In the above section, two illustrated examples are proposed to explain how to satisfy rate-compatible rule between two different puncturing period rows. However, the examples which we show are limited in the number of output coded bits n = 2. Now, a general form of hybrid puncturing which composed of two puncturing tables corresponding to n output coded bits.
In A1 and A1, the corresponding puncturing period of each rows are (p11, p12, · · · , p1n) and (p21, p22, · · · , p2n) respectively as shown in Fig. 3.3. Therefore, the rate-compatible problem of two puncturing table with different puncturing period is extended to a more general view.
The detail procedure of proof is omitted temporarily and will be presented in appendix later.
Here, we demonstrate the concept about how to achieve rate-compatible situation in general
form briefly. In fact, we do not need to be confused with increasing of the number of output coded bits. When rate-compatible situation is needed to be satisfied, the only thing that we have to do is take each pair of row p1j and row p2j out and discuss the relations between the elements in these two rows just like previous section. A result which we found in the procedure of proof is the elements in row p1j and row p2j can be absolutely divided into gcd(p1j, p2j) groups. In this way, we only need to consider the different rate-compatible situations in each corresponding pair of groups and form the possible puncturing patterns according to the combined result of each groups. With the same procedure, we notice that the hybrid skill between rows with different periods is not only for regular and irregular case but also regular and regular case and irregular and irregular case. This discovery allows us to establish a RCPC family with more choices of puncturing methods and the corresponding puncturing periods. Here, we present a practical example.
Example:
period=(4,4) A1 =
⎛
⎝ 0 1 1 1 1 1 1 1
⎞
⎠ rc = 4/7 ∗
A2 =
⎛
⎝ 0 1 0 1 1 1 1 1
⎞
⎠ rc = 4/6
A3 =
⎛
⎝ 0 1 0 0 1 1 1 1
⎞
⎠ rc = 4/5 ∗∗
mark period A possible code rates
∗ (6,6)
⎛
⎝ × 1 × 1 × 1
1 1 1 1 1 1
⎞
⎠ 12/18, 12/20, 12/22, 12/24
∗∗ (4,3) (0101)/(0100)/(0001) 12/14, 12/15, 12/18
(001)/(010)/(100)/(011)/(101)/(110)/(111)
This example shows us a period (4, 4) RCPC family and the mark ∗ and ∗∗ indicates that
A1 and A3 can be substituted by other puncturing tables with different puncturing periods.
The table below lists the candidates which can be used to replace A1 and A3 respectively and the puncturing periods and the possible code rates are also listed in the table. From the example, a simple and practical application of hybrid puncturing technique is presented and the choices of puncturing period and code rates which can be used to form a RCPC family increase certainly. More examples of the application of hybrid puncturing techniques will be presented in chapter 4.
In the process of our discussion, some useful results such as the above example are discovered. However, there is an important issue which is deserved to pay our attention.
This issue is that some unreasonable code rates might occur when we try to build a RCPC family with puncturing tables which have different periods by using hybrid puncturing techniques. Here, we propose two constructions which are already known that might result in unreasonable code rates during the process of our discussion.
Figure 3.4: Construction 1 which might cause unreasonable code rates
In Figure 3.4, there are three puncturing tables A1,A2 and A3 which are arranged from lower code rate to higher code rate and the arrow line connects the two different period rows which achieve rate-compatible rule. To illustrate this kind of condition, we let p1 = 6 and p2 = 4 and the proper puncturing tables and the possible code rates can be obtained
under this assumption.
no. puncturing tables
A1
⎛
⎝ × 1 × 1 × 1
1 0 1 1 /1 1 1 0
⎞
⎠ or
⎛
⎝ 1 × 1 × 1 ×
0 1 1 1 /1 1 0 1
⎞
⎠
possible code rates: 12/15, 12/17,12/19,12/21 A2
⎛
⎝ × 1 × 1 × 1
× 0 × 0 × 0
⎞
⎠ or
⎛
⎝ 1 × 1 × 1 ×
0 × 0 × 0 ×
⎞
⎠
possible code rates: 12/14,12/16,12/18 A3
⎛
⎝ 0 0 0 1 /0 1 0 0
× 0 × 0 × 0
⎞
⎠ or
⎛
⎝ 0 0 1 0 /1 0 0 0
0 × 0 × 0 ×
⎞
⎠
possible code rates: 12/9,12/7,12/5
We notice that the possible code rates of table A3 are all bigger than 1 and these are all unreasonable rates. As we know, these unreasonable code rates cannot be used in the practical application due to the bad error protection ability. Furthermore, another construction which may also cause a unreasonable rate is shown in Figure 3.5.
Figure 3.5: Construction 2 which might cause unreasonable code rates
In the construction which we present in Figure 3.5, it is an example that we form a RCPC family by taking regular puncturing tables with different periods together. For convenience, we let p1 = 6 and p2 = 4, too.
possible code rates: 12/14,12/16,12/18 A2
possible code rates: 12/12 A3
possible code rates: 12/12,12/10,12/8
Obviously, not only table A3 but also A2 would cause unreasonable code rates in this construction. The above two constructions remind us that the unreasonable code rates is an important issue while using hybrid puncturing techniques because some undesired rates may be produced under the specific constructions. Due to the amounts of the constructions which can use hybrid puncturing techniques are huge, we cannot list the all possibilities of
the constructions which may result in unreasonable rates. Therefore, the only thing we have to do to prevent unreasonable code rates is to discuss the possible rates of the puncturing tables in the construction first. In this way, we can own the benefit of hybrid puncturing techniques without tolerating the risk of generating the undesired code rates.
Chapter 4
Simulation Results
In this chapter, we will present the simulation results which correspond to what we mentioned before. First of all, some examples of better free distance and BER performance under the irregular puncturing method are presented. Second, new searching results of irregular RCPC family will be listed. Finally, some results which are based on hybrid puncturing techniques are presented.