1. Introduction
The demand for high-quality broadcasting services has increased rapidly, resulting in significant progress in the digital broadcasting and communication industries. To cater to the evolving needs, the Advanced Television Systems Committee (ATSC) has developed ATSC 3.0 as the standard for next-generation broadcasting systems [
1,
2]. The physical layer of the ATSC 3.0 system has been designed to enhance flexibility, robustness, and spectral efficiency compared to previous standards, enabling the simultaneous support of a diverse range of services. These include ultra-high definition (UHD) and mobile HD services [
3,
4,
5,
6,
7].
The physical layer of ATSC 3.0 includes various technologies, such as channel coding and symbol mapping through bit-interleaved coded modulation (BICM) blocks, low-density parity check, bit-interleaver, and a range of constellations. These constellations span from quadrature phase-shift keying (QPSK) to 4096 quadrature amplitude modulation (QAM), and they also include non-uniform constellations [
8,
9]. The system generates waveforms by performing various interleaving operations based on orthogonal frequency division multiplexing, thus facilitating high-capacity broadcasting services [
1,
2,
3,
4,
5,
6].
The layered division multiplexing (LDM) system adopted within ATSC 3.0 is a non-orthogonal multiple access (NOMA) scheme which allows the concurrent transmission of various signals through distinct layers [
10,
11,
12,
13,
14,
15,
16,
17]. The LDM system efficiently utilizes limited time and frequency resources to provide multiple services. Compared to binary time division multiplexing (TDM) and frequency division multiplexing systems, the two-layer LDM system in ATSC 3.0 demonstrates enhanced transmission efficiency [
10,
11,
12,
13,
14,
15,
16,
17].
Figure 1 illustrates a block diagram of the LDM and TDM systems utilizing two physical layer pipes (PLPs) [
18]. In the LDM system, channel capacity is determined by adjusting the power ratio for each PLP using the injection level parameter
α. The parameter α scales the power level of the lower layer relative to the upper layer. Additionally, the power normalizing factor
β is multiplied by the combined LDM signal to maintain a constant total transmit power. In contrast, the channel capacity of each PLP within a TDM system is determined by the time ratio [
5].
The LDM system has different objectives as compared to traditional NOMA systems. Traditional NOMA systems aim to maximize spectrum efficiency by allowing multiple users to share the same time and frequency resources. Therefore, the goal is to provide equal channel capacity to all users, considering high fairness among them [
19,
20,
21,
22]. However, the LDM system in ATSC 3.0 aims to provide various services, such as emergency messaging, mobile communication, and fixed rooftop broadcasting, within limited time and frequency resources [
3,
4,
5,
6,
7]. The objective is to optimize channel capacity allocation among these services to offer a diverse range of services effectively. In LDM systems, the channel capacity provided for each service is different, so it may be challenging to obtain the fairness and maximum sum rate used in NOMA performance analysis.
In this article, we proposed a method for analyzing the efficiency of the LDM system by normalizing the allocated channel capacities for each service. Through this normalization process, we can evaluate both the normalized fairness and the sum of normalized channel capacities. This approach allows for an equitable comparison of efficiency and resource allocation among various services within the LDM system. In NOMA, maximizing the total channel capacity while upholding fairness is essential. Therefore, we propose a new method for resource allocation in the LDM system that determines where the sum of normalized channel capacities is maximized while preserving normalized fairness. This is achieved by using the Lagrange multiplier method to determine the maximum point. Using this approach, we analyze and compare the channel capacity efficiency and reception performance of implementable LDM, TDM, layered-time division multiplexing (LTDM), and time-layered division multiplexing (TLDM) systems in a 3-PLP configuration.
1.1. Research Background
In previous LDM research, many studies have been conducted to prove the superiority of LDM systems over TDM systems by setting specific injection levels and time ratios [
5,
23].
Figure 2 compares the LDM system with an injection level
α and the TDM system with a time ratio
τ [
11,
24]. The graph shows that the LDM system (represented by the red-line area) outperforms the TDM system (indicated by the red dot) regarding channel capacity for both PLP_1 and PLP_2 at a specific
τ. However, determining the most efficient
α is not straightforward as the channel capacities of the two PLPs are different. No research has been conducted to determine which LDM system is superior under identical conditions (time duration, bandwidth, etc.), that is, which α value is optimal [
5,
23]. Therefore, we proposed a new analytical method to determine the optimal value of α and configure the most efficient LDM system. Shannon channel capacities
and
can be defined based on the target received signal-to-noise ratio (SNR) of each PLP [
11]. By employing the given SNR, the channel capacity of each PLP in the LDM system can be expressed as a function of
α. To determine the optimal value of
α, we define the normalized channel capacities of the two PLPs. Normalizing the channel capacity of each PLP allows for a fair comparison of system efficiencies while considering various parameters. The sum of the normalized channel capacities is used to determine the most advantageous combination. Therefore, the most efficient value of
α for the LDM system is identified by maximizing the sum of normalized channel capacities.
The channel capacity of multi-PLP transmission schemes has been studied in the literature [
5,
18,
25,
26]. Efficient channel capacity allocation is necessary for providing various services, such as breaking news in disaster situations, UHD broadcasting services, and mobile HD services [
3,
4,
5,
6,
7]. Like the LDM system for 2-PLP, the power ratio of each layer in the 3-PLP LDM system is determined by two injection levels. The TDM system transmits using a time ratio divided into three from the total duration [
5,
18]. The LTDM and TLDM systems combine TDM and LDM principles in a 3-PLP transmission scheme [
5,
26]. In literature, investigations have shown that performance analyses for particular injection levels and time ratios demonstrate their superiority over TDM, similar to the 2-PLP system [
5,
26].
Similar to the NOMA system, the LDM system shares the same bandwidth and time resources as all PLPs. In a recent study of the NOMA system, resource allocation has been designed to ensure fairness among multiple users while providing equal channel capacity. This design approach allows NOMA to determine where the sum of the channel capacities of all users is maximized while maintaining fairness [
19,
20,
21,
22]. However, the primary objective of the LDM system is to provide various services [
3,
4,
5,
6,
7]. Consequently, it offers a different channel capacity to each PLP. Therefore, when NOMA’s fairness analysis is applied to the LDM system, the sum of channel capacities cannot be maximized. Likewise, maximizing the sum of channel capacities results in an unfair system. To address this, we proposed a new fairness index in LDM to make it comparable. We also introduce a method to analyze whether the sum of normalized channel capacities is maximized while maintaining fairness among PLPs.
Regarding wireless transmission system, the channel capacity is affected by two key factors: bandwidth and signal power, which are considered specifications of a transmission system [
27]. These specifications are crucial when comparing different multiplexing methods. We assume the bandwidth and total signal power are equal to ensure a fair comparison. Specifically, in the LDM system, the sum of power of the PLPs is equal to the total power of the TDM system [
5,
18,
26]. By imposing this constraint, the Lagrange multiplier method [
28] can be employed to determine the most efficient combination of
α, representing the LDM system’s power allocation. The goal is to find the value of
α that maximizes the sum of normalized channel capacities in the 3-PLP system. The channel capacity efficiency and reception performance of the TDM, LTDM, and TLDM systems can be analyzed and compared by utilizing the Lagrange multiplier method and comparing normalized channel capacities.
1.2. Contribution
The key contributions of this study to the analysis of the LDM system are as follows:
Overall, this study provides valuable insights into the effective resource allocation for the LDM system, enabling a fair comparison of its efficiency and performance with other transmission techniques. The proposed method advances next-generation broadcasting systems by optimizing resource allocation strategies in multi-service environments.
1.3. Organization
The remainder of this article is organized as follows:
Section 2 explains the structures of the LDM, TDM, LTDM, and TLDM used in the 3-PLP system.
Section 3 discusses the issues related to fairness and performance measures in the LDM system.
Section 4 proposes a method to determine the optimal normalized channel capacity using the Lagrange multiplier.
Section 5 analyzes the efficiency of various transmission combinations of the 3-PLP system and compares the reception performance of each PLP using the parameters obtained through the Lagrange multiplier. Finally,
Section 6 concludes the article.
3. Problem Description
LDM shares time, space, and frequency resources to transmit various services, and its transmission method is similar to NOMA. However, there are differences between the two transmission methods in terms of purpose and application. While NOMA focuses on maximizing spectral efficiency with the goal of fair channel capacity allocation to all users, the LDM system aims to efficiently transmit multiple services simultaneously [
30,
31,
32,
33]. In wireless communication systems, Jain’s fairness index (JFI) is commonly used to evaluate fairness [
19]. The JFI ranges from 0 to 1, and when it is close to 1, it indicates a fair allocation of resources. The JFI index
is:
where
and
denote the number of users and
i-th user data rate, respectively. In a NOMA system, research is conducted to allocate resources when the JFI is close to 1, ensuring fair distribution of channel capacity among users [
20]. However, in the case of LDM, the fairness index tends to decrease as the number of services varies because each service has a different target channel capacity. Since the objective of the LDM system is to transmit multiple services with varying requirements efficiently, the emphasis is on optimizing resource allocation to meet the specific needs of each service rather than achieving perfect fairness among them.
NOMA generally aims to provide equal data rates to all users in a fair scenario, and the maximum sum rate is used as a performance measure [
19,
20,
21,
22]. However, in the LDM system, each service may have different data rate requirements, and simply using the maximum sum rate as a performance measure may not lead to an efficient allocation of resources among the services. In the given example of a 3-PLP LDM system, the system considers three different types of transmission with their respective minimum data rate requirements:
Radio transmission ( ≥ 0.5 Mbps, QPSK, code rate: 3/15)
Mobile HD transmission ( ≥ 3 Mbps, QPSK, code rate: 6/15)
UHD broadcast ( ≥ 20 Mbps, 64-QAM, code rate: 9/15)
Suppose the maximum sum rate is used as the performance measure in the 3-PLP LDM system. In that case, the resource allocation will prioritize maximizing the data rate of the UHD broadcast service (). In the case of an LDM system, different constellations and code rates are used for each layer based on the specific service requirements. Therefore, if the goal is to achieve the maximum sum rate, the resource allocation strategy is to allocate as many resources as possible to , having the highest transmission rate per unit sample. Allocating most resources to , which has the highest transmission rate, can lead to unfairness between services, particularly for and , which only meet their minimum performance requirements. In the case of an LDM system where each service has different data rate requirements, simply maximizing the sum rate may not align with achieving fairness among the services.
To address the fairness issue, we propose the utilization of normalized channel capacities as a solution. By normalizing the channel capacities provided to each service, the authors aim to establish a fair comparison measure. The normalized channel capacities can be used to evaluate the fairness among services, similar to the approach used in NOMA. Furthermore, the authors suggest maximizing the sum of normalized channel capacities as a performance measure for the LDM system. By maximizing this value, the system aims to achieve an optimal allocation of resources that balances efficiency and fairness. This approach resembles the objective of maximizing the sum rate in NOMA. It leads to maximizing the overall system performance while ensuring fairness among users.
The maximum channel capacity
of PLP_
i is the Shannon channel capacity [
29] when PLP_
i is used alone and can be expressed as follows:
Then, the normalized channel capacity of each system, denoted as
, is defined as follows:
where
denotes the channel capacity of PLP_
i in
M system, which is one of the four multiplexing systems (TDM, LDM, LTDM, and TLDM). The normalized channel capacity measures the efficiency of each PLP in the different multiplexing systems. We present a method that achieves the highest overall efficiency by maximizing the sum of normalized channel capacities for all PLPs. This approach helps to determine the most efficient multiplexing system among the four options (TDM, LDM, LTDM, and TLDM) based on the considered PLPs.
4. Proposed Resource Allocation Method Using the Lagrange Multiplier
In this section, we aim to determine the optimal channel capacity in an LDM system using the Lagrange multiplier. The Lagrange multiplier method is commonly employed to find the optimal solution under constrained conditions [
28]. In the example shown in
Figure 7, the maximum sum of the channel capacities is 10 Mbps when only PLP_2 is utilized without the PLP_1 service. However, if both services need to be provided simultaneously, some allocation should be made to PLP_1. In the case of LDM, where each PLP may have different channel capacity requirements, it becomes crucial to identify an optimal solution that efficiently utilizes the system’s resources. To address this, we propose the use of normalized channel capacity.
Figure 8 shows the normalized channel capacity of each PLP, as shown in
Figure 7. In the 2-PLP LDM system shown in
Figure 8, we can identify the point where the sum of the normalized channel capacities is maximized. Geometrically obtaining the maximum sum involves determining the position where
is maximized in the linear function of
. For the 2-PLP LDM system, the point where
is maximized corresponds to
, which is a tangent point to the magenta line with a slope of −1. Point
represents the normalized position of point
T in
Figure 7, where point
represents the maximum sum of the normalized channel capacities in the 2-PLP LDM system. The point where the perpendicular to the magenta line passes through point
and intersects the channel capacity of (the red straight line) is called
.
The point
can be analytically determined using the Lagrange multiplier method. Extending to the 3-PLP LDM system, point
in
Figure 9 becomes the optimal solution, where the sum of the normalized channel capacities of the three PLPs reaches its maximum. The following subsections describe how the point
can be obtained using the Lagrange multiplier method for the three PLPs.
4.1. Lagrange Method in the 3-PLP LDM System
The LDM system using the three PLPs is described as follows:
Initial channel capacities ;
(the total powers of TDM and LDM are the same);
(when there is power in all layers);
PLP_i has the same additive noise in both TDM and LDM;
Same bandwidth for 3-PLP TDM and 3-PLP LDM systems;
(SIC [
20])
If the above conditions are satisfied, the channel capacities of the LDM and TDM systems can be configured using three PLPs, as shown in
Figure 9. In
Figure 9, the channel capacity of the TDM system is represented by the green plane, whereas the black spherical area represents the LDM system. We used the value of
W = 1 and
P = 1 to find the optimal point of
T. By substituting
and
Equations (2)–(4) can be rearranged as follows:
By using the above equations, the Lagrangian function can be expressed as follows: [
28]
where
denotes the Lagrange multiplier. The Lagrange multiplier method can obtain a simultaneous equation by assuming that
is 0 [
28]. The resulting simultaneous equations are as follows:
Using Equations (18)–(21), we can derive
,
and
using the following equations:
By substituting the values of
,
, and
into Equations (2)–(4), the channel capacity of an LDM system with maximum efficiency can be obtained.
To compare with the LDM system, we explain the method to obtain point
S in the TDM system, which is closest to point
T in
Figure 9. To determine
S, we derived the equation of a straight line passing through point
T and parallel to the normal vector of the TDM channel capacity plane (green plane). The equations of straight-line and the green plane are expressed as follows:
respectively. The solution to the two equations is located at point
S. The coordinate of point
S is represented as [
], where values of
of the TDM system can be derived as a result of the simultaneous Equations (25) and (26). We obtain the following equations:
The method for obtaining the optimal parameters using the Lagrange multiplier method is summarized as follows:
Select the initial values of , , and .
Calculate the noise corresponding to , , and .
Derive , and using the Lagrangian function.
Calculate the channel capacity of the LDM system using , and .
Find the point S by deriving the equation of the straight line and the equation of the plane and solving these simultaneous equations.
Calculate , , and from the coordinate point S.
4.2. Lagrange Method in the 3-PLP LTDM and TLDM Systems
Figure 10 and
Figure 11 illustrate the channel capacities of an LTDM system (blue area) and a TLDM system (red area). The point at which both systems have their optimal solutions when both
and
are equal to zero, which means they are used as a 2-PLP LDM system. To compare the scenarios using three PLPs, we added a condition that
and
are equal to
of the TDM system obtained from Equation (26). The initial conditions for obtaining the optimal
and
of the LTDM and the TLDM systems are described as follows:
Initial channel capacities .
(the total powers of TDM, LDM, and LTDM are the same).
(when there is power in all layers);
PLP_i has the same additive noise in TDM, LTDM, and TLDM;
Same bandwidth of 3-PLP TDM, 3-PLP LTDM, and 3-PLP TLDM systems;
;
(SIC [
20]);
To find
in the TLDM system, assuming
W = 1 and
P = 1, and substituting
and
Equations (9) and (10) are rearranged as follows:
By using the above equations, the Lagrangian function is given by [
28]:
The system of equations using the Lagrange multiplier method is computed as follows [
28]:
Using Equations (31)–(33),
and
can be derived as follows:
By substituting the calculated
and
into Equations (8)–(10), the channel capacity of the TLDM system with the maximum efficiency can be obtained when
.
For the LTDM system, if the Lagrange multiplier method is applied in the same way using Equation (5) (PLP_1) and Equation (7) (PLP_3), the channel capacity of the LTDM system with the maximum efficiency can be obtained when
. The LTDM parameters calculated by applying the Lagrange multiplier method using Equations (5) and (7) are as follows:
5. Simulation Result
In this section, we present the channel capacity analysis results of the four multiplexing systems using the parameters calculated by the Lagrange multiplier method when operated with three PLPs. The efficiency of each system is compared in terms of the normalized channel capacity
. Additionally, we provide the required SNR values to achieve the target bit error rate (BER). The modulation and code rate (ModCod) of the transmitted signal follow the specifications outlined in the physical layer of ATSC 3.0 [
2]. It is assumed that perfect channel state information is available without synchronization errors. The received PLP_
i signal, considering channel gain and noise in the frequency domain can be represented as
where
denotes the received signal of PLP_
i in
M system;
,
, and
represent the transmitted signal in
M system; channel gain and noise of PLP_
i; and
k is a sub-carrier index. Since we assume perfect channel state information, the estimated transmission signal
can be given by
The transmitter signal of PLP_
i in the
M system, denoted as
, needs to be obtained. This signal can be obtained through DEMUX in the case of TDM system or by using the SIC technique in the case of LDM system [
5,
18].
To evaluate the fairness of the system, we can calculate the normalized fairness index (NFI) using the normalized channel capacity. The NFI measures fairness in resource allocation among the different services in an LDM system. The formula to calculate the NFI in 3-PLPs is as follows:
We derived the parameters through the Lagrange multiplier method using the values
presented in
Table 1 (provided in Ref. [
5]) to determine the sum of normalized channel capacities. The parameters provided in [
5] and the calculated using the proposed method are listed in
Table 2.
and
can be computed using the parameters given in
Table 2, and the sum of the normalized channel capacities, denoted as
is defined as follows:
Table 3 and
Table 4 show the calculated results for
,
,
, and
, which are used to evaluate the performance of different multiplexing systems. The results demonstrate that the sum of the normalized channel capacities obtained using the Lagrange multiplier method is higher than the comparison parameters provided in reference [
5]. For the TDM system, since the channel capacity
changes linearly according to
, the normalized channel capacity
is equal to
. In other words,
for any
. Therefore, the result is the same if
is calculated using the Lagrange multiplier method or the comparison parameters provided in [
5]. However, the channel capacities do not change linearly with the injection levels for other multiplexing systems, unlike in the TDM system. Therefore, the sum of the normalized channel capacities differed depending on the injection level. For an LDM system, the sum of the normalized channel capacities obtained using the Lagrange multiplier method is 0.08 higher than that mentioned in [
5]. This means that the channel efficiency has improved by 8% when considering the TDM system’s channel efficiency as 100%. Furthermore, it has been confirmed that the NFI is also higher at 0.9256. Similarly, the Lagrange multiplier method can achieve 15% and 8% better performances in the LTDM and TLDM systems, respectively. However, it should be noted that the normalized fairness indices are lower than those presented in [
5].
Here, we compared the four multiplexing systems; we found that the LDM system is the most efficient technique. In terms of the sum of normalized channel capacities derived using the Lagrange multiplier method, the LDM system is 49% more efficient than the TDM system. The LTDM and TLDM systems were 38% and 24% more efficient than the TDM system, respectively. Furthermore, when resource allocation is carried out using the Lagrange multiplier method, it can be observed that the NFI of the LDM system is 0.9256, achieving the highest NFI compared to other transmission methods. Through this analysis, it can be confirmed that the LDM system utilizing the Lagrange multiplier method achieves the highest sum of normalized channel capacities even under fair conditions.
Table 5 represents the ModCod supported by the physical layer of ATSC 3.0, along with the parameters
obtained using the Lagrange multiplier method.
is calculated by substituting the required SNR at BER = 10
−4 obtained from the simulations into Equation (11). Using the parameters listed in
Table 5, we calculated the
,
,
, and
values as listed in
Table 6 and
Table 7. From the simulation results in
Table 7, we can see the sum of the normalized channel capacities is 1.4342. It shows that the LDM system is more efficient and outperformed other transmission systems.
Unfortunately, it is difficult to find a ModCod supported by ATSC 3.0 that matches channel capacity as
obtained using the Lagrange multiplier method. Therefore, we set the ModCod to perform similarly to
and analyzed the required SNR accordingly.
Figure 12 shows the BER performance of PLP_3 under an additive white Gaussian noise (AWGN) channel using the ModCod, with the closest value to
listed in
Table 6. For example, in the LDM system, the ModCod used for PLP_3 is 64-QAM with a code rate of 9/15, and it is evident that a required SNR of approximately 21.1 dB is needed, which is close to the
= 20.7 dB listed in
Table 5. In this manner, we identified ModCods closest to
for all cases, as listed in
Table 8.
Table 9 lists the required SNR at BER = 10
−4 obtained from simulations conducted in both AWGN and fading channels, using the ModCods listed in
Table 8. It also includes the corresponding data rate. The data rate
is calculated using the following equation:
where
represents the guard interval time,
is the time ratio of
i-th PLP of each system, and
is the number of bits per symbol used for modulation. As an example of
, when using the ModCod for PLP_3 in the LDM system, 64-QAM is employed, which uses 6 bits. Therefore,
= 6. Both PLP_1 and PLP_2 used TU-6 channel [
5,
34], which is a Rayleigh fading channel in a mobile environment, considering mobile reception at 100 km/h. PLP_3 used RL 20 channel [
5,
35,
36], which is a fixed Rayleigh fading channel, considering the fixed UHD service.
Table 9 shows a comparative analysis of the reception performance and data rates for each PLP category. For PLP_1, both the LTDM and LDM systems had the same data rate. However, the LTDM system exhibited approximately 0.3 dB better reception performance under the AWGN channel and about 0.7 dB better SNR performance under the fading channel. Moving on to PLP_2, the TLDM system achieved the highest data rate, approximately twice as much as that of the LDM system. However, the reception performance deteriorated the most among the four transmission methods to 7.2 dB in the fading channel. As for PLP_3, although the data rates of the LDM and TDM systems are similar, the LDM system outperformed by 1 dB in the fading channel. By using these simulations, we can analyze the performance of received signals by allocating resources in each transmission system and understand the focus of operations for different services.
6. Conclusions
We proposed a novel approach for efficient resource allocation in the LDM system. To achieve efficient resource allocation while maintaining a high level of NFI, a new method is introduced that normalizes the channel capacities allocated to different services using the Lagrange multiplier.
The simulation results demonstrated the superiority of the proposed approach over other transmission techniques. The LDM system exhibited the highest NFI of over 0.9 and achieved the highest sum of normalized channel capacities at over 1.4, indicating its efficiency in serving multiple services simultaneously. By comparing the LDM system with other multiplexing systems, it is evident that the LDM system outperformed in fairness and efficiency. Furthermore, our study explored the practical implementation of ModCods and analyzed their performance under AWGN and fading channels. Here, we can analyze the performance of received signals by allocating resources in each transmission system and gain an understanding of the operational focus on different services. In our simulation experiments, we used the ATSC 3.0 system with three PLPs as an example, this method can be applied to future transmission systems that offer various services and multiple PLPs. This approach is expected to be valuable for performance analysis in terms of system efficiency.
Overall, the research provides valuable insights into resource allocation strategies for the LDM system, paving the way for fair comparisons of efficiency and performance within multi-service environments. The proposed approach contributes to the advancement of next-generation broadcasting systems by enhancing resource allocation efficiency and overall system fairness.