Application Examples

Applying damped fixed-point iteration for decoding PCCCs and SCCCs, its decoding per- formance is simulated and analyzed in this part. Note that withσλ = 1, the recursion in

(4.75) is equivalent to that in (4.56). Based on the configurations of e, Λα(e) and Λβ(e)

in PCCC- and SCCC-coded systems, the recursion in (4.56) is formally identical to that of turbo decoding. Therefore, we can label damped fixed-point iteration as damped turbo decoding in the context of decoding PCCCs and SCCCs.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 10−4 10−3 10−2 10−1 100 Eb/N₀[dB] FER σλ= 0.4 σλ= 0.8 σλ= 1

iter.₂ iter.₄ iter.₂₀ iter.₁₅₀₀

(a) Damping Effect on the Decoding Performance

100 ₁₀1 ₁₀2 ₁₀3 10−6 10−5 10−4 10−3 10−2 10−1 100 Number of iterations l rconv erge (l ) σλ= 0.4 σλ= 0.8 σλ= 1

(b) Damping Effect on the Convergence Behavior atEb/N0= 2 dB

Figure 4.7: The performance of damped turbo decoding with different values of the damp- ing factorσλ;

rate1/3-PCCC code with the generator polynomial _{{1, 15/13}}o and information bit se-

PCCC

The performance of damped turbo decoding in a PCCC-coded system is illustrated in Fig. 4.7. Three different choices of the damping factor σλ, i.e., σλ = 0.4, σλ = 0.8 and

σλ = 1 are examined. Fig. 4.7(a) depicts the FERs achieved by damped turbo decoding

at different iterations. Fig. 4.7(b) reflects the convergence behavior of damped turbo de- coding atEb/N0 = 2 dB. Being more specific, the function rconverge(l) plotted in Fig. 4.7(b)

shows the rate of frames for which damped turbo decoding requires at leastl iterations to converge, wherel can take any integer value from 1 to 1.5_{· 10}3_.

In order to achieve a small FER, the decoding process should be able to converge in as many frames as possible. Certainly, this is not enough. The decoding process should also converge to the right fixed-point in as many frames as possible. Based on these two intu- itions, we can explain Fig. 4.7(a) by reference to Fig. 4.7(b). When the number of iterations is smaller than 20, e.g., l = 2 or l = 4, turbo decoding (i.e., σλ = 1) achieves the best

decoding performance, as it converges more frequently than the others. As the number of iterations increases to20, the damping factor σλ = 0.8 starts to replace σλ = 1 as the

optimal choice. As we can observe from Fig. 4.7(b), the heavy-tailed convergence func- tionrconverge(l) associated to σλ = 1 (i.e., the red line) indicates that turbo decoding either

requires very few iterations to converge or needs to experience a long transient behavior before converging to a fixed-point. According to [54], the long transient behavior is caused by the chaotic non-attracting invariant set in the vicinity of that fixed-point. Damping is one solution to reduce the lifetime of the transient behavior. By allowing1500 iterations, turbo decoding without damping eventually can converge almost as frequently as it with damping. However, we can observe FER loss of damped turbo decoding, particularly at the SNR range1.6_{∼ 2.4 dB. This is because damped turbo decoding is more likely to converge} to wrong fixed-points or chaotic limit cycles. Based on the stability analysis in above, one explanation for such observation is that the number of attracting fixed-points increases as σλ decreases. Convergence to wrong attractors cannot be reversed by simply increasing

the number of iterations. As such, after a sufficiently large number of iterations, it becomes the dominant reason for the FER gap betweenσλ < 1 and σλ = 1. However, such FER gap

does not exist at high SNRs, e.g.,Eb/N0 = 3.5 dB. This is because the basin of attraction

of the desired fixed-point grows with the SNR [54]. In other words, as the SNR increases, the chance of converging to a wrong fixed-point reduces, while the stability of an iterative decoding algorithm becomes the key to the decoding performance.

SCCC

Based on the above-mentioned two intuitions, we can analogously explain observations in Fig. 4.8. It is worth to note that the FER gain achieved by damped turbo decoding with20 iterations is more significant than that in the PCCC-coded system. In accordance with this observation, we also note that the tail of rconverge(l) with respect to σλ = 1 in the SCCC-

coded system is heavier than that in the PCCC-coded system. In the following, we focus on interpreting these observations.

First, we conjecture the reason is related to the configuration ofΛα(e) in SCCC-coded

1.5 2 2.5 3 3.5 4 4.5 5 10−4 10−3 10−2 10−1 100 Eb/N₀[dB] FER σλ= 0.4 σλ= 0.8 σλ= 1

iter.₄ iter.₂₀ iter.₁₅₀₀

(a) Damping Effect on the Decoding Performance

100 ₁₀1 ₁₀2 ₁₀3 10−5 10−4 10−3 10−2 10−1 100 Number of iterations l rconv erge (l ) σλ= 0.4 σλ= 0.8 σλ= 1

(b) Damping Effect on the Convergence Behavior atEb/N0= 3 dB

Figure 4.8: The performance of damped turbo decoding with different values of the damp- ing factorσλ;

rate1/4-SCCC code with the generator polynomial_{{15, 13}}oand information bit sequence

1.5 2 2.5 3 3.5 4 4.5 10−5 10−4 10−3 10−2 10−1 100 Eb/N₀[dB] FER relaxation no relaxation

iter.₄ iter.₂₀ iter.₁₅₀₀

Figure 4.9: FER vs. SNR for turbo decoding with and without using the relaxed function Λα(e);

rate1/4-SCCC code with the generator polynomial_{{15, 13}}oand information bit sequence

lengthNm = 250.

positive in noisy Gaussian channels. However, in SCCC-coded systems,Λα(e) represents

the code constraints of the outer CC, meaning thatΛα(e) yields one only for bit sequences

that are members of the codebook of the outer CC, i.e.,_G[1]; otherwiseΛα(e) = 0. Com-

paring these two configurations, the decoding decision made based onΛα(e) is more un-

equivocal in the SCCC-coded system. Treating turbo decoding as a process in which an agreed decision is gradually reached by both component decoders, a consensus is more difficult to be achieved when one component decoder insists its own decision too much, particularly at the initial iterations in which the decision is often unreliable.

Attempting to validate our conjecture, we secondly propose a relaxation method in Ap- pendix A.6. Briefly, at initial iterations, the functionΛα(e) yields a small positive value ξ′

rather than zero for bit sequences not in_G[1]_{. By further reducing the numberξ}′ _{over iter-}

ations, the functionΛα(e) eventually becomes identical to the indicator function IG[1](e).

Exemplarily, we initializeξ′_{with0.5 and reduce it over iterations by multiplying it with a}

factor0.99, i.e., ξ′ _{← 0.99ξ}′_{. By using such relaxedΛ}

α(e), Fig. 4.9 shows the performance

improvement, particularly when the number of iterations equals20.