The performance of Turbo codes can be near Shannon-limit capacity by iteratively passing probabilistic estimates between two decoders with long codeword, i.e. iterative maximum a posteriori (MAP) probability algorithm [13]. Recently, for practical use, techniques called early-stopping criterion are introduced to see if the iteratively decoding process could be terminated to avoid unnecessary iterations and save computational power
as well as processing delay [37], [44], [83], [93]. In this section, the SIC II presented in Chapter 3 is used as the front-end of a receiver in Turbo coded systems. This scheme is shown to outperform the one with PPIC at front end in multipath fading channels. In addition, with the characteristics of logarithm likelihood ratios (LLR) and ordering information from SIC, we propose a high efficient stopping criterion with low complexity.
5.2.1 Turbo Decoding
An important feature of Turbo codes is the iterative decoder with SISO algorithms, such as MAP algorithm [9], log-MAP algorithm [60], Max-Log-MAP algorithm [25], [43], and soft-output Viterbi algorithm (SOVA) [12], [30]. The MAP algorithm, also known as BCJR algorithm, was first presented by Bahl, Colcke, Jelinik and Raviv in 1974 for both convolutional codes and block codes [9]. The algorithm attempts to minimize the BER by estimating the a posteriori probabilities (APP) of the individual bits in the codeword, and it examines every possible path through the convolutional decoder trellis. For the user with index k, we have the following definitions [61]:
S is the set of all 2m constituent encoder states where m is the encoder memory
usk=(uks[1],uks[2],...uks[MK +m])
ukpx=(ukpx[1],ukpx[2],...ukpx[MK +m])
where x, which can be 1 or 2, denotes one of the constituent encoder E1 or E2.
) , ( ( ) ( )
)
( p n
k s n k n
k y y
y =
is the noisy version of (xks(n),xkp(n)),
) ,..., ,
( ( ) ( 1) ( )
) ,
( y
k x
k x k y x
k y y y
y = + ,
yk = yk(1,Mk)=(yk(1),yk(2),....,yk(Mk))
is the noisy received codeword. Without loss of generality, the index k is temporarily omitted for simplicity. We would like to make decision of u[n]as
[
( [ ])]
1)/2 Iterative MAP algorithm
The MAP algorithm provides not only the estimated information bits, but also the probability for each bit. This is essential for the iterative decoding of Turbo codes. With the encoder shown in Fig. 2-4, the corresponding iterative MAP decoder with two
interconnected elementary decoders is shown in Fig. 5-1. The decoder has three inputs. For SISO decoder D1, inputs are the systematically encoded channel output bit ys(n), the parity bit yp1 n( ) and the extrinsic information Li2−ex1(u[n]). For SISO decoder D2, inputs are the
and the initial conditions
1 To avoid numerically unstable results, we define
∑∑
and the Log-APP in (5-2) becomes
⎟⎟ with noise variance N0/2. The intrinsic information is defined as
⎟⎟⎠ The first term in (5-5) is called the channel value, the second term is the a priori information provided by decoder D2 which is equal to 0 at the first inner iteration, and the third term is the extrinsic information L1iex(u[n]) that can be passed on to the subsequent
decoder D2. For D2, the LLR of APP in (5-2) can be written as
Hence, final decision of u[n] in (5-1) at the D2 output of the Ii-th inner iteration thus becomes By iterative decoding, each decoder passes the updated extrinsic information to the other decoder, and we can expect that the BER of the decoded bits tends to become lower and lower. Further improvement in BER decreases as the number of iterations increases. To reduce the decoding computational complexity, data memory and power consumption, dynamically control of the iteration number is inevitable.
Literature overview of stopping criteria
Many criteria have been proposed in recent years for early stopping in turbo decoding.
The variables generally used in literature for decision making are the extrinsic/intrinsic information and the LLR of APPas shown in Fig. 5-1. For the i-th inner iteration, the LLR of APP in decoder D1 and decoder D2 defined in (5-5) and (5-6) are re-written as follows.
])
and of decoder D1 and decoder D2 are as follows.
1 The stopping criteria mentioned in literature are beiefly described as follows.
Cross Entropy (CE) Criterion. Hagenauer et al. [30] used a threshold value on the Cross entropy between the output distributions of the two SISO decoders. For a Turbo decoder, it is shown that the CE of iteration i can be approximated by,
∑
=where Mk is the block size of user k. The decoding process is stopped after iteration i for i>1,
( ) / (1)
T i T < θ where T(1) is the approximated CE after the first iteration.
Sign-Change Ratio (SCR) Criterion. Based on the concept of CE, Shao et al. [62]
presented two simple and effective criteria, known as SCR and hard-decision aided (HDA), respectively. SCR evaluates the number of sign changes in the extrinsic information between successive iterations, and the decoding process is stopped after iteration i for i>1, if
(
⊕)
<θ∑
=k − M n
i ex i
ex k
n u L S n u L M 1 S
1 2
2 ( [ ])) ( ( [ ]))
1 (
where S(x) denotes the sign part of x and ⊕ denotes the XOR operation.
Hard-Decision Aided (HDA) Criterion. This criterion is proposed in [62]. It compares the decoded bits of the two successive iterations. The decoding process is stopped after iteration i for i>1, if
1
2 2
( ( [ ]))i ( i ( [ ]))
S L u n =S L u n− , ∀n∈1....Mk
HDA2 Criterion. The idea of HDA criterion is extended in [47]. The decoding process is stopped after iteration i for i>1, if
1 2
( ( [ ]))i ( ( [ ]))i
S L u n =S L u n , ∀n∈1....Mk
where j =1 or 2. In this way, about half of the iteration number can be saved.
Sign Difference Ratio (SDR) Criterion. Extending the SCR method, a new criterion called SDR is proposed in [24]. SDR evaluates the number of sign differences between the intrinsic information and the extrinsic information for the same SISO decoder in the same
iteration, and the decoding process is stopped after iteration i for i>1, if
(
⊕)
<θ∑
= Mkn
i jex i
jin k
n u L S n u L
M1 1 S( ( [ ])) ( ( [ ]))
Min-LLR Criterion. The minimum of the absolute values of the LLRs is first used in an early stopping criterion in [85], [86] and is later presented in [47], [93].The decoding process is stopped after iteration i for i>1, if
θ
≤ >
≤min 2( [ ])
1 Li u n
M
n k
Mean-LLR and Sum-LLR Criteria. The stopping criterion based on the mean of the absolute values of the LLRs is presented in [32], [47], [94]. The decoding process is stopped after iteration i for i>1, if
θ
∑
>= Mk
n i k
n u M1 1 L2( [ ])
In [29], the sum of the absolute values of the LLRs is calculated to avoid a costly division operation in the Mean-LLR criterion,
∑
== Mk
n i
i L u n
S
1
2( [ ]) The decoding process is stopped after iteration i for i>1, if
1≤0
− i−
i S
S Cyclic Redundancy Check (CRC) Criterion. CRC is introduced as a stopping criterion in several papers. The CRC criterion takes extra resource. In WCDMA, CRC is attached before channel coding. All bits in a decoded CB should be checked to know its correctness.
Recently, it is shown that more than one parameter in stopping criterion provides better performance. An example of this kind of criterion is as follows.
Comb. Min-LLR and Sum-LLR Criterion. Min-LLR and Sum-LLR criteria are combined in [29]. The decoding process is stopped after iteration i for i>1, if
) ]) [ ( min (
||
) 0
( 2
1 ≤ 1 >θ
−S− ≤ ≤ L u n
S i
M i n
i k
where || denotes the OR operation.
5.2.2 Turbo-Coded SIC with Low-Complexity Stopping Criterion
The block diagram of turbo-coded SIC is shown in Fig. 5-4 with the transmitter model shown in Fig. 2-19. The input to the de-interleaver Π−1 followed by K single-user turbo decoder is soft value YˆSIC(<nu)>
of the SIC shown in (3-12). YˆSIC(<nu)>
is separated to systematic
part y<su>(nb), and parity part y<pu1>(nb), y<pu2>(nb) where 0≤nb <M<u> +3. The iterative decoding algorithm of the turbo decoder is described in 5.2.1. Although the output of SIC II and SIC III in different order are different in mean and variance, according to the central limit theorem, whatever of these outputs in different order are independent random variables, the outputs YˆSIC(n),k
can be approximated as Gaussian distributed random variables with ~N(mk, σ2k) in both power-balanced and power-unbalanced systems, especially when K is large. For practical use, estimation of the residual interference plus noise is done with the aid of pilot-channel signal. The block diagram of variance estimation is shown in Fig. 5-7.
In traffic-channel signal remover (TCSR), all user data respread in SIC for data are summed and removed from r t( ), i.e. the output of TCSR is
; ,
1
( ) ( ) K ( )
SIC pilot data k
k
r t r t C t
=
= −
∑
where⎣ ⎦ ˆ [⎣( )/ ⎦] ( ) ( ) thus calculated as follows.
∑
= After ik iterations, the transmission data sequence of the k-th user is recovered by the hard-decision and demapper of the LLR Li2k(u[n]) as shown in (5-6).From 5.2.1, we can find that most stopping criteria in literature take information from all bits in a CB as measurement bases. The processing delay and required data buffer thus become large when the block length increases. According to the characteristic of SIC II &
SIC III, bits in later detected order have smaller soft outputs, or say, smaller SNR than that in the former detected order. In addition, the LLR of APP is a function of soft outputs of SIC, and systematic bit takes the most important part in LLR of APP. As a result, the decoded bit with the minimum LLR of APP is most probably the one in later detected order.
Also, errors are easier to occur in decoded bits with later detected order. According to these properities, we propose to consider only part of the APP LLRs in a CB. The APP LLRs of a CB are devided into K groups, i.e. for the n-th bit of user with index k detected at the u-th
order,
} 0
, :
{ n k
u = n <u> =k ≤n<M
Λ where 0≤u<K and
} 0
, , :
{n n∈Λu u≥x ≤ x<K
=
Λ . For example, if the stopping criterion is the combination of HDA2 and Min-LLR, i.e. the decoding process is terminated at the Ii,k-th inner iteration if
θ
< >
≤ ( [ ]
min 2,
0 LIik uk n
M
n k
& S(L2Ii,k(uk[n]))=S(L1Ii,k(uk[n])) for all n∈Λ
(5-13) The resultant computational complexity and used buffer become x/K times of the original one. The tradeoff between detecting probability of correct CB and computational complexity relies on the setting of parameters θ and x. Impacts of these parameters are examined in the next section.
5.2.3 Simulation Results and Discussions
The simulation parameters listed in Table 3-2 with multipath fading channel Case 3 shown in Table 3-3 is used for performance evaluation. We first compare the performance of our proposed SIC front-end and the one with PPIC [22] with three stages where the partial coefficient is 0.6 at the first two stages. The turbo iteration number is limited to 10 and the CB length is M=196. In Fig. 5-2, the SIC frond-end is shown to outperform PPIC from 0.25dB with BER at 2*10e-2 to more than 1dB with BER at 10e-3.
In Table 5-1, it is shown that our proposed stopping criterion performs almost the same as the full-block-checking criterion with much less computational complexity needed, and about half of the computational complexity can be saved. In addition, the combination of HDA2 and min-LLR can results in less iteration number needed.