Soft-decision decoding of convolutionally encoded codeword

[0001] The present invention relates to maximum a posteriori (MAP) decoding of convolutional codes and in particular to a decoding method and a turbo decoder based on the LOG-MAP algorithm.

[0002] In the field of digital data communication, error-correcting circuitry, i.e. encoders and decoders, is used to achieve reliable communications on a system having a low signal-to-noise ratio (SNR). One example of an encoder is a convolutional encoder, which converts a series of data bits into a codeword based on a convolution of the input series with itself or with another signal. The codeword includes more data bits than are present in the original data stream. Typically, a code rate of ½ is employed, which means that the transmitted codeword has twice as many bits as the original data. This redundancy allows for error correction. Many systems also additionally utilize interleaving to minimize transmission errors.

[0003] The operation of the convolutional encoder and the MAP decoder are conveniently described using a trellis diagram which represents all of the possible states and the transition paths or branches between each state. During encoding, input of the information to be coded results in a transition between states and each transition is accompanied by the output of a group of encoded symbols. In the decoder, the original data bits are reconstructed using a maximum likelihood algorithm e.g. Viterbi Algorithm. The Viterbi Algorithm is a decoding technique that can be used to find the Maximum Likelihood path in the trellis. This is the most probable path with respect to the one described at transmission by the coder.

[0004] The basic concept of a Viterbi decoder is that it hypothesizes each of the possible states that the encoder could have been in and determines the probability that the encoder transitioned from each of those states to the next set of encoder states, given the information that was received. The probabilities are represented by quantities called metrics, of which there are two types: state metrics α (β for reverse iteration), and branch metrics γ. Generally, there are two possible states leading to every new state, i.e. the next bit is either a zero or a one. The decoder decides which is the most likely state by comparing the products of the branch metric and the state metric for each of the possible branches, and selects the branch representing the more likely of the two.

[0005] The Viterbi decoder maintains a record of the sequence of branches by which each state is most likely to have been reached. However, the complexity of the algorithm, which requires multiplication and exponentiations, makes the implementation thereof impractical. With the advent of the LOG-MAP algorithm implementation of the MAP decoder algorithm is simplified by replacing the multiplication with addition, and addition with a MAX operation in the LOG domain. Moreover, such decoders replace hard decision making (0 or 1) with soft decision making (P_k0 and P_k1). See U.S. Pat. No. 5,499,254 (Masao et al) and U.S. Pat. No. 5,406,570 (Berrou et al) for further details of Viterbi and LOG-MAP decoders. Attempts have been made to improve upon the original LOG-MAP decoder such as disclosed in U.S. Pat. No. 5,933,462 (Viterbi et al) and U.S. Pat. No. 5,846,946 (Nagayasu).

[0006] Recently turbo decoders have been developed. In the case of continuous data transmission, the data stream is packetized into blocks of N data bits. The turbo encode provides systematic data bits and includes first and second constituent convolutional recursive encoders respectively providing e1 and e2 outputs of codebits. The first encoder operates on the systematic data bits providing the e1 output of code bits. An encoder interleaver provides interleaved systematic data bits that are then fed into the second encoder. The second encoder operates on the interleaved data bits providing the e2 output of the code bits. The data uk and code bits e1 and e2 are concurrently processed and communicated in blocks of digital bits.

[0007] However, the standard turbo-decoder still has shortcomings that need to be resolved before the system can be effectively implemented. Typically, turbo decoders need at least 3 to 7 iterations, which means that the same forward and backward recursions will be repeated 3 to 7 times, each with updated branch metric values. Since a probability is always smaller than 1 and its log value is always smaller than 0, α, β and γ all have negative values. Moreover, every time γ is updated by adding a newly-calculated soft-decoder output after every iteration, it becomes an even smaller number. In fixed point representation too small a value of γ results in a loss of precision. Typically when 8 bits are used, the usable signal dynamic range is −255 to 0, while the total dynamic range is −255 to 255, i.e. half of the total dynamic range is wasted.

[0008] In a prior attempt to overcome this problem, the state metrics α and β have been normalized at each state by subtracting the maximum state metric value for that time. However, this method results in a time delay as the maximum value is determined. Current turbo-decoders also require a great deal of memory in which to store all of the forward and reverse state metrics before soft decision values can be calculated.

[0009] An object of the present invention is to overcome the shortcomings of the prior art by increasing the speed and precision of the turbo decoder while better utilizing the dynamic range, lowering the gate count and minimizing memory requirements.

[0010] In accordance with the principles of the invention the quantities _j(R_k,s_j′,s)(j=0, 1) used in the recursion calculation employed in a turbo decoder are first normalized. This results in an increase in the dynamic range for a fixed point decoder.

[0011] According to the present invention there is provided a method of decoding a received encoded data stream having multiple states s, comprising the steps of:

• • [0012] recursively determining the value of at least one of the quantities α_k(s) and β_k(s) defined as ${\alpha }_k\left(s\right)=\log \left(\mathrm{Pr}\left\{S_k=s|R_1^k\right\}\right)$ ${\beta }_k\left(s\right)=\log \left(\frac{\mathrm{Pr}\left\{R_{k+1}^N{S}_k=s\right\}}{\mathrm{Pr}\left\{R_{k+1}^N|R_1^N\right\}}\right)$
  • [0013]  where R₁^k represents received bits from time index 1 to k, and S_k represents the state of an encoder at time index k, from previous values of α_k(s) or β_k(s), and from quantities γ′_j(R_k,s_j′,s)(j=0, 1), where γ′_j(R_k,s_j′,s)(j=0, 1) is a normalized value of γ_j(R_k,s_j′,s)(j=0, 1), which is defined as,
    γ_j(R_ks′_j,s)=log(Pr(d_k=j,S_k=s,R_k|S_k−1=s′_j))
  • [0014]  where Pr represents probability, R_k represents received bits at time index k, and d_k represents transmitted data at time k.

[0015] The invention also provides a decoder for a convolutionally encoded data stream, comprising:

• • [0016] a first normalization unit for normalizing the quantity
    γ_j(R_ks′_j,s)=log(Pr(d_k=j,S_k=s,R_k|S_k−1=s′_j))
  • [0017] adders for adding normalized quantities γ′_j(R_k,s_j′,s)(j=0, 1) to quantities α_k−1(s₀′), α_k−1(s₁′), or β_k−1(s₀′), β_k−1(s₁′), where ${\alpha }_k\left(s\right)=\log \left(\mathrm{Pr}\left\{S_k=s|R_1^k\right\}\right)$ ${\beta }_k\left(s\right)=\log \left(\frac{\mathrm{Pr}\left\{R_{k+1}^N{S}_k=s\right\}}{\mathrm{Pr}\left\{R_{k+1}^N|R_1^N\right\}}\right)$
  • [0018]  a multiplexer and log unit for producing an output α_k′(s), or β_k′(s), and
  • [0019]  a second normalization unit to produce a desired output α_k(s), or β_k(s).

[0020] The processor speed can also be increased by performing an Smax operation on the resulting quantities of the recursion calculation. This normalization is simplified with the Smax operation.

[0021] The present invention additionally relates to a method for decoding a convolutionally encoded codeword using a turbo decoder with x bit representation and a dynamic range of 2^x−1 to −(2^x−1), comprising the steps of:

• [0022] a) defining a first trellis representation of possible states and transition branches of the convolutional codeword having a block length N, N being the number of received samples in the codeword;
• [0023] b) initializing each starting state metric α₋₁(s) of the trellis for a forward iteration through the trellis;
• [0024] c) calculating branch metrics γ_k0(s₀′,s) and γ_k0(s₁′,s);
• [0025] d) determining a branch metric normalizing factor;
• [0026] e) normalizing the branch metrics by subtracting the branch metric normalizing factor from both of the branch metrics to obtain γ_k1′(s₁′,s) and γ_k0′(s₀′,s);
• [0027] f) summing α_k−1(s₁′) with γ_k1′(s₁′,s), and α_k−1(s₀′) with γ_k0′(s₀′,s) to obtain a cumulated maximum likelihood metric for each branch;
• [0028] g) selecting the cumulated maximum likelihood metric with the greater value to obtain α_k(s);
• [0029] h) repeating steps c to g for each state of the forward iteration through the entire trellis;
• [0030] i) defining a second trellis representation of possible states and transition branches of the convolutional codeword having the same states and block length as the first trellis;
• [0031] j) initializing each starting state metric β_N-1(s) of the trellis for a reverse iteration through the trellis;
• [0032] k) calculating the branch metrics γ_k0(s₀′,s) and γ_k1(s₁′,s);
• [0033] l) determining a branch metric normalization term;
• [0034] m) normalizing the branch metrics by subtracting the branch metric normalization term from both of the branch metrics to obtain γ_k1′(s₁′,s) and γ_k0′(s₀′,s);
• [0035] n) summing β_k+1(s₁′) with γ_k1′(s₁′,s), and β_k+1(s₀′) with γ_k0′(s₀′,s) to obtain a cumulated maximum likelihood metric for each branch;
• [0036] o) selecting the cumulated maximum likelihood metric with the greater value as β_k(s);
• [0037] p) repeating steps k to o for each state of the reverse iteration through the entire trellis;
• [0038] q) calculating soft decision values P₁ and P₀ for each state; and
• [0039] r) calculating a log likelihood ratio at each state to obtain a hard decision thereof.

[0040] Another aspect of the present invention relates to a method for decoding a convolutionally encoded codeword using a turbo decoder with x bit representation and a dynamic range of 2^x−1 to −(2^x−1), comprising the steps of:

• [0041] a) defining a first trellis representation of possible states and transition branches of the convolutional codeword having a block length N, N being the number of received samples in the codeword;
• [0042] b) initializing each starting state metric α₋₁(s) of the trellis for a forward iteration through the trellis;
• [0043] c) calculating the branch metrics γ_k0(s₀′,s) and γ_k1(s₁′,s);
• [0044] summing α_k−1(s₁′) with γ_k1(s₁′,s), and α_k−1(s₀′) with γ_k0(s₀′,s) to obtain a cumulated maximum likelihood metric for each branch;
• [0045] selecting the cumulated maximum likelihood metric with the greater value as α_k(s);
• [0046] determining a forward normalizing factor, based on the values of α_k−1(s), to reposition the values of α_k(s) proximate the center of the dynamic range;
• [0047] g) normalizing α_k(s) by subtracting the forward normalizing factor from each α_k(s);
• [0048] h) repeating steps c to g for each state of the forward iteration through the entire trellis;
• [0049] i) defining a second trellis representation of possible states and transition branches of the convolutional codeword having the same number of states and block length as the first trellis;
• [0050] j) initializing each starting state metric β_N-1(s) of the trellis for a reverse iteration through the trellis;
• [0051] k) calculating the branch metrics γ_k0(s₀′,s) and γ_k1(s₁′,s);
• [0052] l) summing β_k+1(s₁′) with γ_k1(s₁′,s), and β_k+1(s_o′) with γ_k0(s₀′,s) to obtain a cumulated maximum likelihood metric for each branch;
• [0053] m) selecting the cumulated maximum likelihood metric with the greater value as β_k(s);
• [0054] n) determining a reverse normalizing factor, based on the value of β_k+1(s), to reposition the values of β_k(s) proximate the center of the dynamic range;
• [0055] o) normalizing β_k(s) by subtracting the reverse normalizing factor from each β_k(s);
• [0056] p) repeating steps k to o for each state of the reverse iteration through the entire trellis;
• [0057] q) calculating soft decision values P₁ and P₀ for each state; and
• [0058] r) calculating a log likelihood ratio at each state to obtain a hard decision thereof.

[0059] Another aspect of the present invention relates to a method for decoding a convolutionally encoded codeword using a turbo decoder, comprising the steps of:

• [0060] a) defining a first trellis representation of possible states and transition branches of the convolutional codeword having a block length N, N being the number of received samples in the codeword;
• [0061] b) initializing each starting state metric α₋₁(s) of the trellis for a forward iteration through the trellis;
• [0062] c) calculating the branch metrics γ_k0(s₀′,s) and γ_k1(s₁′,s);
• [0063] d) summing α_k−1(s₁′) with γ_k1(s₁′,s), and α_k−1(s₀′) with γ_k0(s₀′,s) to obtain a cumulated maximum likelihood metric for each branch;
• [0064] e) selecting the cumulated maximum likelihood metric with the greater value as α_k(s);
• [0065] f) repeating steps c to e for each state of the forward iteration through the entire trellis;
• [0066] g) defining a second trellis representation of possible states and transition branches of the convolutional codeword having the same number of states and block length as the first trellis;
• [0067] h) initializing each starting state metric β_N-1(s) of the trellis for a reverse iteration through the trellis;
• [0068] i) calculating the branch metrics γ_k0(s₀′,s) and γ_k1(s₁′,s);
• [0069] i) summing β_k+1(s₁′) with β_k1(s₁′,s), and β_k+1(s_o′) with β_k0(s₀′,s) to obtain a cumulated maximum likelihood metric for each branch;
• [0070] j) selecting the cumulated maximum likelihood metric with the greater value as β_k(s);
• [0071] k) repeating steps i to k for each state of the reverse iteration through the entire trellis;
• [0072] m) calculating soft decision values P₀ and P₁ for each state; and
• [0073] n) calculating a log likelihood ratio at each state to obtain a hard decision thereof;
  • [0074] wherein steps a to f are executed simultaneously with steps g to l; and
  • [0075] wherein step m includes:
    • [0076] storing values of α₋₁(s) to at least α_N/2-2(s), and β_N-1(s) to at least β_N/2(s) in memory; and
    • [0077] sending values of at least α_N/2-1(s) to α_N-2(s), and at least β_N/2-1(s) to β₀(s) to probability calculator means as soon as the values are available, along with required values from memory to calculate the soft decision values P_k0 and P_k1;
  • [0078] whereby all of the values for α(s) and β(s) need not be stored in memory before some of the soft decision values are calculated.

[0079] The apparatus according to the present invention is defined by a turbo decoder system with x bit representation for decoding a convolutionally encoded codeword comprising:

[0080] receiving means for receiving a sequence of transmitted signals;

[0081] first trellis means with block length N defining possible states and transition branches of the convolutionally encoded codeword;

[0082] first decoding means for decoding said sequence of signals during a forward iteration through said first trellis, said first decoding means including:

• • [0083] branch metric calculating means for calculating branch metrics γ_k0(s₀′,s) and γ_k1(s₁′,s);
  • [0084] branch metric normalizing means for normalizing the branch metrics to obtain γ_k1′(s₁′,s) and γ_k0′(s₀′,s);
  • [0085] summing means for adding state metrics α_k−1(s₁′) with γ_k1′(s₁′,s), and state metrics α_k−1(s₀′) with γ_k0′(s₀′,s) to obtain cumulated metrics for each branch; and
  • [0086] selecting means for choosing the cumulated metric with the greater value to obtain α_k(s);

[0087] second trellis means with block length N defining possible states and transition branches of the convolutionally encoded codeword;

[0088] second decoding means for decoding said sequence of signals during a reverse iteration through said trellis, said second decoding means including:

• • [0089] branch metric calculating means for calculating branch metrics γ_k0(s₀′,s) and γ_k1(s₁′,s);
  • [0090] branch metric normalizing means for normalizing the branch metrics to obtain γ_k1′(s₁′,s) and γ_k0′(s₀′,s);
  • [0091] summing means for adding state metrics β_k+1(s₁′) with γ_k1′(s₁′,s), and state metrics β_k+1(s_o′) with γ_k0′(s₀′,s) to obtain cumulated metrics for each branch; and
  • [0092] selecting means for choosing the cumulated metric with the greater value to obtain β_k(s);

[0093] soft decision calculating means for determining the soft decision values P_k0 and P_k1; and

[0094] LLR calculating means for determining the log likelihood ratio for each state to obtain a hard decision therefor.

[0095] Another feature of the present invention relates to a turbo decoder system, with x bit representation having a dynamic range of 2^x−1 to −(2^x−1), for decoding a convolutionally encoded codeword, the system comprising:

[0096] receiving means for receiving a sequence of transmitted signals:

[0097] first trellis means defining possible states and transition branches of the convolutionally encoded codeword;

[0098] first decoding means for decoding said sequence of signals during a forward iteration through said first trellis, said first decoding means including:

• • [0099] branch metric calculating means for calculating branch metrics γ_k0(s₀′,s) and γ_k1(s₁′,s);
  • [0100] summing means for adding state metrics α_k−1(s₁′) with γ_k1′(s₁′,s), and state metrics α_k−1(s₀′) with γ_k0′(s₀′,s) to obtain cumulated metrics for each branch; and
  • [0101] selecting means for choosing the cumulated metric with the greater value to obtain α_k(s);
  • [0102] forward state metric normalizing means for normalizing the values of α_k(s) by subtracting a forward state normalizing factor, based on the values of α_k−1(s), from each α_k(s) to reposition the value of α_k(s) proximate the center of the dynamic range;

[0103] second trellis means with block length N defining possible states and transition branches of the convolutionally encoded codeword;

[0104] second decoding means for decoding said sequence of signals during a reverse iteration through said trellis, said second decoding means including:

• • [0105] branch metric calculating means for calculating branch metrics γ_k0(s₀′,s) and γ_k1(s₁′,s);
  • [0106] summing means for adding state metrics β_k+1(s₁′) with γ_k1′(s₁′,s), and state metrics β_k+1(s_o′) with γ_k0′(s₀′,s) to obtain cumulated metrics for each branch;
  • [0107] selecting means for choosing the cumulated metric with the greater value to obtain β_k(s); and
  • [0108] wherein the second rearward state metric normalizing means for normalizing the values of β_k(s) by subtracting from each β_k(s) a rearward state normalizing factor, based on the values of β_k+1(s), to reposition the values of β_k(s) proximate the center of the dynamic range;

[0109] soft decision calculating means for calculating the soft decision values P_k0 and P_k1; and

[0110] LLR calculating means for determining the log likelihood ratio for each state to obtain a hard decision therefor.

[0111] Yet another feature of the present invention relates to a turbo decoder system for decoding a convolutionally encoded codeword comprising:

[0112] receiving means for receiving a sequence of transmitted signals:

[0113] first trellis means with block length N defining possible states and transition branches of the convolutionally encoded codeword;

[0114] first decoding means for decoding said sequence of signals during a forward iteration through said first trellis, said first decoding means including:

• • [0115] branch metric calculating means for calculating branch metrics γ_k0(s₀′,s) and γ_k1(s₁′,s);
  • [0116] summing means for adding state metrics α_k−1(s₁′) with γ_k1′(s₁′,s), and state metrics α_k−1(s₀′) with γ_k0′(s₀′,s) to obtain cumulated metrics for each branch; and
  • [0117] selecting means for choosing the cumulated metric with the greater value to obtain α_k(s);

[0118] second trellis means with block length N defining possible states and transition branches of the convolutionally encoded codeword;

[0119] second decoding means for decoding said sequence of signals during a reverse iteration through said trellis, said second decoding means including:

• • [0120] branch metric calculating means for calculating branch metrics γ_k0(s₀′,s) and γ_k1(s₁′,s);
  • [0121] summing means for adding state metrics β_k+1(s₁′) with γ_k1′(s₁′,s), and state metrics β_k+1(s_o′) with γ_k0′(s₀′,s) to obtain cumulated metrics for each branch; and
  • [0122] selecting means for choosing the cumulated metric with the greater value to obtain β_k(s);

[0123] soft decision calculating means for determining soft decision values P_k0 and P_k1; and

[0124] LLR calculating means for determining the log likelihood ratio for each state to obtain a hard decision therefor;

[0125] wherein the soft decision calculating means includes:

• • [0126] memory means for storing values of α₋₁(s) to at least α_N/2-2(s), and β_N-1(s) to at least β_N/2(s); and
  • [0127] probability calculator means for receiving values of at least α_N/2-1(s) to α_N-2(s), and at least β_N/2-1(s) to β₀(s) as soon as the values are available, along with required values from memory to calculate the soft decision values;
  • [0128] whereby all of the values for α(s) and β(s) need not be stored in memory before some soft decision values are calculated.

[0129] The invention now will be described in greater detail with reference to the accompanying drawings, which illustrate a preferred embodiment of the invention, wherein:

[0138] With reference to FIG. 1, a traditional turbo decoder system for decoding a convolutional encoded codeword includes an Add-Compare-Select (ACS) unit. The ADD function is carried out by summators 1 and 2 which, respectively, add state metric _k−1(s₀′) to branch metric γ₀(R_k,s₀′,s) and state metric _k−1(s₁′) to branch metric γ₁(R_k,s₁′,s) to obtain two cumulated metrics. The COMPARE to determine which of the aforementioned cumulated metrics is greater, is performed by substractor 3 which subtracts the second sum _k−1(s₁′)γ₁(s₁′,s) from the first sum _k−1(s₀′)γ₀(s₀′,s). In FIG. 1, the output of adder 3 is epread into two directions: its sign controls the MUX 8 and its magnitude controls a small log table 11. In practice, very few bits are need for the magnitude. The sign of the difference between the cumulated metrics indicates which one is greater, i.e. if the difference is negative α_k−1(s₁′)γ₁(s₁′,s) is greater. The sign of the difference controls a 2 to 1 multiplexer 8, which is used to SELECT the survivor cumulated metric having the greater sum. The magnitude of the difference between the two cumulated metrics acts as a weighting coefficient, since the greater the difference the more likely the correct choice was made between the two branches. The magnitude of the difference also is supplied to a long table unit 11 which produces a corresponding correction and applies it to the summator 4. The magnitude of the difference dictates the size of a correction factor, which is added to the selected cumulated metric at summator 4. The correction factor is necessary to account for an error resulting from the MAX operation. In this example, the correction factor is approximated in the log table 11, although other methods of providing the correction factor are possible, such as that disclosed in the Aug. 6, 1998 edition of Electronics Letters in an article entitled “Simplified MAP algorithm suitable for implementation of turbo decoders”, by W. J. Gross and P. G. Gulak. The resulting corrected cumulated metrics α′_k(s) are then normalized by subtracting therefrom the state metric normalization term (Σα_k′(s)) which is the maximum value of α′_k(s), using subtractor 5. The resultant value is α_k(s). This forward iteration is repeated for the full length of the trellis. The same process is repeated for the reverse iteration using the reverse state metrics β_k(s) in place of the state metric α_k(s) as is well known in the prior art.

[0139] As will be understood by one skilled in the art, the circuit shown in FIG. 1 performs the computation ${\alpha }_k\left(s\right)=\log \left(\mathrm{Pr}\left\{S_k=s|R_1^k\right\}\right)$ ${\beta }_k\left(s\right)=\log \left(\frac{\mathrm{Pr}\left\{R_{k+1}^N{S}_k=s\right\}}{\mathrm{Pr}\left\{R_{k+1}^N|R_1^N\right\}}\right)$
where R₁^k represents the received information bits and parity bits from time index 1 to k[1], and

• • [0140] S_k represents the encode state at time index k.

[0141] A similar structure can also be applied to the backward recursion of β_k.

[0142] In FIG. 1, the value α at state s and time instant k (α_k(s) is related with two previous state values which are _k−1(s₀′) and _k−1(s₁′) at time instant k−1. γ_j(R_k,s_j′,s) j=0, 1 represents the information bit defined as
γ_j(R_k,s′_j,s)=log(Pr(d_k=j,S_k=s,R_k|S_k−1=s′_j))
where R_k represents the received information bits and parity bits at time index k and d_k represents the transmitted information bit at time index k[1].

[0143] A trellis diagram (FIGS. 3 & 4) is the easiest way to envision the iterative process performed by the ACS unit shown in FIG. 1. For the example given in FIGS. 3 and 4, the memory length (or constraint length) of the algorithm is 3 which results in 2³=8 states (i.e. 000, 001 . . . 111). The block length N of the trellis corresponds to the number of samples taken into account for the decoding of a given sample. An arrow represents a transition branch from one state to the next given that the next input bit of information is a 0 or a 1. The transition is dependent upon the convolutional code used by the encoder. To calculate all of the soft decision values α_k, α₋₁(s₀) is given an initial value of 0, while the remaining values α₋₁(s_t) (t=1 to 7) are given a sufficiently small initial value, e.g. −128. After the series of data bits making up the message are received by the decoder, the branch metrics γ_k0 and γ_k1 are calculated in the known way. The iterative process then proceeds to calculate the state metrics α_k. Similarly the reverse iteration can be enacted at the same time or subsequent to the forward iteration. All of the initial values for β_N-1 are set at equal value, e.g. 0.

[0144] Once all of the soft decision values are determined and the required number of iterations are executed the log-likelihood ratio (LLR) can be calculated according to the following relationships: $\begin{array}{c}\mathrm{LLR}=\log \frac{P\left(u_k=1❘R_K\right)}{P\left(u_k=-1|R_K\right)}\\ =\log \frac{\sum a_{k-1}\left(s^{\prime }\right)b_k\left(s\right)c_k\left(s^{\prime },s\right)\mathrm{for}{u}_k=+1}{\sum a_{k-1}\left(s^{\prime }\right)b_k\left(s\right)c_k\left(s^{\prime },s\right)\mathrm{for}{u}_k=-1}\end{array}$ $R_K=\mathrm{Received}\mathrm{signals}$ $\alpha =\ln \left(a\right)$ $\beta =\ln \left(b\right)$ $\gamma =\ln \left(c\right)$ $u_k=\mathrm{bit}$
associated with k_th bit $\begin{array}{c}\mathrm{associated}\mathrm{with}{k}_{\mathrm{th}}\mathrm{bit}=\mathrm{Max}\left({\beta }_k+{\alpha }_{k-1}+{\gamma }_k\right)-\\ \mathrm{Max}\left({\beta }_k+{\alpha }_{k-1}+{\gamma }_k\right)\end{array}$ $u_k=1$ $\begin{array}{c}{u}_k=-1\\ =P_{\mathrm{k1}}-P_{\mathrm{k0}}\end{array}$

[0145] FIG. 5 and FIG. 6 illustrate flow charts representing the calculation of P_k1, and P_k0 respectively based on the forward and backward recursions illustrated in FIGS. 3 and 4.

[0146] In the decoder shown in FIG. 1, the time required for Σ_sα_k′(s) to be calculated can be unduly long if the turbo encoder has a large number of states s. A typical turbo code has 8 or 16 states, which means that 7 0r 25 adders are required to compute Σ_sα_k′(s). Even an optimum parallel structure requires 15 adders and a 4 adder delay for a 16 state turbo decoder.

[0147] Also, a typical turbo decoder requires at least 3 to 7 iterations, which means that the same α and β recursion will be repeated 3 to 7 times, each with updated γ_j(R_k,s₀′,s)(j==0, 1) values. Since the probability is always smaller than 1 and its log value is always smaller than zero, α, β are γ are all negative values. The addition of any two negative values will make the output more negative. When γ is updated by adding a newly calculated soft decoder output, which is also a negative value, γ becomes smaller and smaller after each iteration. In fixed point representation, too small value for γ means loss of precision. In the worst case scenario, the decoder could be saturated at the negative overflow value, which is 0×80 for b but implementation.

[0148] With reference to FIG. 2, the decoder in accordance with the principles of this invention includes some of the elements of the prior art decoder along with a branch metric normalization system 13. To ensure that the values of γ₀ and γ₁ do not become too small and thereby lose precision, the branch metric normalization system 13 subtracts a normalization factor from both branch metrics. This normalization factor is selected based on the initial values of γ₀ and γ₁ to ensure that the values of the normalized branch metrics γ₀′ and γ₁′ are close to the center of the dynamic range i.e. 0.

[0149] The following is a description of the preferred branch metric normalization system. Initially, the branch metric normalization system 13 determines which branch metric γ₀ or γ₁ is greater. Then, the branch metric with the greater value is subtracted from both of the branch metrics, thereby making the greater of the branch metrics 0 and the smaller of the branch metrics the difference. This relationship can also be illustrated using the following equation
γ₀′=0, if γ₀>γ₁,
or
γ₀−γ₁, otherwise
γ₁′=0, if γ₁≧γ₀
or
γ₁−γ₀ otherwise

[0150] Using this implementation, the branch metrics γ₀ and γ₁ are always normalized to 0 in each turbo decoder iteration and the dynamic range is effectively used thereby avoiding ever increasingly smaller values.

[0151] In another embodiment of the present invention in an effort to utilize the entire dynamic range and decrease the processing time of the state metric normalization term, e.g. the maximum value of α_k(s), is replaced by the maximum value of α_k−1(s), which is pre-calculated using the previous state α_k−1(s). This alleviates any delay between summator 4 and subtractor 5 while the maximum value of α_k(s) is being calculated.

[0152] Alternatively, according to another embodiment of the present invention, the state metric normalization term is replaced by a variable term NT, which is dependent upon the value of α_k−1(s) (see Box 12 in FIG. 2). The value of NT is selected to ensure that the values of the state metrics are moved closer to the center of the dynamic range, i.e. 0 in most cases. Generally speaking, if the decoder has x bit representation, when any value of α_k−1(s) is greater than zero, then the variable term NT is a small positive number, e.g. between 1 and 8. If all values of α_k−1(s) are less than 0 and any one value of α_k−1(s) is greater than −2^x−2, then the variable term NT is about −2^x−3, i.e. −2^x−3 and is added to all of the values of α_k(s). If all values of α_k−1(s) are less than −2^x−2, then the variable term NT is the bit OR value of each value of α_k−1(s).

[0153] For example in 8 bit representation:

[0154] if any of α_k−1(s) (s=1, 2 . . . M) is greater than zero, then the NT is 4, i.e. 4 is subtracted from all of the α_k(s);

[0155] if all of α_k−1(s) are less than 0 and any one of α_k−1(s) is greater than −64, then the NT is −31, i.e. 31 is added to all of the α_k(s);

[0156] if all of α_k−1(s) are less than −64, then the NT is the bit OR value of each α_k−1(s).

[0157] In other words, whenever the values of α_k−1(s) approach the minimum value in the dynamic range, i.e. −(2^x−1), they are adjusted so that they are closer to the center of the dynamic range.

[0158] The same values can be used during the reverse iteration.

[0159] This implementation is much simpler than calculating the maximum value of M states. However, it will not guarantee that α_k(s) and β_k(s) are always less than 0, which a log-probability normally defines. However, this will not affect the final decision of the turbo-decoder algorithm. Moreover, positive values of α_k(s) and β_k(s) provide an advantage for the dynamic range expansion. By allowing α_k(s) and β_k(s) to be greater than 0, by normalization, the other half of the dynamic range (positive numbers), which would not otherwise be used, will be utilized.

[0160] FIG. 7 shows a practical implementation of the normalization function. γ₀, γ₁ are input two comparator 701, and Muxes 702, 703 whose outputs are connected to a subtractor 704. Output Muxes produced the normalized outputs γ′₀, γ′₁. This ensures γ′₀, γ′₁ that are always normlalized to zero in each turbo decoder iteration and the dynamic range is effectively used to avoid the values becoming smaller and smaller.

[0161] In FIG. 2, the normalization term is replaced with the maximum value of α_k−1(s) which can be precalculated α_k−1(s). There unlike the situation described with reference to FIG. 1, no wait time is required between adder 4 and adder 5.

[0162] To further simplify the operation, “Smax” is used to replace the true “max” operation as shown in FIG. 8. In FIG. 8, b_nm represents the n^th bit of α_k−1(m) (i.e. the value of α_k−1 at state s=m. In FIG. 8, the bits b_nm are fed through OR gates 801 to Muxes 802, 803, which produce the desired output S_max α_k−1(s). FIG. 8 shows represents three cases for 8 bit fixed point implementation.

[0163] If any of α_k−1(s=1, 2, . . . M) is larger than zero, the Smax output will take a value 4 (0×4), which means that 4 should be subtracted from all α_k(s).

[0164] If all α_k−1(s) are smaller than zero and one of α_k−1(s) is larger than −64, the Smax will take a value −31 (0xe1), which means that 31 should be added to all α_k(s).

[0165] If all α_k−1(s) are smaller than −64, the Smax will take the bit OR value of all α_k−1(s).

[0166] The novel implementation is much simpler than the prior art technique of calculating the maximum value of M states, but it will not guarantee that α_k(s) is always smaller than zero. This does not affect the final decision in the turbo-decoder algorithm, and the positive value of α_k(s) can provide an extra advantage for dynamic range expansion. If α_k(s) are smaller than zero, only half of the 8-bit dynamic range is used. By allowing α_k(s) to be larger than zero with appropriate normalization, the other half of the dynamic range, which would not normally be used, is used.

[0167] A similar implementation can be applied to the β_k(s) recursion calculation.

[0168] By allowing the log probability α_k(s) to be a positive number with appropriate normalization, the decoder performance is not affected and the dynamic range can be increased for fixed point implementation. The same implementation for forward recursion can be easily implemented for backward recursion.

[0169] Current methods using soft decision making require excessive memory to store all of the forward and the reverse state metrics before soft decision values P_k0 and P_k1 can be calculated. In an effort to eliminate this requirement the forward and backward iterations are performed simultaneously, and the P_k1 and P_k0 calculations are commenced as soon as values for β_k and α_k−1 are obtained. For the first half of the iterations the values for α₋₁ to at least α_N/2-2, and β_N-1 to at least β_N/2 are stored in memory, as is customary. However, after the iteration processes overlap on the time line, the newly-calculated state metrics can be fed directly to a probability calculator as soon as they are determined along with the previously-stored values for the other required state metrics to calculate the P_k0, the P_k1. Any number of values can be stored in memory, however, for optimum performance only the first half of the values should be saved. Soft and hard decisions can therefore be arrived at faster and without requiring an excessive amount of memory to store all of the state metrics. Ideally two probability calculators are used simultaneously to increase the speed of the process. One of the probability calculators utilizes the stored forward state metrics and newly-obtained backward state metrics β_N/2-2 to β₀. This probability calculator determines a P_{k0 low} and a P_{k1 low}. Simultaneously, the other probability calculator uses the stored backward state metrics and newly-obtained forward state metrics α_N/2-1 to α_N-2 to determine a P_{k1 high} and a P_{k0 high}.