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

  • Kyounghwan Lee1,

    Affiliated with

    • Aylin Yener2Email author and

      Affiliated with

      • Xiang He2

        Affiliated with

        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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq1_HTML.gif channels, there are http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq4_HTML.gif -MBRC.
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Fig1_HTML.jpg
        Figure 1

        http://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 http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ1_HTML.gif
        (1)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq8_HTML.gif and http://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. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq10_HTML.gif and http://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. http://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. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq13_HTML.gif = http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq15_HTML.gif at the relay node. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq16_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq18_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq19_HTML.gif at the destination node. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq20_HTML.gif denotes the transpose operation, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq21_HTML.gif is an http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq23_HTML.gif -MBRC are
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ3_HTML.gif
        (3)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ4_HTML.gif
        (4)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq24_HTML.gif is the mutual information between http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq25_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq26_HTML.gif . The input distribution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq27_HTML.gif is
        http://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 http://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. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq29_HTML.gif is the complement of http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ6_HTML.gif
        (6)
        where http://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 http://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, http://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. http://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. http://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. http://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. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq38_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq39_HTML.gif are given by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ7_HTML.gif
        (7)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ8_HTML.gif
        (8)
        where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq41_HTML.gif . Similarly, http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq44_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq45_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq46_HTML.gif , and http://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]:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ9_HTML.gif
        (9)

        where http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ10_HTML.gif
        (10)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq49_HTML.gif and http://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. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq51_HTML.gif and http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq55_HTML.gif as
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq56_HTML.gif fraction of the source power and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq57_HTML.gif fraction of the bandwidth, simply is http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq60_HTML.gif source bands, the relay power among http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq64_HTML.gif -MBRC are
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ12_HTML.gif
        (12)
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq65_HTML.gif possible schemes. We observe that these http://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, http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq68_HTML.gif .

        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq69_HTML.gif , http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq73_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq75_HTML.gif over the convex set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq76_HTML.gif :
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ14_HTML.gif
        (14)

        Proposition 1.

        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq77_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq78_HTML.gif are nonnegative and concave over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq79_HTML.gif , there must exist http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq81_HTML.gif is equivalent to (14):
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq82_HTML.gif and http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq86_HTML.gif and the corresponding optimal point http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq88_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq89_HTML.gif

        Case  2:If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq90_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq92_HTML.gif , http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ16_HTML.gif
        (16)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq94_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq96_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq97_HTML.gif are concave over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq98_HTML.gif in (16). Define
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq99_HTML.gif is continuous over http://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.

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq101_HTML.gif is concave over http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq103_HTML.gif , we only need to prove that http://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, http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq107_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq108_HTML.gif are
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ18_HTML.gif
        (18)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ19_HTML.gif
        (19)

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

        The Hessian is the http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq111_HTML.gif th diagonal:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ20_HTML.gif
        (20)

        It is readily seen that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq112_HTML.gif is singular. Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq113_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq114_HTML.gif from (18), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq115_HTML.gif is the negative semidefinite. Thus, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq116_HTML.gif is concave over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq117_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq118_HTML.gif is continuous over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq119_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq120_HTML.gif is concave over http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq122_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq123_HTML.gif corresponds to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq124_HTML.gif in (17) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq125_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq126_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq127_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq128_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq129_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq130_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq131_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq132_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq133_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq134_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq135_HTML.gif , the Hessian is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq136_HTML.gif block diagonal matrix. Similarly, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq137_HTML.gif corresponds to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq138_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq139_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq140_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq141_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq142_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq143_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq144_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq145_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq146_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq147_HTML.gif , the Hessian is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq148_HTML.gif block diagonal matrix.

        Remark 1.

        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq149_HTML.gif is concave over the set http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq151_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq152_HTML.gif are concave over http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq156_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq157_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq158_HTML.gif , jointly when http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq160_HTML.gif , are held constant, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq161_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq162_HTML.gif as a function of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq163_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq165_HTML.gif diagonal matrix in which http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq166_HTML.gif th diagonal term is given by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq167_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq168_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq170_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq171_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq172_HTML.gif is strictly concave over all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq173_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq174_HTML.gif , jointly when all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq175_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq176_HTML.gif , are held constant. Similarly, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq177_HTML.gif is strictly concave over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq178_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq179_HTML.gif , jointly when all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq180_HTML.gif , http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq183_HTML.gif , when all http://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 http://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 http://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 http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq189_HTML.gif -MBRC. Secondly, the case with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq190_HTML.gif is similar to the case with http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq193_HTML.gif -MBRC

        For http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq195_HTML.gif -MBRC are
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ21_HTML.gif
        (21)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ22_HTML.gif
        (22)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq196_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq197_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq198_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq199_HTML.gif , and http://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   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq201_HTML.gif : http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq203_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq204_HTML.gif .

        Scheme   II: http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq206_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq207_HTML.gif .

        Scheme   III: http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq209_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq210_HTML.gif .

        Scheme   IV: http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq212_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq213_HTML.gif .

        We define http://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). http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq215_HTML.gif and http://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
        http://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
        http://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 http://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 http://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.

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq219_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq220_HTML.gif maximizes http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ25_HTML.gif
        (25)
        and we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ26_HTML.gif
        (26)
        The received SNRs must satisfy
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ27_HTML.gif
        (27)
        http://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.

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq222_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq223_HTML.gif maximizes http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ29_HTML.gif
        (29)
        and we obtain
        http://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 http://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.

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq226_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq227_HTML.gif maximizes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq228_HTML.gif for a fixed http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq230_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq231_HTML.gif in one stage, followed by optimizing http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq233_HTML.gif or http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq235_HTML.gif , http://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, http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq239_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq240_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq241_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq242_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq243_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq244_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq245_HTML.gif , and assign values to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq246_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq247_HTML.gif .

           
        2. (ii)

          Iteration http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq248_HTML.gif : update http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq250_HTML.gif ; find http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq251_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq252_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq253_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq254_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq255_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq256_HTML.gif .

           
        3. (iii)

          Iteration http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq257_HTML.gif : update http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq258_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq259_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq261_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq262_HTML.gif ; find http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq263_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq264_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq265_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq266_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq267_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq269_HTML.gif is found by http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq271_HTML.gif , http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq276_HTML.gif maximize http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq277_HTML.gif (Case 1) or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq278_HTML.gif (Case 3). There exists a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq279_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq280_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq281_HTML.gif , otherwise http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq283_HTML.gif , we know that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq284_HTML.gif -MBRC, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq285_HTML.gif maximizes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq286_HTML.gif for http://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 http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq292_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq294_HTML.gif and http://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 http://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.
        http://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 http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq298_HTML.gif = 10 dB and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq299_HTML.gif = 5 dB.

        In contrast, when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq300_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq301_HTML.gif are larger than http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq302_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq303_HTML.gif , respectively, the lower bound increases as http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq306_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq307_HTML.gif get larger compared to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq308_HTML.gif and http://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 http://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.
        http://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 http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq312_HTML.gif = 10 dB and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq314_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq316_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq318_HTML.gif is better than that of http://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.
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq320_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq322_HTML.gif is larger than that of http://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 http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq325_HTML.gif for http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq327_HTML.gif for http://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.
        http://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.

        http://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., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq329_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq331_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq334_HTML.gif and http://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 http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ31_HTML.gif
        (A.1)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ32_HTML.gif
        (A.2)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ33_HTML.gif
        (A.3)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ34_HTML.gif
        (A.4)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ35_HTML.gif
        (A.5)
        http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq337_HTML.gif ) is said to be dominated by ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq338_HTML.gif ) if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq339_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq340_HTML.gif . We say that a set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq341_HTML.gif is dominated by the other set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq342_HTML.gif , or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq343_HTML.gif if every point in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq344_HTML.gif is dominated by some point in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq345_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq346_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq347_HTML.gif are concave over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq348_HTML.gif , we realize that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq349_HTML.gif dominates its convex closure http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq351_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq352_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq353_HTML.gif , we have http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq356_HTML.gif has the following properties. ( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq358_HTML.gif : http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq359_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq360_HTML.gif . Then we must have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq361_HTML.gif . ( http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq364_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq365_HTML.gif , then http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq368_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq369_HTML.gif is at the boundary of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq371_HTML.gif since every interior point of http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq376_HTML.gif by the support plane passing through http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq378_HTML.gif is a closed convex set and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq379_HTML.gif is a concave function over http://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 http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq385_HTML.gif :
        http://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: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq386_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq387_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq388_HTML.gif .

        For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq389_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq390_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ38_HTML.gif
        (A.8)
        For http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq392_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq393_HTML.gif , http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq396_HTML.gif by the supporting plane passing through http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq397_HTML.gif .) Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq398_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq399_HTML.gif , we must have http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ39_HTML.gif
        (A.9)
        Since all points in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq401_HTML.gif can be parameterized by http://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). http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq404_HTML.gif :
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq406_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq408_HTML.gif . Depending on the value of http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq411_HTML.gif , pages 219–221]).

        Suppose that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq413_HTML.gif , a Cartesian product of sets http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq414_HTML.gif , where each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq415_HTML.gif is a closed convex subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq416_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq417_HTML.gif . Furthermore, suppose that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq418_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq419_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq421_HTML.gif , when the other components of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq422_HTML.gif are held constant. Let http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq425_HTML.gif maximizes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq426_HTML.gif over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq427_HTML.gif .

        For a fixed http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ41_HTML.gif
        (B.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq429_HTML.gif corresponds to http://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 http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq432_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq433_HTML.gif , which are closed convex subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq434_HTML.gif :
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq436_HTML.gif when http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq438_HTML.gif , http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ43_HTML.gif
        (B.3)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ44_HTML.gif
        (B.4)

        with a small http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq441_HTML.gif for fixed http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq442_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq443_HTML.gif for fixed http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq444_HTML.gif ) and then maximizing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq445_HTML.gif for fixed http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq446_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq447_HTML.gif for fixed http://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 http://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 http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq451_HTML.gif in (B.1)) and Case 2 ( http://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 ( http://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.

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq454_HTML.gif maximizes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq455_HTML.gif . Using http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq456_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq457_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq458_HTML.gif in (21) can be rewritten as
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq459_HTML.gif , we obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq460_HTML.gif . Observe that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq461_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq462_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq463_HTML.gif . Substituting this expression for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq464_HTML.gif into (B.5), we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ46_HTML.gif
        (B.6)
        For fixed http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq466_HTML.gif by
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq467_HTML.gif leads to http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq469_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq471_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq472_HTML.gif . Thus, using http://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.

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq474_HTML.gif maximizes http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq476_HTML.gif maximizes the following.
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_Equ50_HTML.gif
        (C.2)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq477_HTML.gif and http://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. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq479_HTML.gif and http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq481_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq482_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq483_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq484_HTML.gif . For a fixed http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq485_HTML.gif , we start with values of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq486_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq487_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq488_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq489_HTML.gif = 1 and find the optimal http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq490_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq491_HTML.gif . In iteration http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq492_HTML.gif , we update http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq493_HTML.gif by optimizing the objective function over http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq495_HTML.gif . In iteration http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq496_HTML.gif , http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F792410/MediaObjects/13638_2009_Article_2027_IEq500_HTML.gif satisfies (see Proposition 2)
        http://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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