3.2 Iterative Decoding
3.2.2 Decoder
In the BICM-ID system, a SISO decoder is needed to compute soft information passed to detector. In the thesis, BCJR algorithm [14] is utilized.
The log-APP of coded bit cj, L( )Dϕ
( )
cj is written as can be reformulated as( )
( )
Next we show how the joint probability in (3-6) can be evaluated recursively.(
( ))
L the portion of a prior LLRs before after time l. The last equality follows from the fact that the probability of the state at time l depends only on the previous states before time l.
Defining
using the forward recursion method as
state s at time l using the backward recursion method as
( ) (
( )( ) ) (
( )( ) )
( )
( )
( )( ) ( ( ) ( ) ( ) ) ( )
The extrinsic LLRs are then passed to detector for next iteration of processing.
After iteratively processing is terminated, we decode information bits by using a posteriori LLRs with hard decision. The a posteriori LLRs of information bit b is n
( ) ( ) ( ) ( )
In order to simplify the calculations, the following identity is used
( ) ( ) ( ) ( ) ( )
max* x y, ≜log ex+xy =max x y, +log 1+e− −x y 3-15 and we define the log-domain metrics as
( ) ( ( ) )
Finally by using the log-domain metrics, the e.q.（3-13）can be written as
The exponential terms in e.q.（3-13）have been convert to addition terms in e.q.
（3-16）, and the complexity is reduced largely. To reduce computation complexity further, the “ max* “ function can be approximated as
max*
( )
x y, ≈max( )
x y, 3-15( )
the decoder using the function max instead of max* is called max-log-MAP decoder. In the rests of this thesis, we consider only the max-log-MAP decoder for practical reason.
Chapter 4： ： ： ：Extrinsic Information Transfer Curve（ （ （ （EXIT） ） ） ）
Chart
In this chapter, we discuss a powerful tool, the EXIT chart, and use it to analyze the convergence behavior of iterative decoding. EXIT chart was proposed by S.
ten Brink in 2001. EXIT chart depicts the relation between mutual information of intrinsic log–likelihood ratios and coded bits and that of extrinsic log–likelihood ratios and coded bits for soft input soft output（SISO）decoder（or detector）.
Using mutual information transfer characteristics of SISO decoder, we can analyze the convergence behavior and design the system for better performance.
4.1.1： ： ：Transfer Characteristic ：
A SISO system is shown in Fig. 4.1. Where X is the information we transmit, that is, coded bits sequence in BICM-ID system, LA is intrinsic soft information and LE is extrinsic soft information about X.
First, we consider a simple case of BPSK modulation over AWGN channel. The received signal is
y = +x n （4.1）
where “x” is the transmitted BPSK signal and “n” is AWGN noise with zero mean and variance σn2. At receiver, the log-likelihood ratio is calculated as
Fig 7Fig. 4.1 Soft-input-soft-output（SISO）decoder（a）and detector（b）
Two observations in [9] are used：
（1） For large inter-leavers the a priori values LA remain fairly uncorrelated form the respective channel observation over much iteration.
（2） The probability density functions of the extrinsic values LE approach Gaussian-like distributions with increasing number of iterations.
Observations 1 and 2 suggest that the intrinsic soft information input to detector or decoder can be modeled by an independent Gaussian random variable as
(
2)
2, ~ 0, , 2
A A A A A A A
L =µ x+n n N σ µ =σ
therefore, the conditional PDF of LA is：
( )
The mutual information between extrinsic output E and transmission bit x can be
calculated as the intrinsic input as the independent Gaussian random variable in（4.6）. Finally, we measure the mutual information between extrinsic outputs and transmission bits.
4.1.2： ： ：Transfer Curve of Decoder ：
In BICM-ID system, the outer decoder receives the log-likelihood ratios form the detector. Hence, the transfer curve of decoder only influenced by the error correction code we use.
Fig. 4.3 shows several transfer curves of max-log decoders. In EXIT Chart, the mutual information between intrinsic input and coded bits is plot on “X” axis and the mutual information between extrinsic input and coded bits is plot on “Y” axis.
It is obvious for all the decoders that the a more reliable intrinsic inputs to decoder the more reliable extrinsic the decoder outputs. In Fig. 4.3, we can see that the threshold of the extrinsic is more apparent if the code with more powerful error correcting capability. Take turbo code with eight inner iterations for instance, If the mutual information of coded bits and intrinsic is over than 0.5, the mutual information of coded bits and extrinsic increases dramatically.
Fig 9Fig. 4.3 Transfer curves of several convolutional code with distinct constrain length and of 3GPP turbo code
4.1.3： ： ：Transfer Curve of Detector ：
As shown in Fig. 4.4, the detector receives both the log-likelihood ratios（l ）a from the decoder and signals（y1,…,yT）from communication channel. Random variables are written using upper case letters and their realizations by the corresponding lower case ones. Therefore, the mutual information can be written as
where Xk is k-th bit of transmission label s and LA , k is the intrinsic LLR of Xk. The last step is true because that the intrinsic LLRs are assumed as i.i.d.
Gaussian distribution as chapter 4.1.1. The mutual information between extrinsic LLR and transmission label is written as
( )
-1(
,)
-1(
[ ] 1)
impossible to compute the exact value, Monte Carlo simulation（histogram measurements）is utilized.
From equation（4.11）, we can observe that the channel condition and the mapping function are two key parameters that affect the EXIT chart of detector.
Fig. 4.5 shows some results in [10] for AWGN and Rayleigh fading channels.
With the observation that the transfer curves of chart of detector can be approximated to a straight lines, we define the “slope” of a transfer curve as the slope of the straight line form zero prior point（IA,detector = 0）to ideal prior point
（IA,detector = 1）. Mapping function set with various slope has been fund in [10] for many regular modulation scheme.
Fig. 4.6 shows the transfer curves of Gray labeling under distinct channel condition（SNR）. The transfer curves of identical labeling are approximately parallel, and the transfer curve is “higher” when SNR is larger.
Fig 11Fig. 4.5 Transfer curves of several labeling in AWGN and Rayleigh fading channel.
Fig 12Fig. 4.6 Transfer curves of Gray labeling under distinct channel condition（SNR）
4.1.4： ： ：EXIT Chart ：
The EXIT chart is composed of transfer curves of detector and decoder. By EXIT chart, the behavior of extrinsic information during iterative decoding can be traced. Fig. 4.6 shows an example of EXIT Chart.
In Fig. 4.7, the decoder curve and the detector curve separate far enough to introduce a tunnel, the final extrinsic information of the decoder can reach the location with high reliability. In other word, the waterfall region on BER curve of iterative decoding is related to the appearance of the tunnel between detector transfer curve and decoder transfer curve. It is believed that the higher reliability of decoder output the lower bit error rate. In Fig 4.8, transfer curves of two distinct labeling under AWGN channel with identical SNR and the convolutional
point with low decoder extrinsic reliability, and the “Red” one introduce a tunnel between the detector and decoder. It is obvious that the red one has a better performance than the blue one.
Therefore, the criterion of choosing the suitable labeling is that the labeling with the tunnel appearing at as lower SNR as possible.
Fig 13Fig. 4.7 Example of trajectory under iterative decoding
Fig 14Fig. 4.8 Transfer curves of distinct labeling under identical channel condition
（SNR）
Chapter 5： ： ： ：The Proposed Searching Algorithm
In the chapter, we propose an algorithm to search for labeling for Type-I HARQ in order to make throughput as high as possible. First, an analytical model for EXIT Chart is introduced. With the simple model, the detector transfer curve on EXIT could be approximated accurately by a close-form function. Next, we propose a search algorithm using the concept of binary search algorithm（BSA）
to establish a labeling set. Therefore, optimal labeling can be chosen according to code scheme and SNR.