Patent Publication Number: US-6982659-B2

Title: Method and apparatus for iterative decoding

Description:
CROSS-REFERENCE TO RELATED APPLICATION(S) 
   This application is a continuation of U.S. patent application Ser. No. 10/219,858, filed Aug. 15, 2002 which is now a U.S. Pat. No. 6,686,853, entitled “METHOD AND APPARATUS FOR ITERATIVE DECODING” which is a continuation of Ser. No. 09/900,222, now U.S. Pat. No. 6,518,892, filed Jul. 6, 2001, entitled “STOPPING CRITERIA FOR ITERATIVE DECODING” which claims priority of U.S. Provisional Patent Application No. 60/246,425, filed Nov. 6, 2000, entitled “STOPPING CRITERIA FOR DECODING OF TURBO CODE”. 

   FIELD OF THE INVENTION 
   The present invention relates to a decoding method and apparatus. More specifically, the invention relates to an interative decoding method and apparatus. 
   BACKGROUND OF THE INVENTION 
   A significant amount of interest has recently been paid to channel coding. For example a recent authoritative text states: “Channel coding refers to the class of signal transformations designed to improve communications performance by enabling the transmitted signals to better withstand the effects of various channel impairments, such as noise, interference, and fading. These signal-processing techniques can be thought of as vehicles for accomplishing desirable system trade-offs (e.g., error-performance versus bandwidth, power versus bandwidth). Why do you suppose channel coding has become such a popular way to bring about these beneficial effects? The use of large-scale integrated circuits (LSI) and high-speed digital signal processing (DSP) techniques have made it possible to provide as much as 10 dB performance improvement through these methods, at much less cost than through the use of most other methods such as higher power transmitters or larger antennas.” From “Digital Communications” Fundamentals and Applications Second Edition by Bernard Sklar, page 305 © 2000 Prentice Hall PTR. 
   There are multiple modern decoding methods that involve iterative probabilistic decoding methods. Among the list of iterative probabilistic methods are methods such as MAP decoding, soft output Viterbi decoding and others. Because of the use of iterative decoding techniques, there is a need for improved iterative decoding methods in the art. 
   SUMMARY OF THE DISCLOSURE 
   In a first aspect of the invention a method of generating a stopping criteria for an iterative decoder is disclosed. The method includes, performing an Nth iteration of decoding, forming a signature from extrinsic values of the Nth iteration, comparing the signature of the Nth iteration to a signature of the N-1st iteration and stopping the process of iteration decoding if the signature of the N-1st iteration is equal to the signature of the Nth iteration. 
   In a second aspect of the invention a method of generating a stopping criteria for an iterative decoder is disclosed. The method includes performing an Nth iteration of decoding, forming a signature from extrinsic values of the Nth iteration, comparing the signature of the Nth iteration to a signature of the N-2 iteration and stopping the process of iteration decoding if the signature of the N-2 iteration is equal to the signature of the Nth iteration. 
   In a third aspect of the invention a method of generating a stopping criteria for an iterative decoder is disclosed. The method includes, determining the variance (VAR k ) of extrinsic information on a k&#39;th iteration of the iterative decoder and halting the decoder if VAR k &lt;T 1 , where T 1  is a first threshold and D k  (Differential Variance)&lt;T 2 , where T 2  is a second threshold. 
   In a fourth aspect of the invention a method of determining a threshold T 1  for a particular encoding is disclosed. The method includes selecting a value for E b /N 0 , creating a signal having the particular encoding, adding a noise vector to the signal to create a corrupted signal, iteratively decoding the corrupted signal until the iteration converges and assigning a value less than VAR k  to T 1 . 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The features, aspects, and advantages of the present invention, which have been described in the above summary, will be better understood with regard to the following description, appended claims and drawings where: 
       FIG. 1  is a graphical illustration of an environment in which embodiments of the present invention may operate. 
       FIG. 2  is a block diagram of a model of a data transmission system. 
       FIG. 3  is a block diagram of a simulation of the transmission system illustrated in FIG.  2 . 
       FIG. 4  is a block diagram of a portion of a decoder according to an embodiment of the invention. 
       FIG. 5  is a graphical illustration of table 1 through table 3, which illustrate the relationship between decoding iterations to bit errors. 
       FIG. 6  is a block diagram of a signature circuit, according to an embodiment of the invention. 
       FIG. 7  is a graphical illustration of table 4 through table 7, which illustrate the relationship between decoder iterations, signature stopping criteria, variance criteria and decoding errors. 
       FIG. 8  is a graph illustrating bit error rate (BER) verses E b /N 0  for various stopping criteria. 
   

   DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION 
     FIG. 1  is a graphic illustration of an environment in which embodiments of the present invention may operate. The environment illustrated at  101  is a data distribution system, such as may be found in a cable television distribution system. 
   In  FIG. 1  data is provided to the transmission system by an information source  103 . For purposes of illustration, the information source displayed in  FIG. 1  may be considered to be a cable television system head end, which provides video data to end users. Embodiments of the invention are not limited to any particular type of information source and any other data source could be equivalently substituted. A formatter  105  accepts data from the information source  103 . The data provided by information source  103  may comprise analog or digital signals such as (but not limited to) video signals, audio signals, and data signals. Formatter block  105  formats received data into an appropriate form such as the data illustrated at  107 . The formatted data  107  is then provided to a channel encoder  109 . Channel encoder  109  encodes the data  107  provided to it. In some embodiments of the present invention, the channel encoder  109  may provide an encoding, which is configured differently dependent on different goals of the particular system. For example, the encoding may be used to make the signal more robust, to reduce the error probability, to operate the system using less transmission power or to enable a more efficient decoding of the signal. 
   Channel encoder  109  provides encoded data to a transmitter  111 . Transmitter  111  transmits the encoded data provided by the channel encoder  109 , for example, using an antenna  113 . The signal transmitted from antenna  113  is accepted by a relay satellite  115  and then retransmitted to a terrestrial receiving antenna, such as earth station antenna  117 . Earth station antenna  117  collects the satellite signal and provides the collected signal to a receiver  119 . The receiver  119  amplifies and demodulates/detects the signal as appropriate and provides the detected signal to a decoder  121 . 
   Decoder  121  will, essentially, reverse the process of the channel encoder  109  and recreate the data  123 , which should represent a good estimate of the data  107  that had been broadcast. The decoder  121  may use Forward Error Correction (FEC), in order to correct errors in the received signal. The data  123  provided by the decoder are then provided to a formatting unit  125 , which prepares the received data for use by an information sink, such as the television illustrated at  127 . 
     FIG. 2  is a block diagram illustrating a model of a transmission system. In  FIG. 2  data  203  is provided to encoder  205 . Encoder  205  may provide different types of encoding depending on the application. For example, encoder  205  may be a trellis encoder, a parallel concatenated encoder (PCE) a low density parity check type encoder (LDPC) or a variety of other types of encoders. After being encoded by encoder  205 , the encoded data is then provided to channel  207 . Channel  207  comprises a channel driver, the actual channel medium, and a channel receiver. The channel  207  may comprise a variety of different type channel media, such as, but not limited to, radio or fiber optic media. 
   In the transmission system model, channel  207  also receives an input from a noise block  209 . Noise block  209  may comprise a variety of different types of noise from different sources. 
   The noise introduced to the channel  207  serves to corrupt the encoded signal provided by encoder  205 . The result of the addition of noise  209  to the channel  207  is a corrupted data signal  211  representing a combination of the encoded data and added noise. The corrupted data signal  211  is provided to decoder  213 . Decoder  213  attempts to decode the corrupted data signal and recreate the original data  203 . Decoder  213  provides a data output  215 . 
   The transmission system of  FIG. 2  is a model of a real world type communication channel. The decoder illustrated at  213  is a type of decoder known as an “iterative” decoder. Decoder  213  is an iterative decoder because it produces the output data  215  by processing received data and noise multiple times i.e., it makes several iterations through the data. The decoder  213  makes several iterative passes through the received data computing an estimate of the transmitted data, or some other likelihood metric related to the liability of the data estimate produced on each successive pass. 
   Iterative decoding may be used to decode different types of encoding probabilistically by successfully refining estimates of the data. In such iterative decoding, a first iteration estimate may provide a starting point for a second iteration estimate etc. In such types of iterative decoding, data estimates, for example in the form of probabilities, likelihoods or distance metrics, are passed from one iteration to the next and successively refined and hopefully improved. The output of one iteration of data processing becomes the input to the next iteration of processing. 
   Several types of codes are amenable to the iterative type of decoding. For example, serial and parallel concatenated codes, also known as serial and parallel turbo codes may be decoded iteratively. Additionally product codes, low density parity check codes (LDPC), Reed Solomon codes, graph codes, and belief propagation codes may be decoded iteratively. While the methods disclosed herein may be used with all the aforementioned codes. 
   Examples of the inventive concepts herein will be illustrated through the use of parallel concatenated (turbo) codes. Those skilled in the art will realize that the same iterative decoding method that is illustratively applied to turbo codes may be applied equally well to other iterative decodings. The use of turbo codes to illustrate embodiments of the invention is chosen as a matter of convenience, as an example likely to be familiar to those skilled in the art. There is, however, no intent to limit the inventive concepts disclosed herein to turbo codes or any of the example iterative codes mentioned above. The concepts disclosed and explained herein are equally applicable to any iterative decoding method. 
     FIG. 3  is a block diagram of a simulation of the transmission system illustrated in FIG.  2 . The simulation of  FIG. 3  is used to illustrate, study and quantify the iterative decoding methods disclosed herein. The simulation of  FIG. 3  may be programmed entirely on a computer, or may have portions of it realized in a variety of forms. For example, the decoder  313  may be an actual hardware type decoder or a software simulation. For the purposes of simplicity of explanation, the simulation  301  will be treated as a completely software simulation. 
   Input data  303  may comprise multiple blocks of data. The input data  303  for the software simulation may be contained in a computer file, thus, the data values are known. Data  303  is provided to encoder  305 , which will encode the data. A noise vector  309  is added to the encoded data in adder  307 . Because the noise vector  309  is a simulated noise vector, the amount of corruption added to the encoded signal can be controlled by controlling the value of the noise vector added. The result of the addition of the encoded data and noise vector  309  in adder  307  is a corrupted data signal  311 . The noise and data vector  311  can then be decoded by a decoder  313 . Embodiments of the invention may operate within the decoder  313  and may control the decoding of data within decoder  313 . Iterations of decoder  313  may be interrupted at any point to analyze the effectiveness of the embodiments of the invention, which control the decoding. 
   The output of decoder  313  is a data block  315 . Data block  315  can be compared with the original data  303  in a comparison unit  317 , and the results from any number of iterations saved in a results file  319  for analysis. 
   By using the simulation of  FIG. 3  embodiments of the invention may be tested and analyzed. Throughout the present disclosure test results, arrived at through the use of simulations equivalent to the simulation illustrated in  FIG. 3 , are used to illustrate various aspects and embodiments of the present invention. 
     FIG. 4  is a block diagram of a portion of an iterative decoder, according to an embodiment of the invention. In  FIG. 4 , an example decoding system for parallel concatenated (turbo) codes is illustrated, such a decoder within decoding block  313 , may be controlled by embodiments of the invention.  FIG. 4  assumes that the encoder  305  is a (turbo) encoder. 
   In  FIG. 4 , decoder  313  comprises two soft-in soft-out (SISO) component decoders  403  and  405 . Such decoders may implement a MAP (Maximum A Posteriori) Algorithm, and hence the decoder may also alternatively be referred to as a MAP decoder or MAP turbo decoder. A soft output to hard output converter  407  receives the output of SISO decoder  405 . The converter  407  converts the soft values from SISO decoder  405  to hard output values. 
   SISO decoder  403  provides a priori values for SISO decoder  405 . SISO decoder  405  receives the a priori values from SISO decoder  403  and then provides extrinsic soft values to converter  407 , which are converted into hard values. Converter  407  is not a usual part of a turbo decoder. Converter  407  is used to determine the actual data value, which would be decoded if the present decoding iteration were the final iteration. In other words, converter  407  is used to determine how many errors would be present if the current iteration were converted to hard values. The extrinsic values from SISO  405  are also accepted for iterative processing by SISO  403 . Using such an arrangement the result produced by any decoder iteration can be analyzed. 
   Because the output of the SISO  403  and  405  are soft values, they are not merely 0 or 1 values. The soft values produced by the SISO are values that are representative of the value of the signal decoded, and the confidence in the value of the signal decoded as well. For example, the MAP decoders may output values between −7 and +7. A −7 may represent a binary value of 0 with a high confidence. The minus sign indicating a binary 0 and the value of 7 indicating that the value 0 is known with a high degree of confidence. Similarly, a SISO decoder output of −3 would also indicate a digital 0 value, however with less confidence than −7. An output of a −1 would represent a digital 0 with even less confidence than −7 or −3. An output of 0 would indicate that digital values of 1 and 0 are equally likely. In contrast, a +1 would indicate a digital value of 1 with a low level of confidence. A +3 would represent a digital value of 1 with more confidence than a +1, and a value of +7 would represent a digital value of 1 with more confidence than either a +1 or +3. 
   Since the input data  303  to the simulation comprises hard binary values of 0 or 1, the output of the SISO decoder  405  will be converted to hard, i.e., either 1 or 0, digital values before being compared with the input data block  303 . Converter  407  converts the soft output values of SISO  405  into hard digital values. 
   Once the soft values from SISO  405  are converted to hard values and provided to data block  315 , the hard values can be compared with the original data  303 . 
   The simulation of  FIG. 3  is useful because data from successive iterations of the decoder  313  can be compared with the original data  303 . Once the results of an iteration are compared with the input data  303 , a result  319  comprising the number of errors in the data block  315  can be determined. 
   SISOs  403  and  405  respectively decode two constituent convolutional codes of the turbo encoding being generated by encoder  305 . In each iterative decoding cycle, SISOs  403  and  405  output extrinsic information to each other. In each decoder iteration, SISO  405  uses the extrinsic information provided by SISO  403  in the previous iteration. SISO  403  uses the extrinsic information provided by SISO  405  in the previous iteration. SISO  405  also generates a posterior likelihood sequence in each iteration. The posterior likelihood sequence generated by SISO  405  in the k&#39;th iteration can be represented by Lx i   k , where i is the index of the value being decoded. This posterior likelihood sequence is used by soft to hard convertor  407  to generate hard values. If the posterior likelihood sequence in the k&#39;th iteration is equal to the posterior likelihood sequence in the (k-1)th iteration, i.e., (Lx i   k−1 )=(Lx i   k ) then the posterior likelihood sequence has converged. Convergence, however, may not occur for many iterations. In practice, iterative decoding is commonly halted after a fixed number of iterations. 
   The accuracy of hard decisions may be inferred from convergence of the posterior likelihood values. In a k&#39;th iteration soft to hard converter  407  accepts the posterior likelihood values Lx i   k  and produces corresponding hard values x i   k . If the hard values in a k&#39;th decoder iteration x i   k  match the hard values in a (k-1)th or a (k-2)th iteration i.e. (x i   k =x i   k−1  or x i   k−2 ) then the sequence x i   k  is a fixed point. 
   The concept of the fixed point is not new. In an article entitled “The geometry of turboing dynamics” by T. Richardson, published in the IEEE Transactions on Information Theory Vol. 46 January 2000, which is incorporated by reference herein, Richardson defined a fixed point in terms of probability density, i.e. (Lx i   k ). 
   Richardson postulated that if Lx i   k  and Lx i   k−1  have the same “bit wise marginal distribution” then X L   k  represents a fixed point. In other words (BZ here we need to say what “bitwise marginal distribution”. 
   After a number of iterations decoder  313  (See  FIG. 3 ) may converge to a fixed point. There, however, may be several fixed points. A fixed point may not necessarily represent a correct reproduction of the data sent. Additionally, some fixed points may not be stable, that is although a fixed point is reached, the decoded values will change if the decoding iterations are continued. That is if the decoder continues its iterations for an additional n iterations a fixed point of further iteration x L   k+n  may not correspond to the same value as fixed point x i   k . As an example consider table #1 of FIG.  5 . 
   Table #1 is an example of a simulation of a rate ⅔, 8 Phase Shift Keying (PSK) turbo trellis code having a block length of 10,240 bits. The signal to noise ratio, E b /N 0 , used for the simulation is 3.70 dB. This simulation illustrated in table 1 found a non-stable fixed point in the 6 th  iteration. In a sixth iteration, 5 bit errors were found in the decoded block, which is equal to the 5 bit errors found in a fifth iteration of the decoder. The twelfth iteration of the decoder, however, also yielded a stable fixed point. 
   The simulation illustration in table 1 of  FIG. 5  also illustrates, that after the first non-stable fixed point in iteration  6 , the decoder begins to propagate errors until, in the eighth iteration, 180 bit errors are present. Accordingly, a decoder operating as in table 1 will actually produce an inferior output if it is stopped in the eighth iteration versus if it is stopped in the sixth iteration. Such a condition were further decoding iterations produce more errors is termed “error propagation”. In the course of 80,000 simulations 5 such non-stable fixed points were encountered. 
   Even when the sequence x i   k  is equal to the bit sequence sent, the sequence x i   k  may not be a fixed point. Such a case is illustrated in table #2 of FIG.  5 . In the decoding example illustrated in table #2, the 4 th  iteration produced an output sequence having 0 errors. The fourth iteration, however, is not a fixed point as successive iterations produce a decoding having two errors in each decoded block. 
   Table 3, of  FIG. 5  illustrates a case where two fixed points appear alternatively. The odd iterations, after iteration  4 , exhibit 2 errors per decoding, whereas the even iterations, after iteration  4 , exhibit 0 errors per decoding. 
   According to simulations, fixed points are selected to contain less than 10 bit errors. Accordingly, to avoid error propagation, the iterative decoding may be stopped after a fixed point is reached. A mechanism for stopping the decoding on a particular iteration is illustrated in FIG.  6 . 
     FIG. 6  is a block diagram of a signature circuit, according to an embodiment of the invention. 
   In  FIG. 6 , block  601  represents an iterative decoder, illustratively a turbo-decoder executing a map algorithm (MAP decoder). The SISO comprises two constituent soft in soft out (SISO) decoders. Those skilled in the art will realize that any iterative type or probabilistic decoder could be represented by block  601 . Turbo decoding for block  601  has been selected by way of illustration and not limitation. 
   The output of block  601  is a sequence of soft a posteriori values which are provided to a soft to hard converter  603 . The soft to hard converter converts the sequence of soft a posteriori values to a sequence of hard values i.e., 1s and 0s. The sequence of 1s and 0s are the estimate of the sequence sent by the transmitter, as estimated by the current iteration, i.e., of iterative decoder  601 . The estimate of the sequence sent from the k&#39;th decoder iteration is provided serially to a signature circuit  605 . 
   The signature circuit  605  comprises a series of data storage elements  607 A through  607 N, where N is an arbitrary integer, such as  32 . The storage elements are arranged serially. That is, for example, storage element  607 B accepts its input from the output of storage element  607 A. When clocked by clock  613 , the value of storage element  607 A is clocked into storage element  607 B. Storage element  607 B is clocked into storage element  607 C, and so forth. Storage elements  607  may be a variety of storage elements such as, for example, D-type flip flops. The output of the last storage element  607 N is provided to a modulo-2 adder  609 . Adder  609  also receives, as a second input, the estimated hard values of the k&#39;th decoder iteration. The output of adder  609  is provided to the input of the first storage device  607 A of the signature storage chain  607 A through  607 N. 
   After every iteration of the iterative decoder  601  a sequence of soft values, provided by decoder  601 , are converted to a sequence of hard values in converter  603 . The sequence of hard values produced in converter  603  is then provided to signature circuit  605 . The signature of the iteration is the state of the storage device  607 A through  607 N. 
   In the current example of  FIG. 6 , 32 storage devices  607  form the state of the signature circuit  605 , and hence the signature is 32 bits. Signature circuits may comprise more or less than 32 bits depending on the size of the block being decoded, the expected signal to noise ratios, and a variety of other factors. 
   The signature from the k&#39;th iteration is compared to the signature from the K-1, and K-2 iterations. If the signature from the k&#39;th iteration matches the signature from the k-1 or k-2 iteration the iterative decoding stops. 
   Using 32 bits as the length (the number of memory units) of the signature circuit  605 , 80,000 blocks of rate ⅔, 8 phase shift keying (8-psk) Turbo-Trellis Coded Modulation (TTCM) were simulated. The block length of the TTCM code was 10240 symbols of 2 bits each. The E b /N 0  simulated was 3.70 dB. The signature unit was used to generate a stopping criteria for the decoder simulation, as was stopping the decoding after a fixed number (8) of decoding cycles. 
   The signature unit was initialized to all zeros between iterations and the estimated sequence of hard values was provided to the signature unit. If the signature unit exhibited a value in the k&#39;th iteration equal to the signature value in the k-1 or k-2 iteration the decoder was stopped. 
   The result of simulating the decoding of 80,000 blocks, of rate ⅔ TTCM code, as described previously, is summarized in table 4 of FIG. # 6 . 
   The signature criteria yielded more blocks having errors than the decoder having 8 fixed iterations. The signature circuit produced 162 blocks with errors versus 95 for the 8 iteration decoder; however, using the signature criteria produced a smaller number of bit errors, i.e. 401 versus 530, than the 8 iteration decoding. The signature decoding also resulted in a lower bit error rate 2.447e −7  as opposed to 2.325e −7  for the 8 iteration decoding. 
   The signature method only required an average of 5.5 iterations to reach a fixed point. The signature method required a maximum of 9 iterations in 9 of 80,000 blocks decoded. The fixed number of iterations decoder used 8 iterations. The signature method, in addition to being less time consuming, reduced the iterations required from 8 to an average of 5½ iterations. Only 9 of 80,000 blocks required more than 8, i.e. 9, iterations in the decoding. 
   The signature method stopped the decoding on a variety of different iterations. The iteration on which the signature method are listed by percentage in Table 5. The signature method resulted in less errors, and less time (iterations) to decode, thus showing that not only was iterative decoding time shortened, but that the signature decoding lessened the problem of error propagation into future iterations. Error propagation occurs in a fixed number decoder when a fixed point is reached, but due to the maximum number of iterations not being reached the iterative decoding process continues, with the result that the number of errors in the block is increased over the errors at the fixed point. By avoiding error propagation, resulting from iterative decoding beyond where a fixed point is reached, the decoding is improved by the signature method. 
   Other stopping criteria have been proposed. For example, in “Reduction of the Number of Iterations in Turbo Decoding Using Extrinsic Information,” published in IEEE TenCon, pp. 494-496, which is incorporated by reference, B. Kim and H. Lee proposed a stopping criteria using a variance of extrinsic information. Their method does not work in all decoders. A modified method is proposed herein. 
   Let E k x i  denote the extrinsic information of a SISO (Soft In Soft Out) decoder, for example one executing a MAP algorithm, in the k&#39;th iteration. If the mean value, M k , for the k&#39;th iteration is defined as: 
               M   k     =       ∑     i   -   0       N   -   1       ⁢           ⁢         E   k     ⁢     x   i         exp   ⁡     (            E   k     ⁢     x   i            )                   Equation   ⁢           ⁢   1             
 
Then the variance of the extrinsic information is: 
               VAR   k     =       ∑     i   -   0       N   -   1       ⁢           ⁢         (         E   k     ⁢     x   i       -     M   k       )     2       exp   ⁡     (            E   k     ⁢     x   1            )                   Equation   ⁢           ⁢   2             
 
   Where N is the block size of the block being iteratively decoded. 
   Commonly, for a fixed signal to noise ratio a threshold T exists such that if VAR k &lt;T. The posterior likelihood sequence has converged. This rule, however, has exceptions, and so an additional criterion is needed. Such a criterion is the differential variance D k . D k  is defined as:
 
 D   k   =|VAR   k   −VAR   k−1 |  Equation 3
 
A new threshold rule can be stated as follows, halt the iteration of the decoder if:
 
VAR k &lt;T  Equation 4
 
or
 
VAR k &lt;T 1  and D k &lt;T 2   Equation 5
 
Where T 1  and T 2  are threshold values. The values for T, T 1 , and T 2  may be determined through the use of simulation, for example, using a simulation such as illustrated in FIG.  3 . The threshold selected will depend on the signal to noise ratio, throughout needed, and a variety of other implementation details.
 
   One method to determine thresholds T, T 1 , and T 2 , is as follows: A signal to noise ratio is first selected, and a noise vector  309  introduced to accommodate the selected signal to noise ratio. Successive iterations are then examined for number of errors and thresholds T, T 1  and T 2 . The greater the number of simulations, the more accurate the values of T, T 1  and T 2  may be determined. The threshold values determined will of course depend on such factors as final to noise ratio, code rate etc. 
   As an illustrative example, a rate ⅔ 8-phase shift keying turbo trellis code modulation with block length 10240 and an E b /N 0 =3.70 dB was selected. Using a T and T 2  equal to 10 and T 1  equal to 100, 80,000 blocks were simulated. The results are illustrated in table 6 of FIG.  7 . 
   In addition to signature criteria, a cross entropy criterion may be employed in determining a stopping criterion for iterative decoding. For example, in “Suboptimum Decoding Using Kullback Principle,” published in Lecture Notes in Computer Science, No. 313, B. Bouchon et al. Eds., 1988, pp. 93-101, G. Battail and R. Sfes, which is incorporated by reference, the idea of decoding using cross entropy minimization is discussed. Additionally, in “Iterative Decoding of Binary Block and Convolutional Codes,” published in the IEEE, Transactions on Information Theory, Volume 42, March 1996, pp. 429-445, which is hereby incorporated by reference, J. Hagenauer, E. Offer and L. Papke discuss cross entropy. 
   If a decoder, illustratively a turbo decoder comprising 2 SISO units, produces a sequence of extrinsic information, the extrinsic information from the first SISO may be represented as and the second SISO may be represented as E 2   k x i . The cross entropy can then be defined as: 
               T     (   k   )       =       ∑     i   -   0       N   -   1       ⁢           ⁢                  E   2   k     ⁢     x   i       -       E   2     k   -   1       ⁢     x   i              2       exp   ⁡     (              E   1   k     ⁢     x   i       +       E   2   k     ⁢     x   i              )                   Equation   ⁢           ⁢   6             
 
   The decoder can then terminate the decoding process by testing the value of T (k) /T (1)  to see if it is less than some predetermined threshold. As previously, the threshold for a particular signal to noise ratio may be determined through the use of simulations using simulation such as illustrated in FIG.  3 . 
   A comparison of the simulation of 80,000 blocks of rate ⅔, 8 psk turbo trellis coded modulated code, with an E b /N 0 =3.75 dB was simulated. The results are as seen in table 7 of FIG.  7 . 
     FIG. 8  is a graph illustrating bit error rate versus E b /N 0  for various stopping criteria. As can be seen, the signatures criteria produces a bit error rate (BER) superior to the 8 iteration decoding at an E b /N 0  of 3.75 dB. The variance stopping criteria produces a BER superior to the 8 iteration decoding at all tested E b /N 0 .