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Spectral efficient transmission techniques are necessary and promising for future broadband wireless communications, where the quality of service (QoS) and/or max-min fair (MMF) of intended users are often considered simultaneously. In this paper, both the QoS problem and the MMF problem are investigated together for transmit beamforming in broadband multigroup multicast channels with frequency-selective fading characters. We first present a basic algorithm by directly using the results in frequency-flat multigroup multicast systems (Karipidis et al., 2008), namely, the approximation algorithms in this paper, for both problems, respectively. Due to high computational consumption nature of the approximation algorithms, two reduced-complexity algorithms for each of the two problems are proposed separately by introducing the time-frequency correlations. In addition, parameters in the new time-frequency formulations, such as the number of optimization matrix variables and the taps of the beamformer with finite impulse response (FIR) structure, can be used to make a reasonable tradeoff between computational burden and system performance. Insights into the relationship between the two problems and some analytical results of the computational complexity of the proposed algorithms are also studied carefully. Numerical simulations indicate the efficiency of the proposed algorithms.

Targeting for supporting high throughput and link reliability, multiple-antenna transmission techniques have prevailed in the development of terrestrial wireless communication systems, such as Long Term Evolution Advanced (LTE-A) [

To provide performance assurance to each of the intended receivers in multicast systems, the quality of service (QoS) problem [

List of related works.

Problem description | Application scenarios | Solving approach | Source |
---|---|---|---|

QoS | Broadcast | SDR | [ |

MMF | Broadcast | Iterative algorithm | [ |

QoS | Multigroup multicast | SDR + Gaussian randomization | [ |

MMF | Multigroup multicast | SDR + Gaussian randomization + bisection | [ |

QoS | Multigroup multicast | Iterative algorithm | [ |

MMF | Multigroup multicast | Iterative algorithm | [ |

QoS | Broadcast with PACs | Iterative algorithm + duality | [ |

MMF | Broadcast with PACs | Iterative algorithm + duality | [ |

MMF | Multigroup multicast with PACs | SDR + Gaussian randomization + bisection | [ |

As aforementioned, the design of transmit beamformer has been primarily studied over multiple-antenna multicasting and frequency-flat fading channels and, to the authors’ best knowledge, few works have been dedicated to the case of frequency-selective fading multicasting scenario. Motivated by the potential advantages of multiple-antenna transmission over the frequency-selective fading channels, we are concerned about the QoS and MMF problems in this paper for multigroup multicasting. The main contributions of this paper are listed in the following:

For the QoS problem, an approximation algorithm is firstly derived for broadband systems based on the idea of narrowband multigroup multicasting in [

For the MMF problem, the corresponding approximation algorithm and its low-complexity modifications are also proposed in a similar way as that of QoS problem. Furthermore, the relationship between the QoS problem and the MMF problem is discussed carefully followed by a complexity analysis.

Simulation experiments demonstrate the effectiveness of the frequency-domain and the time-domain algorithms for both QoS problem and MMF problem. The main analysis results that the controlled parameters in the proposed algorithms could be used to make a tradeoff between complexity and performance are verified through the numerical examples.

The remainder of this paper is structured as follows. In Section

Consider a multigroup multicast system with one transmitter (base station) and

With an

The schematic diagram of a multigroup multicast system.

For brevity purpose, (

In this regard, the total transmission power of the multigroup multicast system becomes

With the above-mentioned assumptions and definitions, the problem of minimizing the total transmission power under the SINR constraints of each user

Note that problem

Let

Due to the nonconvex nature of problem

It is noteworthy that problem

The optimal solution

In order to prove this proposition, the dual problem of problem

On the other hand, according to Theorem 3.2 in [

From Proposition

Note that although these processed candidates satisfy

When

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More clearly, the approximation algorithm can be treated as a direct use of the algorithm in [

To combat the heavy computation burden of the approximation algorithm, new beamforming algorithms which can make tradeoff between performance and complexity are eagerly demanded in these situations. Towards this end, two beamforming algorithms are proposed from perspectives of frequency domain and time-domain, respectively, in this section.

In fact, the channel coefficient vector

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As a matter of convenience, denote

The following results shed more lights on the influence of the controlling parameter

Assume problem

Assume

Assume

Define the optimal solution to problem

Besides reducing the number of optimization variables in frequency-domain immediately, an alternative way can also benefit the complexity reduction by cutting down the number of the FIR filter taps from time-domain perspective. Assume the time-domain FIR filter has

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From Algorithm

If problem

Assume

Assume

The basic idea of the proof process is similar to that of Proposition

In addition to the QoS problem, another problem always considered in a multigroup multicast system is the MMF problem. The original problem of maximizing the minimum SINR of all users under the total transmission power constraint can be written as

Fortunately, problem

Alike the QoS problem, the reduced-complexity MMF problem can also be considered both in frequency domain and in time-domain, where the frequency-domain version can be formulated as

Block-diagram of the solving framework.

Denote

The following analytical results demonstrate the relationship between the MMF problems with different parameters.

Assume

Assume

Furthermore, from the problem formulation, it appears that the MMF problems are always feasible, while things could be different for the QoS problem. The relationship between QoS and MMF problems for narrowband multigroup multicast case is discussed in [

For a fixed set of channel vectors and noise powers, the QoS problem

Define

In order to prove (

The QoS and MMF problem pairs, that is, problems

First of all, the computational complexities of solving the QoS problems are discussed in this section. For the approximation algorithm derived in Section

For the frequency-domain beamforming algorithm, the SDP problem

For the time-domain beamforming algorithm, the SDP problem

Next, we will analyze the computational complexities of solving the MMF problems which are analyzed. Define

For both QoS problem and MMF problem, it is important to point out that the computational complexities of the corresponding frequency-domain algorithm and the time-domain algorithm are always much lower than that of the approximation one in practical wireless communication systems, because the parameter

The total arithmetic operations for QoS problems.

Since for MMF problems, the computational consumption for both the frequency-domain algorithm and the time-domain algorithm are linear with those of QoS problems, we ignore the illustrative comparison in this section.

In this section, several numerical examples are illustrated to demonstrate the effectiveness of proposed beamforming algorithms. For simplicity, the frequency-selective fading channel between each receiver and the transmitter is built as a discrete channel model with 3 effective paths; that is,

The first step of the proposed algorithms is solving the relaxed SDP problems. From the relaxation process we can see that the feasible set of the relaxed SDP problems is indeed a superset of one of the original QCQP problems, which leads to the following conclusions: If the relaxed problems are not feasible, the original ones are not feasible either. Rather, if the relaxed problems are feasible, the original ones may be not feasible. Therefore, the feasibility of the SDP problems is a necessary condition for the validity of proposed algorithms. In this subsection, the feasibility of the SDP problems is evaluated under conditions of different number of multigroups

Figure

The frequency-domain (or time-domain) SDP problem with less users is feasible with higher probability (just mentioned above).

The frequency-domain (or time-domain) SDP problem with less groups is feasible with higher probability. That is, because the more the groups are, the more the intergroup interference exists.

The frequency-domain (or time-domain) SDP problem with more transmit antennas is feasible with higher probability, since the more the transmit antennas are, the more the spatial degrees of freedom can be exploited.

The frequency-domain SDP problem with smaller

The time-domain SDP problem with larger

The feasible probability of SDP problems.

Next, the performances of the proposed algorithms solving QoS problems are compared. The system parameters are set as

The transmission power performance for QoS problems.

The cumulative distribution function for QoS problems.

Finally, the proposed algorithms are adopted to solve the MMF problems. We set

The performance for MMF problems.

From Figure

In this paper, the downlink beamforming designs for the broadband multigroup multicast QoS and MMF problems are investigated. By means of the traditional SDR and Gaussian randomization methods, two algorithms designed in frequency and time-domains are proposed to solve the QoS problem. Then we extend these algorithms to handle the MMF problem through an iterative bisection search process. Several Monte Carlo simulations indicate the proposed beamforming designs reduce the computational complexity considerably with slight performance loss: the complexity of solving the QoS problem could be reduced to 1/100 by the time-domain algorithm with

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported in part by the National Natural Science Foundation of China (no. 61271272), the Intercollegiate Key Project of Nature Science of Anhui Province (no. KJ2012A283), and the National High Technology Research and Development Program of China (863 Program) with Grant no. 2012AA01A502. The authors would like to thank Professor Xuchu Dai for his helpful discussions.