Patent Publication Number: US-9419656-B2

Title: Decoder and method for decoding an encoded sequence of bits

Description:
TECHNICAL FIELD 
     The present disclosure generally relates to a decoder and a method for decoding an encoded sequence of bits. 
     BACKGROUND 
     Channel coding is commonly used to increase the robustness of a digital communication system. The principle underlying channel coding is to introduce redundancy into the transmitted bit stream so as to facilitate error detection and correction at the decoder. One general class of channel codes is trellis codes, and a commonly-used class of trellis codes is known as convolutional codes. Convolutional codes are used extensively in numerous applications in order to achieve reliable data transfer, including digital video, radio, mobile communication, and satellite communication. 
     According to a mobile communication standard, such as the 3rd Generation Partnership Project (3GPP) specification for Long Term Evolution (LTE), the Physical Downlink Control Channel (PDCCH) is encoded by a rate 1/3 tail biting convolutional code (TBCC), followed by a rate matching unit that produces the final coding rate ranging from 0.04 to 0.99. The rate is defined as m/n, where each m-bit information symbol to be encoded is transformed into an n-bit symbol. Due to this requirement, the user equipment needs to have a TBCC decoder that can perform PDCCH decoding across all supported code rates. This requirement is challenging since the decoder needs to address at least the following three criteria: (i) reasonably good decoding performance; (ii) acceptable decoding complexity for all code rates; and (iii) required timing budget per the 3GPP for all code rates. These criteria are particularly difficult to achieve for the decoding of high code rate TBCC, e.g., having a code rate of 71/72. A convolutional decoder is needed which has acceptable decoding complexity and is within the required timing budget per 3GPP for all code rates, but without sacrificing decoding performance. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  illustrates a schematic diagram of an example of a convolutional decoder. 
         FIG. 1B  illustrates a trellis representation of the convolutional decoder of  FIG. 1A . 
         FIG. 1C  is a flowchart illustrating an example of a method of decoding a convolutionally-encoded sequence of bits using the decoder of  FIG. 1A . 
         FIGS. 1D and 1E  are graphs illustrating decoding performance of the convolutional decoder of  FIG. 1A . 
         FIG. 2A  illustrates a schematic diagram of another example of a convolutional decoder. 
         FIG. 2B  illustrates a trellis representation of the convolutional decoder of  FIG. 2A . 
         FIG. 2C  is a flowchart illustrating an example of a method of decoding a convolutionally-encoded sequence of bits using the decoder of  FIG. 2A . 
         FIGS. 2D  though  2 G are graphs illustrating decoding performance and complexity of the convolutional decoder of  FIG. 2A . 
         FIG. 3  illustrates a schematic diagram of a computing device having the convolutional decoder of  FIG. 1A or 2A . 
         FIG. 4  illustrates a schematic diagram of a system including a convolutional encoder and decoder. 
     
    
    
     DETAILED DESCRIPTION 
     This disclosure is directed to a decoder including a starting state unit configured to store starting states for the respective maximum likelihood survival paths though a trellis representation of a code, and a path forcing unit configured to select a tail path of a maximum likelihood survival path at w states prior to its ending state in the trellis representation such that its ending state equals its stored starting state, where w is equal to a constraint length of the decoder minus one. As an alternative or in addition to the tail path forcing unit, the decoder may have a state consistency unit configured to determine whether a maximum likelihood survival path has an ending state in the trellis representation that is equivalent to its stored starting state, and if so, ending the decoding process. 
       FIG. 4 , by way of overview, illustrates a schematic diagram of a system  400  including a convolutional encoder  410  and decoder  420 . Output code words, that is, encoded bits, [ c   0   c   1  . . .  c   n−1   c   n ] result from the convolution of a sequence of input bits [a 0 a 1  . . . a n−1 a n ] with the impulse response of the encoder  410  which includes a v-stage shift register (not shown). A sequence of n bits, that is, an n-bit code word, [ c   0   c   1  . . .  c   n−1   c   n ] is generated as a function of m input bits [a 0 a 1  . . . a n−1 a n ] and v bits stored in the shift register. The constraint length L of the encoder  410  is defined as m+v, and the rate of the code is given by m/n, where n&gt;m. The transmitted convolutionally-encoded sequence of bits or code word [ c   0   c   1  . . .  c   n−1   c   n ] is received at decoder  420  as [ r   0   r   1  . . .  r   n−1   r   n ]. Ideally the sequence of encoded bits [ c   0   c   1  . . .  c   n−1   c   n ] transmitted by the encoder  410  is the same as the encoded bits [ r   0   r   1  . . .  r   n−1   r   n ] received at the decoder, but in reality there are generally transmission errors. To decode the convolutional code back to the original sequence of input bits [a 0 a 1  . . . a n−1 a n ], Viterbi algorithm is most frequently used, however, other algorithms, such as Fano algorithm, are alternatives. 
       FIG. 1A  illustrates a schematic diagram of an example of a convolutional decoder  100 A. The convolutional decoder  100 A includes an input  110 , a branch metric unit  120 , a path metric unit  130 , a tail path forcing unit  140 , and a trace back unit  150 . 
       FIG. 1B  illustrates a trellis representation of the convolutional decoder  100 A of  FIG. 1A . The m input bits cause a transition from a present state to a next state, and the number of output bits, i.e., code word bits, produced depends on the rate of the code. The transitions from one state to the next state when viewed as a function of time result in a graph commonly known as a “trellis.” This trellis representation illustrated has a rate 1/3 convolutional code with a constraint length L=3; a constraint length is a number of bits used in the encoding process. This code includes four possible states, 00, 01, 10, 11, each corresponding to a group of v=2 bits and designated by one of four circles in each of an “old state” and “new state” column. The diagram shows all of the possible transitions between a given one of the old states and the new states that can be reached from the given old states. Since m=1 in this example, the encoding process dictates that there can be only two transitions out of a state and two transitions into a state. In general, for m input bits, in this case there are 2m transitions out of and into a state. The rate, constraint length number of states, etc. are merely examples; the disclosure is of course not limited in these respects. 
     The state assignment shown in  FIG. 1B  is arbitrary to some degree. The convention adopted in this example is that the input bit shifts into the least significant bit (LSB) of the shift register while the most significant bit (MSB) shifts out of the register. According to this convention, two states differing in the MSB converge onto the same state when an input is shifted into the LSB. For example, the 00 and 10 states both converge to the 00 state when a 0 is shifted into the LSB of the register. 
     The convolutional encoding/decoding process can be viewed as tracing a path through the trellis representation.  FIG. 1B  shows one such path traced through the four-state trellis as a function of time. The vertical axis denotes the state numbers in ascending order, and the horizontal axis represents time, with each stage of the trellis representing a period of time. Typically, the shift register is initialized to start at the 00 state. For each of the transitions shown in  FIG. 1B  of the encoder, n code bits are generated. Thus, the objective of the corresponding decoding process is to retrace this path through the trellis based on the received sequence of convolutionally-encoded input bits. 
     The convolutional decoder  100 A is described herein as a Viterbi decoder, though the disclosure is not necessarily limited in this respect. When there is no knowledge or constraints on the initial and ending states of the convolutional code, that is, the convolutional code could start and end at any state, the Viterbi decoder is optimal in the sense that it minimizes the code word decoding error. A Tail Biting Convolutional Code (TBCC) is a convolutional code with an additional constraint on the initial and ending states of the code word trellis representation. Specifically, when the input to the convolutional encoder is [a 0 a 1  . . . a n−1 a n ], the initial state and the ending state corresponding to a TBCC code word [ c   0   c   1  . . .  c   n−1   c   n ] need to be the same, that is,  S   n+1 = S   0 . This is illustrated in  FIG. 1B , where each of the initial state  S   0  and the ending state  S   n+1  is 01. 
     Referring back to  FIG. 1A , a convolutionally-encoded sequence of received bits [ r   0   r   1  . . .  r   n−1   r   n ] is applied to the branch metric unit  120  at input unit  110 . The branch metric unit  120  is configured to determine branch metrics for respective transitions between states in the trellis representation of the convolutional code used to generate the convolutionally-encoded sequence of bits. 
     A branch metric is a normed distance between the bits of the convolutional code and the actually received bits. There are hard decision and soft decision Viterbi decoders. A hard decision Viterbi decoder uses as a metric known as a Hamming distance, which is the number of bits at which the corresponding sequences of bits are different. A strong match between the received convolutionally-encoded sequence of bits and a hypothesized code word of the convolutional code results in a smaller branch metric, whereas a weaker match between the received convolutionally-encoded sequence of bits and a hypothesized code word results in a larger branch metric. Alternatively, a soft decision Viterbi decoder receives a bit stream containing information about the reliability of the received bits, wherein generally a strong match between the received convolutionally-encoded sequence of bits and a hypothesized code word of the convolutional code results in a larger branch metric, whereas a weaker match between the received convolutionally-encoded sequence of bits and a hypothesized code word results in a smaller branch metric. 
     The path metric unit  130  is configured to determine, based on the branch metrics, path metrics corresponding to maximum likelihood survival paths reaching the respective states in the trellis representation. The path metric unit  130  includes an Add-Compare-Select (ACS) unit  132 , which summarizes the individual branch metrics to obtain path metrics for respective paths, one of which will eventually be chosen as optimal as being the maximum likelihood path through the trellis representation. The ACS unit  132  is recursive since the new path metrics depend on the path metrics computed for the previous stage and the branch metric corresponding to the transition from the previous state to an ending state. 
     More specifically, at each time index k in the trellis, for each state, the ACS unit  132  computes and records the metric of the most likely path {circumflex over (t)} k  originated from any state  S   k  that ended at state  S   k+1 . The metric of the most likely path ending at state  S   k+1  is computed as follows: 
               M   ⁡     (       S   _       k   +   1       )       =       max     {       t   k     |       t   k     ⁢   ends   ⁢           ⁢   at   ⁢           ⁢       S   _       k   +   1           }       ⁢     {       M   ⁡     (         S   _     k     ⁡     (     t   k     )       )       +     BM   ⁡     (         r   _     k     ,     t   k       )         }             
where the maximization is over all possible paths {t k } that end at state  S   k+1 . Here, M( S   k (t k )) is the metric corresponding to the starting state of path {t k }, and BM( r   k ,t k ) is the branch metric of path {t k }. The Viterbi decoder also records the surviving path {circumflex over (t)} k  of each state  S   k+1  to be used by the following step. In Viterbi decoding, the problem of exponential growth in the number of paths is solved by selecting, at each time step, one of two converging paths. As a result, the number of paths under consideration remains constant with time. This elimination of paths at each time step, i.e., at each trellis stage, is referred to as the ACS operation. The ACS operation is done in a recursive manner starting from k=0 to k=n.
 
     A typical TBCC decoder will process the received code word sequence for multiple iterations (S 0  to S n+1 ) to allow convergence and to obtain better decoding performance. Before the first iteration of the Viterbi algorithm, the state metrics are generally initialized to all zeroes for each of the states. The first iteration is performed from time index 0 through to n+1. After the first iteration is completed, instead of initializing the states to all zeroes again, the states are initialized with the previously accumulated state metrics from the first iteration, and then the second iteration is performed. The iterations are performed multiple times, normally until convergence, but practically not more than five or six times depending on the desired decoding quality. 
     The path metric unit  130  not only includes ACS unit  132 , but additionally includes starting state unit  134 . The starting state unit  134  is configured to store starting states for the respective maximum likelihood survival paths. At time index k, and for each state S k+1 , in addition to updating and storing the state metric M( S   k+1 ), the starting state unit  134  records the starting state  S   0  of the survival path that reaches  S   k+1 . 
     The tail path forcing unit  140  is configured to select a tail path of a maximum likelihood survival path at w states prior to its ending state in the trellis representation such that the ending state equals its stored starting state, where w is equal to a constraint length of the decoder minus one. More specifically, for each state  S   n-L+2 , based on the stored starting state  S   0  as described above, the tail path forcing unit  140  is configured to select the “tail path” such that the survival path at state  S   n+1  satisfies  S   n+1 = S   0 . Here, L is the constraint length of the TBCC. 
     After completing the tail path forcing operation, the output of the tail path forcing unit  140  is supplied to the trace back unit  150 , which reconstructs the most likely survival path {circumflex over (t)} n , corresponding to the most likely code word, recursively in a backward fashion starting from state  S   n+1  in order to reconstruct the sequence of input bits. This process is generally called the trace back operation. 
       FIG. 1C  is a flowchart  100 C illustrating an example of a method of performing convolutional decoding using the decoder  100 A of  FIG. 1A . 
     First, in Step  110 C, a convolutionally-encoded sequence of bits is received the decoder input  110 . 
     At Step  120 C, the branch metric unit  120  determines branch metrics for respective transitions between states in a trellis representation of a convolutional code used to generate the convolutionally-encoded sequence of bits. This step of determining branch metrics may include using the Hamming distance, that is determining the branch metrics such that a strong match between the received convolutionally-encoded sequence of bits and a hypothesized code word of the convolutional code results in a smaller branch metric, and a weaker match between the received convolutionally-encoded sequence of bits and a hypothesized code word results in a larger branch metric. Alternatively, this step may include determining the branch metrics such that a strong match between the received convolutionally-encoded sequence of bits and a hypothesized code word of the convolutional code results in a larger branch metric, whereas a weaker match between the received convolutionally-encoded sequence of bits and a hypothesized code word results in a smaller branch metric. 
     Next, at Step  130 C, the path metric unit  130  determines, based on the branch metrics, path metrics corresponding to maximum likelihood survival paths reaching the respective states in the trellis representation. 
     The starting state unit  134 , at Step  134 C, stores starting states for the respective maximum likelihood survival paths. 
     At Step  140 C, tail path forcing unit  140  selects a tail path of a maximum likelihood survival path at w states prior to its ending state in the trellis representation such that its ending state equals its stored starting state, where w is equal to a constraint length of the decoder minus one. 
     Finally, at Step  150 C Reconstructing the most likely path in a trace back operation 
     This disclosure includes a computer program product embodied on a non-transitory computer-readable medium having program instructions configured such that when executed by processing circuitry cause the processing circuitry to implement this method described with respect to  FIG. 1C . 
       FIGS. 1D and 1E  are graphs illustrating simulated decoding performances of the convolutional decoder  100 A of  FIG. 1A .  FIG. 1D  illustrates the decoding performance for rate 67/72 TBCC, and  FIG. 1E  illustrates the decoding performance for rate 71/72 TBCC. The vertical axes denote Block Error Rates (BLER), and the horizontal axes represent the energies per bit to noise power spectral density ration (E b /N o ) in decibels. 
     The graphs compare the decoding performance between three different types of decoders: a traditional decoder, the decoder  100 A described herein, and an optimal decoder. The optimal decider is provided only as a baseline for performance comparison, and is not a practical decoder due to its extremely high complexity. 
     As can be seen in  FIG. 1D , performance-wise, the decoder  100 A outperforms the traditional decoder by 0.5 dB, and is only less than 0.1 dB different from the optimal maximum likelihood decoding. Complexity-wise, the method  100 C described herein has 20% less complexity as compared to the traditional decoder. Similar conclusions can be drawn from  FIG. 1E  as well, with even more complexity reduction, that is, 64% less complexity. 
       FIG. 2A  illustrates a schematic diagram of another example of a convolutional decoder  200 A. The convolutional decoder  200 A includes an input  110 , a branch metric unit  120 , a path metric unit  130 , a state consistency unit  240 , and a trace back unit  150 . The input  110 , branch metric unit  120 , path metric unit  130 , and trace back unit  150  were described above with respect to  FIG. 1A , and thus for the sake of brevity, their descriptions will not be repeated here. A main difference between decoder  100 A of  FIG. 1A  and decoder  200 A of  FIG. 2A  is that decoder  200 A replaces tail path forcing unit  140  with state consistency unit  240 . 
     By way of overview, the state consistency unit  240  determines whether at the end of a decoding iteration, an optimal survival path has been found. Once the survival optimal path has been found, there is no need to performing an additional iteration of the Viterbi algorithm through the trellis representation. The trace back unit  150  can immediately start the trace back process. This early termination scheme provides a significant reduction in decoding time and complexity, especially for a TBCC with a high code rate. 
       FIG. 2B  illustrates a trellis representation of the convolutional decoder of  FIG. 2A . The portion of the trellis representation between lines X and Y represent the iterations of the Viterbi process ( r (0),  r (1), . . .  r (l−1)) prior to the early termination criteria being satisfied, whereas the portion between lines Y and Z represents the last iteration ( r ) when the early termination criteria is satisfied. 
     As discussed above, the starting state unit  134  is configured to store starting states for the respective maximum likelihood survival paths. At time index k, and for each state  S   k+1 , in addition to updating and storing the state metric M( S   k+1 ), the starting state unit  134  records the starting state S 0  of the survival path that reaches  S   k+1 , denoted by P( S   k+1 ). 
     Memory registers, P( 0 ), P( 1 ), P( 2 ), P( 3 ), store respective survival path starting states  S   0 . The Viterbi algorithm is processed all the way through to ending state S n+1 , then for each ending state 00, 01, 10, 11, there is one survival path. Each survival path has a respective, stored starting state (P( 0 ), P( 1 ), P( 2 ), P( 3 )). For example, the survival path at time index n+1 ending at state 00 is has a stored starting state of 00 (P( 0 )=[00]). Similarly, the survival path ending at state 11 has a stored starting state of 11 (P( 3 )=[01]). 
     The Viterbi algorithm, as part of the ACS operation performed by ACS unit  132  selects the maximum likelihood survival path of the four possible paths. For each state 00, 01, 10, 11, there is a survival path, and the one with the maximum likelihood is selected. More specifically, the ACS unit  132  determines at time index n+1, which state  S   n+1 =q has a state metric M(q) that indicates the maximum likelihood path among {M(i)|i=0, 1, . . . , 2 L-1 −1}. This selection is accomplished by comparing the path metrics 180, 32, 47, and 156. In this case the path metrics represent soft decisions as discussed above, so the larger the path metric is the higher the likelihood the path is the maximum likelihood survival path. In this example state 00 has the largest path metric, 180, so the path ending at this state is the maximum likelihood survival path. 
     The state consistency unit  240  is configured to determine, at the end of each iteration, whether the maximum likelihood survival path has an ending state in the trellis representation that is equivalent to its stored starting state, as require by TBCCs. If the determination is negative, that is, P( S   n+1 =q)≠q, the ACS unit  132  is configured to perform on the received convolutionally-encoded sequence of bits  r  an additional iteration of the Viterbi algorithm until early termination is triggered or until the maximum allowed number of iterations is reached; one example of the maximum number of iterations may be three. On the other hand, if the determination by the state consistency unit  240  is positive, that is, if P( S   n+1 =q)=q, then the best path is found. The ACS unit  132  is configured to end determining the branch and path metrics through the trellis representation as no further iterations of the Viterbi algorithm are needed, and the trace back unit  150  can begin tracing back from state  S   n+1 =q. 
     Once the best path is found at the end of a Viterbi algorithm iteration, even if the decoding process is not terminated early and further iterations are performed, the resulting maximum likelihood path would still be the same, there is no performance gain for performing additional iterations. The survival path is in fact the maximum likelihood survival path as the stored starting state equals the ending state. 
       FIG. 2C  is a flowchart  200 C illustrating an example of a method of performing convolutional decoding using the decoder  200 A of  FIG. 2A . 
     Steps  110 C,  120 C,  130 C, and  134 C were described above with respect to  FIG. 1C , and thus there descriptions will not be repeated here. 
     In Step  240 C, the state consistency unit  240  determines whether a maximum likelihood survival path has an ending state in the trellis representation that is equivalent to its stored starting state. 
     If the determination by the state consistency unit  240  is positive, at Step  244 C the determinations of the branch and path metrics through the trellis representation is ended, and then the trace back unit  150  can begin tracing back the maximum likelihood path. Step  150 C was described above with respect to  FIG. 1C , and thus its description will not be repeated here. Alternatively, if the determination by the state consistency unit  240  is negative, at Step  2424 C the received convolutionally-encoded sequence of bits is processed for an additional iteration through the trellis representation. 
     This disclosure also includes a computer program product embodied on a non-transitory computer-readable medium having program instructions configured such that when executed by processing circuitry cause the processing circuitry to implement this method described with respect to  FIG. 2C . 
       FIGS. 2D  though  2 G are graphs illustrating decoding performance of the convolutional decoder of  FIG. 2A .  FIGS. 2D and 2F  illustrate the performance of the decoder  200 A as compared to an optimal decoder for different code rates; the vertical axes denote Block Error Rate (BLER), and the horizontal axes represent the energy per bit to noise power spectral density ratio (E b /N o ) in decibels.  FIGS. 2E and 2G  illustrate the corresponding complexity and decoding time reduction achieved with the method  200 C described with respect to  FIG. 2C ; the vertical axes denote the average number of iterations, and the horizontal axes represents the energy per bit to noise power spectral density ratio (E b /N o ) in decibels. 
       FIG. 2D  illustrates the decoding performance of the decoder  200 A for rate 67/72 TBCC.  FIG. 2E  illustrates the complexity and decoding time for rate 71/72 TBCC.  FIG. 2F  illustrates the decoding performance of the TBCC decoder  200 A for rate 67/576 TBCC.  FIG. 2G  illustrates the complexity and decoding time for rate 71/576 TBCC. 
     As can be seen from these figures, the decoding performance is the same with or without early termination. From these figures, it can be seen that in the normal operating point (e.g., BLER=10 −2 ), there is approximately 43% decoding complexity reduction for a TBCC rate equal to 67/72, which is a high code rate condition, and approximately a 57% decoding complexity reduction for a TBCC rate equal to 67/576, which is a low code rate condition. It can thus bee seen that the early termination method  200 C described with respect to FIG.  2 C provides a significant reduction in decoding complexity without compromising decoding performance. 
       FIG. 3  illustrates a schematic diagram of a computing device  300  having a receiver  310  including the convolutional decoder  314  of  FIG. 1A or 2A . This computing device  300  may be a wireless communication device, such as a base station or a handheld mobile device. 
     The computing device also includes a de-rate matching unit  312 , which as the name implies, in the inverse of a rate matching unit of an encoder. A rate matching unit is also known as a puncturing unit. Puncturing, as is known, increases a transmission rate by deleting certain bits out of a code word to be output by an encoder according to a puncturing matrix. The de-rate matching unit  312  takes the puncturing by the encoder into consideration. 
     The convolutional decoders  100 A,  200 A, and its corresponding methods  100 C,  200 C, described herein are advantageous in numerous respects. The convolutional decoders  100 A  200 A and methods  100 C,  200 C more often output a valid TBCC code word. Moreover, when operating at high code rates, the decoders  100 A,  200 A converge much more quickly to the correct maximum likelihood survival path, leading to significant reduction in both complexity and decoding time. 
     Example 1 is a decoder comprising an input configured to receive an encoded sequence of bits; a branch metric unit configured to determine a plurality of branch metrics for a plurality of respective transitions between a plurality of states in a trellis representation of a code used to generate the encoded sequence of bits; a path metric unit configured to determine, based on the branch metrics, a plurality of path metrics corresponding to a plurality of maximum likelihood survival paths reaching the plurality of respective states in the trellis representation; a starting state unit configured to store starting states for the respective maximum likelihood survival paths; and a tail path forcing unit configured to select a tail path of a maximum likelihood survival path at a w number of states prior to this maximum likelihood survival path&#39;s ending state in the trellis representation to result in this maximum likelihood survival path&#39;s stored starting and ending states being the same, wherein w is equal to a constraint length of the decoder minus one. 
     In Example 2, the subject matter of Example 1, wherein the decoder is a tail biting convolutional decoder (TBCC) in which the starting and ending states of the maximum likelihood survival path are equivalent. 
     In Example 3, the subject matter of Example 1, wherein the path metric unit comprises an add-compare-select unit. 
     In Example 4, the subject matter of Example 1, wherein the branch metric unit is further configured to determine the branch metrics such that a strong match between the received encoded sequence of bits and a hypothesized code word of the code results in a smaller branch metric, whereas a weaker match between the received encoded sequence of bits and a hypothesized code word results in a larger branch metric. 
     In Example 5, the subject matter of Example 1, wherein the branch metric unit is further configured to determine the branch metrics based on a Hamming distance between the received encoded sequence of bits and the hypothesized code word of the code. 
     In Example 6, the subject matter of Example 1, wherein the branch metric unit is further configured to determine the branch metrics such that a strong match between the received encoded sequence of bits and a hypothesized code word of the code results in a larger branch metric, whereas a weaker match between the received encoded sequence of bits and a hypothesized code word results in a smaller branch metric. 
     In Example 7, the subject matter of Example 1, wherein the decoder is a Viterbi decoder. 
     In Example 8, a computing device comprising the subject matter of Example 1. 
     Example 9 is a method of decoding an encoded sequence of bits, the method comprising receiving, at an input, the encoded sequence of bits; determining, by a branch metric unit, a plurality of branch metrics for a plurality of respective transitions between a plurality of states in a trellis representation of a code used to generate the encoded sequence of bits; determining, by a path metric unit and based on the branch metrics, a plurality of path metrics corresponding to a plurality of maximum likelihood survival paths reaching the respective states in the trellis representation; storing, by a starting state unit, a plurality of starting states for the plurality of respective maximum likelihood survival paths; and selecting, by a tail path forcing unit, a tail path of a maximum likelihood survival path at a w number of states prior to this maximum likelihood survival path&#39;s ending state in the trellis representation to result in this maximum likelihood survival path&#39;s stored starting and ending states being the same, wherein w is equal to a constraint length of the decoder minus one. 
     In Example 10, the subject matter of Example 9, wherein the step of determining branch metrics further comprises determining the branch metrics such that a strong match between the received encoded sequence of bits and a hypothesized code word of the code results in a smaller branch metric, and a weaker match between the received encoded sequence of bits and a hypothesized code word results in a larger branch metric. 
     Example 11 is a computer program product embodied on a non-transitory computer-readable medium comprising program instructions configured such that when executed by processing circuitry cause the processing circuitry to implement the subject matter of Example 9. 
     Example 12 is a decoder, comprising an input means for receiving an encoded sequence of bits; a branch metric means for determining a plurality of branch metrics for a plurality of respective transitions between a plurality of states in a trellis representation of a code used to generate the encoded sequence of bits; a path metric means for determining, based on the branch metrics, a plurality of path metrics corresponding to a plurality of maximum likelihood survival paths reaching the respective states in the trellis representation; a starting state means for storing a plurality of starting states for the plurality of respective maximum likelihood survival paths; and a tail path forcing means for selecting a tail path of a maximum likelihood survival path at a w number of states prior to this maximum likelihood survival path&#39;s ending state in the trellis representation to result in this maximum likelihood survival path&#39;s stored starting and ending states being the same, wherein w is equal to a constraint length of the decoder minus one. 
     Example 13 is a decoder comprising an input configured to receive an encoded sequence of bits; a branch metric unit configured to determine a plurality of branch metrics for a plurality of respective transitions between a plurality of states in a trellis representation of a code used to generate the encoded sequence of bits; a path metric unit configured to determine, based on the branch metrics, a plurality of path metrics corresponding to a plurality of maximum likelihood survival paths reaching the plurality of respective states in the trellis representation; a starting state unit configured to store a plurality of starting states for the plurality of respective maximum likelihood survival paths; and a state consistency unit configured to determine whether a maximum likelihood survival path has an ending state in the trellis representation that is equivalent to its stored starting state, wherein if the state consistency unit&#39;s determination is positive, the branch metric unit and path metric units are configured to end the determinations of the branch and path metrics. 
     In Example 14, the subject matter of Example 13, wherein if the state consistency unit&#39;s determination is negative, the decoder is configured to process the received encoded sequence of bits for an additional iteration through the trellis representation. 
     In Example 15, the subject matter of Example 14, wherein the decoder is configured to end processing of the received encoded sequence of bits through the trellis representation after a predetermined number of iterations through the trellis representation. 
     In Example 16, the subject matter of Example 13, wherein the decoder is a tail biting convolutional decoder (TBCC) in which the starting and ending states of the maximum likelihood survival path are equivalent. 
     In Example 17, the subject matter of Example 13, wherein the path metric unit comprises an add-compare-select unit. 
     In Example 18, the subject matter of Example 13, wherein the branch metric unit is further configured to determine the branch metrics such that a strong match between the received encoded sequence of bits and a hypothesized code word of the code results in a smaller branch metric, whereas a weaker match between the received encoded sequence of bits and a hypothesized code word results in a larger branch metric. 
     In Example 19, the subject matter of Example 13, wherein the branch metric unit is further configured to determine the branch metrics such that a strong match between the received encoded sequence of bits and a hypothesized code word of the code results in a larger branch metric, whereas a weaker match between the received encoded sequence of bits and a hypothesized code word results in a smaller branch metric. 
     In Example 20, the subject matter of Example 13, wherein the decoder is a Viterbi decoder. 
     Example 21 is a wireless communication device comprising the decoder of Example 13. 
     Example 22 is a computing device comprising the decoder of Example 13. 
     Example 23 is a method of decoding an encoded sequence of bits, the method comprising receiving, at an input, the encoded sequence of bits; determining, by a branch metric unit, branch metrics for respective transitions between states in a trellis representation of a code used to generate the encoded sequence of bits; determining, by a path metric unit and based on the branch metrics, path metrics corresponding to maximum likelihood survival paths reaching the respective states in the trellis representation; storing, by a starting state unit, starting states for the respective maximum likelihood survival paths; and determining, by a state consistency unit, whether a maximum likelihood survival path has an ending state in the trellis representation that is equivalent to its stored starting state, wherein if the determination by the state consistency unit is positive, ending the determinations of the branch and path metrics. 
     In Example 24, the subject matter of Example 23, wherein if the determination by the state consistency unit is negative, processing the received encoded sequence of bits for an additional iteration through the trellis representation. 
     Example 25 is a computer program product embodied on a non-transitory computer-readable medium comprising program instructions configured such that when executed by processing circuitry cause the processing circuitry to implement the method of Example 23. 
     In Example 26, the subject matter of any of Examples 1-2, wherein the path metric unit comprises an add-compare-select unit. 
     In Example 27, the subject matter of any of Examples 1-3, wherein the branch metric unit is further configured to determine the branch metrics such that a strong match between the received encoded sequence of bits and a hypothesized code word of the code results in a smaller branch metric, whereas a weaker match between the received encoded sequence of bits and a hypothesized code word results in a larger branch metric. 
     In Example 28, the subject matter of any of Examples 1-4, wherein the branch metric unit is further configured to determine the branch metrics based on a Hamming distance between the received encoded sequence of bits and the hypothesized code word of the code. 
     In Example 29, the subject matter of any of Examples 1-5, wherein the branch metric unit is further configured to determine the branch metrics such that a strong match between the received encoded sequence of bits and a hypothesized code word of the code results in a larger branch metric, whereas a weaker match between the received encoded sequence of bits and a hypothesized code word results in a smaller branch metric. 
     In Example 30, the subject matter of any of Examples 1-6, wherein the decoder is a Viterbi decoder. 
     Example 31 is a computing device comprising the decoder of any of Examples 1-7. 
     Example 32 is a computer program product embodied on a non-transitory computer-readable medium comprising program instructions configured such that when executed by processing circuitry cause the processing circuitry to implement the method of any of Examples 9-10. 
     Example 33 is a decoder, comprising an input means for receiving an encoded sequence of bits; a branch metric means for determining branch metrics for respective transitions between states in a trellis representation of a code used to generate the encoded sequence of bits; a path metric means for determining, based on the branch metrics, path metrics corresponding to maximum likelihood survival paths reaching the respective states in the trellis representation; a starting state means for storing starting states for the respective maximum likelihood survival paths; and a state consistency means for determining whether a maximum likelihood survival path has an ending state in the trellis representation that is equivalent to its stored starting state, wherein if the state consistency unit&#39;s determination is positive, the branch metric unit and path metric units are configured to end the determinations of the branch and path metrics. 
     In Example 34, the subject matter of any of Examples 13-15, wherein the decoder is a tail biting convolutional decoder (TBCC) in which the starting and ending states of the maximum likelihood survival path are equivalent. 
     In Example 35, the subject matter of any of Examples 13-16, wherein the path metric unit comprises an add-compare-select unit. 
     In Example 36, the subject matter of any of Examples 13-17, wherein the branch metric unit is further configured to determine the branch metrics such that a strong match between the received encoded sequence of bits and a hypothesized code word of the code results in a smaller branch metric, whereas a weaker match between the received encoded sequence of bits and a hypothesized code word results in a larger branch metric. 
     In Example 37, the subject matter of any of Examples 13-18, wherein the branch metric unit is further configured to determine the branch metrics such that a strong match between the received encoded sequence of bits and a hypothesized code word of the code results in a larger branch metric, whereas a weaker match between the received encoded sequence of bits and a hypothesized code word results in a smaller branch metric. 
     In Example 38, the subject matter of any of Examples 13-19, wherein the decoder is a Viterbi decoder. 
     Example 39 is a wireless communication device comprising the decoder of any of Examples 13-20. 
     Example 40 is a computing device comprising the decoder of any of Examples 13-20. 
     Example 41 is a computer program product embodied on a non-transitory computer-readable medium comprising program instructions configured such that when executed by processing circuitry cause the processing circuitry to implement the method of any of Examples 23-24. 
     Example 42 is an apparatus substantially as shown and described. 
     Example 43 is a method substantially as shown and described. 
     While the foregoing has been described in conjunction with exemplary embodiment, it is understood that the term “exemplary” is merely meant as an example, rather than the best or optimal. Accordingly, the disclosure is intended to cover alternatives, modifications and equivalents, which may be included within the scope of the disclosure. 
     Although specific embodiments have been illustrated and described herein, it will be appreciated by those of ordinary skill in the art that a variety of alternate and/or equivalent implementations may be substituted for the specific embodiments shown and described without departing from the scope of the present application. This application is intended to cover any adaptations or variations of the specific embodiments discussed herein.