The goal of the coding parameter optimization problem of the proposed scheme is to minimize the SNR threshold under a given PUPE (in
Section 2). However, due to the complicated intra/inter-slot encoding structure, the relationship between the coding parameters and the performance indicator is not explicit. Therefore, this section addresses the problem by error event decomposition [
16]. By breaking the system-level PUPE down to module-level error rates, the error contribution of each module can be analyzed and correlated with their parameters and eventually form a system-level parameter optimization problem.
4.1. Error Rate Analysis by Decomposition
According to the decoder structure described in
Section 3.2, the PUPE in (
1) can be decomposed into four parts, as depicted in
Figure 5.
Although, as described in Algorithm 1, the decoding structure consists of the compound intra/inter-slot iteration, it can nonetheless be regarded as a three-stage process when analyzing errors due to its successive style. The first stage is the CS pilot recovery, where the pilot resource collision
occurs at the transmitter side and the support recovery error
at the receiver side. The pilot plays three significant roles: the first part of the user payload, the user-specific IDMA interleaver, and the pointer used to reconstruct the packet-level factor graph. Thus, the pilot error will cause not only packet loss of its user but also the chain effect spreading to the correlated packets of other users. The IDMA decoding error
and the remaining error after the intra-slot SIC process
are both conditioned on the successfully recovered pilots. Then the modular errors are expressed as the functions of their corresponding coding parameters in (
12).
The CS pilot parameters
and
are restricted by the collision and detection conditions.
is the resource collision avoiding condition in [
16]. The length of the separated pilot info part from user payload
should be large enough to provide non-collision patterns. As for
, the dimension of the sensing matrix is bounded by the Restricted Isometry Property (RIP) [
31] condition. In other words, when
is fixed, large row dimension
can reduce the
. Thus, we can conclude that
and
.
The IDMA block error rate is the function . According to the classical analysis of IDMA, is based on the single-user performance of the inner rate- FEC code and degrades with the gradually severe interference caused by the increase of the number of superimposed users . The SNR cost of at a required level can be extracted on the performance curve of a given threshold by simulation.
4.2. Inter-Slot Degree Distribution Analysis
As the SNR cost of the IDMA system increases with , the SIC decoding on the compound factor graph of the inter-slot code can further reduce the threshold to reach the same level of PUPE, resulting in the reduction of SNR threshold. Thus, degree distribution optimization aims to minimize .
First, we start with an idealized asymptotic investigation. Previously the T-Fold IRSA was modeled as a factor graph, on which the user node degree distribution
determines the slot node degree distribution
:
where
denotes the binomial distribution. The probability of high-degree slot nodes increases with the average packet repetition rate
and decreases with more slots
V, representing intensified superposition.
Figure 6a gives an example of this rule.
Moreover, the convergence behavior of the SIC on the factor graph can be characterized by density evolution (DE), which is similar to the LDPC message passing procedure under the erasure channel [
32]. At the
t-th iteration, the erasure probability of the slot nodes is
and
for the user nodes:
where
at the beginning without a priori knowledge of user nodes, and
can be derived from
as in (
13). Under appropriate
, the erasure probability of slot nodes
gradually descends with iteration. As
is a positive-coefficient polynomial function,
is monotonically increasing, indicating that
also converges to 0 with descending
. Meanwhile, the higher the threshold
is, the higher the decodable probability of slot nodes
is. Notice that the two-probability expression corresponds to the two message-passing procedures in Algorithm 1, respectively. Therefore, the SIC iteration can make as many overload user nodes decodable as possible. An example of the SIC procedure simulated by density evolution is depicted in
Figure 6b. When
V and
are fixed, the cost of carrying more users
is the increase of threshold
to guarantee convergence under limited iterations, i.e., the rise of IDMA’s SNR requirement.
The tool to determine the convergence condition of density evolution iteration is its EXIT (EXtrinsic Information Transfer) chart [
33]. Under given
,
V and
, plot the erasure probability curves
and
in (
14) on one chart, and find a trajectory between those two curves starting from
. If the trajectory reaches
, this
enables the SIC iteration to converge, conversely not.
Figure 6c,d shows a boundary condition case where the iteration tunnel is about to close for
. Since
,
is always concave. Moreover, this effect becomes stronger when the weights of higher-order coefficients increase. However, it is not easy to explicitly express the boundary condition. We adopt the numerical method, differential evolution [
34], to solve the implicit equation and search the range of suitable
:
Then we move further to practical consideration. An important preassumption for density evolution analysis is that ensures the isotropy of distribution, while in practice, they are limited. It is challenging for the user-independent random schedulers to guarantee when is relatively small, especially for the repetition times with low probability (high-order degrees). On the other hand, the finite length effect causes short cycles and trapping loops in the randomly formed factor graph. In some worse cases, the SIC iteration cannot start or converge. Although the effect of can be ignored when considering boundary conditions, the remaining errors on the underload slot nodes may cause error propagation. Nonetheless, gives a basic scope, which just needs to be narrowed.
We use Monte-Carlo simulation to practically examine the effectiveness of candidates in
. Under finite
,
V, and
, the post-iteration packet loss rate of user nodes obtained by simulation is the actual PUPE. Besides, the cost of the packet diversity gain is the additional energy spent on repeated packets. The SNR threshold after inter-slot encoding is:
where
is the SNR when
reaches an effective level under threshold
. After that, the trade-off of intra-slot encoding should also be considered.
4.3. Joint Parameter Optimization Algorithm
Integrating the analysis on the above modules, the complete parameter optimization procedure is proposed as Algorithm 2.
Algorithm 2 Joint optimization of coding parameters |
Require: the legnth of user payload B, number of actove users , frame length , and target PUPE . Ensure: the length of CS pilot info part , length of CS pilot , LDPC code rate , intra-slot repetition rate , and inter-slot packet spreading distribution .
- 1:
Initialization: Determine and by the CS decoding conditions, randomly chose rate configuration , and calculate V; - 2:
while
do - 3:
for from to do - 4:
Obtain and by IDMA simulation; - 5:
Search the convergable range using density evolution by ( 15); - 6:
for from to do - 7:
Perform Monte-Carlo simulation to validate under the residual error ; - 8:
Output the post-SIC-iteration error rate ; - 9:
if then - 10:
Reserve and break; - 11:
end if - 12:
end for - 13:
end for - 14:
if SNR threshold ( 16) increases then - 15:
Choose the pair with higher total rate R, adjust V accordingly; - 16:
else - 17:
Lower the total rate R, adjust V accordingly. - 18:
end if - 19:
end while - 20:
Calculate the SNR threshold by ( 16) and output the optimal coding parameters.
|
Overall, we use a heuristic bootstrap method to jointly optimize the coding parameters analyzed above. The CS pilot configurations control
and
, basically determined by
. To tackle the contradiction between intra-slot coding gain and inter-slot diversity gain, the IDMA rate
R increases with iterations, while the inter-slot diversity decreases in each iteration. The initial rate
R is randomly chosen, and then
V can be determined.
at
is extracted on simulation curves. The convergence condition is ensured by (
15). Then the effectiveness of
with increasing energy cost is checked by simulation until it satisfies the post-SIC-iteration PUPE requirement. If the SNR threshold raises compared with the last iteration after optimization,
R should be increased to enhance the inter-slot code. If not, lower
R to reduce the multi-user interference.