Improving energy efficiency through multimode transmission in the downlink MIMO systems

Abstract

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 article. 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, in which 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 into the preferred mode under given scenarios. Mode switching compromises different power parts, exploits the trade-off between the multiplexing gain and the imperfect CSIT caused inter-user interference and improves the BPJ-EE significantly.

1. 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 multiple-output (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 article.

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]. Miao et al. [4–6] considered the optimal rate and resource allocation problem in OFDMA SISO channels. 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. Zhang et al. [7] extended the energy efficiency problem to a bandwidth variable system 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 that compares the energy efficiency of Alamouti MIMO systems with two antennas and SIMO systems in the sensor networks. Kim et al. [9] presented the energy-efficient mode switching between SIMO and two antenna MIMO systems. A more general link adaptation strategy was proposed in [10] and the system parameters including the 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 works 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 studies 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 article. 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 on 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 trade-offs here. For one thing, more transmit antennas would increase the spatial multiplexing and diversity gain that leads to transmit power saving, while more transmit antennas would increase "Dyn-I" and "Dyn-III" leading 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 trade-off among PC, static and dynamic power needs to be resolved and the trade-off 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 article to meet the challenge.

The contributions of this paper are listed as follows. By mode, we refer to the 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 close-form 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 off-line. 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 article is organized as follows. Section 2 introduces the system model, power model and two transmission schemes and then Section 3 gives the problem definition. Optimal bandwidth, transmit power derivation for each dedicated mode and capacity estimation under imperfect CSIT are presented in Section 4. The ergodic capacity-based mode switching is proposed in Section 5. The simulation results are shown in Section 6 and, finally, section 7 concludes this article.

Regarding the notation, boldface letters refer to vectors (lower case) or matrices (upper case). Notation $\mathbf{E}\left(\mathbf{A}\right)$ 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.

2. Preliminaries

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 k th user at time n is H _k [n] ∈ ℂ^N×M, k = 1,..., K, which can be denoted as

$${\mathbf{H}}_k\left[n\right]={\zeta }_k{\widehat{\mathbf{H}}}_k\left[n\right]=\Phi d_k^{-\lambda }\Psi {\widehat{\mathbf{H}}}_k\left[n\right].$$

(1)

${\zeta }_k=\Phi d_k^{-\lambda }\Psi$ is the large-scale fading including path loss 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 path loss parameter to further adapt the model, which accounts for the BS and MS antenna heights, carrier frequency, propagation conditions and reference distance. ${\widehat{\mathbf{H}}}_k\left[n\right]$ denotes the small-scale fading channel. We assume that the channel experiences flat fading and ${\widehat{\mathbf{H}}}_k\left[n\right]$ is well modeled as a spatially white Gaussian channel, with each entry $C\mathcal{N}\left(0,1\right)$.

For the k th user, the received signal can be denoted as

$${\mathbf{y}}_k\left[n\right]={\mathbf{H}}_k\left[n\right]\mathbf{x}\left[n\right]+{\mathbf{n}}_k\left[n\right],$$

(2)

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 $C\mathcal{N}\left(0,N_{0}W\right)$, where N₀ 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 2-C.

As one objective of this article 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 got 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 ${\mathbf{H}}_k\left[n\right]={\zeta }_k{\widehat{\mathbf{H}}}_k\left[n\right]$ and its delayed version ${\mathbf{H}}_k\left[n-D\right]={\zeta }_k{\widehat{\mathbf{H}}}_k\left[n-D\right]$ are jointly Gaussian with zero mean and are related in the following manner [16].

$${\widehat{\mathbf{H}}}_k\left[n\right]={\rho }_k{\widehat{\mathbf{H}}}_k\left[n-D\right]+{\widehat{\mathbf{E}}}_k\left[n\right],$$

(3)

where ρ _k denotes the correlation coefficient of each user, ${\widehat{\mathbf{E}}}_k\left[n\right]$ is the channel error matrix, with i.i.d. entries $C\mathcal{N}\left(0,{\epsilon }_{e,k}^2\right)$ and it is uncorrelated with ${\widehat{\mathbf{H}}}_k\left[n-D\right]$. Meanwhile, we denote ${\mathbf{E}}_k\left[n\right]={\zeta }_k{\widehat{\mathbf{E}}}_k\left[n\right]$. The amount of delay is τ = DT _s , where T _s is the symbol duration. ρ _k = J₀(2πf _d,k τ) with Doppler spread f _d,k , where J₀(·) is the zeroth order Bessel function of the first kind, and ${\epsilon }_{e,k}^2=1-{\rho }_k^2$ [16]. Therefore, both ρ _k and ε _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 [19, 7, 10], 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 power

$$P_{\mathsf{\text{PC}}}=\frac{P_{\mathsf{\text{t}}}}{\eta },$$

(4)

in which η is the PC 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 on 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.

$$\begin{array}{c}{P}_{\mathsf{\text{Dyn}}}=P_{\mathsf{\text{Dyn}}-\mathsf{\text{I}}}+P_{\mathsf{\text{Dyn}}-\mathsf{\text{II}}}+P_{\mathsf{\text{Dyn}}-\mathsf{\text{III}}},\\ P_{\mathsf{\text{Dyn}}-\mathsf{\text{I}}}=M_{\mathsf{\text{a}}}{P}_{\mathsf{\text{cir}}},\\ P_{\mathsf{\text{Dyn}}-\mathsf{\text{II}}}=P_{\mathsf{\text{ac,bw}}}W,\\ P_{\mathsf{\text{Dyn}}-\mathsf{\text{III}}}=M_{\mathsf{\text{a}}}{p}_{\mathsf{\text{sp}},\mathsf{\text{bw}}}W,\end{array}$$

(5)

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:

$$P_{\mathsf{\text{tota}}1}=P_{\mathsf{\text{PC}}}+P_{\mathsf{\text{Dyn}}}+P_{\mathsf{\text{Sta}}}.$$

(6)

Although the above power model is simple and abstract, it captures the effect of the key parameters such as P_t, M_a,s and W and coincides with the previous literature [19, 7, 10]. 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 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 article.

C. Transmission schemes

Single user (SU)-MIMO with SVD and MU-MIMO with BD are considered in this article 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.^a 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 in the comparison between SU-MIMO and MU-MIMO fairly [16]. The SVD of H[n] is denoted as

$$\mathbf{H}\left[n\right]=\mathbf{U}\left[n\right]\Lambda \left[n\right]\mathbf{V}{\left[n\right]}^H,$$

(7)

in which Λ[n] is a diagonal matrix, U[n] and V[n] are unitary. The precoding matrix is designed as V[n] at the transmitter in the perfect CSIT scenario. However, when only the delayed CSIT is available at the BS, the precoding matrix is based on the delayed version, which should be V[n - D]. After the MS preforms MIMO detection, the achievable capacity can be denoted as

$$R_{\mathsf{\text{s}}}\left(M_{\mathsf{\text{a}}},P_{\mathsf{\text{t}}},W\right)=W\sum _{i=1}^{N_{\mathsf{\text{s}}}}log\left(1+\frac{P_{\mathsf{\text{t}}}}{N_{\mathsf{\text{s}}}{N}_{0}W}{\lambda }_i^2\right),$$

(8)

where λ _i is the i th singular value of H[n]V[n - D].

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 $N_{\mathsf{\text{a}}}=\sum _{i=1}^{K_{\mathsf{\text{a}}}}{N}_{\mathsf{\text{a}},i}$. 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 k th user is T _k [n] and the desired data for the k th user is s _k [n], then $x\left[n\right]=\sum _{i=1}^{K_{\mathsf{\text{a}}}}{T}_i\left[n\right]s_i\left[n\right]$. The transmission model is

$${\mathbf{y}}_k\left[n\right]={\mathbf{H}}_k\left[n\right]\sum _{i=1}^{K_{\mathsf{\text{a}}}}{\mathbf{T}}_i\left[n\right]{\mathbf{s}}_i\left[n\right]+{\mathbf{n}}_k\left[n\right].$$

(9)

In the perfect CSIT case, the precoding matrix is based on ${\mathbf{H}}_k\left[n\right]\sum _{i=1,i\ne k}^{K_{\mathsf{\text{a}}}}{\mathbf{T}}_i\left[n\right]=0$. The detail of the design can be found in [11]. Define the effective channel as H_eff,k[n] = H _k [n]T _k [n]. Then the capacity can be denoted as

$$\begin{array}{c}{R}_{\mathsf{\text{b}}}^P\left(M_{\mathsf{\text{a}}},K_{\mathsf{\text{a}}},N_{\mathsf{\text{a}},1},\dots ,N_{\mathsf{\text{a}}},{\mathsf{\text{K}}}_{\mathsf{\text{a}}},P_{\mathsf{\text{t}}},W\right)=\\ W\sum _{k=1}^{K_{\mathsf{\text{a}}}}log\mathsf{\text{det}}\left(\mathbf{I}+\frac{P_{\mathsf{\text{t}}}}{N_{\mathsf{\text{s}}}{N}_{0}W}{\mathbf{H}}_{\mathsf{\text{eff}},k}\left[n\right]{\mathbf{H}}_{\mathsf{\text{eff}},k}^H\left[n\right]\right).\end{array}$$

(10)

In the delayed CSIT case, the precoding matrix design is based on the delayed version, i.e., ${\mathbf{H}}_k\left[n-D\right]{\sum }_{i=1,i\ne k}^{K_{\mathsf{\text{a}}}}{\mathbf{T}}_i^{\left(D\right)}\left[n\right]=0$. Then define the effective channel in the delayed CSIT case as ${\widehat{\mathbf{H}}}_{\mathsf{\text{eff}},k}\left[n\right]={\mathbf{H}}_k\left[n\right]{\mathbf{T}}_k^{\left(D\right)}\left[n\right]$. The capacity can be denoted as [16]

$$\begin{array}{c}{R}_{\mathsf{\text{b}}}^D\left(M_{\mathsf{\text{a}}},K_{\mathsf{\text{a}}},N_{\mathsf{\text{a}},1},\dots ,N_{\mathsf{\text{a}}},{\mathsf{\text{K}}}_{\mathsf{\text{a}}},P_{\mathsf{\text{t}}},W\right)=\\ W\sum _{k=1}^{K_{\mathsf{\text{a}}}}logdet\left(\mathbf{I}+\frac{P_{\mathsf{\text{t}}}}{N_{\mathsf{\text{s}}}}{\widehat{\mathbf{H}}}_{\mathsf{\text{eff}},k}\left[n\right]{\widehat{\mathbf{H}}}_{\mathsf{\text{eff}},k}^H\left[n\right]R_k^{-1}\left[n\right]\right),\end{array}$$

(11)

in which

$${\mathbf{R}}_k\left[n\right]=\frac{P_t}{N_{\mathsf{\text{s}}}}{\mathbf{E}}_k\left[n\right]\left[\sum _{i\ne k}{\mathbf{T}}_i^{\left(D\right)}\left[n\right]{\mathbf{T}}_i^{\left(D\right)H}\left[n\right]\right]{\mathbf{E}}_k^H\left[n\right]+N_{0}W\mathbf{I}$$

(12)

is the inter-user interference plus noise part.

3. Problem definition

The objective of this article 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

$$\begin{array}{c}max\xi =\frac{R_{\mathsf{\text{m}}}\left(M_{\mathsf{\text{a}}},K_{\mathsf{\text{a}}},N_{\mathsf{\text{a,}}1},\dots ,N_{\mathsf{\text{a,}}{K}_{\mathsf{\text{a}}}},P_{\mathsf{\text{t}}},W\right)}{P_{\mathsf{\text{tota}}1}}\\ \mathsf{\text{s}}.\mathsf{\text{t}}.P_{\mathsf{\text{TX}}}\ge 0,\\ 0\le W\le W_{max}.\end{array}$$

(13)

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.

4. 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 in the derivation.

Lemma 1: For optimization problem

$$\begin{array}{c}max\frac{f\left(x\right)}{ax+b},\\ \mathsf{\text{s}}\mathsf{\text{.t}}\mathsf{\text{.}}x\ge 0\end{array}$$

(14)

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

$$x^{*}=\frac{f\left(x^{*}\right)}{f^{\prime }\left(x^{*}\right)}-\frac{b}{a},$$

(15)

where f'(x) is the first derivative of function f(x).

Proof: See Appendix A.

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

$$W^{*}=\frac{\left(P_{\mathsf{\text{PC}}}+P_{\mathsf{\text{Sta}}}+M_{\mathsf{\text{a}}}{P}_{\mathsf{\text{cir}}}\right)+\left(M_{\mathsf{\text{a}}}{p}_{\mathsf{\text{sp,bw}}}+P_{\mathsf{\text{ac,bw}}}\right)R\left(W^{*}\right)}{\left(M_{\mathsf{\text{a}}}{p}_{\mathsf{\text{sp,bw}}}+P_{\mathsf{\text{ac,bw}}}\right)R^{\prime }\left(W^{*}\right)}$$

(16)

to maximize ξ, in which R(W) denotes the achievable capacity with a dedicated mode. If the transmit power scales as P_t = p_tW, ξ is monotonically increasing as a function of W.

Proof: See Appendix B.

This theorem provides helpful insights into 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_tW, 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, and choosing W* = W_max as the optimal bandwidth can maximize ξ. Therefore, W* = W_max is applied in the rest of this article.

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 [21].

B. Optimal energy-efficient transmit power

After determining the optimal bandwidth, we should derive the optimal $P_{\mathsf{\text{t}}}^{*}$ 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 $P_{\mathsf{\text{t}}}^{*}$ of the BPJ-EE optimization problem given by

$$P_{\mathsf{\text{t}}}^{*}=\frac{R\left(P_{\mathsf{\text{t}}}^{*}\right)}{R^{\prime }\left(P_{\mathsf{\text{t}}}^{*}\right)}-\eta \left(P_{\mathsf{\text{Sta}}}+P_{\mathsf{\text{Dyn}}}\right).$$

(17)

Proof: See Appendix C.

Therefore, the optimal bandwidth and transmit power are derived based on Theorems 1 and 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 needs 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 the next subsection.

C. Capacity estimation under imperfect CSIT

1) SU-MIMO

SU-MIMO with SVD is relatively robust to the imperfect CSIT [16], and using the delayed version of CSIT directly is a simple and direct way. The following proposition shows the capacity estimation of SVD mode.

Proposition 1: The capacity estimation of SU-MIMO with SVD is directly estimated by:

$$R_{\mathsf{\text{s}}}^{\mathsf{\text{est}}}=W\sum _{i=1}^{N_{\mathsf{\text{s}}}}log\left(1+\frac{P_{\mathsf{\text{t}}}}{N_{\mathsf{\text{s}}}{N}_{0}W}{\tilde{\lambda }}_i^2\right),$$

(18)

where ${\tilde{\lambda }}_i$ 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 inter-user interference in the MU-MIMO systems, simply using the delayed CSIT cannot accurately estimate the capacity any longer. We should take the impact of inter-user interference into account. Zhang et al. [16] first considered the performance gap between the perfect CSIT case and the imperfect CSIT case, which is described as the following lemma.

Lemma 2: The rate loss of BD with the delayed CSIT is upper bounded by [16]:

$$\begin{array}{c}\Delta R_{\mathsf{\text{b}}}=R_{\mathsf{\text{b}}}^P-R_{\mathsf{\text{b}}}^D\le \Delta R_{\mathsf{\text{b}}}^{\mathsf{\text{upp}}}=\\ W\sum _{k=1}^{K_{\mathsf{\text{a}}}}{N}_{\mathsf{\text{a}},k}{log}_2\left[\sum _{i=1,i\ne k}^{K_{\mathsf{\text{a}}}}{N}_{\mathsf{\text{a}},i}\frac{P_{\mathsf{\text{t}}}{\zeta }_k}{N_{0}W{N}_{\mathsf{\text{s}}}}{\epsilon }_{e,k}^2+1\right].\end{array}$$

(19)

As the BS can get the statistic variance of the channel error ${\epsilon }_{e,k}^2$ due to the Doppler frequency estimation, the BS can obtain the upper bound gap $\Delta R_{\mathsf{\text{b}}}^{\mathsf{\text{upp}}}$ through some simple calculation. According to Proposition 1, we can use the delayed CSIT to estimate the capacity with perfect CSIT $R_{\mathsf{\text{b}}}^P$ and we denote the estimated capacity with perfect CSIT as

$$R_{\mathsf{\text{b}}}^{\mathsf{\text{est}},P}=W\sum _{k=1}^{K_{\mathsf{\text{a}}}}logdet\left(\mathbf{I}+\frac{P_{\mathsf{\text{t}}}}{N_{\mathsf{\text{s}}}{N}_{0}W}{\mathbf{H}}_{\mathsf{\text{eff}},k}\left[n-D\right]{\mathbf{H}}_{\mathsf{\text{eff}},k}^H\left[n-D\right]\right),$$

(20)

in which H_eff,k[n - D] = H _k [n - D]T _k [n - D]. Combining (20) and Lemma 2, a lower bound capacity estimation is denoted as the perfect case capacity $R_{\mathsf{\text{b}}}^{\mathsf{\text{est}},P}$ minus the capacity upper bound gap $\Delta R_{\mathsf{\text{b}}}^{\mathsf{\text{upp}}}$, which can be denoted as [18]

$$R_{\mathsf{\text{b}}}^{\mathsf{\text{est}}-\mathsf{\text{Zhang}}}=R_{\mathsf{\text{b}}}^{\mathsf{\text{est}},P}-\Delta R_{\mathsf{\text{b}}}^{\mathsf{\text{upp}}}.$$

(21)

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) is tighter than $R_{\mathsf{\text{b}}}^{\mathsf{\text{est,Zhang}}}$ in (21).

$$R_{\mathsf{\text{b}}}^{\mathsf{\text{est}},\mathsf{\text{low}}}=W\sum _{k=1}^{K_{\mathsf{\text{a}}}}logdet\left(\mathbf{I}+\frac{P_{\mathsf{\text{t}}}/N_{\mathsf{\text{s}}}}{N_{0}W+\sum _{i=1,i\ne k}^{K_{\mathsf{\text{a}}}}{N}_{\mathsf{\text{a}},i}\frac{P_{\mathsf{\text{t}}}{\zeta }_k}{N_{\mathsf{\text{s}}}}{\epsilon }_{e.k}^2}{\mathbf{H}}_{\mathsf{\text{eff}},k}\left[n-D\right]{\mathbf{H}}_{\mathsf{\text{eff}},k}^H\left[n-D\right]\right)$$

(22)

$$R_{\mathsf{\text{b}}}^{\mathsf{\text{est}},\mathsf{\text{upp}}}=W\sum _{k=1}^{K_{\mathsf{\text{a}}}}\left\{logdet\left(\mathbf{I}+\frac{P_{\mathsf{\text{t}}}/N_{\mathsf{\text{s}}}}{N_{0}W+\sum _{i=1,i\ne k}^{K_{\mathsf{\text{a}}}}{N}_{\mathsf{\text{a,}}i}\frac{P_{\mathsf{\text{t}}}{\zeta }_k}{N_{\mathsf{\text{S}}}}{\epsilon }_{e,k}^2}{\mathbf{H}}_{\mathsf{\text{eff}},k}\left[n-D\right]\underset{\mathsf{\text{eff}},k}{\overset{H}{\mathbf{H}}}\left[n-D\right]\right)+\left(N_{\mathsf{\text{a}},k}/M_{\mathsf{\text{a}}}\right){log}_2\left(e\right)\right\}$$

(23)

Proposition 2 is motivated by [22]. It is illustrated as follows. Rewrite the transmission mode of user k of (9) as

$${\mathbf{y}}_k\left[n\right]={\mathbf{H}}_k\left[n\right]{\mathbf{T}}_k\left[n\right]{\mathbf{s}}_k\left[n\right]+{\mathbf{H}}_k\left[n\right]\sum _{i\ne k}{\mathbf{T}}_i\left[n\right]{\mathbf{s}}_i\left[n\right]+{\mathbf{n}}_k\left[n\right].$$

(24)

With delayed CSIT, denote

$$B_k\left[n\right]={\mathbf{H}}_k\left[n\right]\sum _{i\ne k}{\mathbf{T}}_i^{\left(D\right)}\left[n\right]{\mathbf{s}}_i\left[n\right]={\mathbf{E}}_k\left[n\right]\sum _{i\ne k}{\mathbf{T}}_i^{\left(D\right)}\left[n\right]{\mathbf{s}}_i\left[n\right],$$

then ${\mathbf{A}}_k\left[n\right]=B_k\left[n\right]B_k^H\left[n\right]$ and the covariance matrix of the interference plus noise is then

$${\mathbf{R}}_k\left[n\right]=\frac{P_{\mathsf{\text{t}}}}{N_{\mathsf{\text{S}}}}{\mathbf{A}}_k\left[n\right]+N_{0}W\mathbf{I}\left[n\right].$$

(25)

The expectation of R _k [n] is [16]

$$\mathbf{E}\left({\mathbf{R}}_k\left[n\right]\right)=\sum _{i=1,i\ne k}^{K_{\mathsf{\text{a}}}}{N}_{\mathsf{\text{a}},i}\frac{P_{\mathsf{\text{t}}}{\zeta }_k}{N_{\mathsf{\text{s}}}}{\epsilon }_{e,k}^2\mathbf{I}+N_{0}W\mathbf{I}$$

(26)

Based on Proposition 1, we use H_eff,k[n - D] with the delayed CSIT to replace the ${\widehat{\mathbf{H}}}_{\mathsf{\text{eff}},k}\left[n\right]$ in (11). Then the capacity expression of each user is similar to the SU-MIMO channel with inter-stream interference. The capacity lower bound and upper bound with a point to point MIMO channel with channel estimation errors in [22] is applied here. Therefore, the lower bound estimation (22) and upper bound estimation (23) can be verified according to the lower and upper bounds in [22] and (26).

We can get $R_{\mathsf{\text{b}}}^{\mathsf{\text{est}},\mathsf{\text{low}}}-R_{\mathsf{\text{b}}}^{\mathsf{\text{est}},\mathsf{\text{Zhang}}}>0$ after some simple calculation, so $R_{\mathsf{\text{b}}}^{\mathsf{\text{est}},\mathsf{\text{low}}}$ is tighter than $R_{\mathsf{\text{b}}}^{\mathsf{\text{est}},\mathsf{\text{Zhang}}}$. ξ

According to Propositions 1 and 2, the capacity estimation for both SVD and BD can be performed. In order to apply Propositions 1 and 2 to derive the optimal bandwidth and transmit power, it is necessary to prove that the capacity estimation (18) for SU-MIMO and (22, 23) for MU-MIMO are all strictly concave and monotonically increasing. At first, as $R_{\mathsf{\text{s}}}^{\mathsf{\text{est}}}$ in (18) is similar to 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), the proof of strictly concave and monotonically increasing is similar with the proof procedure in Theorem 2. If we denote g _k,i > 0, i = 1,..., N_a,kas the eigenvalues of ${\mathbf{H}}_{\mathsf{\text{eff}},k}\left[n-D\right]{\mathbf{H}}_{\mathsf{\text{eff}},k}^H\left[n-D\right]$, (22) and (23) can be rewritten as

$$R_{\mathsf{\text{b}}}^{\mathsf{\text{est}},1\mathsf{\text{ow}}}=W\sum _{k=1}^{K_{\mathsf{\text{a}}}}\sum _{i=1}^{N_{\mathsf{\text{a,}}k}}log\left(1+\frac{P_{\mathsf{\text{t}}}/N_{\mathsf{\text{s}}}}{N_{0}W+{\sum }_{i=1,i\ne k}^{K_{\mathsf{\text{a}}}}{N}_{\mathsf{\text{a,}}i}\frac{P_{\mathsf{\text{t}}}{\zeta }_k}{N_{\mathsf{\text{s}}}}{\epsilon }_{e\mathsf{\text{,}}k}^2}{g}_{k,i}\right)$$

and

$$R_{\mathsf{\text{b}}}^{\mathsf{\text{est}},\mathsf{\text{upp}}}=W\sum _{k=1}^{K_{\mathsf{\text{a}}}}\left\{\left[\sum _{i=1}^{N_{\mathsf{\text{a,}}k}}log\left(1+\frac{P_{\mathsf{\text{t}}}/N_{\mathsf{\text{s}}}}{N_{0}W+{\sum }_{i=1,i\ne k}^{K_{\mathsf{\text{a}}}}{N}_{\mathsf{\text{a}},i}\frac{P_{\mathsf{\text{t}}}{\zeta }_k}{N_{\mathsf{\text{s}}}}{\epsilon }_{e,k}^2}{g}_{k,i}\right)\right]+\left(N_{\mathsf{\text{a}},k}/M_{\mathsf{\text{a}}}\right){log}_2\left(e\right)\right\},$$

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 Propositions 1 and 2, the optimal bandwidth and transmit power can be derived at the BS.

5. Energy-efficient mode switching

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 $M_{\mathsf{\text{a}}}^{*}$, optimal user number $K_{\mathsf{\text{a}}}^{*}$ each with optimal receive antenna number $N_{\mathsf{\text{a,}}i}^{*}$ 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 $P_{\mathsf{\text{t}}}^{*}$ and the corresponding BPJ-EE according to the bandwidth W* = W_max and capacity estimation based on Propositions 1 and 2.

Step 2. Choose the optimal transmission mode m* with optimal $M_{\mathsf{\text{a}}}^{*}$, $K_{\mathsf{\text{a}}}^{*}$ and $N_{\mathsf{\text{a}},i}^{*}$ with the maximum BPJ-EE. ξ

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: For a point to point channel when M_a ≥ N_a, denote $\beta =\frac{M_{\mathsf{\text{a}}}}{N_{\mathsf{\text{a}}}}$ and $\gamma =\frac{P_{\mathsf{\text{t}}}{\zeta }_k}{N_{0}W}$ [16, 23]. The capacity is approximated as

$$R_{\mathsf{\text{s}}}^{\mathsf{\text{appro}}}\approx W{C}_{\mathsf{\text{iso}}}\left(\beta ,\beta \gamma \right)$$

(27)

in which ${\mathcal{C}}_{\mathsf{\text{iso}}}$ is the asymptotic spectral efficiency of the point to point channel, and ${\mathcal{C}}_{\mathsf{\text{iso}}}$ can be denoted as

$$\begin{array}{c}\frac{{\mathcal{C}}_{\mathsf{\text{iso}}}\left(\beta ,\gamma \right)}{N_{\mathsf{\text{a}}}}={log}_2\left[1+\gamma -\mathcal{F}\left(\beta ,\frac{\gamma }{\beta }\right)\right]\\ +\beta {log}_2\left[1+\frac{\gamma }{\beta }-\mathcal{F}\left(\beta ,\frac{\gamma }{\beta }\right)\right]-\beta \frac{{log}_2\left(e\right)}{\gamma }\mathcal{F}\left(\beta ,\frac{\gamma }{\beta }\right)\end{array}$$

(28)

with

$$\mathcal{F}\left(x,y\right)=\frac{1}{4}{\left[\sqrt{1+y{\left(1+\sqrt{x}\right)}^2}-\sqrt{1+y{\left(1-\sqrt{x}\right)}^2}\right]}^2.$$

As SVD is applied in the SU-MIMO systems, and the transmission is aligned with the maximum N_s singular vectors. When M_a < N_a, the achievable capacity approximation is modified as

$$R_{\mathsf{\text{s}}}^{\mathsf{\text{appro}}}\approx W{\mathcal{C}}_{\mathsf{\text{iso}}}\left(\widehat{\beta },\widehat{\beta }\gamma \right),$$

(29)

where $\widehat{\beta }=\frac{1}{\beta }=\frac{N_{\mathsf{\text{a}}}}{M_{\mathsf{\text{a}}}}$.

Therefore, according to Proposition 1, the following proposition can be get directly.

Proposition 3: The ergodic capacity of SU-MIMO with SVD is estimated by:

$$R_{\mathsf{\text{s}}}^{\mathsf{\text{Ergodic}}}=R_{\mathsf{\text{s}}}^{\mathsf{\text{appro}}}.$$

(30)

Although Zhang et al. [16] give another accurate approximation for the MU-MIMO systems with BD, it is only applicable in the scenario in which ${\sum }_{i=1}^{K_{\mathsf{\text{a}}}}{N}_{\mathsf{\text{a}},i}=M_{\mathsf{\text{a}}}$. We develope the ergodic capacity estimation with BD based on Proposition 2.

As T _k [n - D] is designed to null the inter-user interference, it is a unitary matrix independent of H _k [n - D]. So H _k [n - D]T _k [n - D] is also a zero-mean complex Gaussian matrix with dimension N_a,k× M_a,k, where $M_{\mathsf{\text{a}},k}=M_{\mathsf{\text{a}}}-{\sum }_{i=1,i\ne k}^{K_{\mathsf{\text{a}}}}{N}_{\mathsf{\text{a}},i}$. The effective channel matrix of user k can be treated as a SU-MIMO channel with transmit antenna number M_a,kand receive antenna number N_a,k. Combining Propositions 1, 2, and 3, we have the following Proposition.

Proposition 4: The lower bound of the ergodic capacity estimation of MU-MIMO with BD is given by

$$R_{\mathsf{\text{b}}}^{\mathsf{\text{Ergodic}}-1\mathsf{\text{ow}}}\approx W\sum _{k=1}^{K_{\mathsf{\text{a}}}}{\mathcal{C}}_{\mathsf{\text{iso}}}\left({\widehat{\beta }}_k,{\widehat{\beta }}_k{\widehat{\gamma }}_k\right),$$

(31)

while the upper bound of the ergodic capacity estimation of MU-MIMO with BD is given by

$$R_{\mathsf{\text{b}}}^{\mathsf{\text{Ergodic}}-\mathsf{\text{upp}}}\approx W\sum _{k=1}^{K_{\mathsf{\text{a}}}}\left[{\mathcal{C}}_{\mathsf{\text{iso}}}\left({\widehat{\beta }}_k,{\widehat{\beta }}_k{\widehat{\gamma }}_k\right)+\frac{1}{{\widehat{\beta }}_k}{log}_2\left(e\right)\right],$$

(32)

where

$$\begin{array}{c}{\widehat{\beta }}_k=M_{\mathsf{\text{a}},k}/N_{\mathsf{\text{a}},k},\\ {\widehat{\gamma }}_k=\frac{P_{\mathsf{\text{t}}}{\zeta }_k}{N_{0}W+{\sum }_{i=1,i\ne k}^{K_{\mathsf{\text{a}}}}{N}_{\mathsf{\text{a}},i}\frac{P_{\mathsf{\text{t}}}{\zeta }_k}{N_{\mathsf{\text{s}}}}{\epsilon }_{e,k}^2}.\end{array}$$

For comparison, the ergodic capacity lower bound based on (21) is also considered. As shown in (19), the expectation can be denoted as

$$\mathbf{E}\left(R_{\mathsf{\text{b}}}^P-R_{\mathsf{\text{b}}}^D\right)\le \mathbf{E}\left(\Delta R_{\mathsf{\text{b}}}^{\mathsf{\text{upp}}}\right).$$

As $\Delta R_{\mathsf{\text{b}}}^{\mathsf{\text{upp}}}$ is a constant, we have $\mathbf{E}\left(\Delta R_{\mathsf{\text{b}}}^{\mathsf{\text{upp}}}\right)=\Delta R_{\mathsf{\text{b}}}^{\mathsf{\text{upp}}}$, and then

$$\mathbf{E}\left(R_{\mathsf{\text{b}}}^P\right)-\mathbf{E}\left(R_{\mathsf{\text{b}}}^D\right)\le \Delta R_{\mathsf{\text{b}}}^{\mathsf{\text{upp}}}.$$

(33)

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]

$$\mathbf{E}\left(R_{\mathsf{\text{b}}}^{\mathsf{\text{est}},P}\right)=W\sum _{k=1}^{K_{\mathsf{\text{a}}}}{\mathcal{C}}_{\mathsf{\text{iso}}}\left({\widehat{\beta }}_k,{\widehat{\beta }}_k\gamma \right),$$

(34)

the low bound ergodic capacity estimation can be denoted as

$$R_{\mathsf{\text{b}}}^{\mathsf{\text{Ergodic}}-\mathsf{\text{Zhang}}}\approx W\sum _{k=1}^{K_{\mathsf{\text{a}}}}{\mathcal{C}}_{\mathsf{\text{iso}}}\left({\widehat{\beta }}_k,{\widehat{\beta }}_k\gamma \right)-\Delta R_{\mathsf{\text{b}}}^{\mathsf{\text{upp}}}.$$

(35)

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 $P_{\mathsf{\text{t}}}^{*}$ and the corresponding BPJ-EE according to the bandwidth W* = W_max and ergodic capacity estimation based on Propositions 3 and 4.

Step 2. Choose the optimal m* with optimal $M_{\mathsf{\text{a}}}^{*}$, $K_{\mathsf{\text{a}}}^{*}$ and $N_{\mathsf{\text{a,}}i}^{*}$ with the maximum BPJ-EE. ξ

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.

6. 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 path loss is considered for the large-scale fading model and the path loss model is set as 128.1+37.6 log₁₀ d _k dB (d _k in kilometers). Carrier frequency is set as 2 GHz and D = 1 ms. Noise density is N₀ = -174 dBm/Hz. The power model is modified according to [19], which is set as η = 0.38, P_cir = 66.4 W, P_Sta = 36.4 W, p_sp,bw = 3.32 µ W/Hz, and p_ac,bw = 1.82 µ W/Hz. W_max = 5 MHz. For simplification, "SU-MIMO (M_a, N_a)" denotes SU-MIMO mode with M_a active transmit antennas and N_a active receive antennas, "SIMO" denotes SU-MIMO mode with one active transmit antennas and N active receive antennas and "MU-MIMO (M_a, N_a, K_a)" denotes MU-MIMO mode with M_a active transmit antennas and K_a users each N_a active receive antennas. Seven transmission modes are considered in the simulation, i.e., SIMO, SU-MIMO (2,2), SU-MIMO (4,2), SU-MIMO (6,2), MU-MIMO (4,2,2), MU-MIMO (6,2,2), MU-MIMO (6,2,3). In the simulation, the solution of (15)-(17) is derived by the Newton's method, as the close-form solution is difficult to obtain.

Figure 1 depicts the effect of capacity estimation on the optimal BPJ-EE under different moving speed. The optimal estimation means that the BS knows the channel error during calculating $P_{\mathsf{\text{t}}}^{*}$ and the precoding is still based on the delayed CSIT. In the left figure, SU-MIMO is plotted. The performance of capacity estimation and the optimal estimation are almost the same, which indicates that the capacity estimation of the SU-MIMO systems is robust to the delayed CSIT. Another observation is that the BPJ-EE is nearly constant as the moving speed is increasing for SIMO and SU-MIMO (2,2), while it is decreasing for SU-MIMO (4,2) and SU-MIMO (6,2). The reason can be illustrated as follows. The precoding at the BS cannot completely align with the singular vectors of the channel matrix under the imperfect CSIT. But when the transmit antenna number is equal to or greater than the receive antenna number, the receiver can perform detection to get the whole channel matrix's degree of freedom. However, when the transmit antennas are less than the receive antenna, the receiver cannot get the whole degree of freedom only through detection, so the degree of freedom loss occurs. The center and right figures show us the effect of capacity estimation with MU-MIMO modes. The three estimation schemes all track the effect of imperfect CSIT. From the amplified sub-figures, the upper bound capacity estimation is the closest one to the optimal estimation. 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.

Figure 1

The effect of capacity estimation on the energy efficiency of SU-MIMO and MU-MIMO under different speed.

Figure 2 compares the BPJ-EE derived by ergodic capacity estimation schemes and the one by simulations. The left figure demonstrates the SU-MIMO modes. The estimation of SIMO, SU-MIMO (4,2) and SU-MIMO (6,2) is 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) cannot 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 with 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 uses 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 Figures 1 and 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.

Figure 2

Comparison of energy efficiency based on ergodic capacity and instant capacity with SU-MIMO and MU-MIMO.

Figure 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.

Figure 3

Performance of mode switching.

Figure 4 demonstrates the preferred transmission mode under the given scenarios. The optimal mode under different moving speed and distance is depicted. This figure provides insights into the PC power/dynamic power/static power trade-off and the multiplexing gain/inter-user interference compromise. When the moving speed is low, MU-MIMO modes are preferred and vice versa. This result is similar to the spectral efficient mode switching in [15–18]. Inter-user interference is small when the moving speed is low, so there is 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 1.7 km, SU-MIMO (2,2) is the optimal one, while the distance is equal to 2.1 and 2.5 km, the SIMO mode is suggested. When the distance is larger than 2.5 km, the active transmit antenna number increases as the distance increases. The reason of the preferred mode variation can be explained as follows. 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 dominate the total power and then a more active antenna number is preferred. In the short and medium distance scenario, the second and third part dominate the total power and the trade-off between the two parts should be met. Above all, the above mode switching trends of Figure 4 externalize the two trade-offs.

Figure 4

Optimal mode under different scenario. ○: SIMO, ×: SU-MIMO (2,2), +: SU-MIMO (4,2),ξ: SU-MIMO (6,2),◊: MU-MIMO (4,2,2),∇: MU-MIMO (6,2,2),⊲: MU-MIMO (6,2,3).

7. Conclusion

This article 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 $x^{*}=arg\underset{x}{max}\frac{f\left(x\right)}{ax+b}=arg\underset{g\left(y\right)}{max}\frac{y}{ag\left(y\right)+b}$. Denote y* = f(x*). Since f(x) is monotonically increasing, $y^{*}=arg\underset{y}{max}\frac{y}{ag\left(y\right)+b}$. According to [4], there exists a unique globally optimal y* given by

$$y^{*}=\frac{b+ag\left(y^{*}\right)}{a{g}^{\prime }\left(y^{*}\right)}$$

(36)

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 $g^{\prime }\left(y\right)=\frac{1}{f^{\prime }\left(x\right)}$ 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 got through (15), which is given by (16).

Look at the second part. Taking P_t = p_tW into (8), (10) and (11), the capacity is $R\left(P_{\mathsf{\text{t}}},W\right)=W{\widehat{R}}_{\mathsf{\text{m}}}\left(p_{\mathsf{\text{t}}}\right)$, where $\widehat{R}\left(p_{\mathsf{\text{t}}}\right)$ is independent of W. We have that

$$\xi =\frac{W\widehat{R}\left(p_{\mathsf{\text{t}}}\right)}{\left(M_{\mathsf{\text{a}}}{p}_{\mathsf{\text{sp,bw}}}+p_{\mathsf{\text{ac,bw}}}\right)W+M_{\mathsf{\text{a}}}{P}_{\mathsf{\text{cir}}}+P_{\mathsf{\text{PC}}}+P_{\mathsf{\text{Sta}}}}.$$

(37)

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.

Denoting ${\mathbf{A}}_k={\mathbf{E}}_k\left[n\right]\left[{\sum }_{i\ne k}{\mathbf{T}}_i^{\left(D\right)}\left[n\right]{\mathbf{T}}_i^{\left(D\right)H}\left[n\right]\right]{\mathbf{E}}_k^H\left[n\right]$, then rewrite (11) as follows:

$$\begin{array}{c}{R}_{\mathsf{\text{b}}}^D\left(P_{\mathsf{\text{t}}}\right)=W\sum _{k=1}^{K_{\mathsf{\text{a}}}}\left\{logdet\left({\mathbf{R}}_k\left[n\right]+\frac{P_t}{N_{\mathsf{\text{s}}}}{\widehat{\mathbf{H}}}_{\mathsf{\text{eff}},k}\left[n\right]\underset{\mathsf{\text{eff}},k}{\overset{H}{\widehat{\mathbf{H}}}}\left[n\right]\right)-logdet{\mathbf{R}}_k\left[n\right]\right\}\\ =W\sum _{k=1}^{K_{\mathsf{\text{a}}}}\left\{logdet\left[\mathbf{I}+\frac{P_{\mathsf{\text{t}}}}{N_{0}WN\mathsf{\text{s}}}\left({\mathbf{A}}_k+{\widehat{\mathbf{H}}}_{\mathsf{\text{eff}},k}\left[n\right]\underset{\mathsf{\text{eff}},k}{\overset{H}{\widehat{\mathbf{H}}}}\left[n\right]\right)\right]-logdet\left(\mathbf{I}+\frac{P_{\mathsf{\text{t}}}}{N_{0}WN\mathsf{\text{s}}}{\mathbf{A}}_k\right)\right\}\\ =W\sum _{k=1}^{K_{\mathsf{\text{a}}}}\sum _{i=1}^{N_{\mathsf{\text{a,}}k}}\left\{log\left(1+\frac{P_{\mathsf{\text{t}}}}{N_{0}WN\mathsf{\text{s}}}{c}_{k,i}\right)-log\left(1+\frac{P_{\mathsf{\text{t}}}}{N_{0}WN\mathsf{\text{s}}}{g}_{k,i}\right)\right\}.\end{array}$$

(38)

c _k,i and g _k,i are the eigenvalue of ${\mathbf{A}}_k+{\widehat{\mathbf{H}}}_{\mathsf{\text{eff}},k}\left[n\right]{\widehat{\mathbf{H}}}_{\mathsf{\text{eff}},k}^H\left[n\right]$ and A _k , respectively. Sorting c _k,i and g _k,i as $c_{k,1}\ge ...\ge c_{k,N_{\mathsf{\text{a,}}k}}$ and $g_{k,1}\ge ...\ge g_{k,N_{\mathsf{\text{a,}}k}}$. Since A _k and ${\widehat{\mathbf{H}}}_{\mathsf{\text{eff}},k}\left[n\right]{\widehat{\mathbf{H}}}_{\mathsf{\text{eff}},k}^H\left[n\right]$ 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.