Energy-Bandwidth Efficiency Tradeoff in MIMO
Multi-Hop Wireless Networks
Chih-Liang Chen, Student Member, IEEE, Wayne E. Stark, Fellow, IEEE, and Sau-Gee Chen Member, IEEE
Abstract—This paper considers a MIMO multi-hop network
and analyzes the relationship between its energy consumption and bandwidth efficiency. Its minimum energy consumption is formulated as an optimization problem. By taking both transmit antennas (TAs) and receive antennas (RAs) into consideration, the energy-bandwidth efficiency tradeoff in the networks is investigated. Moreover, the minimum energy of an equally-spaced relaying strategy is investigated for various numbers of antennas. In addition, the minimum energy over all possible antenna pairs is derived. Finally, the effect of the number of hops on the energy-bandwidth efficiency tradeoff is considered. For a fixed antenna pair, the minimum energy over all possible rates and hop numbers are obtained. Generally, the routes with more hops minimize the energy consumption in the low effective rate region. On the other hand, in the high effective rate region, the routes with fewer hops minimize the energy consumption.
Index Terms—Communication networks, MIMO,
Communi-cation systems
I. INTRODUCTION
T
HE DEMAND for portable devices with wirelessreal-time and high-rate mulreal-timedia services has been growing rapidly in this decade. Recent research shows that multiple-input and multiple-output (MIMO) communication systems are one of many solutions to achieving high data rate for these applications. Besides, more efficient bandwidth usage than before is also constantly an important factor to consider in order to fulfill the required transmission rate. However, compared to a single-input and single output (SISO) system, a major disadvantage of the MIMO communication system is that it needs more radio-frequency (RF) circuitry, digital-to-analog converters (DACs) and analog-to-digital converters (ADCs) which accordingly consume more power. Therefore, for portable devices, it is crucial to understand the tradeoff be-tween the efficient bandwidth usage and the available battery energy, in MIMO multi-hop networks. As such, in this work, the relationship between bandwidth efficiency and energy is investigated.
For a long-range wireless communication system, the com-munication between a cellular phone and its base station, the power consumption at the receiver is usually much less than the energy at the transmitter, because the power amplifier (PA) at the transmitter needs to output very high power to send
Manuscript received 1 October 2010; revised 15 February 2011. C. L. Chen is with the Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, Taiwan. (e-mail: clchen.nctu@gmail.com)
W. E. Stark is with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, USA.
S. G. Chen is with the Department of Electronics Engineering and Institute of Electronics, National Chiao Tung University, Taiwan.
Digital Object Identifier 10.1109/JSAC.2011.110904.
the signal to a distant receiver. This is a very high portion of the consumed power in such a system. Therefore, the energy consumed at the receiver is usually ignored for long-range communication systems. However, for wireless ad hoc networks, the distance from the source to the destination via relay nodes is relatively short. As such, the power consumption of the transmitter might not be as dominant as that in the single-hop case. The energy consumed at the receiver then be-comes an important factor in the bandwidth-energy efficiency optimization problem. Thus, it is important to explore the tradeoff between power consumption and the data rate when the multi-hop transmissions occur and receiver processing energy is incorporated in the total energy consumption.
Aspects of the energy-bandwidth efficiency tradeoff prob-lem have been investigated previously. In [1], the energy-bandwidth tradeoff under some optimal signaling methods is considered. Minimizing the transmission energy with different packet intervals is described in [2]. In [3], [4] and [5], the authors take the transmission and signal processing energy into consideration but the performance measures do not include the bandwidth efficiency or the end-to-end data rate. In [6] and [7], the authors discuss the energy-bandwidth efficiency and end-to-end throughput in linear multi-hop networks. The case of equally-spaced multi-hop networks is explored in [8] and [9]. Subsequently, works, [10] and [11], show the energy-bandwidth efficiency tradeoff by considering transmission energy, signal processing, and end-to-end throughput at the same time. However, those mentioned works mainly discuss these issues in multi-hop single-antenna networks. For MIMO systems, the works in [12], [13] and [14] characterize the system capacity, and the work in [15] provides the system throughput in a Gaussian broadcast channel. Moreover, the relationship between the transmission power and bandwidth in the MIMO channel is investigated in [16] without considering the signal processing energy.
In this paper, we optimize the overall system performance by taking the variables, the end-to-end throughput and energy consumption in a MIMO multi-hop network as well as the transmission and receiving signal processing energy, into consideration. The energy-bandwidth efficiency optimization problem also includes the factor of different numbers of transmit and receive antennas. Because of the downside of the increased processing energy for multiple receive antennas, we also derive analytical results for the optimization of antenna numbers for minimum energy consumption under the capacity-achieving end-to-end rate. By considering the number of hops and the end-to-end rate jointly, the optimal number of hops as well as the end-to-end data rate can be obtained.
The rest of this paper is organized as follows. The system model is described in Section II. In Section III the performance metrics of energy and bandwidth efficiency are described and optimized with respect to the transmitted energy on different hops. We also derive the energy-bandwidth efficiency tradeoff as well. The minimum total energy consumption for varying number of antennas is derived in Section IV. In this section, we also derive the optimal number of hops and the optimal end-to-end rate for the minimum energy consumption. By presenting the numerical results, the effects of the number of antennas on the tradeoff are discussed and summarized in Section V. Finally, we conclude in Section VI.
II. SYSTEMMODEL
In this paper, a network which has k − 1 relay nodes between a source node and a destination node is considered. All nodes operate in half-duplex mode. Fig. 1 shows the system architecture for the MIMO multi-hop wireless network.
Node i has Mi+1 transmit antennas (TAs) and Ni receiver
antennas (RAs). That is, there is a Mi×Nimultiple-input and multiple-output (MIMO) channel between node i−1 and node i. The input-output relationship of the i-th MIMO channel can be represented as
Yi= HiXi+ ni, (1)
where Hi is the channel matrix, Xi the transmitted data on TAs,Yithe received data on RAs, andniis the additive white Gaussian noise (AWGN). The distance between node i−1 and node i is di, while the end-to-end distance is de. The distance ratio αi of each node is defined as the proportion of its hop node distance to end-to-end distance, αi = ddei. Each relay node is assumed to be in the transmitter’s far-field region. Hence, the relationship between the received power Pr and the transmitted power Pt is given by
Pr= β
dηPt, d > 1, (2)
where β is related to the antenna properties, η is the path-loss parameter, and d is the distance between the transmitter and receiver. In this paper, β is assumed to be one for simplicity. For each MIMO channel, there is a transmission rate limitation for reliable communication, namely the channel capacity. The effective end-to-end rate can be represented in the form of a combination of all hops’ transmission rates. A higher effective end-to-end rate is possible with higher energy consumption. This tradeoff between the effective transmission rate and the total energy consumption will be investigated in the following section.
III. PROBLEMFORMULATION
This section will provide the end-to-end rate and the total energy consumption per information bit for evaluating the system performance. The minimization of the total energy consumption for a given effective end-to-end rate will be found. One can obtain the energy-bandwidth efficiency trade-off through the solution of an optimization problem.
A. Channel Capacity of a MIMO Fading Channel and End-to-end Rate
The channel capacity of each MIMO channel will depend on the knowledge of the channel condition at the transmit-ter, receiver or both. Channel information knowledge at the transmitter will affect the adopted power allocation strategy. Indirectly, it will affect the channel capacity as well. If each transmitter can obtain the full channel state information (CSI) of the MIMO channel, an optimal power allocation strategy, namely water-filling strategy, can be adopted among the transmitter antennas. However, obtaining full CSI at either the transmitter or receiver in practice is not possible due to estimation and quantization noises. On the other hand, if the receiver is required to feedback the estimated CSI to the transmitter, then the bandwidth efficiency will be reduced accordingly. Moreover, since the feedback CSI is generally noisy, it may cause error propagation problems in practice, even when optimized water-filling power allocation strategies in the transmitter are adopted. Due to these concerns, this work only considers the system model in which transmitter does not know CSI. We assume that the channel is sufficiently random which means that the elements of channel matrixHi are independent naturally. In this scenario, the optimal power allocation strategy for the i-th hop is to use equal amounts of energy on all TAs [17]. Therefore, for the i-th Mi
-by-Ni MIMO fading channel with full rank, the upper bound
of channel capacity for the real dimension (coming from combining equation 8.15 and 8.17 in [17]) when transmitters do not know the CSI is given by
C(γi) ≤ nmin2 log2(1 + 2γi Minmin nmin j=1 ˆλ2 j), (3)
where nmin = min(Mi, Ni) represents the channel rank, γi is the ratio of received energy per channel use to noise power spectral density, and ˆλ1 ≥ ˆλ2 ≥ ... ≥ ˆλnmin are the ordered
singular values of the channel matrix Hi. Furthermore, the expectation value ofnmin
j=1 ˆλ2j can be represented as E{ nmin j=1 ˆλ2 j} = E{T r{HiHTi }} = E{ mi,ni h2mi,ni}. (4) Note thatHT
i denotes the transpose matrix ofHi. For further simplifying (3), we assume that the term ofnmin
j=1 ˆλ2j in (3) will be close to its expectation value as shown in (4).
As mentioned, we consider that the channel matrix is suf-ficiently random, and assume the normalized E{h2mi,ni} = 1,
where hmi,ni is the element of channel matrixHi. Equality
holds in (3) when the channel matrix Hi is sufficiently
random and statistically well-conditioned [17]. As such, with the assumption of sufficiently random and statistically well-conditioned channel matrix, the channel capacity becomes
C(γi) ≈ nmin2 log2(1 +2γniNi
min
). (5)
The rate of the i-th MIMO channel within nodes i − 1 and i is given by Ri. Then, the end-to-end rate [8][9] (which is named as the effective rate in [11]) is given by
Re= 1
k
i=1R−1i
Fig. 1. The system architecture for the MIMO multi-hop wireless network
For reliable communications, the transmission rate Ri for the i-th MIMO channel should not be larger than the capacity
C(γi). We assume that the capacity-achieving codes are
applied at each transmitter. As such, the maximum end-to-end rate corresponding to a set of received energy-to-noise ratios (i.e., γi, for i = 1, ..., k) will be
Re=
1
k
i=1(C(γi))−1
. (7)
B. Total Energy Consumption
The energy consumption Ei for the i-th MIMO channel
consists of transmission energy consumption (Et,i per coded symbol on each TA) and receiver processing energy
consump-tion (Ep,i per coded symbol on each RA). Therefore, the
energy consumption per information bit for the i-th Mi
-by-Ni MIMO channel is
Ei= MiEt,i+ NiEp,i
Ri .
(8) Further, let the total energy consumption be represented as
Etot= k
i=1
Ei. (9)
The total energy consumption will be normalized by the noise power spectral density N0. Moreover, in order to compare
the energy consumption with the results of single-link reliable communications, the total energy consumption will be nor-malized by the end-to-end propagation loss as well. Hence, the metric for evaluating the energy consumption is given by
Γ = Etot
N0dηe.
(10) By substituting (8) and (9) into (10), the energy metric can be rewritten as Γ = k i=1 MiEt,i+ NiEp,i N0dηeRi (11) = k i=1 Miαηiγi+ Niγp Ri , (12) where γi ≡ NE0t,idη i and γp ≡ Ep,i
N0dηe. Note that γi is the ratio
of received energy per channel use to noise power spectral density.
C. Optimization Problem
Before formulating the optimization problem, a set of received energy-to-noise ratios (γ1, γ2, ..., γk) can be one-to-one mapped to a set of capacity-achieving transmission rates (C(γ1), ..., C(γk)). Therefore, a set of rates which minimizes the energy metric under a given end-to-end transmission rate can be formulated as follows,
min(C(γ1),...,C(γk))k i=1
Miαηiγi+Niγp
C(γi)
subject to: R−1e =ki=1(C(γi))−1.
(13) One can apply the method of Lagrange multiplier to find the optimal solution of the above problem. The corresponding Lagrange function is defined as
L(λ) = k i=1 Miαηiγi+ Niγp C(γi) − λ(R−1 e − k i=1 (C(γi))−1). (14) where λ is Lagrange multiplier. By setting ∇C(γ∇L(λ)i)to zero (i.e., the Karush-Kuhn-Tucker (KKT) condition), one can obtain
λ = Miαηi(C(γi) ∂γi
∂C(γi) − γi) − Niγp.
(15) By substituting (5) into (15), (15) can be expressed as λ = Miαηi{ nmin 2Ni (1 − 22C(γi)nmin ) + C(γi) ln 2 Ni 22C(γi)nmin )} − Niγp. (16) Further, (16) can be rewritten as
2Ni Miαηinmine(λ+Niγp)− 1 e = ( 2C(γi) ln 2 nmin −1)e (2C(γi) ln 2nmin −1) . (17) Then, an useful lemma is introduced for solving (17).
• Lemma A: Lambert ω-function [18]
The Lambert ω-function is the inverse function of f (X) = XeX,
where X is any complex number. Moreover, if there is another equation in the form of (namely Lambert’s transcendental equation)
ln X = ρXθ, the solution is given by
X = e−ω{−θρ}θ .
By using the Lambert ω-function in (17), the channel capacity C(γi) in the form of λ is given by
C(γi) = n2 ln 2min(1 + ω{ 2Ni
Miαηinmine(λ + Niγp) − 1
e}), (18) where ω{.} is the principal branch of the Lambert ω-function. Hence, all the performance metrics in the form of λ are listed as in (19), where Si,λ= Miα2Nη i
inmine(λ + Niγp) − 1
e. Finally, by varying λ, one can obtain the tradeoff between the energy metric Γ and the effective rate Re.
IV. PERFORMANCEEVALUATION
The performances of MIMO multi-hop wireless networks under the equally-spaced hops will be investigated. We can ob-tain an analytical expression for the performance as a function of the number of antennas. More number of antennas generally leads to higher bandwidth efficiency, while increasing the number of antennas also increases the energy consumption. Therefore, we derive the optimal number of antennas (both TAs and RAs) for the lowest energy consumption for the MIMO channel. The analyses provide the tradeoff between energy and bandwidth efficiency. In fact, the number of an-tennas affects the rank of the MIMO channel and the channel capacity depends on the channel rank. Hence, the following discussion will be divided the energy metric which relates with the channel capacity into two parts for each hop-allocation condition, Ni ≤ Mi and Mi ≤ Ni. Then, nmin of (5) is Ni when Ni≤ Mi, while nmin is Mi when Mi≤ Ni.
First, the equally-spaced hops are given by α1 = ... =
αk = α. We assume that all hops are the same which means
they all have the same number of TAs and RAs. As such, it will result in C(γ1) = ... = C(γk) = C(γ). Note that the following discussion will ignore the suffix i on Mi, Ni, and αi. Therefore, the end-to-end rate can be represented as
Re= C (γ)
k . (20)
By substituting (5) and (20) into (12), the energy metric can be rewritten as Γ(M, N) = M αη(n2Nmin)(e 2kRe ln 2 nmin − 1) + Nγp Re . (21) In this scenario, (21) represents the energy consumption of the capacity-achieving effective rate. In the following discussion, we will optimize numbers of TAs M and RAs N to find the minimum energy consumption with the maximum effective rate.
1) When N ≤ M : When nmin = N, the energy metric in (21) will become Γ(M, N)|N ≤M = Mα η(e2kRe ln 2N − 1) + 2Nγ p 2Re . (22) According to (22), in the low Re region, the term of 2N γp dominates the energy metric and this results in that increasing number of RAs N is expected to increase the energy consump-tion. On the other hand, in the high Re region, the term of e2kRe ln 2N dominates the energy metric. As such, increasing N
will decrease the energy consumption. The simulation results in Section V will exhibit this phenomenon. As for the number of TAs M , increasing M will always increase the energy consumption.
For finding the minimum energy consumption, by treat-ing M and N as continuous variables, the derivatives of Γ(M, N)|N ≤M concerning M and N , respectively, are
∂Γ(M,N ) ∂M = αη(e2kRe ln 2N −1) 2Re ∂Γ(M,N ) ∂N = −Mα ηk ln 2 N2 e 2kRe ln 2 N + γp Re. (23) Note that there are two natural restrictions of M ≥ 1 and N ≥ 1. As such, for M , since the energy metric (22) is a linear increasing function and M ≥ N ≥ 1, the minimum value of energy metric is located at M∗ = N. Then, setting (23) to zero for N will yield N∗. Hence, the optimal M∗and N∗ (when the other variable is fixed) is
M∗= N N∗= (MαηkReln 2 γp e 2ω{(γpkRe ln 2Mαη )12})1 2. (24)
By substituting (24) into (22), the energy metric with
individual optimal N∗ and M∗ for given M and N can be
respectively represented as Γ(M, N∗)| N∗≤M =Mα η 2Re (e 2ω{Q}− 1) + γpk ln 2 ω{Q} Γ(M∗, N )| N ≤M∗ =N α η(e2kRe ln 2N −1)+2Nγ p 2Re , (25) where Q = (γpkRMαeηln 2) 1
2. The above energy metrics
Γ(M, N∗)|
N∗≤M and Γ(M∗, N )|N ≤M∗ are the minimum
energy consumptions with the maximum effective rates for given numbers of TAs and RAs, respectively. That is, for a given antenna number either TA or RA, the energy-bandwidth efficiency tradeoff is bounded by (25).
Moreover, for an arbitrary antenna pair when N ≤ M , setting the derivatives of Γ(M∗, N ) concerning N to zero yields
M∗= N∗= 2kReln 2 ω{2γp
αηe−1e} + 1
. (26)
Then, by substituting (26) into (22), the energy metric with the joint optimal M∗ and N∗ can be obtained by
Γ(M∗,N∗)| N∗≤M∗=α ηkln 2(eω{2γpαη e−1 e}+1−1)+2kγpln 2 ω{α2γηpe−1e} + 1 .(27)
⎧ ⎪ ⎨ ⎪ ⎩
Channel Capacity: C(γi) = n2 ln 2min(1 + ω{Si,λ}) End-to-End Rate: Re= n2 ln 2min(
k
i=1 1+ω{S1i,λ})
−1
Energy Metric: Γ = ln 2ki=1Miαηinmin(e1+ω{Si,λ}−1)+2Ni2γp
Ninmin(1+ω{Si,λ}) ,
(19)
2) When M ≤ N : In this condition, nmin is equivalent to M . Therefore, the energy metric with capacity-achieving effective rate in (21) becomes
Γ(M, N)|M≤N = M
2αη(e2kRe ln 2M − 1) + 2N2γp
2NRe .
(28) Under the condition of N ≤ M , increasing the number of RAs N will increase the energy consumption in the low Reregion, while increasing N in the high Reregion results in decreased energy consumption. Furthermore, for obtaining the minimum
energy consumption, the derivatives of Γ(M, N )|M≤N
con-cerning M and N are ∂Γ(M,N ) ∂M = [(2Mαη−2αηkR eln 2)e2kRe ln 2M −2Mαη] 2NRe ∂Γ(M,N ) ∂N = 4ReN2γp−2ReM2αη(e2kRe ln 2M −1) (2NRe)2 . (29) Hence, by setting (29) to zero, one can obtain the optimal M∗
and N∗ based on Lemma A as
⎧ ⎨ ⎩ M∗= 2kReln 2 2+ω{−2 e2} N∗= [M2αη(e 2kRe ln 2 M −1) 2γp ] 1 2. (30) Further, the energy metric with individual optimal M∗ and N∗, respectively, are Γ(M∗, N )| M∗≤N = (2kRe ln 2 2+ω{ −2 e2} )2αη(e2+ω{ −2 e2}−1)+2N2γ p 2NRe Γ(M, N∗)| M≤N∗ = M[2α ηγ p(e2kRe ln 2M −1)]12 Re , (31)
where M∗ and N∗ are defined in (30). Hence, (31) is the minimum energy consumption which is also a performance bound of the energy-bandwidth efficiency tradeoff for an arbitrary antenna pair when M ≤ N .
On the other hand, substituting M∗into M of N∗yields the joint optimal M∗ and N∗ for the energy metric. The energy metric with the joint optimal solution is represented as
Γ(M∗, N∗)| M∗≤N∗ = 2 √ 2k ln 2 2 + ω{−2 e2} αηγp(e2+ω{−2e2}− 1). (32) Besides, for satisfying the initial condition of M ≤ N , the
optimal M∗ should be less or equal to the optimal N∗.
Therefore, by substituting (30) into M∗≤ N∗, one can obtain that the energy metric with joint optimal solution in (32) is subject to
γp≤ α
η(e2+ω{−2e2}− 1)
2 . (33)
Finally, by comparing (27) with (32), the energy metric with joint optimization whenever the relationship of M and N is
Γ(M∗,N∗) =min{Γ(M∗,N∗)|
N∗≤M∗,Γ(M∗,N∗)|M∗≤N∗}.(34)
Consequently, according to the detailed derivation in Appendix A, the energy metric with joint optimization is
Γ(M∗, N∗) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Γ(M∗, N∗)| N∗≤M∗, as shown in (27), when γp>α η(e2+ω{ −2e2}−1) 2 Γ(M∗, N∗)| N∗≤M∗ = Γ(M∗, N∗)|M∗≤N∗, as shown in (27) and (32), when γp=α η(e2+ω{ −2e2}−1) 2 Γ(M∗, N∗)| M∗≤N∗, as shown in (32), when γp<α η(e2+ω{ −2e2}−1) 2 , (35)
3) Optimization of the number of hops: So far, previous discussions are under the given number of hops. Further, this section explores the energy-bandwidth efficiency tradeoff with the optimal number of hops and the optimal end-to-end rate. By setting the derivative of (21) concerning the number of hops k to zero, the optimal number of hops can be obtained as
k∗= n2Rmin eln 2(ω{
−η
eη } + η). (36)
By substituting (36) into (21), the energy metric with the optimal k∗ is Γ(k∗, R e) = P MR η−1 e N nη−1min + γpN Re , (37) where P ≡ [eω{ −ηeη }+η−1 2 ][(ω{2 ln 2−ηeη}+η)η].
Next, setting the derivate of (37) respect to Reequal to zero yields the optimal end-to-end rate Re∗ as
R∗e= ( γ
pN2nη−1min P (η − 1)M)
1
η. (38)
Hence, by substituting the optimal R∗ein (38) into the optimal k∗ in (36), the optimal number of hops k∗ can be rewritten as k∗= ( M(e ω{−ηeη}+η− 1) 2N2nη−1 min )1η( γp η − 1) −1 η (39) = ( M(eω{ −η eη}+η− 1) 2N2nη−1 min )1η( Ep,i N0(η − 1)) −1 η de. (40)
0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 25 30 35 40 45 50 R e E (dB) 10x1 10x2 10x3 10x4 10x5 10x6 10x7 10x8 10x9 10x10 Increasing number of RA Increasing number of RA Analytical result with individual optimal N*
Fig. 2. The energy-bandwidth characteristic of 4 equally-spaced hops with a fixed number of TA and different numbers of RAs whenN ≤ M, and the analytical result in (25). can be represented as Γ(k∗, R∗ e) =M αη(n2Nmin)(e 2k∗R∗e ln 2 nmin − 1) + Nγp R∗e (41) = { M P N nη−1min( N 2nη−1 min P (η − 1)M) η−1 η +N( N 2nη−1 min P (η − 1)M) −1 η } ×γη−1η p (42) = { M P N nη−1min ( N2nη−1min P (η − 1)M) η−1 η +N( N 2nη−1 min P (η − 1)M) −1 η } ×(Ep,i N0 ) η−1 η (de)1−η. (43)
From the above results, one can obtain The optimal number of hops k∗∝ de
The optimal energy metric Γ(k∗, R∗e) ∝ de1−η The optimal energy metric Γ(k∗, R∗e) ∝ Ep,i
η−1 η .
(44)
Note that de is the end-to-end distance and Ep,i is the
processing energy. Furthermore, according to (10), the total energy consumption Etotal (which is the un-normalized en-ergy metric) increases linearly with de.
Besides, if the number of TAs is equivalent to the number of RAs, namely M = N , the energy metric with optimal k∗ and R∗e in (43) can be reduced to
Γ(k∗, R∗ e)|M=N = {P ( 1 P (η − 1)) η−1 η +( 1 P (η − 1)) −1 η } ×(Ep,i N0 ) η−1 η (de)1−η (45) = η η − 1( 1 P (η − 1)) −1 η ( Ep,i N0 ) η−1 η (de)1−η. (46) The above result shows that the minimum energy consump-tions with the optimal k and Re, in the scenario of M = N ,
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 5 10 15 20 25 30 R e E (dB) 1x1 2x1 3x1 4x1 5x1 6x1 7x1 8x1 9x1 10x1 Increasing number of TA Analytical result
with individual optimal M*
Fig. 3. The energy-bandwidth characteristic of 4 equally-spaced hops with different numbers of TAs and a fixed number of RA whenN ≤ M, and the analytical result in (25).
are independent of the antenna numbers. Surely, the propor-tional relationships in (44) still hold in this scenario.
V. NUMERICALRESULTS
This section provides the numerical results of the tradeoff between energy and bandwidth for various configurations in the number of transmitting and receiving antennas. The adopted system parameters are as follows,
• Path-loss exponent: η = 4 • End-to-end distance: de= 3000m
• Signal processing energy: Ep,i = 0.95μJ/symbol unless otherwise specified
In the figures, each curve represents the energy-bandwidth efficiency tradeoff for a specific antenna pair. For instance, 1x10 denotes that there are one TA and ten RAs in each MIMO channel.
Fig. 2 shows the tradeoff of energy and bandwidth with a fixed number of TAs and different numbers of RAs when N ≤ M , for the case of four equally-spaced hops. As shown in Fig. 2, for low end-to-end transmission rates, increasing number of RAs will result in increasing energy consumption. However, increasing number of RAs will lead to decreasing energy consumption in the high end-to-end rate region. As explained in Section IV-1, this phenomenon comes from that the term of 2N γp in (22) dominates the energy metric (i.e.,
(22)) in the low Re region, while the term of e
2kRe ln 2
N
dominates the energy metric in the high Reregion.
Fig. 3 shows the tradeoff in different numbers of TAs and a fixed number of RAs when N ≤ M . With a fixed energy consumption, Fig. 2 shows that increasing the number of RAs can lead to higher end-to-end transmission rate, while Fig. 3 shows that decreasing the number of TAs can result in higher end-to-end transmission rate. For explaining why more TAs lead to smaller rates as shown in Fig. 3, we can start from (22). In (22), with a fixed energy consumption Γ(M, N )|N ≤M and a given number N of RAs, increasing M will result in de-creasing Renaturally. The analytical results Γ(M, N∗)|N∗≤M
0 0.5 1 1.5 2 2.5 0 5 10 15 20 25 30 35 R e E (dB) 1x1 1x2 1x3 1x4 1x5 1x6 1x7 1x8 1x9 1x10 Analytical result with individual optimal N*
Increasing number of RA
Increasing number of RA
Fig. 4. The energy-bandwidth characteristic of 4 equally-spaced hops with a fixed number of TAs and different numbers of RAs whenM ≤ N, and the analytical result in (31). 0 2 4 6 8 10 12 5 10 15 20 25 30 35 40 45 50 55 R e E (dB) 1x10 2x10 3x10 4x10 5x10 6x10 7x10 8x10 9x10 10x10 Analytical result with individual optimal M*
Increasing number of TA
Fig. 5. The energy-bandwidth characteristic of 4 equally-spaced hops with different numbers of TAs and a fixed number of RAs whenM ≤ N, and the
analytical result in (31).
and Γ(M∗, N )|N ≤M∗ in (25) are also shown in Fig. 2 and Fig. 3, respectively. Note that M in (25) for Fig. 2 and N in (25) for Fig. 3 are set to the fixed number of TAs and RAs, namely 10 and 1, respectively. The analytical results Γ(M, N∗)|
N∗≤M and Γ(M∗, N )|N ≤M∗ in (25) provide a
very good approximation for the energy-bandwidth efficiency tradeoff for Fig. 2 and Fig. 3, respectively.
The following discussion focuses on the numerical results corresponding to M ≤ N . For this scenario, Fig. 4 shows the energy-bandwidth efficiency tradeoff of four equally-spaced hops with a single transmitting antenna and different numbers of RAs. For the tradeoff between energy and rate, Fig. 4 exhibits the same phenomenon as Fig. 2. Besides, for plotting the analytical result in Fig. 4, the term M of the analytical result Γ(M, N∗)|M≤N∗ with individual optimal N∗(i.e., (31)) is set to a fixed number of TAs which is one here.
On the other hand, Fig. 5 shows the effect of the number of TAs on the energy-bandwidth efficiency tradeoff of four
1 2 3 4 5 6 7 8 9 10 5 10 15 20 25 30 35 40 45 50 R e E (dB) 1x1 2x2 3x3 4x4 5x5 6x6 7x7 8x8 9x9 10x10 Increasing number of antennas Increasing number of antennas
Fig. 6. The energy-bandwidth characteristic of 4 equally-spaced hops with different numbers of TAs whenM = N.
TABLE I
EFFECTS OF ANTENNA NUMBERS ON THE EFFECTIVE RATES FOR THE FIXED ENERGY CONSUMPTION AND FOR THE FIXED EFFECTIVE RATE
FixedE N < M N = M N > M Increasing Re↑ Re↑ Re↑ number of RA (N ↑) Increasing Re↓ Re↑ Re↑ number of TA (M ↑) Fixed Re N < M N = M N > M
Increasing LowRe:E ↑ LowRe:E ↑ LowRe:E ↑
number of RA (N ↑) HighRe:E ↓ HighRe:E ↓ HighRe:E ↓
Increasing E ↑ LowRe:E ↑ E ↓
number of TA (M ↑) HighRe:E ↓
equally-spaced hops. In Fig. 5, increasing the number of TAs can decrease the energy consumption. Besides, with a fixed energy consumptions, more TAs lead to higher transmission rates. Also, as described in Section IV-2, the analytical result Γ(M∗, N )|
M∗≤N in (31) forms a good approximation to the
energy-bandwidth efficiency tradeoff in Fig. 5, when N is set to a fixed number of RAs.
Fig. 6 shows the energy-bandwidth efficiency tradeoff when M = N . For M = N , increasing the number of antennas results in increasing end-to-end rate with a fixed energy con-sumption. With a fixed effective rate, increasing the number of antennas leads to increasing energy consumption in the low effective rate region and decreasing energy consumption in the high effective rate region. Further, the effects of different numbers of antennas are summarized in Table I for the fixed energy consumption and for the fixed effective rate. According to this table, one can choose a suitable antenna pair for effective transmissions.
Then, according to (35), since the lower bound of the energy metric for an arbitrary antenna pair relates to quantity of the normalized signal processing energy γp, the following three figures (i.e., Fig. 7, Fig. 8, and Fig. 9) show all possible antenna pairs for up to four antenna numbers and
Fig. 7. The energy-bandwidth characteristic of 4 equally-spaced hops with all possible antenna pairs for up to four antenna numbers, and the analytical results in (27) and (35) whenγp>αη(e
2+ω{ −2
e2}−1)
2 .
Fig. 8. The energy-bandwidth characteristic of 4 equally-spaced hops with all possible antenna pairs for up to four antenna numbers, and the analytical results in (27), (32) and (35) whenγp=αη(e
2+ω{ −2
e2}−1)
2 .
the analytical results with joint optimal M∗ and N∗. Fig. 7, Fig. 8, and Fig. 9 correspond to γp= 2.946 (which is larger than αη(e2+ω{ −22e2}−1)), γp = 0.0077 (which is equivalent to
αη(e2+ω{ −2e2}−1)
2 ), and γp = 7.6593 × 10−4 (which is less
than αη(e2+ω{ −22e2}−1)), respectively. In the high end-to-end rate region, the curves fall into four groups and this phenomenon is more obvious in the high γpcondition especially. Because the energy metric in (19) relates to nmin, each group corresponds to a value of nmin. For instance, nmin value corresponding to the curve of 2x3 is two. This curve will be in the curve group of 2x2, 2x4, 3x2, and 4x2 because of the same nmin. Moreover, the relationships between Γ(M∗, N∗)|N∗≤M∗ and
Γ(M∗, N∗)|
M∗≤N∗ in (35) for different conditions of γpcan be verified in those figures.
Fig. 9. The energy-bandwidth characteristic of 4 equally-spaced hops with all possible antenna pairs for up to four antenna numbers, and the analytical results in (27), (32) and (35) whenγp<αη(e
2+ω{ −2 e2}−1) 2 . 0 0.5 1 1.5 2 2.5 3 3.5 4 0 2 4 6 8 10 12 14 16 18 Re Antenna Number
A: Individual opt. M* (When given N=6, N M*) B: Individual opt. N* (When given M=6, N* M) C: Joint opt. M*=N* (When N* M*) D: Individual opt. M* (When given N=6, M* N) E: Individual opt. N* (When given M=6, M N*) F: Joint opt. M* (When M* N*)
G: Joint opt. N* (When M* N*)
Fig. 10. The optimized antenna numbers versus the effective rate when
γp=αη(e
2+ω{ −2
e2}−1)
2 .
Fig. 10 shows the results of the optimized antenna
num-bers in Section IV versus the effective rate, when γp =
αη(e2+ω{ −2e2}−1)
2 . Because of γp = α
η(e2+ω{ −2
e2}−1)
2 , as shown
in (35), the results of joint optimization corresponding to N∗≤ M∗and M∗≤ N∗will be the same. Hence, in Fig. 10, the curves of C, D, F, and G overlap with each other. Curve A is a constant line and its value is equal to the given N . Curve B will not exceed the given M due to the restriction of N∗ ≤ M, while curve E will tower above the given M because of M ≤ N∗.
Furthermore, for the case with the optimal number of hops k∗, the energy-bandwidth efficiency characteristic for the four equally-spaced hops are shown in Fig. 11. The analytical result with the optimal k∗ forms a low bound of the energy con-sumption as a function of rate. As shown in Fig. 11, increasing number of hops leads to increasing energy consumption in the
Analytical result with optimal k* Increasing number of hops Increasing number of hops
Fig. 11. The energy-bandwidth characteristics under the 1x1 antenna pair for the four equally-spaced hops withk = 1, ..., 10, and the analytical result in (37).
high end-to-end rate region, while increasing number of hops results in decreasing energy consumption in the low end-to-end rate region.
VI. CONCLUSION
The energy-bandwidth efficiency tradeoff in MIMO multi-hop wireless networks is derived in this paper and the effects of different number of antennas on the energy-bandwidth efficiency tradeoff are investigated at the same time. Besides, we also optimize over the number of antennas to find the minimum energy consumption with the maximum effective rate. The joint optimization over the numbers of TAs and RAs are performed as well. In addition, by optimizing over the number of hops and the effective rate, the optimal number of hops and the optimal effective rate can be obtained. According to the results, one can choose suitable system parameters with considering the energy and bandwidth in the meantime.
APPENDIXA
THE ENERGY METRIC WITH JOINT OPTIMIZATION
First of all, the following discussion is divided into
three parts according to the relationship between γp and
αη(e2+ω{ −2e2}−1)
2 .
1) When γp>α
η(e2+ω{ −2e2}−1)
2 : Since Γ(M∗, N∗)|M∗≤N∗
holds only when γp ≤ α
η(e2+ω{ −2 e2}−1) 2 , the Γ(M∗, N∗) in (34) become Γ(M∗, N∗) = Γ(M∗, N∗)| N∗≤M∗, (47) in this scenario. 2) When γp = α η(e2+ω{ −2 e2}−1) 2 :: By letting
A ≡ 2 + ω{−2e2} and substituting γp into (32), γp and Γ(M∗, N∗)| M∗≤N∗ can be rewritten as γp= α η(eA− 1) 2 (48) and Γ(M∗, N∗)| M∗≤N∗ = αηk ln 2 ×2(e A− 1) A , (49) respectively.
Besides, with A, one can represent B ≡ ω{2γp
αηe−1e} + 1 in (27) as
B = ω{e A− 2
e } + 1. (50)
Note that eB = eω{eA−2e }+1 holds as well because of (50). Then, eB = eω{eA−2e }+1 can be rewritten to match the form
of X = e−ω{−θρ}θ in Lemma A as
eB−1= eω{eA−2e } (51)
Hence, X, θ, and ρ in Lemma A are equivalent to eB−1,−1, and eA−2
e , respectively. As such, by substituting X = eB−1, θ = −1, and ρ = eAe−2 into ln X = ρXθ in Lemma A, one can obtain
eB= e
A+ eB− 2
B . (52)
Then, Γ(M∗, N∗)|N∗≤M∗ in (27) can be represented as
Γ(M∗, N∗)| N∗≤M∗ = αηk ln 2(e B− 1 B ) + k ln 2( 2γp B ) = αηk ln 2(eB− 1 B ) + α ηk ln 2(eA− 1 B ) = αηk ln 2 × eB = αηk ln 2 × eω{eA−2e }+1. (53) Finally, with (49) and (53), the difference between Γ(M∗, N∗)| M∗≤N∗ and Γ(M∗, N∗)|N∗≤M∗is represented as Γ(M∗, N∗)| N∗≤M∗− Γ(M∗, N∗)|M∗≤N∗ = αηk ln 2 × (eω{eA−2e }+1−2(eA− 1) A ). (54)
Since A is a constant, the result of (eω{eA−2e }+1− 2(eA−1)
A )
can be proved as zero with the numerical methods. It means that Γ(M∗, N∗)|N∗≤M∗ and Γ(M∗, N∗)|M∗≤N∗ are the same
when γp = α η(e2+ω{ −2 e2}−1) 2 . As such, Γ(M∗, N∗) in (34) is equivalent to Γ(M∗, N∗)|N∗≤M∗ and Γ(M∗, N∗)|M∗≤N∗ in this scenario. 3) When γp < α η(e2+ω{ −2e2}−1) 2 :: Due to (27) and (32), Γ(M∗, N∗)|
N∗≤M∗ decreases with γp approximately and
Γ(M∗, N∗)|
M∗≤N∗ decreases with √γp, respectively. More-over, since Γ(M∗, N∗)|N∗≤M∗ and Γ(M∗, N∗)|M∗≤N∗ are
the same when γp = α
η(e2+ω{ −2
e2}−1)
2 , Γ(M∗, N∗)|M∗≤N∗
will be less than Γ(M∗, N∗)|N∗≤M∗ when γp <
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Chih-Liang Chen was born in Tainan, Taiwan. He
received the B.S.E.E. degree from National Chung Hsing University, Taiwan, in 2005. He is currently a Ph.D. canidate in the Department of Electron-ics Engineering and Institute of ElectronElectron-ics in Na-tional Chiao Tung University. His research interests include digital signal processing, synchronization, channel estimation and equalization.
Wayne E. Stark received the B.S. degree (highest
honors), and the M.S. and Ph.D. degrees in electrical engineering from the University of Illinois, Urbana-Champaign, in 1978, 1979, and 1982, respectively. Since September 1982, he has been a Faculty Mem-ber in the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Ar-bor, where he is currently a Professor. His research interests include the areas of coding and commu-nication theory, especially for spread-spectrum and wireless communications networks. Dr. Stark is a Member of Eta Kappa Nu, Phi Kappa Phi, and Tau Beta Pi. He was involved in the planning and organization of the 1986 International Symposium on Information Theory, held in Ann Arbor, MI. From 1984 to 1989, he was Editor for Communication Theory of the IEEE TRANSACTIONS ON COMMUNICATIONS in the area of spread-spectrum communications. He was selected by the National Science Foundation as a 1985 Presidential Young Investigator. He is Principal Investigator of an Army Research Office Multidisciplinary University Research Initiative (MURI) Project on low-energy mobile communications.
Sau-Gee Chen received his B.S. degree from
Na-tional Tsing Hua University, Taiwan, in 1978, M.S. degree and Ph.D. degree in electrical engineering, from the State University of New York at Buffalo, NY, in 1984 and 1988, respectively. Currently, he is a professor at the Department of Electronics Engi-neering, National Chiao Tung University, Taiwan. He was the director of Institute of Electronic at the same organization from 2003 to 2006. During 2004-2006, he served as an associate editor of IEEE Transactions on Circuits and Systems I. His research interests include digital communication, multi-media computing, digital signal processing, and VLSI signal processing. He has published more than 100 conference and journal papers, and holds several US and Taiwan patents.
