Open Access

Resource Allocation for the Multiband Relay Channel: A Building Block for Hybrid Wireless Networks

EURASIP Journal on Wireless Communications and Networking20102010:792410

DOI: 10.1155/2010/792410

Received: 1 June 2009

Accepted: 17 February 2010

Published: 29 March 2010

Abstract

We investigate optimal resource allocation for the multiband relay channel. We find the optimal power and bandwidth allocation strategies that maximize the bounds on the capacity, by solving the corresponding max-min optimization problem. We provide sufficient conditions under which the associated max-min problem is equivalent to a supporting plane problem, which renders the solution for an arbitrary number of bands tractable. In addition, the sufficient conditions derived are general enough so that a class of utility functions can be accommodated with this formulation. As an example, we concentrate on the case where the source has two bands and the relay has a single band available and find the optimal resource allocation. We observe that joint power and bandwidth optimization always yields higher achievable rates than power optimization alone, establishing the merit of bandwidth sharing. Motivated by our analytical results, we examine a simple scenario where new channels become available for a transmitter to communicate; that is, new source to relay bands are added to a frequency division relay network. Given the channel conditions of the network, we establish the guidelines on how to allocate resources in order to achieve higher rates, depending on the relative quality of the available links.

1. Introduction

Future wireless networks are expected to enable nodes to communicate over multiple technologies and hops. Recent advances in the development of software defined radios support the vision where agile radios are employed at each node that utilize multiple standards and communicate seamlessly. Indeed, an intense research effort is directed towards having multiple communication standards coexist within one system, for example, the cellular network and IEEE 802.11 WLAN as in [1, 2]. We refer to a group of nodes capable of employing a number of communication technologies to find the best multihop route between the source-destination pairs, as a hybrid wireless network.

In this paper, we consider a simple hybrid wireless network with a source destination pair and aim at understanding its performance limits, that is, information theoretic rates with optimal resource allocation. In particular, we consider a scenario where a source node can communicate over multiple frequency bands to its destination, and a node that overhears the source transmission acts as a relay. We assume that the frequency bands that the source utilizes as well the ones used by the relay node are mutually orthogonal. The different bands are envisioned to represent links that operate with different wireless communication standards.

There has been considerable research effort up to date towards characterizing the information theoretic capacity of relay channels [37]. Most of the earlier work on relay channel capacity assumes that simultaneous transmission and reception at the relay is possible [4]. Since this is difficult to implement, recent work considers employing orthogonality at the relay via time-division [5, 810], frequency-division [11, 12], or code-division [13, 14]. To compensate the loss of spectral efficiency caused by this architecture and to increase the capacity, optimal resource allocation has been considered in [5, 8, 10, 11, 15, 16]. The optimal power and time slot duration allocation for the time-division relay channel has been considered in [5]. The work in [8] investigates three half-duplex time-division protocols that vary in the method of broadcasting they employ and the existence of receiver collision. The optimal power and time-slot allocation has been investigated for the protocol with the maximum degree of broadcasting and no receiver collision in [5].

We note that resource allocation in wireless relay networks is employed by utilizing the received SNR and the channel state information which are typically assumed to be available at the source and the relay node [5, 8, 10, 11, 16]. Notably, [16] studies optimal power and bandwidth allocation strategies for collaborative transmit diversity schemes for the situation when the source and the relay know only the magnitudes of the channel gains. The outage minimization and the corresponding optimal power control are considered when the network channel state is available at the source and the relay [10]. The model considered in this paper is in accordance with previous work and utilizes the received SNRs that are available at the source and the relay in order to find optimal resource allocation strategy.

In this paper, we investigate the optimal resource allocation strategies that maximize the capacity bounds for a simple hybrid wireless relay network. The channel model in this work can be traced back to a class of orthogonal relay networks first proposed in [11]. The three-node relay network in [11] is composed of two parts: a broadcast channel from the source node to the relay and destination node, and a separate orthogonal link from the relay node to the destination node. The parallel channel counterpart of [11] is later examined in [15]. A sum power constraint is imposed on the source node, and the relay node is restricted to perform a partial decode and forward operation. The sum rate from the source to the destination is then maximized by performing power allocation among different subchannels and the time sharing factor between the two parts of the network. A supporting plane technique is proposed in [15] to solve the associated max-min optimization problem. The results for the parallel network are then applied to the block fading model [15].

The model considered in this work is similar to the parallel relay network in [15]; yet, for the hybrid wireless network considered, the rate maximization leads to a different optimization problem than [15]: in a hybrid network, in addition to power allocation among different bands, it is conceivable to consider bandwidth allocation as well, and we find that the joint optimal power and bandwidth allocation yields higher rate than power optimization only. It is worth mentioning that dynamic bandwidth allocation is beneficial for a hybrid wireless network even in a scenario of a flat overall band. This is because different systems (standards) may exhibit different received SNR behavior even if the underlying channel gain and noise level are the same. This can be caused, for example, by different coding schemes or different requirements on feedback. Thus, one system will not, in general, be invariably better than all the others over all links.

At the outset, the joint power and bandwidth optimization appears challenging. Luckily, the resulting max-min optimization problem, we show, conforms to a set of sufficient conditions that render the solution manageable, even for an arbitrarily large number of bands. The technique that we can use under these sufficient conditions is the supporting plane technique used in [15]. We remark that the sufficient conditions are general enough that a class of utility functions can be optimized using the technique although our focus is on the information theoretic rates. This implies that the optimization technique used in this paper can be incorporated as a building block in a variety of resource allocation settings.

Lastly, in order to gain insight into the impact of optimal resource allocation on the construction of a hybrid wireless network, we examine a scenario where new wireless links can be added to the classical frequency division relay network to form a simple hybrid wireless network. Given the channel conditions between nodes, we study how to allocate resources to achieve the higher achievable rate. We observe that the source node is encouraged to communicate over the best network by dedicating all resources exclusively when condition of source-to-relay (SR) link and source-to-destination (SD) link of the new network is better (or worse) than that of SD link and SR link of the current network. Otherwise, it is beneficial to share resource between the current network and the new network to achieve a higher rate.

2. The Multiband Relay Channel

We consider the multiband relay channel (MBRC), which models a three-node hybrid wireless network where multiple frequency bands available from the source and the relay are mutually orthogonal. In particular, the situation where, among total https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq1_HTML.gif channels, there are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq2_HTML.gif channels available for the source node and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq3_HTML.gif for the relay node, shown in Figure 1, is termed the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq4_HTML.gif -MBRC.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Fig1_HTML.jpg
Figure 1

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq5_HTML.gif Multiband Relay Channel.

The source node transmits information over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq6_HTML.gif orthogonal channels to the relay and the destination node. The relay node uses a decode-and-forward scheme [4]. The https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq7_HTML.gif -MBRC input-output signal model is thus given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ1_HTML.gif
(1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq8_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq9_HTML.gif are the transmitted signal vectors from the source node and the relay node, respectively. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq10_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq11_HTML.gif are the received signal vectors at the destination node and the relay node when the signal is transmitted from the source node. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq12_HTML.gif is the received signal vector at the destination from the relay. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq13_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq14_HTML.gif is the zero-mean independent additive white Gaussian noise (AWGN) vector with covariance matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq15_HTML.gif at the relay node. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq16_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq17_HTML.gif are the zero-mean independent AWGN vectors with covariance matrices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq18_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq19_HTML.gif at the destination node. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq20_HTML.gif denotes the transpose operation, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq21_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq22_HTML.gif diagonal matrix. Since channels are independent, the channel transition probability mass function is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ2_HTML.gif
(2)

and we have the following theorem.

Theorem 1.

The upper and lower bounds for the capacity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq23_HTML.gif -MBRC are
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ3_HTML.gif
(3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ4_HTML.gif
(4)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq24_HTML.gif is the mutual information between https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq25_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq26_HTML.gif . The input distribution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq27_HTML.gif is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ5_HTML.gif
(5)

Proof.

The lower bound is obtained by taking the maximum of all possible transmission rates given the total number of bands; that is, the lower bound includes all possible transmission schemes which depend on whether the transmission from the source band(s) is decoded at the relay.

We define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq28_HTML.gif as the set of bands in which the transmission from the source is decoded at the relay. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq29_HTML.gif is the complement of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq30_HTML.gif and includes the set of bands for direct communication. For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq31_HTML.gif -MBRC, the lower bound is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ6_HTML.gif
(6)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq32_HTML.gif is the transmitted signal vector from the source and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq33_HTML.gif is the transmitted signal vector from the source intended for direct transmission. Similarly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq34_HTML.gif is the transmitted signal from the relay. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq35_HTML.gif is the received signal vector at the relay. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq36_HTML.gif is the received signal vector at the destination. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq37_HTML.gif is the received signal vector at the destination as a result of direct transmission. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq38_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq39_HTML.gif are given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ7_HTML.gif
(7)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ8_HTML.gif
(8)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq40_HTML.gif is the input joint distribution with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq41_HTML.gif . Similarly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq42_HTML.gif is the input joint distribution with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq43_HTML.gif . We note that (7) can be readily obtained by using the results in [4] by taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq44_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq45_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq46_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq47_HTML.gif . Applying the same approach, we obtain the following from the cut set bound [17]:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ9_HTML.gif
(9)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq48_HTML.gif . Following a similar approach to [11], (5) can be shown to maximize the mutual information in (7)–(9), and the optimization over (5) leads to (3)-(4).

3. Capacity Bounds and Optimal Resource Allocation

In the remainder of the paper, we will consider optimal resource allocation on the bounds obtained for the MBRC, that is, for hybrid wireless networks where the source node has access to distinct bands (standards) and a second node that overhears the source information relays to the destination using additional orthogonal bands. We consider the Gaussian case, where all the transmitted signals are corrupted by additive white Gaussian noises.

We have the input-output signal model given by (1) under source and relay power constraints:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ10_HTML.gif
(10)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq49_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq50_HTML.gif are the total available power at the source and relay node. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq52_HTML.gif are the nonnegative power allocation parameters for each orthogonal band at the source and relay node, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq53_HTML.gif . Unlike [5, 10], we do not have a total power constraint between the source and the relay and assume that each has its own battery.

We assume that the system has total bandwidth https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq54_HTML.gif . We define the received SNRs at the relay and the destination over channel https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq55_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ11_HTML.gif
(11)

Note that the actual received SNR values are the scaled versions of (11) depending on the power and bandwidth allocation. For example, the actual received SNR at the relay from channel 1, which is allocated https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq56_HTML.gif fraction of the source power and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq57_HTML.gif fraction of the bandwidth, simply is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq58_HTML.gif . Given the received SNRs which are available at the source and relay, our aim is to find the optimal resource allocation parameters that maximize capacity lower bound in terms of the transmitted power and the total bandwidth for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq59_HTML.gif -MBRC, which leads to optimally allocating the source power among https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq60_HTML.gif source bands, the relay power among https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq61_HTML.gif relay bands, and the total bandwidth among https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq62_HTML.gif bands. We can obtain the capacity lower and upper bounds of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq63_HTML.gif -MBRC from Theorem 1 as follows.

Theorem 2.

The upper and lower bounds for the capacity of the Gaussian https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq64_HTML.gif -MBRC are
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ12_HTML.gif
(12)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ13_HTML.gif
(13)

We omit the proof for Theorem 2 since the derivation for each mutual information follows directly from [15]. For each broadcast channel, if the relay node sees a higher received SNR than the destination node, then a superposition coding scheme [17] is used to convey independent information to the relay node, which cannot be decoded by the destination directly. The relay node then collects this information from all the channels where superposition coding is used, and transmits it to the destination at the appropriate rate.

Based on whether the relay node is utilized by a certain channel (band), we note that there are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq65_HTML.gif possible schemes. We observe that these https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq66_HTML.gif schemes are not exclusive to each other, since a superposition coding scheme may be reduced to a direct source-to-destination transmission if no band is allocated to the relay-to-destination link. We also note that which scheme yields the largest rate is completely decided by the SNR relationship, namely, the componentwise relationship between the received SNRs of the source-to-relay links, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq67_HTML.gif and the received SNRs of the source-to-destination links, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq68_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq69_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq70_HTML.gif , then for any bandwidth allocation, the signal received by the relay over this broadcast channel can be viewed as a degraded version of the signal received by the destination. Therefore, direct link transmission should be used for this band, regardless of what scheme is used for the other bands. On the other hand, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq71_HTML.gif , then the relay node can always learn something more than the destination node over this band and uses the superposition code scheme, and although the superposition scheme may be reduced to a direct link transmission scheme, optimizing under this scheme does not incur any rate loss. Based on these observations, we conclude that there is no need to examine all the schemes to find the best rate and the corresponding resource allocation. That is, practically, the system checks the received SNRs and chooses one of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq72_HTML.gif schemes satisfying the relationship of the received SNRs to communicate and the rate with optimized resource allocation for the chosen scheme is the maximum achievable rate, and the corresponding resource allocation is the globally optimal solution.

Next, we maximize the capacity lower bound in (12). To achieve this goal, we introduce the following general max-min optimization problem. We define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq73_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq74_HTML.gif as any utility function with any resource allocation vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq75_HTML.gif over the convex set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq76_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ14_HTML.gif
(14)

Proposition 1.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq77_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq78_HTML.gif are nonnegative and concave over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq79_HTML.gif , there must exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq80_HTML.gif such that maximizing the following equation with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq81_HTML.gif is equivalent to (14):
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ15_HTML.gif
(15)

Proof.

See Appendix A.

Note that the optimization problem in (14) corresponds to finding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq82_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq83_HTML.gif maximizing the minimum of two end points in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq84_HTML.gif . One possible technique to solve the max-min optimization problem in (14) is given by the following proposition [15], which we will also utilize.

Proposition 2 ([15, Proposition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq85_HTML.gif ]).

The relationship between optimal resource allocation parameters https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq86_HTML.gif and the corresponding optimal point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq87_HTML.gif is given by the following.

Case  1:If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq88_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq89_HTML.gif

Case  2:If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq90_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq91_HTML.gif .

Case  3:Neither case 1 nor 2 occurs; under this case, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq92_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq93_HTML.gif .

Now, one can restate our max-min optimization problem given in Theorem 2 as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ16_HTML.gif
(16)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq94_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq95_HTML.gif are the first and the second terms of max-min optimization problem in (12). Next, one needs to prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq97_HTML.gif are concave over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq98_HTML.gif in (16). Define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ17_HTML.gif
(17)

It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq99_HTML.gif is continuous over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq100_HTML.gif . Then, one has the following proposition.

Proposition 3.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq101_HTML.gif is concave over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq102_HTML.gif .

Proof.

First, note that due to the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq103_HTML.gif , we only need to prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq104_HTML.gif is concave over the interior of the region, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq105_HTML.gif . This is done by examining the Hessian, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq106_HTML.gif , of (17). The second-order derivatives of (17) with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq107_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq108_HTML.gif are
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ18_HTML.gif
(18)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ19_HTML.gif
(19)

We note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq109_HTML.gif .

The Hessian is the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq110_HTML.gif block diagonal matrix with the following matrix in its https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq111_HTML.gif th diagonal:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ20_HTML.gif
(20)

It is readily seen that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq112_HTML.gif is singular. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq113_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq114_HTML.gif from (18), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq115_HTML.gif is the negative semidefinite. Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq116_HTML.gif is concave over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq117_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq118_HTML.gif is continuous over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq119_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq120_HTML.gif is concave over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq121_HTML.gif .

We note that for any choice of set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq122_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq123_HTML.gif corresponds to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq124_HTML.gif in (17) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq125_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq126_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq127_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq128_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq129_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq130_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq131_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq132_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq133_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq134_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq135_HTML.gif , the Hessian is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq136_HTML.gif block diagonal matrix. Similarly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq137_HTML.gif corresponds to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq138_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq139_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq140_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq141_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq142_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq143_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq144_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq145_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq146_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq147_HTML.gif , the Hessian is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq148_HTML.gif block diagonal matrix.

Remark 1.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq149_HTML.gif is concave over the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq150_HTML.gif , it is also concave over any convex subset of it. Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq151_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq152_HTML.gif are concave over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq153_HTML.gif . (It is readily seen that the sum constraints define a convex set.) This establishes that the local optimal for (16) is also the global optimal [18, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq154_HTML.gif , page 125-126].

Remark 2.

We further find that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq155_HTML.gif is strictly concave over any convex subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq156_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq157_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq158_HTML.gif , jointly when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq159_HTML.gif , are held constant. Note that when all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq160_HTML.gif , are held constant, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq161_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq162_HTML.gif as a function of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq163_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq164_HTML.gif . In this case, it is easily seen that the Hessian is the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq165_HTML.gif diagonal matrix in which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq166_HTML.gif th diagonal term is given by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq167_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq168_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq169_HTML.gif . Since now all of the diagonal terms are strictly negative when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq170_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq171_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq172_HTML.gif is strictly concave over all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq173_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq174_HTML.gif , jointly when all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq175_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq176_HTML.gif , are held constant. Similarly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq177_HTML.gif is strictly concave over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq178_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq179_HTML.gif , jointly when all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq180_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq181_HTML.gif , are held constant. Since if a function is strictly concave over a set, it is also strictly concave over any convex subset of that set, the preceding argument implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq182_HTML.gif is strictly concave over any convex subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq183_HTML.gif , when all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq184_HTML.gif , are held constant. This fact will be useful in the sequel.

Based on Proposition 1 and Proposition 3, the methodology given in Proposition 2 can be applied to our max-min optimization problem in (16) for an arbitrary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq185_HTML.gif . That said, in the remainder of the paper, we will examine the optimal resource allocation for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq186_HTML.gif -MBRC where the source has two bands and the relay has a single band available to communicate and uses its own full power https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq187_HTML.gif . We find this network model representative and meaningful because of the following two observations. First, if there is more than one band available for the link between the relay and the destination, then only the best band among them will be used. This can be seen by fixing the overall band for this link and performing joint power and bandwidth optimization. Therefore, as long as the relay-to-destination SNRs are different, which is usually the case in practice, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq188_HTML.gif -MBRC will have the same resource allocation parameters as those of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq189_HTML.gif -MBRC. Secondly, the case with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq190_HTML.gif is similar to the case with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq191_HTML.gif except that there are more schemes to choose from. Therefore, we focus on the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq192_HTML.gif -MBRC in the sequel.

3.1. Maximization of Capacity Bounds for the Gaussian https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq193_HTML.gif -MBRC

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq194_HTML.gif -MBRC, there are four schemes to choose from. Let us label them Schemes I through IV. From Theorem 2, upper and lower bounds for the capacity of the Gaussian https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq195_HTML.gif -MBRC are
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ21_HTML.gif
(21)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ22_HTML.gif
(22)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq196_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq197_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq198_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq199_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq200_HTML.gif for schemes I to IV, respectively. Each scheme materializes as a function of the received SNRs as follows.

Scheme   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq201_HTML.gif : https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq202_HTML.gif , the scenario where transmission from the source node over both links is decoded at the relay node. This scheme is chosen if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq203_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq204_HTML.gif .

Scheme   II: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq205_HTML.gif , the scenario where transmission from the source node over band 1 is decoded at the relay node while band 2 is used for direct transmission only. This scheme is chosen if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq206_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq207_HTML.gif .

Scheme   III: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq208_HTML.gif , the scenario where transmission from the source node over band 2 is decoded at the relay node while band 1 is used for direct transmission only. This scheme is chosen if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq209_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq210_HTML.gif .

Scheme   IV: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq211_HTML.gif , the scenario where transmissions from the source node from both bands are used only for direct transmission. This scheme is chosen if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq212_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq213_HTML.gif .

We define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq214_HTML.gif as the optimal resource allocation parameters for (21). https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq215_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq216_HTML.gif are the first and second terms in (21). From (21), we note that the capacity for scheme IV is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ23_HTML.gif
(23)
In this case, the max-min optimization problem reduces to a maximization problem and it is readily shown that the optimal resource allocation for the rate of scheme IV is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ24_HTML.gif
(24)

For schemes I, II, III, once the appropriate scheme is decided upon, parameters https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq217_HTML.gif can be substituted accordingly and we can examine https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq218_HTML.gif for each of the cases in Proposition 2.

Case 1.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq219_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq220_HTML.gif maximizes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq221_HTML.gif .

This case holds if the following condition is satisfied:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ25_HTML.gif
(25)
and we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ26_HTML.gif
(26)
The received SNRs must satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ27_HTML.gif
(27)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ28_HTML.gif
(28)

Proof.

See Appendix B.

Case 2.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq222_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq223_HTML.gif maximizes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq224_HTML.gif .

This case holds if the following condition is satisfied:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ29_HTML.gif
(29)
and we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ30_HTML.gif
(30)

Proof.

See Appendix B.

Remark 3.

By substituting the appropriate parameters for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq225_HTML.gif for each scheme into (30), we observe that Case 2 does not ever materialize for schemes I, II, III.

Case 3.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq226_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq227_HTML.gif maximizes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq228_HTML.gif for a fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq229_HTML.gif .

This case occurs when (25) or (29) doES not hold. The closed form solution for this optimization problem does not exist. Thus, we have to rely on an iterative algorithm. We propose to use alternating maximization algorithm that calls for optimizing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq230_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq231_HTML.gif in one stage, followed by optimizing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq232_HTML.gif in the next stage. The iterations are obtained by finding KKT points of the corresponding optimization problem with the variable vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq233_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq234_HTML.gif . We note that the objective function is not differentiable at the boundary of the feasible region, that is, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq235_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq236_HTML.gif and the corresponding KKT points are not defined. Thus, we need to introduce a small positive value, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq237_HTML.gif , and define a modified feasible region as illustrated in (B.3) and (B.4) that excludes the boundary point. Every time an iteration reaches the boundary of the new feasible region, we expand the feasible region by successively reducing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq238_HTML.gif so that we can continue with the iterations until convergence. The detailed description of the following proposed iterative algorithm and proof of its convergence to the global optimal solution is given in Appendix C.

Step 1.
  1. (i)

    Initialization: for initial values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq239_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq240_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq241_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq242_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq243_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq244_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq245_HTML.gif , and assign values to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq246_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq247_HTML.gif .

     
  2. (ii)

    Iteration https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq248_HTML.gif : update https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq249_HTML.gif by finding the solution of KKT condition of (C.2) with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq250_HTML.gif ; find https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq251_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq252_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq253_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq254_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq255_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq256_HTML.gif .

     
  3. (iii)

    Iteration https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq257_HTML.gif : update https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq258_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq259_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq260_HTML.gif by finding the solution of KKT condition of (C.2) with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq261_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq262_HTML.gif ; find https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq263_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq264_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq265_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq266_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq267_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq268_HTML.gif .

     
  4. (iv)

    Repeat step (ii) until the optimal https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq269_HTML.gif is found by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq270_HTML.gif in (C.3).

     

Step 2.

If the iteration does not reach the boundary of the feasible region of (B.3) and (B.4), the algorithm terminates.

Step 3.

Otherwise, set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq271_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq272_HTML.gif in (B.3) and (B.4) and repeat Steps (1) to (2) by using the KKT points from the previous iteration as the initial points. (For numerical results, we use https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq273_HTML.gif .)

We reiterate that based on the scheme at hand, we would substitute the correct parameters for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq274_HTML.gif to find the optimal resource allocation strategy.

3.2. Upper Bound on Capacity

Recall that the upper bound given by (13) is obtained by the max-flow min-cut theorem. The maximization for the upper bound follows same steps to that of the lower bound, details of which we will omit here. In general, the upper bound is not tight. One exception is that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq275_HTML.gif -MBRC, since Case 2 for schemes I, II, and III is not possible, the optimal resource allocation parameters https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq276_HTML.gif maximize https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq277_HTML.gif (Case 1) or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq278_HTML.gif (Case 3). There exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq279_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq280_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq281_HTML.gif , otherwise https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq282_HTML.gif . Since the first term of the upper bound in (22) is the same as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq283_HTML.gif , we know that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq284_HTML.gif -MBRC, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq285_HTML.gif maximizes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq286_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq287_HTML.gif and the resulting optimized rate is the capacity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq288_HTML.gif -MBRC. A similar observation was made for the frequency division relay network, that is, when one band exists from the source in [11]. It is interesting to observe that the same observation extends to the multiband case.

4. Numerical Results and Discussion

4.1. Capacity Bounds

In this section, we present numerical results to support our analysis described in Section 3. Specifically, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq289_HTML.gif -MBRC, we plot the capacity lower bound (LB) obtained by optimal resource allocation as well as the capacity upper bound (UB) with the same resource allocation parameters. For comparison purposes, we also consider the case where overall bandwidth https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq290_HTML.gif is equally divided between the three bands and only optimal power allocation is done.

Figure 2 shows the capacity UB and LB for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq291_HTML.gif -MBRC with optimal power allocation only. When the source-to-relay (SR) SNRs https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq292_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq293_HTML.gif are smaller than or equal to the source-to-destination (SD) SNRs https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq294_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq295_HTML.gif , respectively, the lower bound does not increase and saturate even if the relay-to-destination (RD) SNR https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq296_HTML.gif increases. This is expected, since using the relay is not beneficial when the source-to-relay channel is worse than the source-to-destination channel.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Fig2_HTML.jpg
Figure 2

Upper and lower bounds of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq297_HTML.gif -MBRC with power optimization only: SNRs at SD, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq298_HTML.gif = 10 dB and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq299_HTML.gif = 5 dB.

In contrast, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq300_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq301_HTML.gif are larger than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq302_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq303_HTML.gif , respectively, the lower bound increases as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq304_HTML.gif increases and saturates after a certain threshold of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq305_HTML.gif . This threshold becomes larger as the quality of the SR links improves as compared to the SD links, that is, as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq306_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq307_HTML.gif get larger compared to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq308_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq309_HTML.gif . Indeed, the fact that we can achieve higher rates when the SR channel is better than the SD channel is intuitively pleasing as the power allocation becomes more effective when we have a better SR channel. It is noticeable that the upper and lower bounds approach each other as the SR link quality improves as compared to that of SD.

Figure 3 shows the capacity UB and LB for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq310_HTML.gif -MBRC with joint optimal power and bandwidth allocation. We observe that the lower bound does not saturate when the SR links are better than the SD links. This additional improvement is thanks to the dynamic bandwidth allocation. By comparing Figure 2 and 3, we observe that the achievable rate of MBRC with joint optimal power and bandwidth is always larger than that of power optimization only, sometimes by a significant margin. This points to the advantage of joint power and bandwidth optimization, promoting the idea of different wireless technologies lending each other frequency resources to improve capacity.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Fig3_HTML.jpg
Figure 3

Upper and lower bounds of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq311_HTML.gif -MBRC with joint power and bandwidth optimization: SNRs at SD, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq312_HTML.gif = 10 dB and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq313_HTML.gif = 5 dB.

4.2. Guidelines for Hybrid Network Design

When a new wireless link becomes available at the source in addition to the existing single band relay network, a hybrid wireless network can be formed. In this case, a meaningful question is how to allocate resources between links in order to maximize the data rate. It is evident that the resource allocation strategy is a function of the channel quality of the available links (SD/SR/RD). To answer this question, we compare the achievable rates with optimal resource allocation for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq314_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq315_HTML.gif and observe the effect of adding a new link on the maximum achievable rate.

Figure 4 shows the achievable rates when the new SR ink is better than the current SR link, and the new SD link is better than the current SD link. Comparing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq316_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq317_HTML.gif , we observe that the achievable rate of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq318_HTML.gif is better than that of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq319_HTML.gif . This is because quality of the new link is better than that of the current link, and all resources are allocated to the new link. If the new links were worse, the maximum achievable rates would stay the same since all resources would be allocated to the current link.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Fig4_HTML.jpg
Figure 4

Comparison of achievable rates: the new SR ink is better than the current SR link, and the new SD link is better than the current SD link.

Figure 5 shows the achievable rates when the new SR link is better than the current SR link, and the new SD link is worse than the current SD link. We observe that the achievable rates for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq320_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq321_HTML.gif are almost same for low RD SNR. This is because when the RD link is poor, the relay becomes less useful, and most of bandwidth and power are allocated into channel with the best direct link. As the RD SNR increases, we observe that the achievable rate for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq322_HTML.gif is larger than that of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq323_HTML.gif . This is because it is optimal resource allocation that we allocate more bandwidth and power to the new link with the best SR link. The observation is justified by examining bandwidth allocation (the power allocation follows a similar pattern) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq324_HTML.gif shown in Figure 6. We see that more bandwidth is allocated to the current link ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq325_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq326_HTML.gif ) for low received RD SNR. More bandwidth is allocated to the new link ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq327_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq328_HTML.gif ) when the RD link becomes better. We also observe that Case 1 and Case 3 of our proposed optimal resource allocation occur depending on the RD SNR: with both SR SNRs better than both SD SNRs, Case 1 occurs at low RD SNR (from 0 dB to 10 dB); otherwise, the optimal resource allocation corresponds to Case 3. We note that the optimal resource allocation scheme would be reversed if the new SR link were worse than the current SR link, and the new SD link were better than the current SD link.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Fig5_HTML.jpg
Figure 5

Comparison of achievable rates: the new SR link is better than the current SR link, and the new SD link is worse than the current SD link.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Fig6_HTML.jpg
Figure 6

Optimal bandwidth allocation: the new SR link is better than the current SR link, and the new SD link is worse than the current SD link.

We note that the given received SNRs in the numerical results correspond to scheme I (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq329_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq330_HTML.gif ). Similarly, we can examine the effect of adding a new link under different received SNR relationship between https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq331_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq332_HTML.gif which corresponds to scheme II or scheme III, and we could readily apply the optimal resource allocation solution found in Section 3.1.

5. Conclusions

In this paper, we have investigated the optimal resource allocation for a hybrid three-node relay network where the source, with the help of a relay node, communicates to the destination via multiple orthogonal channels (MBRCs). In particular, we have studied joint optimal power and bandwidth allocation strategies that maximize the bounds on the capacity, which results in a max-min optimization problem. We have solved this problem using a supporting plane technique [15]. In particular, we have provided sufficient conditions for when this max-min optimization problem can be solved using this technique. It is worthwhile to mention that these sufficient conditions are general enough so that other utility functions that rely on SNR can be considered as well as the information theoretic rates considered in this paper.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq333_HTML.gif -MBRC, we have found the joint power and bandwidth allocation. We have observed that the upper and lower bounds approach each other as the source-to-relay channel condition improves as compared to the source-to-destination channel condition, and joint power and bandwidth optimization always yields better performance than power optimization only.

Our numerical results have also investigated the scenario where a new link at the source becomes available for an existing frequency division relay network, and the power and bandwidth resources are to be reallocated. We have observed that the source node is encouraged to communicate over the best link by dedicating all resources when the new SR link and SD link are better (or worse) than the current SD link and SR link. Otherwise, it is beneficial to share resources between the current link and the new link to achieve the higher rate.

The simple MBRC investigated in this paper can be considered as a building block for more complex hybrid wireless networks. From the system design point of view, we conclude that, for this two-hop, simple network, higher achievable rates can be obtained by optimally allocating resources between multiple standards. It would be of interest to gain an understanding of the set of conditions under which using multiple communication links (standards) and optimal sharing of resources would be beneficial for multihop hybrid wireless networks.

Appendices

A. Proof of Proposition 1

Proof.

Suppose that both https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq334_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq335_HTML.gif are nonnegative and concave over convex set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq336_HTML.gif . Then, we claim that the optimization problem (14) can be relaxed as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ31_HTML.gif
(A.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ32_HTML.gif
(A.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ33_HTML.gif
(A.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ34_HTML.gif
(A.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ35_HTML.gif
(A.5)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ36_HTML.gif
(A.6)

To see that, we devise the following notion of dominance: pair ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq337_HTML.gif ) is said to be dominated by ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq338_HTML.gif ) if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq339_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq340_HTML.gif . We say that a set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq341_HTML.gif is dominated by the other set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq342_HTML.gif , or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq343_HTML.gif if every point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq344_HTML.gif is dominated by some point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq345_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq346_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq347_HTML.gif are concave over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq348_HTML.gif , we realize that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq349_HTML.gif dominates its convex closure https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq350_HTML.gif . Furthermore, from the definition of dominance closure in (A.5), it is easy to see https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq351_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq352_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq353_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq354_HTML.gif . We note that adding dominated points to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq355_HTML.gif does not change the value of optimization problem (14), which allows us to consider problem (A.1)–(A.6) instead.

Set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq356_HTML.gif has the following properties. ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq357_HTML.gif ) It is a closed convex set. To see that, consider two points in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq358_HTML.gif : https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq359_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq360_HTML.gif . Then we must have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq361_HTML.gif . ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq362_HTML.gif ) Consider any supporting plane of this set, which is a line in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq363_HTML.gif in this case. The slope of this line cannot be both positive and finite. Otherwise, suppose the supporting plane passes through point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq364_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq365_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq366_HTML.gif , defined by (A.6), will not be in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq367_HTML.gif .

We then observe that (A.1)–(A.6) must be solved when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq368_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq369_HTML.gif is at the boundary of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq370_HTML.gif . The maximum of (A.1)–(A.6) should be attained at the boundary of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq371_HTML.gif since every interior point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq372_HTML.gif must be dominated by some point at its boundary. Also, there must be such a point on the boundary with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq373_HTML.gif . We then show that the point with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq374_HTML.gif must be a local maximal point. This is because any improvement over this point would require increasing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq375_HTML.gif simultaneously. Suppose such improved point exists. Then it will be strictly separated from set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq376_HTML.gif by the support plane passing through https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq377_HTML.gif , since no supporting plane has finite positive slope. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq378_HTML.gif is a closed convex set and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq379_HTML.gif is a concave function over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq380_HTML.gif , any local maximum must be globally optimal [18,Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq381_HTML.gif , page 125-126]. This completes our claim that optimality must be attained at the diagonal line. We observe that this point may not be in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq382_HTML.gif . Therefore, it may not be parameterizable by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq383_HTML.gif . This necessitates consideration of the three cases as we will show later.

Since optimality must be attained at the boundary of the convex set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq384_HTML.gif , we observe that (A.1)–(A.6) is equivalent to (A.7) for a certain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq385_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ37_HTML.gif
(A.7)

(A.7) can be solved by examining three cases: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq386_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq387_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq388_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq389_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq390_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ38_HTML.gif
(A.8)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq391_HTML.gif , we prove below that (A.7) is equivalent to (A.9). Let the optimal solution of (A.7) be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq392_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq393_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq394_HTML.gif cannot be dominated by any point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq395_HTML.gif other than itself. (If such a point exists, it would be separated from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq396_HTML.gif by the supporting plane passing through https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq397_HTML.gif .) Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq398_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq399_HTML.gif , we must have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq400_HTML.gif . This means that we can solve (A.9) instead:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ39_HTML.gif
(A.9)
Since all points in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq401_HTML.gif can be parameterized by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq402_HTML.gif , problem (A.9) is equivalent to (A.10). https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq403_HTML.gif is adjusted until the solution to (A.10) yields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq404_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ40_HTML.gif
(A.10)

In summary, we proved that there must exist a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq405_HTML.gif such that (14) is equivalent to (A.7) if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq406_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq407_HTML.gif are nonnegative and concave over convex set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq408_HTML.gif . Depending on the value of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq409_HTML.gif , we find that there are three cases for the max-min optimization (A.7) as given in Proposition 2.

B. Optimal Resource Allocation

Proof.

We provide the proof for optimal resource allocation for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq410_HTML.gif -MBRC. To do so, we utilize the following theorem which we restate here with our notation for convenience.

Theorem 3 ([19, Proposition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq411_HTML.gif , pages 219–221]).

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq412_HTML.gif is continuously differentiable and concave on the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq413_HTML.gif , a Cartesian product of sets https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq414_HTML.gif , where each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq415_HTML.gif is a closed convex subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq416_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq417_HTML.gif . Furthermore, suppose that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq418_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq419_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq420_HTML.gif is a strictly concave function of each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq421_HTML.gif , when the other components of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq422_HTML.gif are held constant. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq423_HTML.gif be the sequence generated by the iterative algorithm obtained by optimizing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq424_HTML.gif over one vector variable at a time. Then, every limit point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq425_HTML.gif maximizes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq426_HTML.gif over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq427_HTML.gif .

For a fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq428_HTML.gif , the objective function of our max-min problem corresponds to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ41_HTML.gif
(B.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq429_HTML.gif corresponds to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq430_HTML.gif .

Based on Proposition 3, we know that the objective function (B.1) of our max-min optimization problem given in (21) is concave over the constraint set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq431_HTML.gif , which is a Cartesian product of the following two sets, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq432_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq433_HTML.gif , which are closed convex subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq434_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ42_HTML.gif
(B.2)
Also, for the interior points of the feasible region of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq435_HTML.gif , we note that (B.1) is strictly concave over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq436_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq437_HTML.gif are fixed; see Remark 2. We note that the objective function (B.1) is not differentiable at the boundary of the feasible region, that is, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq438_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq439_HTML.gif . Therefore, we cannot directly apply Theorem 3. We define a modified feasible region as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ43_HTML.gif
(B.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ44_HTML.gif
(B.4)

with a small https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq440_HTML.gif . Since (B.1) is continuously differentiable over the new feasible region given by (B.3) and (B.4), we can now apply Theorem 3. Thus, we can devise an iterative algorithm to maximize the function, by maximizing over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq441_HTML.gif for fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq442_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq443_HTML.gif for fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq444_HTML.gif ) and then maximizing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq445_HTML.gif for fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq446_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq447_HTML.gif for fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq448_HTML.gif ). The iterative method, by Theorem 3, will converge to the maximizer of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq449_HTML.gif over (B.3) and (B.4), which is the global optimal. If the iterative algorithm converges to a point that is at the boundary of the new feasible region given by (B.3) and (B.4), we need to reduce https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq450_HTML.gif further and repeat the iteration using the KKT points from the previous iteration as the initial point. By repeating this procedure, the iterative algorithm converges to the global optimal solution.

For Case 1 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq451_HTML.gif in (B.1)) and Case 2 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq452_HTML.gif in (B.1)), we provide the closed form solution below. For Case 3 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq453_HTML.gif in (B.1)), we provide the iterative algorithm in Appendix C.

Case 1.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq454_HTML.gif maximizes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq455_HTML.gif . Using https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq456_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq457_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq458_HTML.gif in (21) can be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ45_HTML.gif
(B.5)
Differentiating (B.5) with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq459_HTML.gif , we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq460_HTML.gif . Observe that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq461_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq462_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq463_HTML.gif . Substituting this expression for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq464_HTML.gif into (B.5), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ46_HTML.gif
(B.6)
For fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq465_HTML.gif , we can maximize (B.6) in terms of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq466_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ47_HTML.gif
(B.7)
Substituting (B.7) into (B.6), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ48_HTML.gif
(B.8)

Maximizing (B.8) with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq467_HTML.gif leads to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq468_HTML.gif . The optimal power allocation parameter in (B.7) indicates that one of channels whose received SNR at the destination is smaller is not used. Thus, for the case where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq469_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq470_HTML.gif . On the other hand, for the case where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq471_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq472_HTML.gif . Thus, using https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq473_HTML.gif , the optimal resource allocation parameter for Case 1 is given by (26). By Remark 1, this is the global optimal solution. The condition of the received SNRs given by (27) and (28) for Case 1 to occur can be readily found by substituting (26) into (25).

Case 2.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq474_HTML.gif maximizes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq475_HTML.gif . By applying the same technique as in Case 1, we find the optimal resource allocation parameters given in (30). By Remark 1, this is the global optimal solution.

C. Iterative Algorithm for Case 3

Proof.

For Case 3, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq476_HTML.gif maximizes the following.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ49_HTML.gif
(C.1)
Since the closed form solution does not exist for case 3, we rely on the iterative algorithm given in Theorem 3. As we noted in Appendix B, (C.1) is not differentiable at the boundary of the feasible region. Thus, we start with the new feasible region given in (B.3) and (B.4). Then, the Lagrangian is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ50_HTML.gif
(C.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq477_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq478_HTML.gif are Lagrange multipliers corresponding to sum constraints for bandwidth and power allocation, respectively. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq479_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq480_HTML.gif are inequality constraints for bandwidth and power allocation, respectively, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq481_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq482_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq483_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq484_HTML.gif . For a fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq485_HTML.gif , we start with values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq486_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq487_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq488_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq489_HTML.gif = 1 and find the optimal https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq490_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq491_HTML.gif . In iteration https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq492_HTML.gif , we update https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq493_HTML.gif by optimizing the objective function over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq494_HTML.gif while keeping the total power constraints satisfied, and fixing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq495_HTML.gif . In iteration https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq496_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq497_HTML.gif is found by optimizing the objective function with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq498_HTML.gif while keeping the bandwidth constraints satisfied, and fixing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq499_HTML.gif . By Remark 1 and 2, and Theorem 3, this algorithm converges to the global optimal solution. Note that the optimal solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq500_HTML.gif satisfies (see Proposition 2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ51_HTML.gif
(C.3)

Declarations

Acknowledgments

This work was supported in part by NSF Grants CNS-0626905 and CNS-0721445 and DARPA ITMANET Program via Grant W911NF-07-1-0028. An earlier version of this work was presented in part in Conference on Information Sciences and Systems (CISS), 2005, and in International Conference on Wireless Networks, Communications, and Mobile Computing (WirelessCom), 2005.

Authors’ Affiliations

(1)
Reverb Networks
(2)
Wireless Communications and Networking Laboratory, Department of Electrical Engineering, Pennsylvania State University

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Copyright

© Kyounghwan Lee et al. 2010

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