Patent Publication Number: US-7219292-B2

Title: Cyclic redundancy check modification for length detection of message with convolutional protection

Description:
DESCRIPTION OF THE INVENTION 
   1. Field of the Invention 
   This invention is in general related to a cyclic redundancy check method for message length detection in a variable-length communications system and, more particularly, to a modified cyclic redundancy check method that has a low false detection probability for a variable-length communications system utilizing a convolutional coding method. 
   2. Background of the Invention 
   In a digital communications system, such as a CDMA (Code Division Multiple Access) system, a data bit stream comprising a stream of binary bits representing a message is transmitted by a transmitter, travels through a data channel, and is received by a receiver. The data bit stream is generally composed of a number of message blocks. If the length of a message block is not fixed, the system may be referred to as a variable-length communications system. In such a variable-length system, additional length information of each message block is generally required for the receiver to identify each message block and extract the message from the received data bit stream. A conventional method of a variable-length communications system designates a separate channel as a control channel for transmitting length information of each message block. Thus, when the receiver receives both the length information and the data bit stream, the receiver identifies the corresponding message blocks based on the length information, and de-blocks the data bit stream. 
   The conventional method also generally utilizes cyclic redundancy check (CRC) bits for error detection purposes. In particular, a fixed number of CRC bits are appended to the end of each message block and have a predetermined relationship with the corresponding message block. The receiver receives both the message block and the CRC bits following that message block, and tries to re-establish the relationship therebetween. If the relationship is satisfied, the message block is considered without error. Otherwise, an error has occurred during the transmission of that block. This method is further explained in greater detail next. 
   First, a CRC generating polynomial, g l (x), of order l, is chosen. A common way of choosing the CRC generating polynomial is that g l (x) should satisfy gcd(g l (x),x i )=1 for each and every i between 0 and l, inclusive, wherein l and i are integers, and the function gcd(A(x),B(x)) is defined as the greatest common divider of polynomials A(x) and B(x). Examples of suitable g l (x) include g 4 (x)=x 4 +x 3 +x 2 +x+1 for l=4; g 7 (x)=x 7 +x 6 +x 4 +1 for l=7; g 8 (x)=x 8 +x 7 +x 4 +x 3 +x+1 for l=8; and g 12 (x)=x 12 +x 11 +x 3 +x 2 +x+1 for l=12. The information regarding the CRC generating polynomial is stored in both the transmitter and the receiver. 
   For illustrative purposes, a binary polynomial is defined for each binary bit stream as follows: if a binary bit stream A includes t binary bits, a t−1 , a t−2 , . . . , a 0 , wherein t is an integer, then the binary polynomial of A is denoted as A(x) and A(x)=a t−1 x t−1 +a t−2 x t−2 + . . . +a 0 . Also for illustrative purposes, binary bit stream A is said to satisfy the CRC condition if A(x) is divisible by g l (x). Two binary bit streams A and B are said to satisfy the CRC condition if x 5 A(x)+B(x) is divisible by g l (x), wherein s is the number of bits contained in bit stream B. It is well known to one skilled in the art that when, for example, a polynomial A(x) is divisible by another polynomial g l (x), the remainder of A(x) divided by g l (x) is 0, and it may be said that g l (x) divides A(x), denoted as g l (x)|A(x). 
   Next, for a message block M containing k bits of binary information, m k−1 , m k−2 , . . . , m 0 , a parity check bit stream P including l parity check bits, or CRC bits, p l−1 , p l−2 , . . . , p 0 , is generated such that M and P satisfy the CRC condition, or g l (x)|(x l M(x)+P(x)). A parity check bit stream may also be called a parity block, a parity check block, or a CRC block. For each message block M, it may be proved that there is only one corresponding parity check bit stream P. The proof is understood by one skilled in the art and is not discussed in detail herein. 
   According to the standard CRC method, parity check bit stream P may be generated using either hardware or software. Examples of hardware implementations are shown in  FIGS. 1–2 , and an example of a software implementation is shown in  FIG. 3 . Both  FIGS. 1 and 2  assume l=8 and CRC generating polynomial g l (x)=x 8 +x 7 +x 4 +x 3 +x+1. 
     FIG. 1  illustrates a first hardware implementation of generating parity check bit stream P. Referring to  FIG. 1 , a feedback shift register circuit  100  is used for generating parity check bit stream P based on CRC generating polynomial g l (x)=x 8 +x 7 +x 4 +x 3 +x+1. Circuit  100  includes a plurality of delay circuits  102 , which may be implemented as flip-flops. The number of delay circuits  102  is equal to the order of g l (x), i.e., l=8. Thus, in  FIG. 1 , there are 8 delay circuits,  102   1 ,  102   2 , . . . ,  102   8 . Several XOR gates  104  are inserted between delay circuits  102 . Each XOR gate  104  corresponds to a coefficient of CRC generating polynomial g l (x). For example, as shown in  FIG. 1 , an XOR gate  104   1  to the left side of the first delay circuit  102   1  indicates that the coefficient of x 0 =1 of g l (x) is 1; the absence of XOR gate  104  between delay circuits  102   2  and  102   3  indicates that the coefficient of x 2  of g l (x) is 0; and an XOR gate  104   5  between delay circuits  102   7  and  102   8  indicates that the coefficient of x 7  of g l (x) is 1. A clock signal (not shown) shifts register circuit  100  from left to right one bit at a time. Also shown in  FIG. 1  is that the output of delay circuit  102   8  is fed back to each of XOR gates  104   1 – 104   5 . The parity check bit stream P is generated by feeding into the left side of circuit  100  the message block M followed by eight bits of 0. The output of delay circuit  102   8  then comprises message block M followed by its corresponding parity check bit stream P. 
   A second hardware implementation of generating a parity check bit stream is illustrated in  FIG. 2 . Similarly, a feedback shift register circuit  200  includes a plurality of delay circuits  202 , each of which may be implemented as a flip-flop circuit. Several XOR gates  204  are inserted between delay circuits  202  according to CRC generating polynomial g l  (x). However, in contrast to  FIG. 1 , an XOR gate  204  is added to the right end of circuit  200 , rather than the left end of circuit  200 , and the message block M is input into the rightmost XOR gate  204 . A switch  206  switches the output of feedback shift register circuit  200  between message block M and the output of the rightmost XOR gate  204 . Feedback shift register circuit  200  first outputs message block M and then outputs the parity bits by switching switch  206  to the output of the rightmost XOR gate  204 . 
     FIG. 3  illustrates a software implementation of generating parity check bit stream P. Rather than generating the parity check bit stream P bit by bit, a lookup table is used in the software implementation. The lookup table contains an entire list of CRC bit streams for all possible messages of a certain length. For example, when l=8, the lookup table includes 2 8 =256 entries of CRC bit streams, each bit stream containing eight binary bits. As shown in  FIG. 3 , a message including 3 bytes (24 bits), Byte  1 , Byte  2 , and Byte  3 , is encoded using the lookup table. At step  302 , Byte  1  is considered and the lookup table is searched for a matching entry for Byte  1 . An XOR operation is performed on the outcome of the search and Byte  2  at step  304  to generate an intermediate CRC bit stream CRC  2 . An entry that matches CRC 2  is looked up in the lookup table (step  306 ) and is XOR&#39;ed with Byte  3  (step  308 ), to generate the CRC bit stream, CRC  3 , of the message. 
   The above three implementations will be readily understood by one skilled in the art and, therefore, the details thereof are not further discussed herein. 
   After the parity check bit stream P is generated, the parity check bits thereof are appended to the end of the message block M to form a concatenated bit stream C including k+l bits, m k−1 , m k−2 , . . . , m 0 , p l−1 , p l−2 , . . . , p 0 . In view of the above conditions, g l (x) divides C(x)=x l M(x)+P(x). 
   For each message block contained in the message, the above encoding process is repeated to generate a corresponding concatenated bit stream, and a data bit stream including the concatenated bit streams and length information of each message block are then transmitted through a data channel and a control channel, respectively. 
   At the receiver side, both the data bit stream and the length information are received. A received message block M′ and a parity check bit stream P′ are extracted based on the length information, wherein M′ includes k bits, m′ k−1 , m′ k−2 , . . . , m′ 0 , and P′ includes l bits, p′ l−1 , P′ l−2 , . . . , p′ 0 . The receiver then performs a so-called CRC test to determine if M′ and P′ satisfy the CRC condition. If the condition is satisfied, then the message block is considered received with no error. 
   A system using a separately designated control channel for transmitting length information as discussed above could be very inefficient when the data rate is slow. For example, in a standard UMTS (universal mobile telecommunications system) WCDMA (wideband code division multiple access) mode, AMR (Adaptive Multi-Rate) 12.2 kbps mode, the overhead for transmitting the length information may be as large as 3 kbps, or almost 25% of the total transmission rate of 12.2 kbps. 
   To reduce the overhead incurred by separately transmitting the length information, there has been proposed a CRC method (hereinafter “standard CRC method”) that uses the CRC bits for message length detection, rather than transmitting the length information of each message block through a separate channel. According to the standard CRC method, the transmitter only transmits the data bit stream, and the receiver receives the data bit stream with no length information. Thus, the receiver may not directly identify the message blocks or extract the message. Instead, the receiver repeats a trial-and-error step to search the received data bit stream for a pair of a message block and a parity check bit stream that satisfy the CRC condition. First, the receiver guesses a number, for example, {circumflex over (k)}, as the block length, and treats the first {circumflex over (k)} bits of the received bit stream as the message block, and the following l bits as a parity check bit stream. The receiver then performs the CRC test to determine if the guessed message block and the guessed parity check bit stream satisfy the CRC condition. If the result is affirmative, the receiver has successfully identified a message block and continues to identify the next message block. Otherwise, the message block has not been identified, the guessed block length {circumflex over (k)} is increased by 1, and the CRC test is repeated. Theoretically, after a few trials, the correct message block will be identified. 
   However, the standard CRC method has an inherent problem of probable false detection. Assuming a noise-free transmission and a uniformly distributed message, the probability of a false detection by the standard CRC method is given by Expression (1): 
                     P   F     ⁡     (   i   )       =     {           0   ,               ⁢         for   ⁢           ⁢   i     =   0     ;                   2     -   i       ,               ⁢         for   ⁢           ⁢   1     ≤   i   ≤     l   -   1       ;                   2     -   l       ,               ⁢         for   ⁢           ⁢   i     ≥   l     ,                       (   1   )               
wherein i=k−{circumflex over (k)} is the message length offset. A brief explanation of Expression (1) is provided next.
 
   Since the transmission is assumed noise-free, all the bits transmitted are received without error. Therefore, if i=k−{circumflex over (k)}=0, the CRC condition is satisfied and a correct message block is identified. No false detection occurs, i.e., P F (0)=0. 
   If i=k−{circumflex over (k)}=1, the wrongly guessed message block M′ includes k−1 bits, m k−1 , m k−2 , . . . , m 1 , and the guessed parity block P′ includes l bits, m 0 , p l−1 , p l−2 , . . . , p 1 . The CRC test therefore decides whether g l (x) divides 
               C   ′     ⁡     (   x   )       =           x   l     ⁢       M   ′     ⁡     (   x   )         +       P   ′     ⁡     (   x   )         =         m     k   -   1       ⁢     x     l   +   k   -   2         +       m     k   -   2       ⁢     x     l   +   k   -   3         +   ⋯   +       m   1     ⁢     x   l       +       m   0     ⁢     x     l   -   1         +       p     l   -   1       ⁢     x     l   -   2         +       p     l   -   2       ⁢     x     l   -   3         +   ⋯   +       p   1     .               
Because gcd(g l (x),x)=1, deciding whether g l (x)|C′(x), is equivalent to deciding whether g l (x)|xC′(x). Comparing C′(x) with C(x), there is xC′(x)=C(x)−p 0 . Thus, if p 0 =0, because g l (x)|C(x), then g l (x)|xC′(x), and g l (x)|C′(x). The receiver regards the wrong message block M′ as the correct message block, and there is a false detection. Otherwise, if p 0 =1, the CRC condition is not satisfied, the receiver concludes that M′ is not the correct message block, and there is no false detection. For a uniformly distributed message, the probability of p 0 =0 is ½, and, therefore, the probability of a false detection is ½.
 
   Similarly, if 1&lt;i≦l−1, the wrongly guessed message block M′ includes k−i bits, m k−1 , m k−2 , . . . , m i , and the wrongly guessed parity block P′ includes l bits, m i−1 , m i−2 , . . . , m 0 , p l−1 , p l−2 , . . . , p i . The CRC test therefore decides whether g l (x) divides 
               C   ′     ⁡     (   x   )       =           x   l     ⁢       M   ′     ⁡     (   x   )         +       P   ′     ⁡     (   x   )         =         m     k   -   1       ⁢     x     l   -   i   +   k   -   1         +       m     k   -   2       ⁢     x     l   -   i   +   k   -   2         +   ⋯   +       m   0     ⁢     x     l   -   i         +       p     l   -   1       ⁢     x     l   -   i   -   1         +       p     l   -   2       ⁢     x     l   -   i   -   2         +   ⋯   +       p   i     .               
Comparing C′(x) with C(x), there is
 
               x   i     ⁢       C   ′     ⁡     (   x   )         =       C   ⁡     (   x   )       -       ∑     j   =   0       i   -   1       ⁢         p   j     ⁡     (   x   )       .               
Because the order of g l (x), l, is greater than i, g l (x) does not divide
 
               ∑     j   =   0       i   -   1       ⁢       p   j     ⁡     (   x   )         ,         
unless p 0 =p 1 = . . . =p i−1 =0. Further because g l (x)|C(x), and gcd(g l (x),x i )=1, g l (x)|C′(x) is satisfied only when p 0 =p 1 = . . . =p i−1 =0. Thus, the probability of a false detection when 1&lt;i≦l−1 is equal to the probability of p 0 =p 1 = . . . =p i-1 =0, which, for a uniformly distributed message, is 2 −i .
 
   Finally, if i≧l, the guessed message block M′ includes k−i bits, m k−1 , m k−2 , . . . , m i , and the guessed CRC bit stream P′ includes, m i−1 , m i−2 , . . . , m i−l . Since there is only one possible CRC bit stream that corresponds to M′, the probability of P′ satisfying the CRC condition of g l (x)|(x l M′(x)+P′(x)), i.e., the probability of a false detection, is 2 −l  for a uniformly distributed message block. 
     FIG. 4  shows a simulation result of the probability of passing the CRC test for the standard CRC method with different guessed message lengths. The simulation conditions include that the order of the CRC generating polynomial is 8 and that the actual message length is 15. As shown in  FIG. 4 , as the estimated message size approaches the actual message length, i.e., the length offset i approaches 0, the probability of passing the CRC test increases exponentially. 
   In view of the large probability of false detection for a standard CRC method, a modified CRC method (“conventional modification”) with a reduced probability of false detection has been proposed. According to the conventional modification of the CRC method, after the parity check bit stream P is generated, the parity check bits are appended to the message block in a reversed order, to form a concatenated bit stream, m k−1 , m k−2 , . . . , m 0 , p 0 , p 1 , . . . , P l−1 .  FIG. 5  shows simulation results for the conventional modification as compared to the standard CRC method. The conditions are the same as those in  FIG. 4 , i.e., the order of the CRC generating polynomial is 8 and the actual message length is 15. As shown in  FIG. 5 , the probability of passing the CRC test, i.e., the probability of false detection is reduced to 2 −l  for all message length offset i&gt;0. 
   If the data channel is noisy, errors will occur during the transmission. To protect the safe transmission of data, a method called convolutional coding may be applied to encode the data prior to the transmission thereof. At the receiver side, a corresponding decoding method is applied to decode the received data. 
   Conceptually, the convolutional coding method encodes the data to generate redundant bits of information and therefore sacrifices bit rate for improved transmission accuracy. According to the convolutional coding method, a convolutional coder receives message blocks to be transmitted and through an encoding process generates a bit stream including a plurality of portions each corresponding to a message block. Each portion may be referred to as a convolutional code word, or code word. The convolutional code word is then transmitted by the transmitter. The convolutional coder may at one time receive t bits of a message block and generate n bits of output, wherein n is generally greater than t. Each of the n bits of output may be a linear combination of the t bits of input and one or more prior bits preceding the t bits of input. The convolutional coder includes a number of memory registers to remember such prior bits and to receive the t bits of input and a plurality of logic gates connected to the memory registers in a manner consistent with an encoding algorithm to generate the n bits of output. A memory order of the convolutional coder is defined as the number of memory registers for each of the input t bits. A convolutional coder having an order j, receiving t bits of input, and outputting n bits of output may be referred to as an (n, t, j) coder. Apparently, an (n, t, j) coder has tj memory registers for storing the prior bits and t memory registers for receiving the one bit of input. In a special case, an (n, 1, j) coder has j memory registers for storing the prior bits and one memory register for receiving the one bit of input. Because each memory register stores either a 0 or a 1, there are 2 j  possible states of such memory registers, i.e., there are 2 j  possible states of the coder. 
   Consider a message block A having k bits. On the transmitter side, prior to coding message block A, an (n, 1, j) coder is at a beginning state. After coding message block A, the coder is at an end state. Apparently, given a beginning state, there is only one corresponding end state. It is common to set the beginning state as an all-zero state, in which all memory registers, except the one for receiving the one bit of input, have a bit of 0 stored therein. As a result of the encoding process, a code word including n(k+j) bits is generated. 
   On the receiver side, a decoder such as a Viterbi decoder may be used to decode a received code word including n(k+j) bits. Generally, the decoder knows in advance the beginning state of the coder. For a k-bit message block, there are 2 k  possible code words. The decoder compares these 2 k  possible code words with the received code word and finds the best match. In an error-free system, there should be one code word that completely matches the received code word. However, in a noisy system, it is possible no one of the 2 k  possible code word completely matches the received code word. A path metric of a possible code word is defined as the number of bits of the possible code word matching those of the received code word. The decoder tries to find the possible code word having the best path metric. The message block corresponding to that code word with the best path metric is considered as the decoded message block. Depending on which decoding algorithm is applied, different decoders may have different efficiencies in finding the best path metric. For example, compared with a sequential decoder, a Viterbi decoder generally has a better efficiency. The sequential decoder and Viterbi decoder are well-known to one skilled in the art and are not described herein. 
   In a variable-length system, the length of each message block and the length of the corresponding code word are unknown. Therefore, the convolutional decoder does not know when to stop the decoding process. Accordingly, extra bits may be inserted to the message block before the convolutional encoding process for that message block. For example, for an (n, 1, j) coder, j bits of 0 may be appended to the end of each message block. As a result, the coder returns to the all-zero state at the end of the encoding process. In the decoding process, the decoder guesses a length {circumflex over (k)} of the message block, and determines if the one of the 2 {circumflex over (k)}  possible code words having the best path metric corresponds to an end state of all-zero. If affirmative, {circumflex over (k)} is considered as the correct length of the message block. Otherwise, the decoder increases {circumflex over (k)} by 1 and repeats the above process. 
   A message block may be encoded using both the CRC method (either the standard CRC method or the conventional modification thereof) and the convolutional method. For example, after the standard CRC encoding process discussed above, a concatenated bit stream C including a message block M including k bits, m k−1 , m k−2 , . . . , m 0 , and a corresponding parity check bit stream P including l bits, p l−1 , p l−2 , . . . , p 0 , may be sent to a convolutional (n, 1, j) coder, which generates a code word including n(k+l) bits. 
   A decoder on the receiver side also decodes the received code word using both the convolutional method and the CRC method. Particularly, a Viterbi decoder convolutionally decodes the received code word and finds a putative concatenated bit stream Ĉ including {circumflex over (k)}+l bits. Then the putative concatenated bit stream Ĉ is sent to a CRC decoder where the CRC test is performed. If the CRC test is passed, the putative concatenated bit stream Ĉ is considered as containing the correct message block M. Otherwise, {circumflex over (k)} is increased by 1 and the above process is repeated. 
   SUMMARY OF THE INVENTION 
   The present invention provides for a modified CRC method for length detection of a message with convolutional protection. 
   Consistent with the present invention, there is provided a method for a variable-length communications system, wherein messages to be transmitted are divided into variable-length message blocks. The method includes providing a cyclic redundancy check (CRC) generating polynomial, providing a binary flip polynomial, and encoding a message block of a message to be transmitted. A message block of the message is encoded by generating a parity check bit stream, flipping the parity check bit stream, appending the flipped parity check bit stream and a number of 0&#39;s to the end of the message block, and convolutionally encoding the resultant bit stream. 
   Consistent with the present invention, there is also provided a method for a variable-length communications system, wherein the system includes a receiver. The method includes storing in the receiver information of a cyclic redundancy check (CRC) generating polynomial and information of a flip polynomial, receiving a data bit stream including a plurality of code words, each code word corresponding to a concatenated bit stream consisting of a message block and a corresponding flipped parity check bit stream; and decoding a first message block in the data bit stream. The first message block is decoded by (a) guessing a message block length and generating a concatenated bit stream including a guessed message block and a guessed flipped bit stream, (b) generating a parity check bit stream for the guessed message block using the CRC generating polynomial, (c) flipping the parity check bit stream using flip polynomial to generate a flipped parity check bit stream, and (d) if the flipped parity check bit stream and the guessed flipped parity check bit stream are different, increasing {circumflex over (k)} by 1 and repeating steps (a)–(c). 
   Consistent with the present invention, there is further provided a method for a variable-length communications system including encoding a message and decoding a data bit stream, wherein the message includes a plurality of message blocks. A message block of the message is encoded by generating a parity check bit stream, flipping the parity check bit stream, appending the flipped parity check bit stream and a number of 0&#39;s to the end of the message block, and convolutionally encoding the resultant bit stream. When a data bit stream is received, a guessed message block and a guessed flipped parity check bit stream are extracted based on a guessed message block length. A parity check bit stream is generated for the guessed message block and then flipped. If the flipped parity check bit stream is the same as the guessed flipped parity check bit stream, the message block has been identified. Otherwise, the guessed message block length is increased by 1 and the above step is repeated. 
   Consistent with the present invention, there is also provided a variable-length communications system that includes a transmitter for encoding messages into a data bit stream and then transmitting the data bit stream, the messages being divided into variable-length message blocks, a data channel for passing the data bit stream comprising the encoded messages; and a receiver for receiving the data bit stream and decoding the messages. The encoding of the messages includes encoding a message block of the messages and encoding the message block includes generating a parity check bit stream using a CRC generating polynomial, flipping the parity check bit stream to generate a flipped parity check bit stream using a flip polynomial, appending the flipped parity check bit stream to the end of the corresponding message block to create a concatenated bit stream, and convolutionally encoding the concatenated bit stream to generate a code word. Also the decoding of the messages includes decoding a message block of the messages contained in the received data bit stream and decoding the message block includes guessing a message block length, generating a concatenated bit stream from the received data bit stream, the concatenated bit stream including a guessed message block and a guessed flipped parity check bit stream, generating a parity check bit stream using the CRC generating polynomial, flipping the parity check bit stream using the flip polynomial to generate a flipped parity check bit stream, if the flipped parity check bit stream and the guessed flipped parity check bit stream are different, increasing the guessed message block length by 1 and returning to the generating of the concatenated bit stream, and if the flipped parity check bit stream and the guessed flipped parity check bit stream are the same, removing the code word of the corresponding message block from the data bit stream. 
   Additional features and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. The features and advantages of the invention will be realized and attained by means of the elements and combinations particularly pointed out in the appended claims. 
   It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the invention, as claimed. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and, together with the description, serve to explain the objects, advantages, and principles of the invention. 
     In the drawings, 
       FIG. 1  illustrates a first hardware implementation of generating a parity check bit stream according to the standard CRC method; 
       FIG. 2  illustrates a second hardware implementation of generating a parity check bit stream according to the standard CRC method; 
       FIG. 3  illustrates a software implementation of generating a parity check bit stream according to the standard CRC method; 
       FIG. 4  shows simulation results of the standard CRC method; 
       FIG. 5  shows simulation results of a conventional modification of the CRC method; 
       FIG. 6  illustrates a first hardware implementation of generating a flipped parity check bit stream consistent with embodiments of the present invention; 
       FIG. 7  illustrates a second hardware implementation of generating a flipped parity check bit stream consistent with embodiments of the present invention; 
       FIG. 8  illustrates a software implementation of generating a flipped parity check bit stream consistent with embodiments of the present invention; 
       FIG. 9  shows a communication system that utilizes the flip-bit CRC coding method consistent with the present invention; 
       FIG. 10A  shows the probability of passing both a path metric test and a CRC test with respect to a message length offset for the CRC method consistent with the present invention, assuming a flip polynomial that satisfies a condition consistent with the present invention; 
       FIG. 10B  shows the probability of passing both the path metric test and the CRC test with respect to the message length offset for the CRC method consistent with the present invention, assuming a flip polynomial that does not satisfy a condition consistent with the present invention; 
       FIGS. 11A–11C  compares the performance of the CRC method consistent with the present invention and the conventional modification method; 
       FIG. 12A  shows simulation results of the probability of failing to detect the correct message block for the CRC method consistent with the present invention and the conventional modification method; and 
       FIG. 12B  shows simulation results of the probability of a false detection for the flip-bit CRC method consistent with the present invention and the conventional modification method. 
   

   DESCRIPTION OF THE EMBODIMENTS 
   Reference will now be made in detail to preferred embodiments of the invention, examples of which are illustrated in the accompanying drawings. Wherever possible, the same reference numbers will be used throughout the drawings to refer to the same or like parts. 
   Embodiments consistent with the present invention provide a modified CRC method for length detection of a message having convolutional protection. 
   Particularly, methods consistent with the present invention are suitable for use in a variable-length communications system, which includes a transmitter and a receiver. A message to be transmitted may include a number of message blocks having non-fixed lengths. Each message block is encoded with both the CRC method and the convolutional method and transmitted by the transmitter. When the receiver receives the encoded message blocks, each message block is decoded and the message is extracted. Since the method of encoding and decoding is the same for all the message blocks, only one message block M including k bits, m k−1 , m k−2 , . . . , m k−1 , m 0 , is considered in the following description. 
   For illustrative purposes, a binary polynomial is defined for each binary bit stream as follows: if a binary bit stream A includes t binary bits, a t−1 , a t−2 , . . . , a 0 , wherein t is an integer, the binary polynomial of A is denoted as A(x) and A(x)=a t−1 x t−1 +a t−2 x t−2 + . . . +a 0 . It is assumed that, unless otherwise indicated, hereinafter, when two binary polynomials are added, coefficients of the two polynomials corresponding to the same power are added according to a modulo-2 addition operation. A modulo-2 addition is defined as a binary addition with no carry, for example, 0+1=1, and 1+1=0. Thus, if bit stream B includes s binary bits, b s−1 , b s−2 , . . . , b 0 , then, assuming s&lt;t, A(x)+B(x)=a t−1 x t−1 +a t−2 x t−2 + . . . +a s x s +(a s−1 +b s−1 )x s−1 +(a s−2 +b s−2 )x s−2 + . . . +(a 0 +b 0 ), wherein a i +b i  gives the result of modulo-2 addition of a i  and b i , for 0≦i≦s−1. It is also assumed that, unless otherwise indicated, hereinafter, when two binary bit streams are added, the corresponding bits of the two bit streams are added according to modulo-2 addition operation. It is also well known to one skilled in the art that, according to the definition of modulo-2 addition, there are a+b+b=a, A+B+B=A, and A(x)+B(x)+B(x)=A(x), wherein a and b are binary bits, and A and B are binary bit streams. 
   A method consistent with the present invention starts by choosing two binary polynomials, a CRC generating polynomial (hereinafter “CRC polynomial”), g l (x), and a flip polynomial f l (x). CRC polynomial g l (x) has an order of l, and flip polynomial f l (x) has an order of l−1, wherein l is an integer. In one aspect, gcd(g l (x),x i )=1 for each and every 0≦i≦l, wherein i is an integer, and gcd(g l (x),x i ) is the greatest common divider of g l (x) and x i . Examples of suitable g l (x) include g 4 (x)=x 4 +x 3 +x 2 +x+1 for l=4; g 7 (x)=x 7 +x 6 +x 4 +1 for l=7; g 8 (x)=x 8 +x 7 +x 4 +x 3 +x+1 for l=8; and g 12 (x)=x 12 +x 11 +x 3 +x 2 +x+1 for l=12. Flip polynomial f l (x) may be expressed as f l (x)=f l−1 x l−1 +f l−2 x l−2 + . . . +f 0 , wherein f i ε{0,1} for 0≦i≦l. The coefficients of flip polynomial f l (x), namely, f l−1 , f l−2 , . . . , f 0 , may be referred to as flip bits. The information of CRC polynomial g l (x) and flip polynomial f l (x) is stored in both the transmitter and the receiver. 
   For illustrative purposes, a binary bit stream A is said to satisfy the CRC condition if g l (x) divides A(x), or g l (x)|A(x); and two binary bit streams A and B are said to satisfy the CRC condition if g l (x) divides x s A(x)+B(x), or g l (x)|(x s A(x)+B(x)), wherein s is the number of bits contained in bit stream B. 
   On the transmitter side, an encoding process first generates a parity check bit stream P including l parity check bits, or CRC bits, p l−1 , p l−2 , . . . , p 0 , such that M and P satisfy the CRC condition, or g l (x)|(x l M(x)+P(x)), wherein M(x)=m k−1 x k−1 +m k−2 x k−2 + . . . +m 0 , and P(x)=p l−1 x l−1 +p l−2 x l−2 + . . . +p 0 . Parity check bit stream P may also be referred to as a parity check block, a parity block, or a CRC block. One skilled in the art will now appreciate that each message block M corresponds to only one unique parity check bit stream P. 
   The encoding process then flips the parity check bits according to flip polynomial f l (x), or, specifically, by performing a modulo-2 addition of each bit in the parity check bit stream P and a corresponding flip bit. The resultant flipped parity check bit stream  P  thus includes l flipped parity check bits:  p l−1   =p l−1 +f l−1 ,  p l−2   =p l−2 +f l−2 , . . . ,  p 0   =p 0 +f 0 . In effect, if f i =1, then  p i    is the flip of p i ; if f 0 =1, then  p i    is the same as p i . 
   Then, the flipped parity check bits are appended to the end of the message block to form a concatenated bit stream C including k+l bits, m k−1 , m k−2 , . . . , m 0 ,  p l−1   ,  p l−2   , . . . ,  p 0   . 
   Consistent with the present invention, the flipped parity check bits may be generated through hardware or software.  FIG. 6  illustrates a first hardware implementation for generating the flipped parity check bits according to an embodiment consistent with the present invention. Referring to  FIG. 6 , a feedback shift register circuit  600  is used for generating flipped parity check bit stream  P  based on CRC generating polynomial g l (x)=x 8 +x 7 +x 4 +x 3 +x+1. Circuit  600  includes a plurality of delay circuits  602 , which may be implemented as flip-flops. The number of delay circuits  602  is equal to the order of g l (x), i.e., l=8. Thus, in  FIG. 6 , there are 8 delay circuits,  602   1 ,  602   2 , . . . ,  602   8 . Several XOR gates  604  are inserted between delay circuits  602 . Each XOR gate  604  corresponds to a coefficient of CRC generating polynomial g l (x). For example, as shown in  FIG. 6 , an XOR gate  604   1  to the left side of the first delay circuit  602   1  indicates that the coefficient of x 0 =1 of g l (x) is 1; the absence of XOR gate  604  between delay circuits  602   2  and  602   3  indicates that the coefficient of x 2  of g l (x) is 0; and an XOR gate  604   5  between delay circuits  602   7  and  602   8  indicates that the coefficient of x 7  of g l (x) is 1. An XOR gate  604   6  is also coupled to XOR the output of delay circuit  602   8  with the message block M followed by flip bits f l−1 , f l−2 , . . . , f 0 . According to the expression for f l (x) previously described and as shown in  FIG. 6 , f 8 (x)=x 7 +1. Thus, the corresponding 8 flip bits are 10000001. A clock signal (not shown) shifts register circuit  600  from left to right one bit at a time. Also shown in  FIG. 6  is that the output of XOR gate  604   6  is fed back to each of XOR gates  604   1 – 604   5 . A switch  606  switches the output of feedback shift register circuit  600  between message block M and the output of XOR gate  604   6 . Feedback shift register circuit  600  first outputs message block M and then outputs the flipped parity bits by switching switch  606  to the output of XOR gate  604   6 . 
   A second hardware implementation for generating flipped parity check bit stream  P  consistent with an embodiment of the present invention is illustrated in  FIG. 7 . As shown in  FIG. 7 , a feedback shift register circuit  700  includes a plurality of delay circuits  702 , each of which may be implemented as a flip-flop circuit. Several XOR gates  704  are inserted between delay circuits  702  according to CRC generating polynomial g l (x). Two XOR gates  704   1  and  704   2  are added to the left and right ends of circuit  700 , respectively. The message block M is input into XOR gate  704   1 , and the rightmost delay circuit  702  outputs message block M and its corresponding parity check bit stream P. XOR gate  704   2  then flips the parity check bit stream P using the flip bits, f l−1 , f l−2 , . . . , f 0 , to generate the flipped parity check bit stream  P . It is also assumed in  FIG. 7  that the flip polynomial is f 8 (x)=x 7 +1, and therefore the flip bits are 10000001. 
     FIG. 8  diagrammatically illustrates a software implementation of generating flipped parity check bit stream  P , wherein a lookup table is used in the software implementation. The lookup table contains an entire list of CRC bit streams for all possible messages of a certain length. For example, when l=8, the lookup table includes 2 8 =256 entries of CRC bit streams, each bit stream containing eight binary bits. As shown in  FIG. 8 , a message including 3 bytes (24 bits), Byte  1 , Byte  2 , and Byte  3 , is encoded using the lookup table. At step  802 , Byte  1  is considered and the lookup table is searched for a matching entry for Byte  1 . An XOR operation is performed on the outcome of the search and Byte  2  at step  804  to generate an intermediate CRC bit stream CRC  2 . An entry that matches CRC 2  is looked up in the lookup table (step  806 ) and is XOR&#39;ed with Byte  3  (step  808 ), to generate the CRC bit stream, CRC  3 , of the message. Further, CRC  3  is flipped using the flip bits. It is also assumed in  FIG. 8  that the flip polynomial is f 8 (x)=x 7 +1, and therefore the flip bits are 10000001. 
   After the above CRC encoding process, the concatenated bit stream C is further encoded by an (n, t, j) convolutional coder, where n is an integer representing how many bits are output at a time by the coder, t is an integer indicating the number of inputs the coder receives, and j is the memory order of the coder. For simplicity of illustration, it is assumed that t=1. First, the concatenated bit stream C is appended with j bits of 0 to create a 0-terminated bit stream B, which includes k+l+j bits, m k−1 , M k−2 , . . . , m 0 ,  p l−1   ,  p l−2   , . . .  p   0 , 0, 0, . . . , 0 (j bits of 0 at the end). 0-terminated bit stream B is then passed through the (n, 1, j) convolutional coder to generate a convolutional code word D including n(k+l+j) bits. The convolutional encoding process is well-known to one skilled in the art and is not described in detail herein. 
   The same encoding process as above is performed to generate a convolutional code word for each of the other message blocks in the message, and a data bit stream including the resultant convolutional code words is transmitted. 
   When the receiver receives a data bit stream including at least a convolutional code word, a decoding process is performed to identify the first message block in the data bit stream. After the first message block is identified, the corresponding code word is removed from the data bit stream, and the receiver continues to identify the first message block in the resultant data bit stream. Thus, when the receiver starts to decode message block M, the data bit stream includes the convolutional code word D corresponding to message block M, followed by the corresponding convolutional code word of the next message blocks. 
   The decoding process includes a convolutional decoding process and a CRC decoding process. First, a message block M′ having a length {circumflex over (k)} is guessed and the decoder convolutionally decodes the first n({circumflex over (k)}+l+j) bits in the received data bit stream. In one aspect, {circumflex over (k)} is chosen to be smaller than the length k of message block M. The decoder determines if the one of the 2 {circumflex over (k)}+l+j  possible code words with the best path metric, or the best code word, corresponds to an end state of all-zero. If negative, {circumflex over (k)} is increased by 1 and the above process is repeated. If affirmative, {circumflex over (k)} is considered as the correct length of the message block and a putative concatenated bit stream Ĉ including {circumflex over (k)}+l bits, m k−1 , m k−2 , . . . , m k−{circumflex over (k)} , m k−{circumflex over (k)}−1 , m k−{circumflex over (k)}−2 , . . . , m 0 ,  p l−1   ,  p l−2   , . . . , p k−{circumflex over (k)} , is extracted and is subjected to the CRC test in the CRC decoding process. 
   In the CRC decoding process, a parity check bit stream {circumflex over (P)} including l parity check bits, {circumflex over (p)} l−1 , {circumflex over (p)} l−2 , . . . , {circumflex over (p)} 0 , is first generated for the guessed message block M′ such that g l (x)|(x l M′(x)+{circumflex over (P)}(x)). Second, using flip polynomial f l (x), the parity check bit stream {circumflex over (P)} is flipped to generate a flipped parity check bit stream {circumflex over (P)}′, which includes l flipped parity check bits, {circumflex over (p)}′ l−1 ={circumflex over (p)}′ l−1 +f l−1 , {circumflex over (p)}′ l−2 ={circumflex over (p)}′ l−2 +f l−2 , . . . , {circumflex over (p)}′ 0 ={circumflex over (p)} 0 +f 0 . Finally, the receiver compares the flipped parity check bit stream {circumflex over (P)}′ with the guessed flipped parity check bit stream  P′ . If {circumflex over (P)}′≠{circumflex over (P)}{circumflex over (′)}, the CRC test fails and no message block has been identified; the estimated length {circumflex over (k)} is increased by 1, and the above path metric test and CRC test are repeated. If, otherwise, {circumflex over (P)}′=  P′ , the CRC test is passed and it is considered that a message block has been correctly identified. The first n({circumflex over (k)}+l+j) bits constituting the code word corresponding to message block M′ are removed from the data bit stream, and the receiver continues to decode the first message block in the resultant data bit stream. 
   A false detection occurs when first n({circumflex over (k)}+l+j) bits pass both the path metric test and the CRC test while {circumflex over (k)} is not the correct length of message block M. In the following description, it is assumed that the guessed message block M′ corresponds to a concatenated bit stream C′ including {circumflex over (k)}+l bits, a 0-terminated bit stream B′ including {circumflex over (k)}+l+j bits, and a code word D′ including the first n({circumflex over (k)}+l+j) bits. 
   First, as discussed above, to pass the path metric test in an error-free channel, 1) D′ must have the best path metric among those 2 {circumflex over (k)}+l+j  possible code words each corresponding to a message block length of {circumflex over (k)}, and 2) D′ must correspond to an all-zero end state of the coder, i.e., the coder returns to the all-zero state after encoding C′. For these two conditions to be satisfied, the last j bits of bit stream B′ must all be 0&#39;s. 
   Second, for the putative concatenated bit stream C′ to pass the CRC test, there must be g l (x)|(C′(x)+f l (x)). 
   By choosing a proper flip polynomial f l (x), the CRC method of the present invention may have a low probability of false detection. In one aspect, flip polynomial f l (x) is chosen such that 
                     deg   ⁡     (     remainder   ⁢           ⁢     of   ⁡     (         (     1   +     x   i       )     ⁢       f   l     ⁡     (   x   )             g   l     ⁡     (   x   )         )         )       ≥     i   -   j       ,       for   ⁢           ⁢   1     ≤   i   ≤     l   +   j   -   1.               (   2   )               
For example, when l=8, g l (x)=x 8 +x 7 +x 4 +x 3 +x+1, and a (2, 1, 8) convolutional coder is used, there are 66 different flip polynomials f l (x) that satisfy condition (2), an example of which is f l (x)=x 4 +x.
 
   Under condition (2), and assuming both a uniformly distributed message and an error-free transmission, the probability of D′ passing both the path metric test and the CRC test is given in Expression (3): 
                     P   F     ⁡     (   i   )       =     {               ⁢     0   ,                 ⁢         for   ⁢           ⁢   0     ≤   i   ≤     l   +   j   -   1       ;                     ⁢       2     -     (     l   +   j     )         ,                 ⁢         for   ⁢           ⁢   i     ≥     l   +   j       ,                       (   3   )               
wherein i=k−{circumflex over (k)} is the message length offset. A brief proof of Expression (3) is given next.
 
   When i=0, message block M′ includes k bits, m k−1 , m k−2 , . . . . m 0 , and the corresponding flipped parity check block  P′  includes l bits,  p l−1   ,  p l−2   , . . .  p 0   . B′ includes M′ followed by  P′  and j bits of 0&#39;s. Both the path metric test and the CRC test are passed, the correct message block is identified, and there is no false detection. 
   When 0&lt;i≦j, M′ includes {circumflex over (k)} bits, m k−1 , m k−2 , . . . , m i ,  P′  includes l bits, m i−1 , m i−2 , . . . , m 0 ,  p l−1   ,  p l−2   , . . .  p i   , and B′ includes M′ and  P′ , followed by j bits,  p i−1   ,  p i−2   , . . . ,  p 0   , and j−i 0&#39;s. For the path metric test to pass, the last j bits of bit stream B′ must all be 0, i.e.,  p i−1   ,  p i−2   , . . . ,  p 0    are all 0&#39;s. For the concatenated bit stream C′ to pass the CRC test, there must be g l (x)|(C′(x)+f l (x)), where
 
 C ′( x )= m   k−1   x   l+{circumflex over (k)}−1   +m   k−2   x   l+{circumflex over (k)}−2   + . . . +m   0   x   l−i +    p   l−1     x   l−i−1 + . . . +    p   i   .
 
Comparing C′(x) with C(x)=x l M(x)+P(x), there is
 
             C   ⁡     (   x   )       =         (         C   ′     ⁡     (   x   )       +       f   l     ⁡     (   x   )         )     ⁢     x   i       +       ∑     s   =   0       i   -   1       ⁢         p   s     _     ⁢     x   s         +         f   l     ⁡     (   x   )       .             
Because g l (x)|C(x) and gcd(g l (x),x i )=1, g l (x)|C′(x) is satisfied if and only if
 
               g   l     ⁡     (   x   )       ⁢            (         (     1   +     x   i       )     ⁢       f   l     ⁡     (   x   )         +       ∑     s   =   0       i   -   1       ⁢         p   s     _     ⁢     x   s           )     .             
Further,
 
             deg   ⁡     (     remainder   ⁢             ⁢             ⁢     of   ⁡     (         (     1   +     x   i       )     ⁢       f   l     ⁡     (   x   )             g   l     ⁡     (   x   )         )         )       ≥     i   -   j           
(condition (2) above), and
 
               ∑     s   =   0       i   -   1       ⁢         p   s     _     ⁢     x   s         =   0         
because  p i−1   ,  p i−2   , . . . ,  p 0    must all be 0 for the path metric test to pass. Therefore,
 
               deg   (     remainder   ⁢             ⁢             ⁢     of   (       (         (     1   +     x   i       )     ⁢       f   l     ⁡     (   x   )         +       ∑     s   =   0       i   -   1       ⁢         p   s     _     ⁢     x   s           )         g   l     ⁡     (   x   )         )       )     ≥     i   -   j       ,         
and g l (x) does not divide
 
               (     1   +     x   i       )     ⁢       f   l     ⁡     (   x   )         +       ∑     s   =   0       i   -   1       ⁢         p   s     _     ⁢       x   s     .               
Therefore, the probability of both the path metric test and the CRC test being passed, i.e., the probability of a false detection, is 0.
 
   When j&lt;i≦l+j−1, M′ includes {circumflex over (k)} bits, m k−1 , m k−2 , . . . , m i ,  P′  includes l bits, m i−1 , m i−2 , . . . , m 0 ,  p l−1   ,  p l−2   , . . . ,  p i   , and B′ includes M′ and  P′ , followed by j bits,  p i−1   ,  p i−2   , . . . ,  p i-j   . For the path metric test to pass, the last j bits of bit stream B′ must all be 0, i.e.,  p i−1   ,  p i−2   , . . . ,  p i−j    are all 0&#39;s. For the concatenated bit stream C′ to pass the CRC test, there must be g l (x)|(C′(x)+f l (x)), where
 
 C ′( x )= m   k−1   x   l+{circumflex over (k)}−1   +m   k−2   x   l+{circumflex over (k)}−2   + . . . +m   0   x   l−i +    p   l−1     x   l−i−1 + . . . +    p   i   .
 
Comparing C′(x) with C(x)=x l M(x)+P(x), there is
 
             C   ⁡     (   x   )       =         (         C   ′     ⁡     (   x   )       +       f   l     ⁡     (   x   )         )     ⁢     x   i       +       ∑     s   =   0       i   -   1       ⁢         p   s     _     ⁢     x   s         +         f   l     ⁡     (   x   )       .             
Because g l (x)|C(x) and gcd(g l (x), x i )=1, g l (x)|C′(x) is satisfied if and only if
 
               g   l     ⁡     (   x   )       ⁢     ❘     ⁢       (         (     1   +     x   i       )     ⁢       f   l     ⁡     (   x   )         +       ∑     s   =   0       i   -   1       ⁢         p   s     _     ⁢     x   s           )     .           
Further,
 
             deg   ⁡     (     remainder   ⁢             ⁢             ⁢   of   ⁢           ⁢     (         (     1   +     x   i       )     ⁢       f   l     ⁡     (   x   )             g   l     ⁡     (   x   )         )       )       ≥     i   -   j           
(condition (2) above), and
 
             deg   ⁡     (       ∑     s   =   0       i   -   1       ⁢         p   s     _     ⁢     x   s         )       &lt;     i   -   j           
because  p i−1   ,  p i−2   , . . . ,  p i−j   must all be 0 for the path metric test to pass. Therefore,
 
               deg   (     remainder   ⁢             ⁢             ⁢     of   (       (         (     1   +     x   i       )     ⁢       f   l     ⁡     (   x   )         +       ∑     s   =   0       i   -   1       ⁢         p   s     _     ⁢     x   s           )         g   l     ⁡     (   x   )         )       )     ≥     i   -   j       ,         
and g l (x) does not divide
 
               (     1   +     x   i       )     ⁢       f   l     ⁡     (   x   )         +       ∑     s   =   0       i   -   1       ⁢         p   s     _     ⁢       x   s     .               
Therefore, the probability of a false detection is 0.
 
   In view of the above, the probability of a false detection is 0 when 0&lt;i≦l+j−1. 
   If i≧1+m, then the guessed message block M′ includes k−i bits, m k−1 , m k−2 , . . . , m i , the guessed flipped parity check block  P′  includes, m i−1 , m i−2 , . . . , m i−l , B′ includes M′ and  P′ , followed by j bits, m i−l−1 m i−l−2 , . . . , m i−l−j . For the path metric test to pass, the last j bits of bit stream B′ must all be 0, i.e., m i−l−1 , m i−l−2 , . . . , m i−l−j  are all 0&#39;s. Further, because there is only one possible parity check block corresponding to a certain message block M′, there is only one possible flipped parity check block  P′  that corresponds to message block M′. Therefore, assuming message block M is uniformly distributed, the probability of m i−1 , m i−2 , . . . , m i−l  composing the flipped check parity block  P′  that corresponds to M′ and m i−l−1 , m i−l−2 , . . . , m i−l−j  are all 0&#39;s is 2 −(l+j) . 
     FIG. 9  shows a communication system  900  that utilizes the flip-bit CRC coding method consistent with the present invention. System  900  includes a transmitter  902  and a receiver  904 . Transmitter  902  includes a flip-bit CRC encoder  906  and a convolutional encoder  908 . Receiver  904  includes a convolutional decoder  910  and a flip-bit CRC decoder  912 . Message blocks are sequentially encoded by flip-bit CRC encoder  906  and convolutional encoder  908 , transmitted by transmitter  902 , passed through a data channel  914 , and sequentially decoded by convolutional decoder  910  and flip-bit CRC decoder  912 . 
   Computer simulation has been carried out and simulation results are shown in  FIGS. 10A–10B  and  11 A– 11 C. 
     FIG. 10A  shows the probability of passing both the path metric test and the CRC test with respect to the message length offset for the CRC method consistent with the present invention, assuming an initial signal-to-noise ratio (SNR) of 2.0 dB, 4.0 dB, and 6.0 dB. It is assumed in  FIG. 10A  that a (2, 1, 8) convolutional coder is used, the order of the CRC generating polynomial is 8, g 8  (x)=x 8 +x 7 +x 4 +x 3 +x+1, f 8  (x)=x 4 +x, and the actual message length is 30. f 8  (x)=x 4 +x satisfies condition (1). As shown in  FIG. 10A , when the SNR is high such as 6.0 dB, no false detection occurs when the message length offset is less than l+j=16. Even with a poor SNR such as 2.0 dB or 4.0 dB, the probability of a false detection when the message length offset is less than 16 is significantly lower than 2 −(l+j) =2 −16 . 
     FIG. 10B  illustrates the effect when the flip polynomial fails to satisfy condition (1).  FIG. 10B  is the simulation result based on the same assumptions as  FIG. 10A  except that the flip polynomial is f 8 (x)=x 7 +1, which does not satisfy condition (1). Consequently, the probability of false detection is much higher. 
     FIGS. 11A–11C  compare the performance of the CRC method consistent with the present invention and the conventional modification method, where the circles represent the flip-bit CRC method consistent with the present invention, and the cross symbols represent the conventional modification method. It is assumed in  FIGS. 11A–11C  that a (2, 1, 8) convolutional coder is used, the order of the CRC generating polynomial is 8, g 8  (x)=x 8 +x 7 +x 4 +x 3 +x+1, f 8 (x)=x 4 +x, and the actual message length is 30.  FIG. 11A  shows the comparison when the SNR is 2.0 dB.  FIG. 11B  shows the comparison when the SNR is 4.0 dB.  FIG. 11C  shows the comparison when the SNR is 6.0 dB. As shown in  FIGS. 11A–11C , the CRC method consistent with the present invention has a better performance than the conventional modification method when the message length offset is smaller than l+j. 
   As discussed above, when a message block M′ is guessed, the convolutional decoder determines whether D′, which includes the first n({circumflex over (k)}+l+j) bits of the received data bit stream, passes the path metric test, which requires 1) that D′ must have the best path metric among those 2 {circumflex over (k)}+l+j  possible code words each corresponding to a message block length of {circumflex over (k)}, and 2) D′ must correspond to an all-zero end state of the coder, i.e., the coder returns to the all-zero state after encoding C′. The path metric test under these requirements is very strict and may result in a failure to detect a correct message block, as a code word D corresponding to a correct message block M may fail this test. Therefore, it is sometimes desirable to relax the test, as discussed next. 
   To quantize the relaxation of the path metric test, a relative path metric d is defined as 
             d   =         λ   0     -     λ   min           λ   max     -     λ   min           ,         
and a pre-determined threshold D init  is selected such that 0≦D init ≦1, wherein λ 0  is the path metric of the code word corresponding to an all-zero end state, λ max  is the maximum path metric, and λ min  is the minimum path metric. According to the relaxed path metric test, if D′ corresponds to an all-zero end state of the coder and d≧D init , then D′ is considered to contain the correct message block. Apparently, the strict path metric test is the special instance of D init =1.  FIG. 12A  shows simulation results of the probability of failing to detect the correct message block (“Block Error Rate”) with respect to the initial SNR (“Uncoded SNR”) and a comparison between the flip-bit CRC method consistent with the present invention and the conventional modification method for different values of D init  including 0.0, 0.5, and 1.0.  FIG. 12B  shows simulation results of the probability of a false detection (“Undetected Error Rate”) with respect to the initial SNR (“Uncoded SNR”) and a comparison between the flip-bit CRC method consistent with the present invention and the conventional modification method for different values of D init  including 0.0 and 0.5. As shown in  FIG. 12A , when D init  is set to 1 (corresponding to the strict path metric test), there is a significantly high probability of the flip-bit CRC method failing to find any message block that satisfies both the path metric test and the CRC test.  FIG. 12A  also shows that the probability of the flip-bit CRC method failing to find any message block is slightly higher than the case when the message block length is known (represented by the circles) when D init  is 0.0 or 0.5. As shown in  12 B, when D init  is set to smaller numbers such as 0.0 or 0.5, the probability of a false detection rises to prohibitive levels when the SNR is moderate (such as 4.0 dB) or lower. Thus, by choosing an appropriate CRC generating polynomial, an appropriate flip polynomial, and an appropriate D init , the flip-bit CRC method consistent with the present invention may achieve error detection capabilities in a variable-length system commensurate with that in a system where the length of message blocks is known.
 
   It will be apparent to those skilled in the art that various modifications and variations can be made in the disclosed process without departing from the scope or spirit of the invention. Other embodiments of the invention will be apparent to those skilled in the art from consideration of the specification and practice of the invention disclosed herein. It is intended that the specification and examples be considered as exemplary only, with a true scope and spirit of the invention being indicated by the following claims.