoder and Decoder used in the cable modem system of  FIG. 1 ; 
       FIG. 3  shows an example of a convolutional encoder; 
       FIGS. 4 and 5  show examples of convolutional encoders for producing Turbo convolutional codes; 
       FIG. 6  shows a concatenation of outputs of two convolutional codes in a Turbo encoder; 
       FIG. 7  shows a structure of a pseudo-random interleaver for a Turbo encoder; 
       FIG. 8  shows an alternative structure of a Turbo encoder; 
       FIG. 9  illustrates an exemplary structure of a constituent encoder of  FIG. 8 ; 
       FIG. 10  illustrates a Turbo Decoder of  FIG. 2 ; 
       FIG. 11  illustrates an alternative structure of the Turbo Decoder of  FIG. 2 ; 
       FIG. 12  illustrates the performance improvement due to the use of Turbo encoding in a cable modem system. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   Reference will now be made in detail to the preferred embodiments of the present invention, examples of which are illustrated in the accompanying drawings. 
     FIG. 1  is a block diagram of a cable modem communication system  100 , including a headend baseband modulator/demodulator (“headend”)  102  that communicates with a plurality of cable modulators/demodulators (“modems”)  104  through a primary cable  106 , which branches to user cables  108 . The cable modems  104  demodulate data from the headend  102 , and modulate data to be transmitted to the headend  102 . One or more optional intermediate power amplifiers  110  can be placed along the cables  106  and/or  108  to boost signal strength. The cables  106  and  108  have less impairments compared to wireless communication systems. The relatively low noise, the optional intermediate power amplifiers  110 , and relatively short distances involved, provide the cable modem communication system  100  with a relatively high signal-to-noise ratio (“SNR”). 
   The communication paths from the headend  102  to the users  104  are called down-stream paths or channels. The communication paths from the users  104  to the headend are called up-stream paths or channels. The protocol commonly used to send data upstream and downstream is known as DOCSIS, although the invention is not limited to any particular protocol. 
   In conventional DOCSIS systems, upstream channels are time division multiple access (“TDMA”) channels, where multiple cable modems share an upstream channel. The headend assigns bandwidth to the cable modems by means of time-slot mapping (“MAP”) messages that are broadcast to users of a given upstream channel. The MAP messages contain information allowing each user to burst an appropriate type of data on the upstream channel at an appropriate time. In conventional DOCSIS systems, the upstream data bursts are typically encoded with Reed Solomon (RS) forward error correction (“FEC”), to increase the reliability of the data reception at the headend. In conventional DOCSIS systems, upstream signals to the headend are transmitted at relatively low frequencies (e.g., in the range of 5-65 MHz). 
   The flexibility and high performance of Turbo Codes make them a potentially attractive technology for sophisticated data communications services, such as cable modem communications systems, though they have never been applied to cable modem systems before for a number of reasons. 
     FIG. 2  illustrates the cable modem transmitter and receiver system of the present invention. As shown in  FIG. 2 , input data is fed into an MPEG framer  201  (e.g., for MPEG 2 or MPEG 4 framing). After the MPEG framer  201 , the MPEG frames go into an outer FEC (Forward Error Correction) encoder  202 . Generally, the FEC encoder  202  may be either a block type or a trellis type (sometimes known as convolution type. Block type encoders are well known in the art, and include, e.g., Reed Solomon, Reed-Muller, Hamming, and a number of others. 
   Further with reference to  FIG. 2 , the data from the outer FEC encoder  202  goes into an interleaver  203 , whose primary purpose is to spread the data out temporally to reduce the effect of errors at the decoder. These errors are due to impulse noise or bursts of errors produced by a Viterbi decoder. 
   From the interleaver  203 , the signal enters a Turbo Encoder  204 , which will be described in further detail below. It then enters a modulator  205 . The modulator  205 , e.g., may be a QAM modulator (e.g., a 16 QAM modulator, a 64 QAM modulator, 256 QAM or 1024 QAM modulator), or it may be a QPSK modulator. 
   The modulator  205  outputs the signal onto the channel  108 , which may, for example, be a coaxial cable or a fiber optic cable. 
   On the receiver end, the signal is received by a demodulator  207 , and is inputted into a Turbo Decoder  208 , which will be discussed in additional detail below. A deinterleaver  209  reverses the interleaving operation of the interleaver  203 , and an outer FEC decoder applies the appropriate error correction, scheme matching the FEC encoder  202 . An MPEG deframer  211  (e.g., for MPEG 2 or MPEG 4 deframing) then outputs data out to the rest of the receiver system. 
   Error correcting (FEC) codes are normally classified according to whether they employ memory in the encoding process. This classification process results in codes being classified as either convolutional codes or block codes. The present invention is applicable to both block codes and trellis codes. 
   Block codes (e.g., RS, RM, Hamming) transform a block of k bits into an n-bit codeword by adding n-k redundant bits that are algebraically related to the k message bits. The channel encoder for an (n,k) linear block code generates bits at the rate: R 0 =(n/k)·R s  where R s  is the information rate of the source r=k/n is known as the code rate, and R o  is the channel data rate. 
   Block codes in which the message bits are transmitted unaltered are known as systematic codes. A systematic structure divides the codeword into two parts, the k message bits and the (n-k) parity bits. The (n-k) parity bits are linear sums of the k message bits, where each of the (n-k) equations are linearly independent (that is, no equation in the set can be expressed as a linear combination of the remaining equations). 
   As an example, for MPEG 2 frame format, the Reed-Solomon code becomes a (188, 204) code, i.e., the frames includes 16 parity bytes and 188 data bytes. FEC overhead tends to be higher for trellis codes than for block codes such as Reed-Solomon. Note further that block codes such as Reed-Solomon, error connection is done in a single pass. 
   The convolutional encoding process (trellis encoding) is a discrete-time convolution of the input sequence with the impulse response of the encoder. A convolutional encoder operates on the incoming message sequence continuously in a serial manner, and can be modeled as a finite-state machine consisting of an M-stage shift register. An L-bit message sequence produces a coded output sequence of length of n(L+M) bits. The code rate is given by 
   
     
       
         
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     FIG. 3  shows an example of a (2, 1) convolutional encoder, with constraint length M=3. In theory, as convolution codes are not block codes, this encoder should have a code rate of ½. However, convolutional encoders are often forced into a block structure, due to periodic truncation. This occurs as the convolutional encoder flushes the remaining bit out of the register by appending zeros. In this example, 3 zeros would be appended, which brings the effective code rate down. However, as the number of bits before the periodic truncation increases, the code rate approaches ½. 
   The two generators for this code are G 1 =7 O  (octal) and G 2 =5 O  (octal). With an input sequence  101 , the following output sequence results: 11 10 00 10 11. These are pairs of outputs from G 1  and G 2  respectively. In this example, two extra zeros have been inputted, to flush the register, and ensure a full code. In the case shown above, the two 6-bit codewords are 11 10 11 for an input bit 1, and 00 00 00 for an input bit 0. To encode an input of  101 , the output becomes 11 (10+00) (11+00+11) (00+10) 11, which gives the same result as above, 11 10 00 10 11. 
   A convolution code (i.e., trellis code) may be decoded by applying the principle of maximum likelihood decoding to minimum distance decoding by choosing a path in a code tree whose coded sequence differs from the received sequence in the fewest number of places. 
   Turbo encoders are generally described in Valenti, Matthew C., “Turbo Codes and Iterative Processing,” Mobile and Portable Radio Research Group, Virginia Polytechnic Institute and State University, Blacksburg, Va.; “Research and Development: Communications/Turbo Coding,” Xenotran, http://xenotran.com/turbo_tech_error_turbo.html, Mar. 11, 2002; W. E. Ryan, “A Turbo Code Tutorial,” Proc. IEEE Globecom&#39;98, 1998; “Telecommunications and Mission Operations Directorate—DSN Technology Program: Communications Systems Analysis: Turbo Codes,” http://www331.jpl.nasa.gov/public/TurboForce.GIF, Mar. 3, 2002; and Luke Hebbes and Ron Malyan, “Comparative Performance Modelling of Turbo, Block and Convolutional Coding over very noisy channels,” (http://technology.kingston.ac.uk/ncg/Research/Publications/1998/Comp_Mod el_TC/Comp_Model_TC.htm), all of which are hereby incorporated by reference in their entireties. 
   Turbo encoders typically use at least two convolutional component encoders. Turbo encoders can also be based on block encoding techniques, such as Reed Solomon, Reed Muller, or Hamming codes. Turbo codes include, for example, and without limitation, Parallel Concatenated Convolutional Codes (PCCC), Serial Concatenated Convolutional Codes (SCCC), and Hybrid Concatenated Convolutional Codes (HCCC). 
   Turbo codes are parallel or serial concatenated, Recursive Systematic Convolutional (RSC) codes. RSC codes can perform better than the best Non-Systematic Convolutional (NSC) codes at any Signal-to-Noise Ratio (SNR). Turbo codes, therefore, can provide significant performance improvements over more conventional coding schemes. 
   An RSC code is obtained by employing a feedback loop in a NSC code, and setting one of the outputs to be the input bit sequence. This can be more easily seen in  FIGS. 4 and 5 , described below, which show alternative embodiments for producing an NSC code ( FIG. 4 ) or an RSC code ( FIG. 5 ). The example used here shows encoders with memory M=4, with generators G1=37 O  and G2=21 O . 
   The memory is provided by the four delay blocks T shown in  FIG. 4 . For instance, if the initial state of the memories were to be 0000, in the example above, then an input sequence of 1001 would produce the following memory states after each bit has been presented: 1000, 1100, 0110, 1011. The combination of these memories is then taken according to the generator. In the above case, the two generators are 37 O  and 21 O . It can be seen from the figure that the two generators can be represented in binary by 1111 b  and 10001 b  respectively, or 31 d  and 17 d  respectively, in decimal. They are, however, usually quoted in octal. 
   In the Turbo code, two identical RSC codes are combined in parallel or serially. A parallel concatenation of the two constituent codes can be seen in  FIG. 6 .  FIG. 6  shows two (37, 21) RSC codes with memory M=4. It can be seen from  FIG. 6  that two outputs are taken: one is the actual input bit, and the other is either a bit from one RSC encoder or the other. 
   The Turbo encoder  208  includes an interleaver  602  (interleavers are usually designated by “π”). The interleaver  602  permutes the block of input bits to the second encoder. Although both of the constituent RSC encoders  501  are working on the same block of bits, they are in a different order. Thus, it is likely that when one encoder  501  produces a low-weight codeword, the other encoder  501  may produce a high-weight codeword. This combination of weak codes can, therefore, produce a powerful combined code. 
   The equations governing Turbo codes will now be discussed. A binary rate R=½ convolutional encoder has a constraint length K and memory M=K−1. The rate is calculated from the number of information bits transmitted divided by the total number of bits transmitted. The input to this encoder at time k is then the data bit d k , and the corresponding codeword C k  is the binary couple (X k , Y k ) where 
   
     
       
         
           
             
               
                 
                   
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   In the case of the RSC code, however, the feedback loop needs to be taken into account. If the code is as in  FIG. 6 , the X output is the input data and the feedback feeds the Y output. Equations 3 and 4 result, which are modified from (1) and (2) above. 
   
     
       
         
           
             
               
                 
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   A pseudo-random interleaver  602  may be used. Interleaver  602  with length N=2 m −1 can be produced by using a shift-register with feedback connections made according to a primitive polynomial of degree m. This is then loaded with a non-zero codeword, and cycled through all 2 m −1 different binary words. The resultant order can then be used to permute blocks of data bits. 
   For example, with a polynomial D 3 +D 2 +1, the structure of this pseudo-random generator is shown in  FIG. 7 . Starting with the codeword  101 , we then have the result: 101, 010, 001, 100, 110, 111 &amp; 011. The permutation [1234567]→[5214673] is obtained. 
     FIG. 8  shows an alternative structure of the Turbo Encoder  204 . The input bits are fed into a constituent encoder  802 A (discussed below with reference to  FIG. 9 ), and, alternatively, to an interleaver  801  and then to an identical constituent encoder  802 B. The letters T and B refer to “top” and “bottom” which are formed into a combined data stream  803 . The data stream  803  is fed through a symbol mapper, which may be, e.g., a QAM constellation mapper (part of  205  in  FIG. 2 ). 
     FIG. 9  illustrates the structure of the constituent encoder  802  of  FIG. 8 . Some of the encoded bits u k  remain uncoded, as shown in  FIG. 9 . Other encoded bits are fed into a convolutional encoder  301 , as discussed above. The convolutional encoder  301  outputs coded bits i k  and redundant bits r k . 
     FIG. 10  shows one implementation of a Turbo Decoder  208 . Data comes in from the channel  108 , and enters a soft decoder  1001 , which outputs a soft decision of the input symbol. Two soft input, soft output (SISO) blocks  1002 A,  1002 B, are used to arrive at a better estimate of the received symbol through a number of iterations. The output of the SISO  1002 A is fed into an interleaver  1003 , and then to the SISO  1002 B, the output of the SISO  1002 B is fed to a de-interleaver  1004 , and to a hard or soft decision block  1005 . Similarly, the output of deinterleaver  1004  is fed back to the SISO  1002 A, and optionally to the hard or soft decision block  1005 . The hard or soft decision block  1005  can output either the best estimate of the symbol, or the probabilities and the weights obtained through the Turbo Decoder  208  to the subsequent processing logic. 
   An alternative structure of the Turbo Decoder  208  is shown in  FIG. 11 . The Turbo Decoder  208  uses a soft-input/soft-output algorithm that makes a decision about the output based on weights. The highest weight codeword becomes the output word. The actual structure of the Turbo Decoder  208  is a serial concatenation of two identical elementary decoders  1102 A,  1102 B, separated by an interleaver  1103  and a de-interleaver  1104 A. The decoder also has feedback between the two elementary decoders  1102 A,  1102 B. Decoded output passes through a de-interleaver  1104 B. The Turbo Decoder  208  takes the form shown in  FIG. 11 . 
   The input of the Turbo Decoder  208  is the binary couple (X k , Y k ). The Y k  is the combination of Y 1 k and Y 2 k from the Turbo Encoder  204  discussed above. The input is switched from the first decoder  1104 A to the second decoder  1104 B depending on the constituent encoder. When the input is switched to one decoder  1102 , the input to the other decoder is set to zero. The decision is made after a set number of iterations. The number of iterations performed affects the BER. 
   The decision process made by the symbol-by-symbol MAP decoder  1102  may be based on the sign of the Log A Posteriori Probability (LAPP) ratio. The decision is made as follows: u k =+1 if P(u k =+1|y)&gt;P(u k =−1|y), and u k =−1 otherwise. 
   Each constituent decoder  1102  must have full knowledge of the trellis of the corresponding encoders. Input bits and parity bits for all possible state transitions must be known, and can be stored in an array or matrix. Also, the interleaver  1103  and de-interleavers  1104 A,  1004 B must be matched to the Turbo Encoder  204 . 
   The iterative process will now be described. The two constituent decoders  1102  are initialized separately. Starting with decoder  1102 A: 
                     α   0     (   1   )       ⁡     (   s   )       =       1   ⁢           ⁢   for   ⁢           ⁢   s     =   0                 =       0   ⁢           ⁢   for   ⁢           ⁢   s     ≠   0                     ρ   N     (   1   )       ⁡     (   s   )       =       1   ⁢           ⁢   for   ⁢           ⁢   s     =   0                 =       0   ⁢           ⁢   for   ⁢           ⁢   s     ≠   0                       L   21   e     ⁡     (     u   k     )       =       0   ⁢           ⁢   for   ⁢           ⁢   k     =   1       ,   2   ,   …   ⁢           ,   N               
where L(u k ) is the LAPP ratio. To set the initial state of the decoder  1102 B we get the following:
 
                     α   0     (   2   )       ⁡     (   s   )       =       1   ⁢           ⁢   for   ⁢           ⁢   s     =   0                 =       0   ⁢           ⁢   for   ⁢           ⁢   s     ≠   0                       ρ   N     (   2   )       ⁡     (   s   )       =       α   N     (   2   )       ⁡     (   s   )         ,     ∀     s   .                   
This is set in the first iteration L 12   e (u k )that is determined after the first half-iteration from decoder  1102 A. The following explains the n th  iteration. Again, the two decoders  1102 A,  1102 B are considered separately. For decoder  1102 A, we have, for k=1,2, . . . , N
         get y k =(y k   5 ,y k   1y ), where is y k   5  the source bit, and y k   1y  is the parity bit from encoder  204 .       
   
     
       
         
           
               
           
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is the de-permuted extrinsic information from the previous decoder  1002 B iteration, and E c  is the energy per channel bit
 
         
       
     
  
                         ⁢             compute   ⁢           ⁢       α   k     (   1   )       ⁡     (   s   )         =         ∑     s   ′       ⁢         α     k   -   1       (   1   )       ⁡     (     s   ′     )       ·       γ   k     ⁡     (       s   ′     ,   s     )               ∑   s     ⁢       ∑     s   ′       ⁢         α     k   -   1       (   1   )       ⁡     (     s   ′     )       ·       γ   k     ⁡     (       s   ′     ,   s     )                 ,     ∀   s       ⁢     
     ⁢       for   ⁢           ⁢   k     =   N       ,     N   -   1     ,   …   ⁢           ,   2     ⁢                   (   5   )                       ⁢         compute   ⁢           ⁢         ρ   ^     k     (   1   )       ⁡     (   s   )         =         ∑   s     ⁢           ρ   ^     k     (   1   )       ⁡     (   s   )       ·       γ     k   .       ⁡     (       s   ′     ,   s     )               ∑   s     ⁢       ∑     s   ′       ⁢         α     k   -   1       (   1   )       ⁡     (     s   ′     )       ·       γ     k   .       ⁡     (       s   ′     ,   s     )                 ,       ∀     s   ⁢     
     ⁢   for   ⁢           ⁢   k       =   1     ,   2   ,   …   ⁢           ,   N     ⁢                   (   6   )                       ⁢       compute   ⁢           ⁢       L   12   e     ⁡     (     u   k     )         =     log   ⁡     (         ∑     s   +       ⁢         α     k   -   1       (   1   )       ⁡     (     s   ′     )       ·       γ   k   e     ⁡     (       s   ′     ,   s     )       ·         ρ   ^     k     (   1   )       ⁡     (   s   )               ∑     s   -       ⁢         α     k   -   1       (   1   )       ⁡     (     s   ′     )       ·       γ   k   e     ⁡     (       s   ′     ,   s     )       ·         ρ   ^     k     (   1   )       ⁡     (   s   )             )                 (   7   )               
For decoder  1102 B, the iterative process is similar. For k=1,2, . . . , N
         get y k =(y P[k]   5 ,y k   2y )       
   
     
       
         
           
               
           
           ⁢ 
           
             
               
                 compute 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     γ 
                     k 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         s 
                         ′ 
                       
                       , 
                       s 
                     
                     ) 
                   
                 
               
               = 
               
                 
                   exp 
                   ⁡ 
                   
                     [ 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           u 
                           k 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 L 
                                 12 
                                 e 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   u 
                                   
                                     P 
                                     ⁡ 
                                     
                                       [ 
                                       k 
                                       ] 
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 
                                   4 
                                   ⁢ 
                                   
                                     E 
                                     c 
                                   
                                 
                                 
                                   N 
                                   0 
                                 
                               
                               ⁢ 
                               
                                 y 
                                 
                                   P 
                                   ⁡ 
                                   
                                     [ 
                                     k 
                                     ] 
                                   
                                 
                                 5 
                               
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
                 · 
                 
                   exp 
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           2 
                           ⁢ 
                           
                             E 
                             c 
                           
                         
                         
                           N 
                           0 
                         
                       
                       ⁢ 
                       
                         y 
                         
                           P 
                           ⁡ 
                           
                             [ 
                             k 
                             ] 
                           
                         
                         y 
                       
                       ⁢ 
                       
                         x 
                         k 
                         y 
                       
                     
                     ] 
                   
                 
               
             
             , 
           
         
       
     
       
       
         
            state transitions allowed, where u k  is set to the value of the encoder input that caused the transition s′→s and L 12   e (u p[k] ) is the permuted extrinsic information from the previous decoder 1 iteration 
           compute α k   (2) (s), α k-1   (2) (s) and L 21   e (u k ) from (5), (6) &amp; (7). 
         
       
     
  
   Finally, after the last iteration, we must compute the decoded bits are computed by the following iteration: 
   for k=1,2, . . . ,N 
   
     
       
         
           
               
           
           ⁢ 
           
             
               compute 
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   L 
                   1 
                 
                 ⁡ 
                 
                   ( 
                   
                     u 
                     k 
                   
                   ) 
                 
               
             
             = 
             
               
                 
                   
                     4 
                     ⁢ 
                     
                       E 
                       c 
                     
                   
                   
                     N 
                     0 
                   
                 
                 · 
                 
                   y 
                   k 
                   s 
                 
               
               + 
               
                 
                   L 
                   21 
                   e 
                 
                 ⁡ 
                 
                   ( 
                   
                     u 
                     
                       Pin 
                       ⁡ 
                       
                         [ 
                         k 
                         ] 
                       
                     
                   
                   ) 
                 
               
               + 
               
                 
                   L 
                   12 
                   e 
                 
                 ⁡ 
                 
                   ( 
                   
                     u 
                     k 
                   
                   ) 
                 
               
             
           
         
       
     
       
       
         
            if L 1 (u k )&gt;0, decide u k =+1 else decide u k −1. 
         
       
     
  
     FIG. 12  shows the improvement in performance obtained using the present invention. As may be seen in  FIG. 12 , the improvement ranges from approximately to 1 dB to 2 dB. For example, consider the two center graphs, labeled A and B. If the required bit error rate is 10 −4 , and the physical channel (e.g., the actual coax cable connecting the transmitter and receiver) has a signal to noise ratio of 12 dB, it would be impossible to use a 16 QAM conventional trellis coded modulator (TCM) as shown in  FIG. 12  (see graph B), because the signal to noise ratio of the channel is insufficient to effect the appropriate bit error rate. However, it is possible to use a 16 QAM modulation scheme with Turbo Encoding (see graph A), as discussed above. Thus, there is no need to go to a higher constellation QAM modulation scheme for this particular example of BER and SNR. 
   It will be appreciated that although in the example above, eight iterations are used to decode a symbol, more or fewer iterations may be used. It is expected that after approximately 16 iterations, further increases in the number of iterations will not be particularly useful. Generally, as the number of iterations in the Turbo Decoder  108  increases, the demands on the hardware also increase. However, when the lag time due to hardware issues is acceptable, it is expected that the optimal number of iterations will be somewhere between 8 and 16. 
   It will also be appreciated from looking at  FIG. 12  that higher signal to noise ratio systems permit the use of higher QAM constellations, e.g., 256 QAM or 1024 QAM. However, it is believed that, since practical cable systems are limited to about 30 dB signal to noise ratio due to the inherent physical properties of the system, the highest QAM modulation scheme possible is 1024, or, more likely, 256 QAM. 
   It will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined in the appended claims. Thus, the breadth and scope of the present invention should not be limited by any of the above-described exemplary embodiments, but should be defined only in accordance with the following claims and their equivalents.