 Research
 Open Access
Energy optimization in multiuser quantized feedback systems
 Saud Althunibat^{1}Email author,
 Nizar Zorba^{2},
 Fabrizio Granelli^{1} and
 Christos Verikoukis^{3}
https://doi.org/10.1186/16871499201383
© Althunibat et al.; licensee Springer. 2013
 Received: 20 December 2012
 Accepted: 27 February 2013
 Published: 21 March 2013
Abstract
This article explores the effects of quantization of feedback information on energy consumption in multiuser wireless communication systems. In order to optimize the energy consumption of the system, the article concentrates on the amount of transmit energy, the additional energy due to quantization and the probability of power outage. Closed form expressions for such parameters are obtained, where the impact of the number of quantization bits is explicitly outlined. An optimization problem is then formulated to find the optimum number of quantization bits able to minimize the consumption in the energy resources. Simulations demonstrate the good results obtainable with the presented optimization strategy, and provide effective validation of the analytic solution presented in the article.
Keywords
 Probability Density Function
 Power Allocation
 Channel State Information
 Closed Form Expression
 Feedback Process
Introduction
Given the increased diffusion and usage of wireless and mobile networks, energy efficiency represents an open issue of paramount importance in order to support scalability and sustainability of wireless networks. Wireless network interfaces consume significant amount of energy, especially in the case of cellular base stations (BS). Indeed, the energy request for the infrastructure of wireless operators represents a large portion of the overall network costs [1]. As a consequence, decreasing the consumed energy in BSs would reduce the running costs of the whole communication infrastructure.
Channel state information (CSI) at the transmitter side is required to enable efficient resource management. Such information is obtained through a feedback process: each user periodically measures the CSI, and informs the transmitter about it. The performance in resource management by using such feedback process highly depends on the method used to represent the feedback values: where increasing the feedback values for channel state estimation would enhance the efficiency of resource management [2].
The most common approach to feedback the CSI is through the feedback quantization method, where the wide (sometimes infinite) continuous range of the CSI values is represented by finite discrete values [3], thus introducing a quantization error [4]. In multiuser systems, the users quantize their CSI, represented by the signaltonoise ratio (SNR), into a number of bits, and send them back to the BS. As the quantized version of the SNR is represented by from a finite and low number of values, then the corresponding loss of information will negatively impact on the achieved data rate and the consumed energy, especially in cases where dynamic resource allocation is employed.
An increase to the amount of the feedback bits better matches the quantized CSI and the actual (nonquantized) CSI values [5], with a corresponding improvement in the resources distribution mechanism. On the other hand, feedback quantization is a required step in order to limit the number of bits associated to CSI feedback and avoid a potential source of inefficiency in the system. The impact of the feedback load is larger in multiuser systems, as the feedback bits are multiples of the number of active users in the cell [2]. Based on the above, an optimization to find the best amount of feedback bits is required, in order to optimize both the system performance and the energy consumption.
This article analyzes the effects of feedback quantization on energy resources and performance metrics. To this goal, the transmit energy, additional energy due to quantization, feedback energy, and the power outage probability are all derived in closed form expressions, with all the relevant parameters affecting them. Using such values, an optimization problem for defining the appropriate number of feedback bits to increase the energy efficiency of the system is formulated.
Previous work as [6] analyzes the effect of quantization on the throughput of constant rate transmission, and it concludes that few quantization bits are required to achieve a throughput slightly lower than in the nonquantized case. However, the impact on energy consumption is not considered, while time and energy consumed during the feedback process are neglected. In [7], an optimum quantization scheme is derived for opportunistic beamforming systems, which maximizes the average system throughput while ensuring fairness among users. In [8], mathematical expressions for the rate outage probability and the capacity loss due to quantization are derived for a transmission system with finite rate feedback; while, in [9], a new quantization scheme is developed to maximize the overall throughput and minimize the scheduling error in multiuser systems. A convex optimization problem is formulated and solved in [10, 11], in order to find the optimal number of quantization bits that minimizes the reconstruction error in wireless sensor networks, where the constraint in [10] is the energy per sensor and in [11] is the total consumed energy. In [12], a joint channel quantization and resource allocation schemes for singleuser OFDM systems, which use limitedrate feedback, are obtained by solving an optimization problem that maximizes average ergodic rate subject to average power and bit error rate constraints. The effects of the quantization noise on energy measures, for cooperative spectrum sensing in cognitive radio networks, is investigated in [13], and a method to compensate these effects based on a new soft metric at the fusion center is proposed. The asymptotic performance of multiple antenna channels where the transmitter has finite bit CSI is analyzed in [14], where it is demonstrated that channel feedback can fundamentally change the system behavior. To the best of the authors’ knowledge, there are no previous contributions on the energy efficiency for feedback quantization in opportunistic scheduling systems.
The main contributions of this article can be summarized as follows:

Obtaining the probability density function (PDF) of the serving SNR in opportunistic scheduling in quantized feedback systems.

Deriving closed form expressions for the transmit energy of quantized and nonquantized feedback systems, the additional energy due to quantization, and the power outage probability.
The remainder of this article is organized as follows: Section 2 describes the system model, while Section 3 discusses the effects of quantization and derives the closed form expressions, in order to formulate and solve the optimization problem to find the best amount of feedback bits that minimizes the consumed energy. Section 4 provides the numerical results and Section 5 draws conclusions and outlines future work on the subject.
System model
where z_{ i }(t) is an additive i.i.d. complex noise component with zero mean and E{z_{ i }^{2}}=σ^{2}.
where T is the total frame length.
At the BS, after all the quantized SNR values are received from all users, an opportunistic scheduling technique [15–17] is employed to choose the user with best channel characteristics (i.e., maximum SNR). Therefore, the user with best SNR is scheduled, but due to the quantization error, more than one user may appear within the same SNR interval (i.e., the BS understands that they have the same SNR value), and hence, the BS is forced to randomly select one of them, which may yield to an erroneous selection, as the user with the best actual SNR is not selected [9]. Following commercial systems, time separation is accomplished in the feedback channel, so that the feedback is not simultaneous over all the users.
Once a user is selected, the transmit power has to be defined to meet the quality of service (QoS) demand given in terms of a minimum SNR value. If the measured SNR value is (x), then the required transmit power to achieve a QoS demand (q) is $\left(\frac{q}{x}\right)$. In contrast, in quantizedfeedback systems, as the exact value of the measured SNR is unknown at the BS, the worst case approach is followed to allocate the transmit power [18]. The worst case approach implies that as the BS just knows the interval [γ_{ i },γ_{i+1}] where the measured SNR is the lower threshold of the interval (γ_{ i }) is used to calculate the transmit power. Therefore, the transmit power in quantized feedback systems equals to $\left(\frac{q}{{\gamma}_{i}}\right)$.
Notice that the power allocation based on quantized feedback yields a larger amount of transmit energy, a parameter that must be optimized due to the large drawbacks of increasing the amount of transmit energy to the system, the customer and the environment^{b}.
Quantization effects on the consumed energy
To reach the optimum performance of the opportunistic scheduling, the SNR for all users must be available at the BS, but, as we previously discussed, in practical systems the SNR values are quantized inducing a quantization error which affects the performance of the opportunistic scheduler. In this section, we concentrate on the system energy resource, and present the effects of the quantization process on the required transmit energy, additional energy, and power outage probability. After that, we formulate an optimization problem to minimize the overall energy consumption.
Transmit energy
It is proven that the opportunistic scheduling achieves the highest possible data rate among all other scheduling algorithms [15], leading to the minimum amount of required power to satisfy the QoS demand for the customers.
for γ_{ i }<x<γ_{i+1} being the quantization intervals, as previously explained.
where γ_{ i }≤x<γ_{i+1}.
Power outage probability
Due to multipath fading and channel fluctuation experienced by users, the scheduled user may have a low SNR, so that the user needs high power to satisfy its QoS. In this section, we deal with the fact that the total available power in the BS is limited, so the scheduled user may experience some power outage [19] when the user needs an amount of power larger than the available power at the transmitter, hence, there will be an outage for the scheduled user. The probability of power outage due to power limitation exists regardless of the feedback type (i.e., quantized or nonquantized).
where f_{max} and F_{ Q } are the CDFs of f_{max}, given in Equation (5), and F_{ Q }, given in Equation (11), respectively.
Feedback energy
The system performance will be enhanced by increasing the number feedback bits making the quantized version of the SNR to approach the actual value, which yields a better performance of the opportunistic scheduling, and consequently, less additional energy. On the other hand, the negative effect of increasing the feedback bits is the increase in the required reserved resources, time and energy, for the feedback process, so the users consume a portion of their energy to report the SNR, while the whole system waits additional time until the feedback values are received from all users.
It is worthy mentioning that we do not consider the energy consumption during training and synchronization since it is independent of the quantization process. Clearly, the function is nonconvex of B, and hence, the solution of this problem is infeasible in a closed form. In next section, we solve the problem by computer simulations in order to obtain the optimum number of quantization bits that minimizes the overall energy consumption.
Simulation results
A multiuser scenario is considered, where the BS intends to communicate with a single user at each scheduling time T=20 ms. A total of N=30 users are available in the system with i.i.d. channel characteristics. A homogeneous system (i.e., all users have the same application) is assumed, where the QoS can be achieved by a data rate of 2 bit/s/Hz. The average SNR is considered to be SNR = 0 dB, but due to the channel fluctuations, the provided data rate could be lower than the average at some instant. The feedback data rate (D_{ f }) is assumed to be 20 kbps. In all figures in this section, the markers represent the simulation results while the solid lines represent the results obtained by the provided equations.
Conclusions
In this article, we present a deep analysis on the quantization effects on the energy resources, where we obtain a closed form expression for the transmit energy in quantized and nonquantized feedback systems, the power outage probability, and the additional energy due to quantization.
In addition, we formulate an optimization problem, that aids to find the optimum number of quantization bits that minimizes the overall energy consumption.
The obtained equations for each effect are checked by simulation, where the results show a very tight agreement with the mathematical formulation. As a future work, the optimization problem could be reformulated for systems with multiple antenna and variable transmission rates.
Appendix
Derivation of Equation (8)
as given in Equation (8).
Endnotes
^{a}A single cell is tackled, and no interference from other cells is considered. The article’s contribution is not targeted to cellular networks but to the quantization and energy efficiency.^{b}Along the article, all the users are assumed to have the same average channel characteristics, and showing the same distribution for the channel and SNR value. If this is not the case (e.g., heterogeneous users distribution in the cell, with some users far from the BS), then a channel normalization (e.g., division by the path loss) can be accomplished for such a scenario. This is done because the article’s contribution is independent of the path loss.
Declarations
Acknowledgements
This study was funded by the Research Projects GREENET (PITNGA2010264759), CO2GREEN (TEC201020823), and GREENT (TSI020400201116CP8006).
Authors’ Affiliations
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Copyright
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.