Improving Energy Efficiency Through Multimode Transmission in the Downlink MIMO Systems

Adaptively adjusting system parameters including bandwidth, transmit power and mode to maximize the"Bits per-Joule"energy efficiency (BPJ-EE) in the downlink MIMO systems with imperfect channel state information at the transmitter (CSIT) is considered in this paper. By mode we refer to choice of transmission schemes i.e. singular value decomposition (SVD) or block diagonalization (BD), active transmit/receive antenna number and active user number. We derive optimal bandwidth and transmit power for each dedicated mode at first. During the derivation, accurate capacity estimation strategies are proposed to cope with the imperfect CSIT caused capacity prediction problem. Then, an ergodic capacity based mode switching strategy is proposed to further improve the BPJ-EE, which provides insights on the preferred mode under given scenarios. Mode switching compromises different power parts, exploits the tradeoff between the multiplexing gain and the imperfect CSIT caused inter-user interference, improves the BPJ-EE significantly.


I. INTRODUCTION
Energy efficiency is becoming increasingly important for the future radio access networks due to the climate change and the operator's increasing operational cost. As base stations (BSs) take the main parts of the energy consumption [1], [2], improving the energy efficiency of BS is significant. Additionally, multiple-input multipleoutput (MIMO) has become the key technology in the next generation broadband wireless networks such as WiMAX and 3GPP-LTE. Therefore, we will focus on the maximizing energy efficiency problem in the downlink MIMO systems in this paper.
Previous works mainly focused on maximizing energy efficiency in the single-input single output (SISO) systems [3]- [7] and point to point single user (SU) MIMO systems [8]- [10]. In the uplink TDMA SISO channels, the optimal transmission rate was derived for energy saving in the non-real time sessions [3]. G. W. Miao et al. considered the optimal rate and resource allocation problem in OFDMA SISO channels [4]- [6]. The basic idea of [3]- [6] is finding an optimal transmission rate to compromise the power amplifier (PA) power which is proportional to the transmit power and the circuit power which is independent of the transmit power. S. Zhang et al. extended the energy efficiency problem to a bandwidth variable system [7] and the bandwidth-power-energy efficiency relations were investigated. As the MIMO systems can improve the data rates compared with SISO/SIMO, the transmit power can be reduced under the same rate. Meanwhile, MIMO systems consume higher circuit power than SISO/SIMO due to the multiplicity of associated circuits such as mixers, synthesizers, digital-to-analog converters (DAC), filters, etc. [8] is the pioneering work in this area comparing the energy efficiency of Alamouti MIMO systems with 2 antennas and SIMO systems in the sensor networks. H. Kim et al. presented the energy efficient mode switching between SIMO and 2 antenna MIMO systems [9]. A more general link adaptation strategy was proposed in [10] and the system parameters including number of data streams, number of transmit/receive antennas, use of spatial multiplexing or space time block coding (STBC), bandwidth etc. were controlled to maximize the energy efficiency. However, to the best of our knowledge, there are few literatures considering energy efficiency of the downlink multiuser (MU) MIMO systems.
The number of transmit antennas at BS is always larger than the number of receive antennas at the mobile station (MS) side because of the MS's size limitation. MU-MIMO systems can provide higher data rates than SU-MIMO by transmitting to multiple MSs simultaneously over the same spectrum. Previous literatures mainly focused on maximizing the spectral efficiency of MU-MIMO systems, some examples of which are [11]- [18]. Although not capacity achieving, block diagonalization (BD) is a popular linear precoding scheme in the MU-MIMO systems [11]- [14]. Performing precoding requires the channel state information at the transmitter(CSIT) and the accuracy of CSIT impacts the performance significantly. The imperfect CSIT will cause inter-user interference and the spectral efficiency will decrease seriously. In order to compromise the spatial multiplexing gain and the inter-user interference, spectral efficient mode switching between SU-MIMO and MU-MIMO was presented in [15]- [18].
Maximizing the "Bits per-Joule" energy efficiency (BPJ-EE) in the downlink MIMO systems with imperfect CSIT is addressed in this paper. A three part power consumption model is considered. By power conversion (PC) power we refer to power consumption proportional to the transmit power, which captures the effect of PA, feeder loss, and extra loss in transmission related cooling. By static power we refer to the power consumption which is assumed to be constant irrespective of the transmit power, number of transmit antennas and bandwidth. By dynamic power we refer to the power consumption including the circuit power, signal processing power, etc. and it is assumed to be irrespective of the transmit power but dependent of the number of transmit antennas and bandwidth. We divide the dynamic power into three parts. The first part "Dyn-I" is proportional to the transmit antenna number only, which can be viewed as the circuit power. The second part "Dyn-II" is proportional to the bandwidth only, and the third part "Dyn-III" is proportional to the multiplication of the bandwidth and transmit antenna number. "Dyn-II" and "Dyn-III" can be viewed as the signal processing power etc.. Interestingly, there are two main tradeoffs here. For one thing, more transmit antennas would increase the spatial multiplexing and diversity gain which leads to transmit power saving, while more transmit antennas would increase "Dyn-I" and "Dyn-III" which leads to dynamic power wasting. For another, multiplexing more active users with higher multiplexing gain would increase the inter-user interference, in which the multiplexing gain makes transmit power saving but inter-user interference induces transmit power wasting. In order to maximize BPJ-EE, the tradeoff among PC power, static power and dynamic power needs to be resolved and the tradeoff between the multiplexing gain and imperfect CSIT caused inter-user interference also needs to be carefully studied. The optimal adaptation which adaptively adjusts system parameters such as bandwidth, transmit power, use of singular value decomposition (SVD) or BD, number of active transmit/receive antennas, number of active users is considered in this paper to meet the challenge.
Contributions. By mode we refer to choice of transmission schemes i.e. SVD or BD, active transmit/receive antenna number and active user number. For each dedicated mode, we prove that the BPJ-EE is monotonically increasing as a function of bandwidth under the optimal transmit power without maximum power constraint.
Meanwhile, we derive the unique globally optimal transmit power with a constant bandwidth. Therefore, the optimal bandwidth is chosen to use the whole available bandwidth and the optimal transmit power can be correspondingly obtained. However, due to imperfect CSIT, it is emphasized that the capacity prediction is a big challenge during the above derivation. To cope with this problem, a capacity estimation mechanism is presented and accurate capacity estimation strategies are proposed.
The derivation of the optimal transmit power and bandwidth reveals the relationship between the BPJ-EE and the mode. Applying the derived optimal transmit power and bandwidth, mode switching is addressed then to choose the optimal mode. An ergodic capacity based mode switching algorithm is proposed. We derive the accurate closeform capacity approximation for each mode under imperfect CSIT at first and calculate the optimal BPJ-EE of each mode based on the approximation. Then, the preferred mode can be decided after comparison. The proposed mode switching scheme provides guidance on the preferred mode under given scenarios and can be applied offline.
Simulation results show that the mode switching improves the BPJ-EE significantly and it is promising for the energy efficient transmission.
The rest of the paper is organized as follows. Section II introduces the system model, power model and two transmission schemes and then section III gives the problem definition. Optimal bandwidth, transmit power derivation for each dedicated mode and capacity estimation under imperfect CSIT are presented in section IV. The ergodic capacity based mode switching is proposed in section V. The simulation results are shown in section VI and finally, section VII concludes this paper.
Regarding the notation, bold face letters refer to vectors (lower case) or matrices (upper case). Notation E(A) and Tr(A) denote the expectation and trace operation of matrix A, respectively. The superscript H and T represent the conjugate transpose and transpose operation, respectively.

A. System model
The downlink MIMO systems consist of a single BS with M antennas and K users each with N antennas.
M ≥ K × N is assumed. We assume that the channel matrix from the BS to the kth user at time n is H k [n] ∈ C N ×M , k = 1, . . . , K, which can be denoted as ζ k = Φd −λ k Ψ is the large scale fading including pathloss and shadowing fading, in which d k , λ denote the distance from the BS to the user k and the path loss exponent, respectively. The random variable Ψ accounts for the shadowing process. The term Φ denotes the pathloss parameter to further adapt the model which accounts for the BS and MS antenna heights, carrier frequency, propagation conditions and reference distance.Ĥ k [n] denotes the small scale fading channel. We assume that the channel experiences flat fading andĤ k [n] is well modeled as a spatially white Gaussian channel, with each entry CN (0, 1).
For the kth user, the received signal can be denoted as in which x[n] ∈ C M×1 is the BS's transmitted signal, n k [n] is the Gaussian noise vector with entries distributed according to CN (0, N 0 W ), where N 0 is the noise power density and W is the carrier bandwidth. The design of x[n] depends on the transmission schemes which would be introduced in subsection II-C.
As one objective of this paper is to study the impact of imperfect CSIT, we will assume perfect channel state information at the receive (CSIR) and imperfect CSIT here. CSIT is always get through feedback from the MSs in the FDD systems and through uplink channel estimation based on uplink-downlink reciprocity in the TDD systems, so the main sources of CSIT imperfection come from channel estimation error, delay and feedback error [15]- [17].
Only the delayed CSIT imperfection is considered in this paper, but note that the delayed CSIT model can be simply extended to other imperfect CSIT case such as estimation error and analog feedback [15], [16]. The channels will stay constant for a symbol duration and change from symbol to symbol according to a stationary correlation model.
Assume that there is D symbols delay between the estimated channel and the downlink channel. The current channel are jointly Gaussian with zero mean and are related in the following manner [16].Ĥ where ρ k denotes the correlation coefficient of each user,Ê k [n] is the channel error matrix, with i.i.d. entries . The amount of delay is τ = DT s , where T s is the symbol duration. And ρ k = J 0 (2πf d,k τ ) with Doppler spread f d,k , where J 0 (·) is the zeroth order Bessel function of the first kind, and ǫ 2 e,k = 1 − ρ 2 k [16]. Therefore, both ρ k and ǫ e,k are determined by the normalized Doppler frequency f d,k τ .

B. Power model
Apart from PA power and the circuit power, the signal processing, power supply and air condition power should also be taken into account at the BS [19]. Before introduction, assume the number of active transmit antennas is M a and the total transmit power is P t . Motivated by the power model in [7], [10], [19], the three part power model is introduced as follows. The total power consumption at BS is divided into three parts. The first part is the PC in which η is the power conversion efficiency, accounting for the PA efficiency, feeder loss, and extra loss in transmission related cooling. Although the total transmit power should be varied as M a and W changes, we study the total transmit power as a whole and the PC power includes all the total transmit power. The effect of M a and W on the transmit power independent power is expressed by the second part: the dynamic power P Dyn . P Dyn captures the effect of signal processing, circuit power, etc., which is dependent of M a and W but independent of P t . P Dyn is separated into three classes. The first class "Dyn-I" P Dyn−I is proportional to the transmit antenna number only, which can be viewed as the circuit power of the RF. The second part "Dyn-II" P Dyn−II is proportional to the bandwidth only, and the third part "Dyn-III" P Dyn−III is proportional to the multiplication of the bandwidth and transmit antenna number. P Dyn−II and P Dyn−III can be viewed as the signal processing related power. Thus, the dynamic power can be denoted as follows.
P Dyn = P Dyn−I + P Dyn−II + P Dyn−III , P Dyn−I = M a P cir , P Dyn−II = p ac,bw W, The third part is the static power P Sta which is independent of P t , M a and W , including the power consumption of cooling systems, power supply and so on. Combining the three parts, we have the total power consumption as follows.
Although the above power model is simple and abstract, it captures the effect of the key parameters such as P t , M a and W and coincides with the previous literatures [7], [10], [19]. Measuring the accurate power model for a dedicated BS is very important for the research of energy efficiency and the measuring may need careful field test, however, it is out of the scope here.
Note that here we omit the power consumption at the user side, as the users' power consumption is negligible compared with the power consumption of BS. Although any BS power saving design should consider the impact to the users' power consumption, it is beyond the scope of this paper.

C. Transmission Schemes
SU-MIMO with SVD and MU-MIMO with BD are considered in this paper as the transmission schemes. We will introduce them in this subsection.

1) SU-MIMO with SVD:
Before discussion, we assume that M a transmit antennas are active in the SU-MIMO.
As more active receive antennas result in transmit power saving due to higher spatial multiplexing and diversity gain, N antennas should be all active at the MS side 1 . The number of data streams is limited by the minimum number of transmit and receive antennas, which is denoted as N s = min(M a , N ).
In the SU-MIMO mode, SVD with equal power allocation is applied. Although SVD with waterfilling is the capacity optimal scheme [20], considering equal power allocation here helps the comparison between SU-MIMO and MU-MIMO fairly [16]. The SVD of H[n] is denoted as where 2) MU-MIMO with BD: We assume that K a users each with N a,i , i = 1, . . . , K a antennas are active at the same time. Denote the total receive antenna number as N a = Ka i=1 N a,i . As linear precoding is preformed, we have that M a ≥ N a [11], and then the number of data streams is N s = N a . The BD precoding scheme with equal power allocation is applied in the MU-MIMO mode. Assume that the precoding matrix for the kth user is T k [n] and the desired data for the kth user is The transmission model is In the perfect CSIT case, the precoding matrix is based on The detail of the design can be found in [11]. Define the effective channel as . Then the capacity can be denoted as In the delayed CSIT case, the precoding matrix design is based on the delayed version, i.e.
in which is the inter-user interference plus noise part.

III. PROBLEM DEFINITION
The objective of this paper is to maximize the BPJ-EE in the downlink MIMO systems. The BPJ-EE is defined as the achievable capacity divided by the total power consumption, which is also the transmitted bits per unit energy (Bits/Joule). Denote the BPJ-EE as ξ and then the optimization problem can be denoted as According to the above problem, bandwidth limitation is considered. In order to make the transmission most energy efficient, we should adaptively adjust the following system parameters: transmission scheme m ∈ {s, b}, i.e. use of SVD or BD, number of active transmit antennas M a , number of active users K a , number of receive antennas N a,i , i = 1, . . . , K a , transmit power P t and bandwidth W .
The optimization of problem (13) is divided into two steps. At first, determine the optimal P t and W for each dedicated mode. After that, apply mode switching to determine the optimal mode, i.e. optimal transmission scheme m, optimal transmit antenna number M a , optimal user number K a and optimal receive antenna number N a,i , according to the derivations of the first step. The next two sections will describe the details.

IV. MAXIMIZING ENERGY EFFICIENCY WITH OPTIMAL BANDWIDTH AND TRANSMIT POWER
The optimal bandwidth and transmit power are derived in this section under a dedicated mode. Unless otherwise specified, the mode, i.e. transmission scheme m, active transmit antenna number M a , active receive antenna number N a,i , i = 1, . . . , K a and active user number K a , is constant in this section. The following lemma is introduced at first to help the derivation.
in which a > 0 and b > 0. f (x) ≥ 0 (x ≥ 0) and f (x) is strictly concave and monotonically increasing. There exists a unique globally optimal x * given by where f ′ (x) is the first derivative of function f (x).

A. Optimal Energy Efficient Bandwidth
To illustrate the effect of bandwidth on the BPJ-EE, the following theorem is derived.
Theorem 1: Under constant P t , there exists a unique globally optimal W * given by to maximize ξ, in which R(W ) denotes the achievable capacity with a dedicated mode. If the transmit power scales as P t = p t W , ξ is monotonically increasing as a function of W .
Proof: See Appendix B.
This Theorem provides helpful insights about the system configuration. When the transmit power of BS is fixed, configuring the optimal bandwidth helps improve the energy efficiency. Meanwhile, if the transmit power can increase proportionally as a function of bandwidth based on P t = p t W , transmitting over the whole available spectrum is thus the optimal energy efficient transmission strategy. As P t can be adjusted in problem (13) and no maximum transmit power constraint is considered there, choosing W * = W max as the optimal bandwidth can maximize ξ. Therefore, W * = W max is applied in the rest of this paper.
One may argue that the transmit power is limited by the BS's maximum power in the real systems. In that case, W and P t should be jointly optimized. We consider this problem in our another work [23].

B. Optimal Energy Efficient Transmit Power
After determining the optimal bandwidth, we should derive the optimal P * t under W * = W max . In this case, we denote the capacity as R(P t ) with the dedicated mode. Then the optimal transmit power is derived according to the following theorem.
Theorem 2: There exists a unique globally optimal transmit power P * t of the BPJ-EE optimization problem given by Proof: See Appendix C.
Therefore, the optimal bandwidth and transmit power are derived based on Theorem 1 and Theorem 2. That is to say, the optimal bandwidth is chosen as W * = W max and the optimal transmit power is derived according to (17).
However, note that during the optimal transmit power derivation (17), the BS needs to know the achievable capacity based on the CSIT prior to the transmission. If perfect CSIT is available at BS, the capacity formula can be calculated at the BS directly according to (8) for SU-MIMO with SVD and (10) for MU-MIMO with BD.
But if the CSIT is imperfect, the BS need to predict the capacity then. In order to meet the challenge, a capacity estimation mechanism with delayed version of CSIT is developed, which is the main concern of next subsection.

C. Capacity Estimation Under Imperfect CSIT
1) SU-MIMO: SU-MIMO with SVD is relatively robust to the imperfect CSIT [16], using the delayed version of CSIT directly is a simple and direct way. Following proposition shows the capacity estimation of SVD mode.

Proposition 1:
The capacity estimation of SU-MIMO with SVD is directly estimated by: whereλ i is the singular value of H[n − D].
Proposition 1 is motivated by [16]. In Proposition 1, when the receive antenna number is equal to or larger than the transmit antenna number, the degree of freedom can be fully utilized after the receiver's detection, and then the ergodic capacity of (18) would be the same as the delayed CSIT case in (8). When the receive antenna number is smaller than the transmit antenna number, although delayed CSIT would cause degree of freedom loss and (18) cannot express the loss, the simulation will show that Proposition 1 is accurate enough to obtain the optimal ξ in that case.

2) MU-MIMO:
Since the imperfect CSIT leads to interuser interference in the MU-MIMO systems, simply using the delayed CSIT can not accurately estimate the capacity any longer. We should take the impact of interuser interference into account. J. Zhang et al. first considered the performance gap between the perfect CSIT case and the imperfect CSIT case [16], which is described as the following lemma.

Lemma 2:
The rate loss of BD with the delayed CSIT is upper bounded by [16]: Ptζ k N0W Ns ǫ 2 e,k + 1 .
As the BS can get the statistic variance of the channel error ǫ 2 e,k due to the doppler frequency estimation, the BS can obtain the upper bound gap △R upp b through some simple calculation. According to Proposition 1, we can use the delayed CSIT to estimate the capacity with perfect CSIT R P b and we denote the estimated capacity with perfect CSIT as . Combining (20) and Lemma 2, a lower bound capacity estimation is denoted as the perfect case capacity R est,P b minus the capacity upper bound gap △R upp b , which can be denoted as [18] However, this lower bound is not tight enough, a novel lower bound estimation and a novel upper bound estimation are proposed to estimate the capacity of MU-MIMO with BD.

Proposition 2:
The lower bound of the capacity estimation of MU-MIMO with BD is given by (22), while the upper bound of the capacity estimation of MU-MIMO with BD is given by (23). The lower bound in (22) Proposition 2 is motivated by [21]. It is illustrated as follows. Rewrite the transmission mode of user k of (9) as With delayed CSIT ,denote and the covariance matrix of the interference plus noise is then The According to Proposition 1 and Proposition 2, the capacity estimation for both SVD and BD can be performed. In order to apply Proposition 1 and Proposition 2 to derive the optimal bandwidth and transmit power, it is necessary to prove that the capacity estimation (18) (18) is similar with R s (M a , P t , W ) in (8), the same property of strictly concave and monotonically increasing of (18) is fulfilled. About (22) and (23) respectively. Calculating the first and second derivation of the above two equations, it can be proved that (22) and (23) are both strictly concave and monotonically increasing in P t and W . Therefore, based on the estimations of Proposition 1 and Proposition 2, the optimal bandwidth and transmit power can be derived at the BS.

A. Mode Switching Based on Instant CSIT
After getting the optimal bandwidth and transmit power for each dedicated mode, choosing the optimal mode with optimal transmission mode m * , optimal transmit antenna number M * a , optimal user number K * a each with optimal receive antenna number N * a,i is important to improve the energy efficiency. The mode switching procedure can be described as follows.

Energy Efficient Mode Switching Procedure
Step 1. For each transmission mode m with dedicated active transmit antenna number M a , active user number K a and active receive antenna number N a,i , calculate the optimal transmit power P * t and the corresponding BPJ-EE according to the bandwidth W * = W max and capacity estimation based on Proposition 1 and Proposition 2.
Step 2. Choose the optimal transmission mode m * with optimal M * a , K * a and N * a,i with the maximum BPJ-EE.
The above procedure is based on the instant CSIT. As we know, there are two main schemes to choose the optimal mode in the spectral efficient multimode transmission systems. The one is based on the instant CSIT [12]- [14], while the other is based on the ergodic capacity [15]- [17]. The ergodic capacity based mode switching can be performed off line and can provide more guidance on the preferred mode under given scenarios. If applying the ergodic capacity of each mode in the energy efficient mode switching, similar benefits can be exploited. The next subsection will present the approximation of ergodic capacity and propose the ergodic capacity based mode switching.

B. Mode Switching Based on the Ergodic Capacity
Firstly, the ergodic capacity of each mode need to be developed. The following lemma gives the asymptotic result of the point to point MIMO channel with full CSIT when M a ≥ N a .
Lemma 3: [16], [22] For a point to point channel when M a ≥ N a , denote β = Ma Na and γ = Ptζ k N0W . The capacity is approximated as in which C iso is the asymptotic spectral efficiency of the point to point channel, and C iso can be denoted as As SVD is applied in the SU-MIMO systems, the transmission is aligned with the maximum N s singular vectors.
When M a < N a , the achievable capacity approximation is modified as whereβ = 1 β = Na Ma . Therefore, according Proposition 1, the following proposition can be get directly.

Proposition 4:
The lower bound of the ergodic capacity estimation of MU-MIMO with BD is given by while the upper bound of the ergodic capacity estimation of MU-MIMO with BD is given by For comparison, the ergodic capacity lower bound based on (21) is also considered. As shown in (19), the expectation can be denoted as As △R upp b is a constant, we have E(△R upp b ) = △R upp b , and then Therefore, the lower bound estimation in (21) can also be applied to the ergodic capacity case. As the expectation of (20) can be denoted as [16] the low bound ergodic capacity estimation can be denoted as After getting the ergodic capacity of each mode, the ergodic capacity based mode switching algorithm can be summarized as follows.

Ergodic Capacity Based Energy Efficient Mode Switching
Step 1. For each transmission mode m with dedicated M a , K a and N a,i , calculate the optimal transmit power P * t and the corresponding BPJ-EE according to the bandwidth W * = W max and ergodic capacity estimation based on Proposition 3 and Proposition 4.
Step 2. Choose the optimal m * with optimal M * a , K * a and N * a,i with the maximum BPJ-EE. According to the ergodic capacity based mode switching scheme, the operation mode under dedicated scenarios can be determined in advance. Saving a lookup table at the BS according to the ergodic capacity based mode switching, the optimal mode can be chosen simply according to the application scenarios. The performance and the preferred mode in a given scenario will be shown in the next section.

VI. SIMULATION RESULTS
This section provides the simulation results. In the simulation, M = 6, N = 2 and K = 3. All users are assumed to be homogeneous with the same distance and moving speed. Only pathloss is considered for the large scale fading model and the pathloss model is set as 128.1+37.6 log 10 d k dB (d k in kilometers). Carrier frequency is set as 2GHz and D = 1ms. Noise density is N 0 = −174dBm/Hz. The power model is modified according to [19], which is set as η = 0.38, P cir = 66.4W, P Sta = 36.4W, p sp,bw = 3.32µW/Hz and p ac,bw = 1.82µW/Hz. W max = 5MHz.     It indicates that the upper bound capacity estimation is the best one in the BD scheme. Moreover, we can see that BPJ-EE of the BD scheme decreases seriously due to the imperfect CSIT caused inter-user interference. accurate when the moving speed is low. But when the speed is increasing, the ergodic capacity estimation of SU-MIMO(4,2) and SU-MIMO(6,2) can not track the decrease of BPJ-EE. There also exists a gap between the ergodic capacity estimation and the simulation in the SIMO mode. Although the mismatching exists, the ergodic capacity based mode switching can always match the optimal mode, which will be shown in the next figure. For the MU-MIMO modes, the two lower bound ergodic capacity estimation schemes mismatch the simulation more than the upper bound estimation scheme. That is because the lower bound estimations cause BPJ-EE decreasing twice.
Firstly, the derived transmit power would mismatch the exactly accurate transmit power because the derivation is based on a bound and this transmit power mismatch will make the BPJ-EE decrease compared with the simulation.
Secondly, the lower bound estimation use a lower bound formula to calculate the estimated BPJ-EE under the derived transmit power which will make the BPJ-EE decrease again. Nevertheless, the upper bound estimation has the opposite impact on the BPJ-EE estimation during the above two steps, so it matches the simulation much better.
According to Fig. 1 and Fig. 2, the upper bound estimation is the best estimation scheme for the MU-MIMO mode.
Therefore, during the ergodic capacity based mode switching, the upper bound estimation is applied. Fig. 3 depicts the BPJ-EE performance of mode switching. For comparison, the optimal mode with instant CSIT ('Optimal') is also plotted. The mode switching can improve the energy efficiency significantly and the ergodic capacity based mode switching can always track the optimal mode. The performance of ergodic capacity based switching is nearly the same as the optimal one. Through the simulation, the ergodic capacity based mode switching is a promising way to choose the most energy efficient transmission mode.   Inter-user interference is small when the moving speed is low, so higher multiplexing gain of MU-MIMO benefits.
When the moving speed is high, the inter-user interference with MU-MIMO becomes significant, so SU-MIMO which can totally avoid the interference is preferred. Let us focus on the effect of distance on the mode under high moving speed case then. When distance is less than 1.7km, SU-MIMO (2,2) is the optimal one, while the distance is equal to 2.1 km and 2.5km, the SIMO mode is suggested. When the distance is larger than 2.5km, the active transmit antenna number increases as the distance increases. The reason of the preferred mode variation can be explained as follows. As the total power can be divided into PC power, transmit antenna number related power "Dyn-I" and "Dyn-III" and transmit antenna number independent power "Dyn-II" and static power. The first and third part divided by capacity would increase as the active number increases, while the second part is opposite. In the long distance scenario, the first part will dominates the total power and then more active antenna number is preferred. In the short and medium distance scenario, the second and third part dominate the total power and the tradeoff between the two parts should be met. Above all, the above mode switching trends of Fig. 4 externalize the two tradeoffs.

VII. CONCLUSION
This paper discusses the energy efficiency maximizing problem in the downlink MIMO systems. The optimal bandwidth and transmit power are derived for each dedicated mode with constant system parameters, i.e. fixed transmission scheme, fixed active transmit/receive antenna number and fixed active user number. During the derivation, the capacity estimation mechanism is presented and several accurate capacity estimation strategies are proposed to predict the capacity with imperfect CSIT. Based on the optimal derivation, ergodic capacity based mode switching is proposed to choose the most energy efficient system parameters. This method is promising according to the simulation results and provides guidance on the preferred mode over given scenarios.

APPENDIX A PROOF OF LEMMA 1
Proof: The proof of the above lemma is motivated by [4]. Denote the inverse function of y = f (x) as x = g(y), then x * = arg max x f (x) ax+b = arg max g(y) y ag(y)+b . Denote y * = f (x * ). Since f (x) is monotonically increasing, y * = arg max y y ag(y)+b . According to [4], there exists a unique globally optimal y * given by if g(y) is strictly convex and monotonically increasing. (36) is fulfilled since the inverse function of g(y), i.e. f (x) is strictly concave and monotonically increasing. Taking g ′ (y) = 1 f ′ (x) and f (x) = y into (36), we can get (15).

APPENDIX B PROOF OF THEOREM 1
Proof: The first part can be proved according to Lemma 1. Calculating the first and second derivation of R(W ) based on (8), (10) and (11), we can see that R(W ) of both SVD and BD mode is strictly concave and monotonically increasing as a function of W . The optimal W * can be get through (15), which is given by (16).
Look at the second part. Taking P t = p t W into (8), (10) and (11), the capacity is R(P t , W ) = WR m (p t ), wherê R(p t ) is independent of W . We have that ξ = WR(pt) (Map sp,bw +p ac,bw )W +MaPcir+PPC+PSta .
The second part is verified.

APPENDIX C PROOF OF THEOREM 2
Proof: According to Lemma 1, the above theorem can be verified if we prove that R m (P t ) is strictly concave and monotonically increasing for both SVD and BD. It is obvious that the capacity of SVD and BD with perfect CSIT is strictly concave and monotonically increasing based on (8) and (10). If the capacity of BD with imperfect CSIT can also be proved to be strictly concave and monotonically increasing, Theorem 2 can be proved. and A k , respectively. Sorting c k,i and g k,i as c k,1 ≥ . . . ≥ c k,N a,k and g k,1 ≥ . . . ≥ g k,N a,k . Since A k andĤ eff,k [n]Ĥ H eff,k [n] are both positive definite, c k,i > g k,i , i = 1, . . . , N a,k . Calculating the first and second derivation of (38), (11) is strictly concave and monotonically increasing.
Then Theorem 2 is verified.