Power spectrum optimization for interference mitigation via iterative function evaluation
- Hayssam Dahrouj^{1},
- Wei Yu^{1}Email author and
- Taiwen Tang^{2}
https://doi.org/10.1186/1687-1499-2012-244
© Dahrouj et al.; licensee Springer. 2012
Received: 13 February 2012
Accepted: 26 June 2012
Published: 6 August 2012
Abstract
Power spectrum optimization is a well-known difficult nonconvex optimization problem for which only local optimality can typically be assured. This paper unifies several classes of local optimization methods and proposes efficient and new methods for power spectrum optimization by observing that methods for reaching the local optimal points can often be expressed in the form of an iterative function evaluation. This proposed new approach is based on the fact that the gradient of the objective function is zero at a local optimum, and that different manipulations of the optimality condition can then lead to different power update equations. As a practical application, this paper examines the benefit of dynamic power spectrum optimization for interference mitigation in a wireless backhaul network in which remote radio units are deployed to serve mobile users in areas with high data traffic demand. The remote radio units, called remote terminals (RT), are connected to access nodes (AN) via orthogonal frequency division multiple access (OFDMA) over a fixed bandwidth with one RT active in each frequency tone. The system performance is thus limited by internode interference solely, and no intranode interference. This paper shows that iterative function evaluation based methods provide a significant improvement in the overall network throughput in this setting as compared to a conventional network with fixed transmit power spectrum. The proposed methods have computationally fast convergence and can be implemented in a distributed fashion assuming reasonable amount of internode information exchange. Further, some of the proposed methods can be implemented asynchronously at each AN, which makes them amenable to practical utilization.
Introduction
Interference is a major bottleneck in wireless network design. Developing and optimizing advanced, yet practical, interference mitigation techniques is particularly important nowadays, due to the rapid pace of growth of wireless networks with enormous data usage, and the scarcity of the available radio resources, e.g. bandwidth and transmit power. Dynamic power spectrum optimization is an important class of interference mitigation methods that seek to increase the network capacity and reliability via power control. The present paper aims to develop novel, feasible, practical methods for power spectrum optimization.
Dynamic power spectrum optimization is especially important to distributed antenna systems (DAS) where the base-station transmit capability is enhanced by adding multiple remote radio units. This paper focuses on a wireless cellular network comprising multiple remote terminals (RT), each covering a relatively small area, called a picocell, as a means to increase the network capacity for areas with dense data traffic. The mobile users within each picocell are served by the picocell base-station, which are collocated with the RTs. The RTs are then connected to access nodes (ANs) via wireless backhaul links, which are meant to replace the expensive optical fiber links. This paper assumes that the ANs can be coordinated in transmission strategies and radio resource management when transmitting to different RTs. From a design perspective, the interest of this paper is to mitigate the internode interference, thereby maximizing the aggregate data capacity to the RTs via practical power spectrum optimization methods.
This paper adopts a network utility maximization framework and focuses on the weighted rate-sum maximization problem, which is known to be a difficult nonconvex problem. Consequently, most numerical methods focus on finding its local optimum only. The main point of this paper is that under power spectral density (PSD) constraints, the local optimization of weighted rate-sum in an interference environment can often be performed efficiently via a class of methods based on iterative function evaluation. This proposed new approach is based on the observation that local optima of an optimization problem occur where the gradient of the objective function is zero, and that different manipulations of the gradient function can lead to different numerical power update algorithms. The power update equations obtained this way take the form of an iterative function evaluation. The performance and convergence properties of iterative function evaluation methods (IFEM) for power spectrum optimization are the focus of this paper.
While power spectrum optimization methods have been widely proposed in the literature, the practical implementation of power adaptation in wireless applications is often challenging because wireless channels can vary rapidly over time, and because wireless transmitters are often located in geographically separated locations. Thus, computational complexity and speed of convergence are key considerations for power adaptation in wireless applications; further, distributed and asynchronous implementation are often desired. The main advantage of the iterative function evaluation based power adaptation methods proposed in this paper is that they typically have a low computational complexity, while showing a significant performance gain as compared to the conventional maximum power transmission policy. Further, these methods lend themselves to distributed implementation with reasonable amount of internode information exchange. Some of the methods can also be implemented asynchronously, which makes them amenable to practical utilization.
One of the objectives of this paper is to quantify the benefit of power spectrum optimization in a practical wireless backhaul deployment. This paper envisions a setting that makes use of channel measurements done on a per-frequency-tone basis for every AN-RT pair. The measurements are subsequently provided either to a central server for further centralized processing, or to each of the several ANs for distributed processing. To minimize the overhead of such exchange of information, the channel measurements can alternatively take the form of frequency domain average channel gains. The performance of the proposed methods with such average channel gains is also quantified in this paper. The power spectrum optimization methods proposed in this paper are of particular interest in a fixed deployment scenario such as the wireless backhaul network, where the channels are relatively slow varying. Further, the measurements can be done periodically, thus allowing the adaptation of radio resource allocations with the dynamically changing environment.
Related work
The main challenge in dynamic power spectrum management is that of finding computationally efficient methods to allocate powers to different transmitters across the different frequency tones in order to maximize some overall system wide utility. This power spectrum optimization problem is especially well studied in the literature for the digital subscriber line (DSL) system[1–5]. In particular, the works[4, 5] make use of concept known as interference price, which is related to the methods proposed in this paper. Interference price quantifies the effect of interference between the multiple transmitter-receiver pairs. The use of interference pricing for power spectrum adaptation in fact originated from the wireless literature[6–9], where various power control algorithms are studied.
In the wireless application, power control methods need to be incorporated with scheduling. Joint power control and scheduling has been investigated in the literature for both single-cell[10, 11] and multicell systems[12–15]. The overall problem setup of this paper is closely related to that in[14, 15], where power spectrum adaptation is jointly performed with proportionally fair scheduling. The contribution of this paper focuses on power control.
The main focus on this paper is on the efficiency-complexity tradeoff of different power control algorithms. The weighted sum-rate maximization problem studied in this paper has a well-known and difficult nonconvex structure induced by the coupling between the power terms. The solutions referenced above (and the new algorithms proposed in this paper), therefore, only reach local optimal points. Finding the global optimum of the spectrum management problem is typically much more difficult. Toward this end, an algorithm called MAPEL is recently proposed in[16] based on monotonic optimization and fractional programming, and it is guaranteed to converge to the global optimal solution. The computational complexity of implementing MAPEL can however be high. Further, it requires centralized processing; its convergence speed also remains an open problem.
This paper proposes novel, low-complexity, but local algorithms to solve the weighted sum-rate maximization problem. The proposed methods are based on the manipulations of the gradient of the objective function. Although they only guarantee local optimality, as the simulations results show, they already bring in significant improvement as compared to conventional systems with the maximum power transmission strategy. The main advantages of this new class of algorithms are efficiency and potential for distributed and asynchronous implementation. In particular, they do not require step size choices, as traditional subgradient or Newton’s methods in optimization do.
System model and problem formulation
System model
Problem formulation
is the instantaneous rate of the scheduled k th RT for the l th AN at the n th tone, S^{ max } is the maximum power constraint imposed on each AN at each tone, and where the maximization is over the scheduling assignment k = f(l,n), and the power spectral density levels${P}_{l}^{n}$. The Γ is the signal-to-noise ratio (SNR) gap. The weights w_{ lk } here come typically from upper layer considerations that account for queue length, traffic demand and user priorities.
where the maximization is over the set of powers${P}_{l}^{n}$.
is the instantaneous rate of the scheduled ξ th RT for the l th AN at the n th tone. The rest of the paper examines practical numerical methods to solve (4), i.e. under fixed scheduling.
This paper chooses to use a power spectrum density (PSD) constraint which particularly simplifies the numerical optimization procedure. Although the numerical solution of this paper can also adapted to the case where total power constraints are also present, one could argue that PSD constraints occur more naturally in practical settings, and is therefore adopted in this paper.
Power spectrum optimization
The weighted sum-rate maximization problem (4) is a nonconvex optimization problem, whose global optimal solution is not easy to find. Like many previous approaches, this paper also aims at local optimal solutions, but with a focus on reducing computational complexity. Traditional optimization approaches for solving a nonlinear optimization problem include gradient projection or Newton’s method. In contrast, this paper proposes a new class of strategies based on an iterative function evaluation approach. These novel methods, described below, are simple to implement, fast in convergence, and similar in performance to traditional full-blown Newton’s method. In addition, they have the advantages of being amenable to distributed and asynchronously implementation, and are free of step size choices.
The proposed methods all result from different manipulations of the gradient of the objective function. Below, we first present two “full” methods, called IFEM-1 and IFEM-2. We then present simplified versions that offer further reduced computational complexity. Since the objective function in (4) is not necessarily concave, the approach used in deriving IFEM-1 and IFEM-2 only looks for stationary points to the problem. Nevertheless, the proposed solutions are shown to provide a significant performance improvement as compared to conventional systems.
Iterative function evaluation methods
IFEM-1
The point is that it is possible to interpret the above condition as an iterative algorithm for computing the optimal${\left\{{P}_{l}^{n}\right\}}_{l=1}^{L}$. In other words, given the current set of power allocation, one can compute the terms on the right-hand-side of the above equation using the current power, then use the result to update the new power allocation. This step is a simple function evaluation, which can be done iteratively. Hence this method is named IFEM-1.
Note that if (and when) the iterative process converges, it must converge to a point that satisfies the stationarity condition. Thus, the convergence point must be a local stationary point, (and often observed to be a local optimum in practice). This is obvious to see when the converged point is in the interior of the constrained set, but is also true when the converged point is on the boundary. In this latter case, it can be shown that a suitable set of Lagrangian multipliers can be found so that the converged point satisfies the KKT conditions of the optimization problem. A detailed proof can be found in[5]. Note that the${\tau}_{\mathrm{jl}}^{n}$ term defined above is an interference pricing term. It is a function of the channel gain$|{h}_{\mathrm{lj}{k}^{\u2033}}^{n}{|}^{2}$ and indicates the impact of the interference from the l th AN to the scheduled RT of the j th AN at the n th tone. As the interference price${\tau}_{\mathrm{jl}}^{n}$ increases, the l th AN tends to decrease its power${P}_{l}^{n}$. In a distributed implementation, these pricing terms are exactly the information that needs to be exchanged among the ANs.
IFEM-2
where the power constraint is again taken into account though the projection step.
High-SINR IFEM
The above manipulations of the optimality condition are by no means the only ways to derive a fixed-point equation for power optimization. Numerous possibilities exist, but the above two are the most sensible. These IFEM algorithms, if converging, likely converge to a local optimum solution of the original problem. The key differentiating factor is therefore their convergence properties.
We call the resulting algorithm high-SINR IFEM (HSIFEM). As shown later in the paper, the above update algorithm can be shown to always converge, but HSIFEM is no longer guaranteed to converge to a local optimal point due to the omission of SINR terms.
θ-IFEM-1 and θ-IFEM-2
Thus far, we have on one hand IFEM-1 and IFEM-2 algorithms, for which convergence is not easy to establish, but if they do converge, they converge to a local optimum. On the other hand, we have HSIFEM which is guaranteed to converge, but not necessarily to a local optimum. To make further progress, we may consider compromises along the following directions.
Note that HSIFEM is simply θ-IFEM-1 with${\theta}_{\mathrm{jl}}^{n}=1$.
Convergence analysis
As mentioned earlier, the convergence of IFEM-1 and IFEM-2 is difficult to establish in full generality, but θ-IFEM-1 and HSIFEM algorithms always converge. This section establishes this fact using the theory of standard functions.
Proposition 1
Starting from any initial feasible${P}_{l}^{n}\left(0\right)$, θ-IFEM-1 and HSIFEM algorithms respectively converge to unique fixed points. Furthermore, the convergence is still guaranteed under a totally asynchronous model.
Proof
- 1.
If ${P}_{l}^{n}\ge 0$ ∀ l,n, then ${g}_{l}\left({\Psi}^{n}\right)>0$.
- 2.
If ${P}_{l}^{n}\ge {{P}^{\u2033}}_{l}^{n}$∀ l,n, then ${g}_{l}\left({\Psi}^{n}\right)\ge {g}_{l}\left({\Psi {}^{\u2033}}^{n}\right)$.
- 3.
For ρ > 1, we have $\rho {g}_{l}\left({\Psi}^{n}\right)>{g}_{l}\left(\rho {\Psi}^{n}\right)\forall l,n$.
where the variables${P}_{l}^{n}$l = 1,⋯ L, are stacked into one vector Ψ^{ n }. The convergence to the unique fixed point and the asynchronous convergence follow as a consequence.
- 1.
If ${P}_{l}^{n}\ge 0\forall l,n$, then ${f}_{l}\left({\Psi}^{n}\right)>0$.
- 2.If ${P}_{l}^{n}\ge {{P}^{\u2033}}_{l}^{n}\forall l,n$, then$\frac{{w}_{\mathrm{lk}}}{\sum _{j\ne l}\frac{{w}_{j{k}^{\u2033}}|{h}_{\mathrm{lj}{k}^{\u2033}}^{n}{|}^{2}{\theta}_{\mathrm{jl}}^{n}}{{\sigma}^{2}+\sum _{i\ne j}{P}_{i}^{n}|{h}_{\mathrm{ij}{k}^{\u2033}}^{n}{|}^{2}}}\ge \frac{{w}_{\mathrm{lk}}}{\sum _{j\ne l}\frac{{w}_{j{k}^{\u2033}}|{h}_{\mathrm{lj}{k}^{\u2033}}^{n}{|}^{2}{\theta}_{\mathrm{jl}}^{n}}{{\sigma}^{2}+\sum _{i\ne j}{{P}^{\u2033}}_{i}^{n}|{h}_{\mathrm{ij}{k}^{\u2033}}^{n}{|}^{2}}}$(24)
- 3.For ρ > 1, we have:$\begin{array}{ll}\rho {f}_{l}\left({\Psi}^{n}\right)\phantom{\rule{2.77695pt}{0ex}}& =\phantom{\rule{2.77695pt}{0ex}}\frac{{w}_{\mathrm{lk}}}{\sum _{j\ne l}{w}_{j{k}^{\u2033}}\frac{|{h}_{\mathrm{lj}{k}^{\u2033}}^{n}{|}^{2}}{{\sigma}^{2}+\sum _{i\ne j}\rho {P}_{i}^{n}|{h}_{\mathrm{ij}{k}^{\u2033}}^{n}{|}^{2}}{\theta}_{\mathrm{jl}}^{n}}\phantom{\rule{2em}{0ex}}\\ >\phantom{\rule{2.77695pt}{0ex}}{f}_{l}\left(\rho {\Psi}^{\mathbf{n}}\right).\phantom{\rule{2em}{0ex}}\end{array}$(25)
Thus, f_{ l }(.) satisfies the standard function properties. Based on Theorem 7 in[17], the power constrained function${\left[{f}_{l}\left(.\right)\right]}_{0}^{{S}^{\mathrm{max}}}$ is also standard. Note that a fixed point of this function iteration must exist, since the original optimization problem has at least one local optimum. Together with Corollary 1 in[17], this implies that, starting from any initial${P}_{l}^{n}\left(0\right)$θ-IFEM-1 is guaranteed to converge to a unique fixed point.
where the index${\varpi}_{i}^{j}$ point to possibly outdated information of the i th AN that is available at the processor of the j th AN.
Finally, note that High-SINR IFEM (HSIFEM) and θ-IFEM-1 are based on different power update equations. Therefore, they can potentially converge to different unique fixed points. □
Connection with SCALE
In[3], a power control algorithm named SCALE is proposed. The algorithm is motivated by geometric programming. SCALE is a two-stage algorithm, and in the notation of this paper can be thought of as having an inner loop to update the power${P}_{l}^{n}$, and an outer loop to update the coefficients${\theta}_{\mathrm{jl}}^{n}$. The SCALE algorithm defines${\alpha}_{j}^{n}=\frac{{\mathrm{SINR}}_{j}^{n}}{1+{\mathrm{SINR}}_{j}^{n}}$, so that${\theta}_{\mathrm{jl}}^{n}={\alpha}_{j}^{n}/{\alpha}_{l}^{n}$. It runs iterative function evaluation with fixed${\alpha}_{j}^{n}$ in the inner loop to update${P}_{l}^{n}$, then updates${\alpha}_{j}^{n}$ based on the resulting SINRs in the outer loop. IFEM-1 proposed in this paper is essentially a simplification of SCALE. Instead of the two-stage process in which${\theta}_{\mathrm{jl}}^{n}$ is updated in an outer loop, IFEM-1 implicitly updates the power vector and${\theta}_{\mathrm{jl}}^{n}$ at the same time.
In addition, as shown in the simulation results presented later in this paper, the use of a single fixed${\theta}_{\mathrm{jl}}^{n}$ derived from the maximum transmit power level may already be near optimum (leading to θ-IFEM-1), thus${\theta}_{\mathrm{jl}}^{n}$ may not need to be updated at all. Further, at high SINR,${\theta}_{\mathrm{jl}}^{n}$ can be set to 1, leading to HSIFEM. It is interesting to remark that although SCALE is motivated by geometric programming, the algorithm itself can be easily derived without geometric programming, as shown in this paper.
Connection with iterative water-filling
Iterations between the inner and outer loops are performed until convergence. IFEM-2, proposed in this paper, is essentially a simplification of MIWFA. Instead of the two-stage process in which the interference pricing terms${\tau}_{\mathrm{jl}}^{n}$ are updated in an outer loop, IFEM-2 implicitly updates the power vector and${\tau}_{\mathrm{jl}}^{n}$ at the same time.
Comparison with Newton’s method
where${\tau}_{\mathrm{jl}}^{n}$ is the interference price defined in (23).
Distributed implementation
Typically,${P}_{l}^{n}$ is known and${\mathrm{SINR}}_{l}^{n}$ can be easily measured at each AN. A distributed implementation of power allocation (27) is, therefore, possible if ANs exchange the variables${\tau}_{\mathrm{jl}}^{n}$. As mentioned earlier,${\tau}_{\mathrm{jl}}^{n}$ has the interpretation of being the interference price. This exchange of information amounts to reasonable overhead for distributed implementation. Note that θ-IFEM-1 and HSIFEM can, in addition, be implemented asynchronously at each AN using possibly outdated power information, as shown in Proposition 1.
Methods based on average channel gains
All the methods above are based on a per-tone channel measurements. Those measurements are typically provided to either a central server for further centralized processing, or to each of the several access nodes for distributed processing. In practice, to minimize the overhead of such exchange of information, the measurements can alternatively provide the frequency domain average channel gains. The proposed methods above can make use of the average channel gains instead. In the following, we list these “average gain” methods, and subsequently quantify their performance in a typical wireless backhaul network. These methods are of particular interest when the delay spread of the wireless propagation environment is limited.
By substituting the average gains in the equations of the methods presented earlier, the update equations of the resulting methods can be summarized as follows:
Finally, there are situations in which the channel measurement can be done on a per-tone basis, but the physical transmission platforms only permit each AN to allocate one value of power across all frequency tones. In this case, we take the average power (AP) values of IFEM algorithms, and call the resulting method AP algorithms. For example, AP HSIFEM takes the average power value of the HSIFEM.
In practice, the AP algorithms require an additional module which runs the algorithms on a per-tone basis to find the set of powers${P}_{l}^{n}$, and then returns an average value of the power P_{ l } for each AN l, i.e.${P}_{l}=\sum _{n=1}^{N}{P}_{l}^{n}$. This approach for solving the power optimization subject to constant power across tones is obviously suboptimal; nevertheless, this proposed solution always outperforms conventional systems with maximum power transmission, as the simulations suggest.
Simulations
System model parameters
Cellular layout | Hexagonal |
---|---|
Number of ANs | 7 |
Frequency Reuse | 1 |
Number of RTs per AN | 4 |
AN-to-AN Distance | d _{1} |
AN-to-RT Distance | d _{2} |
Duplex | TDD |
Channel Bandwidth | 10 MHz |
AN Max Tx Power per Subcarrier | -32.70 dBw |
SINR Gap | 12 dB |
Total Noise Power Per Subcarrier | -158.61 dBw |
BS Tx Antenna No. | 1 |
MS Rx Antenna No. | 1 |
Sampling Frequency | 11.2 MHz |
FFT Size | 1024 |
Distance-dependent Path Loss | $128.1+37.6\underset{10}{log}\left(d\right)$ |
Multipath Delay Profile | SUI-3 Terrain type B |
We also observe that AP HSIFEM, which allocates equal power in each AN across all the tones, is always superior to the maximum power method. However, there is some performance loss as compared to HSIFEM/HSNM where per-tone adaptation is enabled.
Methods behavior over 7 ANs, 4 RTs per AN
Sum rate in bps/Hz | d_{1} = 0.5km | d_{1} = 1km |
---|---|---|
SIFEM | 60.68 | 91.33 |
AG HSIFEM | 60.67 | 91.33 |
HSNM | 60.68 | 91.33 |
AG HSNM | 60.67 | 91.33 |
IFEM-1 | 62.61 | 91.58 |
AG IFEM-1 | 62.59 | 91.57 |
IFEM-2 | 62.61 | 91.58 |
AG IFEM-2 | 62.59 | 91.57 |
NM | 62.61 | 91.58 |
AG NM | 62.59 | 91.57 |
AP HSIFEM | 56.88 | 88.13 |
Max Power Method | 53.01 | 86.22 |
IFEM-1 Gain | 18.1% | 6.2% |
Methods behavior over 7 ANs, 4 RTs per AN
Sum rate in bps/Hz | Cell-edge | Cell-center |
---|---|---|
(d_{2} = 333m) | (d_{2} = 125m) | |
HSIFEM | 34.84 | 78.39 |
AG HSIFEM | 34.83 | 78.38 |
HSNM | 34.84 | 78.39 |
AG HSNM | 34.83 | 78.38 |
IFEM-1 | 41.11 | 78.77 |
AG IFEM-1 | 41.09 | 78.76 |
IFEM-2 | 41.11 | 78.77 |
AG IFEM-2 | 41.09 | 78.76 |
NM | 41.11 | 78.77 |
AG NM | 41.09 | 78.76 |
AP HSIFEM | 31.54 | 75.00 |
Max Power Method | 30.54 | 71.91 |
IFEM-1 Gain | 34.6% | 9.5% |
It can be observed from Tables2 and3 that HSIFEM and HSNM always have the same performance, as both employ a high-SINR approximation. At the cell-edge of small cells, where the SINR level is not sufficiently large to justify the high-SINR approximation, HSIFEM and HSNM become inferior to IFEM-1, IFEM-2 and NM. This is, however, not the case for cell-center users, e.g. the case of d_{1} = 1km, d_{2} = 150m as shown in Table2 and the case of d_{1} = 0.5km, d _{2} = 125m as shown in Table3, where SINR values are larger, and where HSIFEM, HSNM, IFEM-1,IFEM-2 and NM all have similar performance. In all cases, IFEMs always remain superior to the maximum power method.
Tables2 and3 also show that the methods that rely on the frequency domain average channel gain have a very similar performance to the methods that are based on the per-tone channel gain measurements. This is due to the fact that a typical fixed wireless backhaul channel is relatively flat fading. Our simulation assumes an SUI-3 model with Terrain type B, in which the delay profile has three taps; the k-factors of the three taps are 3, 0, and 0; the relative taps delays are 0, 0.4, and 0.9 in microseconds.
Conclusion
Given the scarcity of the available radio resources, the performance of future wireless networks would be expected to depend on the feasibility of the dynamic power spectrum optimization methods. This paper presents a class of very simple yet effective power control methods to manage interference in wireless systems. The proposed methods represent efficient ways of updating the power spectral density levels for all transmitters, based on either the frequency domain channel gains, or the frequency domain channel average gains. The proposed methods are based on the observation that different manipulations of the gradient of the objective function give different power update equations. These methods have low computational complexity and fast convergence. Their performance is similar to the full-blown Newton’s method, but without the need for step size choices. They can also be implemented in a distributed fashion and asynchronously at each transmitter, and are therefore excellent fits for practical applicability.
Declarations
Acknowledgements
The authors acknowledge Natural Science and Engineering Research Council (NSERC) ENGAGE program and BLiNQ Networks Inc. for funding this work. The authors also wish to thank Mr. Steve Beaudin who was supportive and helpful during the project. This paper has been presented in part at IEEE Globecom First Workshop on Distributed Antenna Systems for Broadband Mobile Communications, Houston, TX, USA, Dec 2011[18], and in part at 26th Biennial Symposium on Communications, Kingston, Ontario, Canada, May 2012[19].
Authors’ Affiliations
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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.