ADAPTIVE power allocation for wireless systems in fading

Transcription

1 Power Allocation in Multi-Antenna Wireless Systems Subject to Simultaneous Power Constraints Mostafa Khoshnevisan, Student Member, IEEE, and J. Nicholas Laneman, Senior Member, IEEE Abstract We address the problem of power allocation to maimize ergodic capacity subject to multiple power constraints assuming that perfect causal channel state information CSI) is available at both the transmitter and the receiver. We characterize the optimal power allocation subject to both long-term and short-term power constraints, which depends upon the ratio of the two power levels. Additionally, we find a suboptimal power allocation if the input power is subject to long-term and per-antenna power constraints. We characterize the conditions for which one power constraint dominates and the other can be ignored. Numerical results suggest that, for the Rayleigh fading case, a short-term power constraint that is larger than a long-term power constraint does not significantly impact the ergodic capacity of the channel. The effect of per-antenna power constraints is also eplored for the case of Rayleigh fading through our numerical results. Inde Terms Adaptive transmission, channel state information CSI), fading channels, multiple-input multiple-output MIMO) systems, power control, resource allocation. I. INTRODUCTION ADAPTIVE power allocation for wireless systems in fading environments attempts to maimize a performance metric, e.g., the ergodic capacity, by allocating power based on the instantaneous channel state information CSI) subject to limitations in terms of power constraints. In real-world wireless communication systems, there are three important limitations on the transmitted signal s power. One limitation results from the battery life of the mobile, which is captured by long-term power constraints. Another limitation results from regulations that prevent the transmitter from having an arbitrary power level due to environmental safety and interference avoidance. According to the Federal Communication Commission FCC), the transmit power in any time duration should not eceed a certain amount depending on the application, frequency, height of the antenna, population of that area per square mile, and so on ]. This regulatory constraint is captured by short-term power constraints. Still another set of limitations result from practical system design, such as a per-antenna power constraint that eeps the amplifier at each transmit antenna in its linear range. In designing the communication system, these types of constraints and others should all be taen into account. A. Related Wor Some wors consider long-term average power constraints only, for which the average is taen over both the codewords This wor was supported in part by NSF grant CCF Mostafa Khoshnevisan and J. Nicholas Laneman are with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN, USA, s: {mhoshne, jnl}@nd.edu). and channel fading coefficients. For eample, ] considers single-antenna systems and shows that the optimal power allocation policy is water-filling in time. Additionally, multipleantenna systems are considered in ]. Both of these papers assume that CSI is available causally at both the transmitter and the receiver. In other wors, only short-term average power constraints are considered, for which the average is taen only over the codewords and the constraint applies to each channel fading coefficient. In 4], the author studies the capacity of a multipleinput-multiple-output MIMO) channel subject to a short-term power constraint for complete CSI, and for CSI at the receiver only. Another eample of evaluating the capacity subject to only a short-term power constraint is 5], in which the authors provide an overview of the results on the Shannon capacity of MIMO channels under different assumptions on availability of CSI or Channel Distribution Information CDI). We use terminology for the power constraints long-term and shortterm) from 6], in which the authors study the delay-limited capacity subject to a long-term power constraint, a short-term power constraint, or both. In this paper, we focus on the single user case. Optimal multiuser power allocation subject to long-term power constraints has been studied in 7] and 8], while 9] studies optimal multiuser power allocations subject to both long- and short-term power constraints in wireless single-input-single-output SISO) systems. Another power constraint that arises in this paper is a perantenna power constraint. In ], an optimization problem for a multiuser downlin channel with multiple transmit antennas at the base station subject to per-antenna power constraints is considered, which is transformed into a dual uplin problem. In ], per-antenna power constraints are considered in MIMO wireless systems in the contet of beamforming with incomplete CSI. Also, ] studies optimal power allocation subject to l p -norm constrained eigenvalues, which can be viewed as a suboptimal power allocation policy under shortterm and per-antenna power constraints. In a recent wor, ] studies the capacity of MIMO channels subject to only perantenna power constraints in which the authors formulate the capacity optimization as a semi-definite program SDP) and introduce an algorithm to find the optimal signaling. In general, per-antenna power constraints can be specified as either long- or short-term constraints. We only consider shortterm per-antenna power constraints, since they are important in practical system design. For succinctness, we refer to this power constraint as the per-antenna power constraint, which should not be confused with the short-term power constraint, by which we mean a short-term constraint on sum power.

2 B. Summary of Contributions Despite of the rich literature on power allocation in wireless systems, power allocation subject to multiple power constraints is not well-studied. In this paper, we study the structure of optimal power allocation to maimize the ergodic capacity in single and multiple-antenna systems considering simultaneous power constraints described in the previous sections, assuming perfect causal CSI at both the transmitter and the receiver. In Section II, we describe the channel model and problem formulation. In Section III, we study the optimal power allocation for the SISO and MIMO channels. Our main contributions in this section can be summarized as follows. For single-antenna systems subject to both long-term and short-term power constraints, power allocations that maimize the ergodic capacity are partially developed in 4]. We characterize the complete structure of the optimal power allocation considering the cases in which one of the power constraints dominates, e.g., the shortterm power constraint can be ignored if the ratio of the short-term power constraint to the long-term power constraint is larger than a certain threshold. We etend the results to MIMO channels. In addition to providing proofs for the results that were initially published in our conference paper 5], we specify an algorithm for computing the optimal power allocation. We study a suboptimal power allocation if the input power is subject to simultaneous long-term and perantenna power constraints. The suboptimality comes from using a more stringent power constraint in order to mae the optimization problem more tractable. For each of the problems described above, we mathematically characterize the conditions for which one power constraint dominates and the other can be essentially ignored. These conditions allow a system designer to simplify the power allocations in several regimes. In Section IV, we specialize many of the results to the case of Rayleigh fading and provide numerical results that illustrate the SNR regimes in which one power constraint can be ignored. We find simpler epressions for the thresholds in the power allocations introduced in Section III, and prove their uniqueness for the Rayleigh fading case. Numerical results for the case of Rayleigh fading suggest that if the input power is subject to long- and short-term power constraints, a short-term power constraint P ma that is larger than a long-term power constraint does not significantly impact the ergodic capacity of the channel, especially for large values of average SNR. We also eplore the impact of per-antenna power constraints on the ergodic capacity for the case of Rayleigh fading through our numerical results. We conclude the paper in Section V. II. CHANNEL MODEL AND PROBLEM STATEMENT The baseband-equivalent discrete-time input-output relationship in our MIMO channel model is yi) = Hi)i) + ni), ) where i is the time inde, yi) is a comple vector of N R received signals, i) is a comple vector of N T transmit signals. Hi) and ni) are random sequences capturing the effect of multipath fading and additive noise, respectively. The noise ni) is a vector of N R zero-mean, circularly symmetric, comple Gaussian random variables with Eni)ni) H ] = N I NR, and ni), i =,,... is a sequence of independent random vectors. The multipath fading Hi) at each time is an N R N T matri of comple fading coefficients. We assume that the matri fading process is stationary and ergodic, and consider codewords that are are long enough to eperience all fading coefficients, and therefore, a suitable metric is the ergodic capacity. We assume that the fading coefficients vary slowly enough that CSI is available to the receiver and transmitter. CSI in this paper is the causal channel matri Hi). For the case of Rayleigh fading in Section IV, we assume that the entries of Hi) are independent and identically distributed IID) comple Gaussian random variables with mean zero and variance / per real dimension. The general optimization problem for maimizing the ergodic capacity subject to long-term, short-term, and perantenna power constraints is ma QH) E H log det I + N HQH)H H )], a) subject to E H trqh))], b) H : trqh)) P ma, H : q H) ˆP, =,,..., N T, c) d) where Q is the input covariance matri, which can be a function of the instantaneous channel matri, and q is the th diagonal entry of the matri Q. The epectations in a) and b) are with respect to the distribution of H. The power constraints are described in b), c), and d); represents the long-term power constraint, P ma represents the shortterm power constraint, and ˆP represents the per-antenna power constraint. For simplicity, we drop the time inde i. This channel model and the optimization problem simplify to a scalar/vector problem for the SISO/MISO cases considered in the sequel. Throughout the paper, we consider the optimization problem subject to simultaneous power constraints b) and c), or simultaneous power constraints b) and d). Interested readers are referred to 6] for a suboptimal power allocation subject to all three constraints. III. GENERAL STRUCTURE OF OPTIMAL POWER ALLOCATION It is conceptually and notationally appealing to treat SISO and MIMO channels separately, since in the SISO case the short-term power constraint coincides with the per-antenna power constraint. A. SISO Channels We first obtain the optimal power allocation in a SISO system subject to both long- and short-term power constraints. The channel model is the scalar form of ) with N T = N R =. Let γ := h N denote the instantaneous received SNR without power adaptation, where h is the scalar fading

3 coefficient, and let fγ) denote the probability density function pdf) of γ. Let P γ) denote the power policy capturing the second moment of the input signal as a function of γ. Then the received SNR with power adaptation is P γ)γ/, and the ergodic capacity is ] C = E γ log + P γ)γ )]. ) Based upon the coding theorem in 4], the ergodic capacity is the solution to the following optimization problem: ma log + P γ)γ ) fγ) dγ, 4) P γ) subject to E γ P γ)], γ : P γ) P ma, which corresponds to a scalar version of ). To determine the general power policy, let α := P ma / be the ratio of the short-term power constraint to the long-term power constraint. The form of the optimal power allocation depends upon the values of α and a certain threshold α th which is obtained at the end of this section), and can be separated into the following three cases: Case, α : In this case, the short-term power constraint dominates and the long-term power constraint can be ignored. Therefore, the optimal power policy in this case is P γ) = P ma, γ. 5) Case, α th α: In this case, the long-term power constraint dominates and the power policy is water-filling in time ] {, γ < γ P γ) = γ γ, γ γ, 6) where the threshold γ is determined from substituting 6) into E γ P γ)] =. 7) Case, < α α th : In this case both long- and short-term power constraints play a role, and the power allocation in 4] provides the solution to the optimization problem. The power allocation in this case becomes, γ < γ P γ) = γ, γ γ < γ, 8) γ P ma, γ αγ γ αγ where the threshold γ is determined by substituting 8) into 7), i.e., the solution to equation γ αγ ) fγ) dγ + P ma fγ) dγ = γ γ γ. γ αγ 9) Optimal power allocation 8) comes from the Karush- Kuhn-Tucer KKT) conditions 7]. As we mentioned previously, 4] determines a similar solution, which is valid only for Case, since the threshold γ in 8) has a valid solution only in this regime. In Section IV, for the Rayleigh fading channel, we show that the threshold γ is uniquely determined from 7) if < α α th Case ). Note that the threshold γ in 8) and γ in 6) are, in general, different. Now, we obtain the value of α th. In order to eliminate the short-term power constraint, the power policy in 6) should always satisfy the short-term power constraint P γ) P ma, γ, but the maimum value of P γ) in 6) occurs as γ and is equal to /γ. Therefore, γ P ma γ α α th = γ. ) In general, the values of γ and α th depend on the distribution of γ and the average SNR of the system. In Section IV, we find these values for a Rayleigh fading channel, both analytically and numerically. It is worth mentioning that the value of α th is important in practical wireless communication systems, since it might be the case that the allowed α is larger than α th. In that case, we can simply ignore the short-term power constraint, and the optimal power allocation policy is water-filling in time. B. MIMO Channels In this section, we assume that there are multiple antennas at the transmitter and the receiver. The channel model and problem statement are described in Section II. Let n := ma N R, N T ) and m := min N R, N T ). The fading matri H can be represented using singular value decomposition SVD) as H = UΛV H, ) where U and V are unitary matrices and Λ is a diagonal matri with entries equal to the square roots of the eigenvalues of the matri { HH H, if N R N T W = H H. H, if N R > N T Denote the eigenvalues of W by λ, m. The equivalent channel model is 4] ỹ = Λ + ñ, ) where we use the transformation ỹ = U H y, = V H, and ñ = U H n. The equivalent channel consists of m parallel channels. Note that the trace of the input covariance matri is invariant with respect to this transformation, e.g., trq ) = trq ), where Q and Q are the covariance matrices of the random vectors and, respectively. Let Λ := mn Λ be the normalized channel with diagonal entries equal to square root of λ = mn λ, m. In the SVD equivalent channel model, the power allocation policy is a function of Λ. Therefore, the covariance matri can be given as QΛ ) or Qλ ), where λ := λ, λ,..., λ m]. To maimize the ergodic capacity, the covariance matri should be diagonal 4]. Let P λ ), m denote the th diagonal entry of the covariance matri. Note that each P is a function of the vector λ, or in other words, a function of all λ l s,, l m.

4 4 ) Long-Term and Short-Term Power Constraints: If the input power is subject to long- and short-term power constraints, we remove the constraint in d). Then, using the SVD equivalent channel model and the above definitions, the optimization problem in ) becomes ma P λ ),=,,...,m C = E λ m = m subject to E λ P λ ) λ : = log + P λ )λ ) ], /m ) ], P λ ) P ma. = The optimal power allocation structure can be found by eamining the KKT conditions. Formally, we state and prove the following theorem. Theorem. The solution to the optimization problem ) for P ma is ) +, P λ v ) = m if = v + Pma ) ) +,, otherwise β+v 4) where ) + := ma, ). The Lagrange multipliers v and β are obtaining by solving = β + v E λ ) + = P ma, 5) m ] P λ ) =. 6) = Proof: See Appendi A. In the power allocation above, v and β are Lagrange multipliers. Note that there is a subtle difference between v and β; v is a constant that is fied for all fading coefficients all λ ) and is obtained such that 6) is satisfied. Although v does not depend on the realization of fading coefficients, it certainly depends on the distribution of the fading process. By contrast, the value of β depends on the current channel fading coefficients λ as we can see from 5). We now give an algorithm to find the thresholds as well as the optimal power allocation for a given realization of the fading coefficients, which gives some intuition about Theorem. Algorithm. ) Computing the constant v: Let βv, λ ) denote the solution of 5). Substituting the power allocation 4) into 6) and averaging over the distribution of the fading coefficients, the only remaining variable, v, can be determined. This step needs to be done only once, and can be performed off-line. ) Water-filling in time and space: Given the realization of fading coefficients, compute allocated powers for the m parallel channels according to water-filling in time and space with the threshold v, that is P = v ) +. Fig.. Illustration of the optimal power allocation for a MIMO system subject to long- and short-term power constraints ) Chec the summation: If the summation of these powers m = P ) is not larger than P ma, then the optimal power allocation is the computed one and we are done. Otherwise, go to step 4. 4) Computing the value of β: Given the fading coefficients and the constant v, find β according to 5). 5) Penalizing the powers: Repeat step, but this time with the threshold ) β + v instead of v, i.e., P = +. β+v Then, we have the optimal power allocation and we are done. Note that this step penalizes the previously computed powers by adding β into the denominator of v such that the short-term power constraint is satisfied with equality. An illustration of the optimal power policy for a MIMO system in Figure gives some insight about the structure of the power allocation and Algorithm. The horizontal ais corresponds to time. For every time slot the value of for =, for the two singular values of the channel) is setched. In fact, the vertical ais corresponds to how wea the channels are, and the arrows indicate the allocated power to the corresponding channels. Note that constant v is chosen such that the long-term power constraint is satisfied with equality, and is obtained from step of Algorithm. More power is allocated to stronger channels, but we do not allow the sum of the allocated powers corresponding to the two singular values of the channel to eceed the short-term power constraint P ma in any time slot step ). As it can be seen, at time slots,, 4, 5, and 7, the power policy is water-filling across time and space step ). However, for time slots and 6, we have to adjust the water level, according to step 5 of Algorithm, such that the sum of the powers allocated to the two singular values of the channel is equal to the short-term power constraint. The value of β depends on the channel fading coefficients, and can be obtained from step 4 of Algorithm. In Section III-A, finding the optimal power allocation was separated into three cases depending on the value of α and a constant α th. An analogous situation arises for MIMO systems, as we now briefly discuss. Case, α : In this case, the short-term power constraint

5 5 dominates and the long-term power constraint can be removed. The optimal power allocation is the well-nown water-filling across antennas, but not across time, as obtained in 4]. Case, α th α: In this case, the power allocation in Theorem is valid, but it simplifies to P = v ) +, 7) where the constant v is determined by substituting 7) into 6). In fact, we can eliminate the short-term power constraint, and the problem reduces to finding the optimal power allocation subject to only a long-term power constraint, which has been eamined in ]. Case, < α α th : Both power constraints play a role and the optimal power allocation is given by Theorem, which was discussed earlier. The only remaining tas is to obtain the value of α th. Note that for Case, we must ensure that the power allocation in 7) does not violate the short-term power constraint for all channel fading coefficients. As a result, = v ) + P ma, λ = v P ma m v P ma. Therefore, we have α th = m. 8) v ) Long-Term and Per-Antenna Power Constraints: If the input power is subject to long-term and per-antenna power constraints, we remove the constraint in c). First, note that for a Hermitian matri Q we have 8] ma q ma eigenvalue Q), 9) N T N T where q is the th diagonal entry of the matri Q, and eigenvalue Q) is the th eigenvalue of the matri Q. If we consider the SVD method discussed earlier, then under the transformation = V H, the eigenvalues of the input covariance matri do not change. On the other hand, the eigenvalues of the matri trq ) are equal to the diagonal entries P λ ), m. Therefore, the constraint P λ ) ˆP, =,..., m, ) is sufficient to satisfy d). Because of inequality 9), this condition is only sufficient and not necessary It is possible that d) is satisfied, but ) is not). Therefore, we can find a suboptimal solution by assuming a more stringent constraint. Specifically, we consider the optimization problem ma P λ ),=,,...,m C = E λ m = m subject to E λ P λ ) = log + P λ )λ ) ], /m ) ], λ : P λ ) ˆP, =,..., m. Theorem. The solution to the optimization problem ) for m ˆP is, if P λ v ) = v, if v > v ˆP. ) ˆP, otherwise Proof: The proof is similar to the proof of Theorem and is omitted due to space considerations. In ), v is a constant that is fied for all fading coefficients, and is determined by substituting the power allocation ) into 6). Let α := m ˆP /. Note that the definition of α is slightly different from Sections III-A and III-B, since there are m per-antenna power constraints here. Again, the general power allocation can be specialized in three cases. Case, α : In this case, the long-term power constraint can be removed and the power policy is given by P λ ) = ˆP. Case, α th α: In this case, the per-antenna power constraint can be removed. The power allocation in Theorem is valid but simplifies to 7), where v is determined from 6). By the same procedure as before, we can find that α th = m/v ). Case, < α α th : In this case, both power constraints play a role and the power policy is given by ). IV. RAYLEIGH FADING CHANNELS In Section III, we obtained the optimal power allocation policies for general channel models subject to various combinations of power constraints. In this section, we consider the IID Rayleigh fading channel model, simplify the power policies to the etent possible, and provide some numerical results. A. Analysis Many of the constants and thresholds for the power allocations developed in Section III are functions of the distribution of the fading process. In this section, we simplify the calculations needed to obtain these constants and thresholds in the IID Rayleigh fading case. ) SISO and MISO: Consider the parameter γ defined in Section III-A. For SISO Rayleigh fading γ is an eponential random variable with epected value /N. Let γ be the average SNR of the system, which in the case of SISO and MISO systems is γ = /N. We denote the probability density function of γ by fγ). Consider the thresholds γ, α th, and γ in the optimal power allocation from Section III-A. In 9], the value of γ is derived for Rayleigh fading channel as the solution to the equation e γ γ γ γ E ) γ = γ, ) γ where E ) is the eponential integral defined by E ) := + t e t dt,. Note that the value of α th is only a function of γ. For the threshold γ, we have the following theorem.

6 6 Theorem. If < α α th Case ), then the threshold γ in 8) for the Rayleigh fading SISO channel can be determined from e ) e α γ E )+E +α γe α γ = γ, α γ 4) where := γ / γ. Furthermore, γ is unique and γ, α ]. Proof: See Appendi B. Note that a MISO system with N T = n can be analyzed just lie a SISO system with different distributions for the equivalent channel gain γ, which depends on n and the multiantenna scheme. We have considered two different schemes, one based on optimal beamforming and referred to as the SVD method, and the other based upon the antenna selection. Interested readers are referred to 6] for analytical results on the thresholds α th and γ in MISO systems with Rayleigh fading. Numerical results are provided in Section IV-B for both SISO and MISO systems. ) MIMO: In the MIMO case, because of the more intricate structure of the power policies, finding a simplified equation for epressing some of the thresholds is challenging. In particular, when the power allocation contains a Lagrange multiplier, which depends on the instantaneous fading coefficients lie β in Sections III-B), there is no closed form. However, we can find α th in the Rayleigh fading channel model. In Sections III-B and III-B, we have α th = m/v ), where v is the constant in the optimal power policy 7). From 6), α th becomes the solution to α th ) f λ ) d =, 5) = p= q= α th where f λ ) is the probability distribution function of a normalized unordered eigenvalue of the Wishart matri 4]. Let λ = mn denote the average SNR per parallel channel. From the calculation of ] and ], 5) reduces to )! ) p+q ) ) n m+ n m+ n m+ )!p!q! p q where G r s) is defined as G r s) := G n m+p+q ) = m λ, 6) λα th Γr +, s) s B. Numerical Results and Discussion Γr, s) integers r. As with the analysis in Section IV-A, all the numerical results in this section are for IID Rayleigh fading channels. For ease of presentation, we split this section into two subsections. ) SISO and MISO: In Figure, we plot α th versus γ for SISO systems, and MISO systems with SVD and antenna selection, each with n = and n = 8. Note that in practical wireless communication systems, it is advantageous to have α th small, since this means that we can ignore the short-term power constraint for a wider range of α. According to Figure, as we increase the average SNR, the value of α th gets smaller α th SISO MISO Selectionn=) MISO SVDn=) MISO Selectionn=8) MISO SVDn=8) Average SNR db) Fig.. The value of α th, the threshold separating Cases and, versus average SNR γ for SISO and MISO systems with Rayleigh fading. and finally approaches for large average SNRs. It can be easily seen that ) has a unique solution and γ always lies in the interval, ] 9]. Therefore, α th = /γ is unique and is always larger than one. The behavior of α th for large average SNR can be intuitively eplained as follows. Power allocation water-filling) is most important at lower SNR, which might require the short-term power to fluctuate significantly with the fading coefficients. As a result, for optimality at low SNR, we need to allow higher short-term powers relative to longterm powers, and correspondingly α th is larger at low SNR. In other words, the short-term power constraint is somehow more relevant at low SNR compared to high SNR. Also, note that for the case of MISO systems, the value of α th is always smaller than for the SISO system. As we can see in Figure, the more antennas at the transmitter, the smaller α th. Net, we eamine the effect of α on the capacity. Figure shows the capacity versus average SNR for each of the two schemes in the MISO channel with the two etreme possibilities for α: α = and α = We are not interested in the case α < for which the long-term power constraint does not affect the ergodic capacity, since average SNR depends on the long-term power constraint). In fact, if α =, the problem reduces to the case in which there is no short-term power constraint it is also the case in which α th α, γ), and if α =, the problem reduces to constant power allocation since the power policy is P γ) =. Note that even in this case of constant power allocation, we use the CSIT because we perform singular value decomposition or antenna selection. For other values of < α < between the two etremes, the capacity versus average SNR curve lies between the two curves for α = and α =. As we can see from Figure, the difference between the capacities for the two etreme values of α becomes negligible at high SNR for both SVD and antenna selection. We observe that the value of the short-term power constraint P ma ) does not significantly impact the ergodic capacity of the channel for large average SNRs, as long as it is larger than the long-term power constraint, i.e., α.

7 7.5 SVD, α= SVD, α= Antenna Selection, α= Antenna Selection, α= 4.5 C α= /C No CSIT 4 System) C α= /C No CSIT System) C α= /C α= System) C α= /C α= 4 System) Capacity Bits).5 Ratio of Capacities Average SNR db) Average SNR db) Fig.. Ergodic capacity versus average SNR for MISO systems with Rayleigh fading, n = and α = and α =. Fig. 5. Ratio of ergodic capacities versus average SNR per parallel channel ) for MIMO systems with Rayleigh fading subject to long- and shortterm power mn constraints. α th MIMO system 4 MIMO system 8 MIMO system Average SNR db) Fig. 4. The value of α th, the threshold separating Cases and, versus Average SNR per parallel channel ) for MIMO systems with Rayleigh mn fading. ) MIMO: Figure 4 shows the value of α th versus the average SNR per parallel channel λ = mn ) for MIMO systems according to 6). This plot suggests similar observations as for MISO systems: Water-filling has more shortterm fluctuations at low SNR, which maes α th larger. Again, the short-term power constraint is more relevant at low SNR compared to high SNR. Also, note that Figures and 4 show the effect of the distribution of the channel singular values on α th. The observation is that, as we increase the number of antennas at the transmitter, the value of α th decreases. In fact, we can conclude that for a well-conditioned Wishart matri in which the number of transmit antennas is much larger than the number of receive antennas ], the value of α th is smaller compared to the case of an ill-conditioned Wishart matri. This suggests that as we increase the number of antennas at the transmitter, the short-term power constraint becomes less relevant, since the value of α th becomes smaller. Figure 5 eamines the effect of P ma on the ergodic capacity of and 4 MIMO systems with IID Rayleigh fading subject to long- and short-term power constraints. In the two lower curves, we compare the capacity of two etreme cases: P ma = and P ma or α = and α ). These curves suggest that if P ma is larger than or equal to, the value of the short-term power level P ma ) has a negligible impact on the ergodic capacity, especially for large values of average SNR. The same result for the case of equal number of antennas at the transmitter and receiver is reported in ], in which the curves for the two cases of space-time waterfilling and spatial water-filling are equivalent to the case with α = and α =, respectively. Therefore, for a fied average SNR, the value of the short-term power constraint has a very small impact on the ergodic capacity at moderate to high SNR. Note that in the case of MISO systems in Figure, P ma has a considerable effect on the ergodic capacity at low SNR. However, for MIMO systems, this effect is reduced. Also, note that the impact of the short-term power constraint on the ergodic capacity again reduces as we increase the number of antennas at the transmitter. This is not surprising, as we have already seen from Figure 4 that the value of α ma becomes smaller for a larger number of transmit antennas. Figure 5 also shows the ratio of the ergodic capacity for α = to the ergodic capacity if there is no CSI at the transmitter two upper curves). For the MIMO system, the loss in the ergodic capacity is negligible at high SNR. This always happens for an m n IID Rayleigh fading MIMO system with m n as discussed in ], since at high SNR, the waterfilling strategy allocates an equal amount of power to all the spatial modes, as well as an equal amount of power over time. However, this is not true for the 4 MIMO system, because if there are more transmit antennas than receive antennas, there is a power boost of N T /N R from having CSIT ]. Now, we turn our attention to the effect of per-antenna power constraint on the ergodic capacity. To better see this impact in the low SNR regime, we plot the ratio of the ergodic capacities versus average SNR for and 4 MIMO

8 8 Ratio of Capacities.5.5 C Power Allocation /C No CSIT 4 System) C Power Allocation /C No CSIT System) C Power Allocation /C No Per antenna 4 System) C Power Allocation /C No Per antenna System) Average SNR db) Fig. 6. Ratio of ergodic capacities versus average SNR for MIMO systems with Rayleigh fading subject to long-term and per-antenna power constraints with α =. systems with IID Rayleigh fading in Figure 6. Consider the problem in Section III-B with α = or ˆP = ). In Figure 6, C Power Allocation is the ergodic capacity obtained from power allocation ), C No CSIT is the ergodic capacity if there is no CSIT, and C No Per-Antenna is the ergodic capacity if there are no per-antenna power constraints and the input is only subject to a long-term power constraint. The two upper curves in Figure 6 are the ratio C Power Allocation /C No CSIT, which shows how much gain one can get from the suboptimal power allocation ) compared to the naive equal power allocation across time and space. This ratio is considerably larger than for low average SNR, and decreases to for large average SNR in the case of MIMO system. This implies that the benefit of CSIT and using power allocation is considerable in the low SNR regime. The two lower curves in Figure 6 are the ratio C Power Allocation /C No Per-Antenna, which shows the impact of a per-antenna power constraint on the ergodic capacity. Note that this ratio is eactly equal to for average SNR larger than 4. db and.45 db for the 4 and MIMO systems, respectively. The reason is that the value of α th is equal to at average SNR equal to 4. db and.45 db for the 4 and MIMO systems, respectively, as can be seen from Figure 4, which means that for average SNR larger than those, we are in Case and the per-antenna power constraint can be ignored. From Figure 6, we can infer that per-antenna power constraints become more important at low SNR. Also, note that the fact that this ratio is close to is consistent with the observation in ] that with equal per-antenna power levels the capacity with per-antenna power constraints can be close to the capacity with sum power constraint. The figure also suggests that as we increase the number of antennas at the transmitter the effect of per-antenna power constraint on the ergodic capacity becomes less important. We stress that the results and observations in this section are only applicable to IID Rayleigh fading. By contrast, the effect of short-term power constraint on the ergodic capacity is considerable if the model is Rayleigh fading with lognormal shadowing. We anticipate such a result based upon our observations in conjunction with results in ], in which the authors demonstrate that with log-normal shadowing, spacetime water-filling achieves significantly higher ergodic capacity than spatial water-filling in low to moderate SNR regimes. V. CONCLUSION In this paper, we considered simultaneous power constraints in multi-antenna wireless systems with perfect causal CSI, and aimed to allocate power in order to maimize the ergodic capacity. Under some conditions, one of the power constraints does not have any impact on the ergodic capacity and can be essentially ignored. These conditions allow a system designer to simplify the power allocations in several regimes. Numerical results for the case of Rayleigh fading suggest that a shortterm power constraint that is larger than a long-term power constraint does not significantly impact the ergodic capacity. In the high SNR regime, both short-term and per-antenna power constraints have only a slight impact on the ergodic capacity, since the water-filling strategy allocates an equal amount of power across time and space. Numerical results also suggest that in the low SNR regime, the benefit of CSIT and power allocation can be considerable. APPENDIX A PROOF OF THEOREM Since ) is a conve optimization problem, we prove the theorem using the KKT conditions 7]. For simplicity, we drop λ from P λ ) and simply denote it by P in the proof of Theorems, and. To maimize the capacity, the long-term power constraint should be satisfied with equality, which is possible since α. Let θ, θ,..., θ m denote the Lagrange multipliers corresponding to the constraints that force the powers to be positive P, P,..., P m, respectively), θ be the Lagrange multiplier corresponding to the short-term power constraint, and v be the Lagrange multiplier corresponding to the long-term power constraint we consider the long-term power constraint with equality). Then, the KKT conditions can be written as θ P =, =,,..., m, θ, =,,..., m, P, =,,..., m, θ P P ma ) =, = θ, P P ma, = E λ 7a) 7b) 7c) 7d) 7e) 7f) mfλ ) mp + θ + θ + vfλ ) =, =,,..., m, λ 7g) m ] P =, 7h) =

9 9 where 7g) is obtained by setting the derivative of the Lagrangian function with respect to P to zero. Now, based on the above conditions, we obtain some restrictions on the solution: Restriction : if P, then from 7a), θ =, so θ = P + / ) v)fλ ) from 7g) and 7e)). Restriction : if v, then P = from Restriction ). Now, consider two different situations: Situation, m = v ) + Pma : In this case, the ) power allocation P = v +, m is a valid solution and satisfies all the KKT conditions note that in this case θ =, and from Restriction and, above power allocation results). ) m Situation, = v + > Pma : In this case, θ and from 7d), we have m = P P ma =. Now, consider two cases:.: If θ /fλ )+v, then P = because if P >, then from 7a) θ =, and from 7g) P = >, which is a contradiction). θ /fλ )+v.: If < θ /fλ )+v, then θ = because if θ >, then from 7a) P =, and from 7g) θ = θ + vfλ ) fλ ) >, so / ) > θ /fλ )+v, which is a contradiction). With the eplanation in. and. cases, and defining β := θ /fλ ), we now can determine the ) power allocation in situation. That is P = β+v +, m, where β is the answer to ) m = β+v + = Pma. The power allocation described in Situation and Situation completes the proof. APPENDIX B PROOF OF THEOREM Putting the eponential distribution fγ) into 9), 4) results. Note that in the power allocation 8), the inequality γ γ αγ should always be satisfied for the power allocation to be valid. Therefore, the threshold γ should lie in the region, /α] γ /α, therefore, / γα). Now, define the function g) as ) g) := e e α γ E ) + E α γ + α γe α γ γ. Then, we have: g) = e α γ e,, ], γα 8) lim g) = α ) γ > since < α), + 9) e lim g) = E ) γ, = γα) γα. ) Note that < α α th for the power allocation 8), and from ) α th = γ, where γ is given by ). Therefore, = γα γ γ, and from 9], we have e E ) γ for = γα. 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