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 maximize 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 charac-terize 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 explored for the case of Rayleigh fading through our numerical results.
Index Terms—Adaptive transmission, channel state infor-mation (CSI), fading channels, multiple-input multiple-output (MIMO) systems, power control, resource allocation.
I. INTRODUCTION
A
DAPTIVE power allocation for wireless systems in fad-ing environments attempts to maximize 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 Communi-cation Commission (FCC), the transmit power in any time duration should not exceed a certain amount depending on the application, frequency, height of the antenna, population of that area per square mile, and so on [1]. 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 keeps the amplifier at each transmit antenna in its linear range. In designing the communication system, these types of constraints and others should all be taken into account.A. Related Work
Some works consider long-term average power constraints only, for which the average is taken over both the codewords
This work was supported in part by NSF grant CCF05-46618.
Mostafa Khoshnevisan and J. Nicholas Laneman are with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN, 46556 USA, (e-mails:{mkhoshne, jnl}@nd.edu).
and channel fading coefficients. For example, [2] considers single-antenna systems and shows that the optimal power al-location policy is water-filling in time. Additionally, multiple-antenna systems are considered in [3]. Both of these papers assume that CSI is available causally at both the transmitter and the receiver.
In other works, only short-term average power constraints are considered, for which the average is taken only over the codewords and the constraint applies to each channel fading coefficient. In [4], the author studies the capacity of a multiple-input-multiple-output (MIMO) channel subject to a short-term power constraint for complete CSI, and for CSI at the receiver only. Another example 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 short-term) 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 per-antenna power constraint. In [10], an optimization problem for a multiuser downlink channel with multiple transmit antennas at the base station subject to per-antenna power constraints is considered, which is transformed into a dual uplink prob-lem. In [11], per-antenna power constraints are considered in MIMO wireless systems in the context of beamforming with incomplete CSI. Also, [12] studies optimal power allocation subject to `p-norm constrained eigenvalues, which can be
viewed as a suboptimal power allocation policy under short-term and per-antenna power constraints. In a recent work, [13] studies the capacity of MIMO channels subject to only per-antenna 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 term constraints. We only consider short-term 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.
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 maximize 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 formu-lation. 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 maximize the ergodic capacity are partially developed in [14]. We characterize the complete structure of the optimal power allocation considering the cases in which one of the power constraints dominates, e.g., the short-term 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 extend the results to MIMO channels. In addition
to providing proofs for the results that were initially published in our conference paper [15], 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 per-antenna power constraints. The suboptimality comes from using a more stringent power constraint in order to make the optimization problem more tractable.
• For each of the problems described above, we mathemat-ically 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 expressions 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 Pmax that is larger than a long-term power
constraintP¯does not significantly impact the ergodic capacity of the channel, especially for large values of average SNR. We also explore 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. CHANNELMODEL ANDPROBLEMSTATEMENT
The baseband-equivalent discrete-time input-output relation-ship in our MIMO channel model is
y(i) =H(i)x(i) +n(i), (1) where i is the time index, y(i) is a complex vector of NR
received signals, x(i) is a complex vector of NT transmit
signals. H(i) and n(i) are random sequences capturing the effect of multipath fading and additive noise, respectively. The noisen(i)is a vector ofNRzero-mean, circularly symmetric,
complex Gaussian random variables with E[n(i)n(i)H] =
N0INR, and n(i), i = 1,2, ... is a sequence of independent
random vectors. The multipath fadingH(i)at each time is an NR×NT matrix of complex fading coefficients. We assume
that the matrix fading process is stationary and ergodic, and consider codewords that are are long enough to experience 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 matrix
H(i). For the case of Rayleigh fading in Section IV, we assume that the entries ofH(i)are independent and identically distributed (IID) complex Gaussian random variables with mean zero and variance 1/2 per real dimension.
The general optimization problem for maximizing the er-godic capacity subject to long-term, short-term, and per-antenna power constraints is
max Q(H)EH log det I+ 1 N0 HQ(H)HH , (2a) subject to EH[tr(Q(H))]≤P ,¯ (2b) ∀H: tr(Q(H))≤Pmax, (2c) ∀H:qkk(H)≤P , kˆ = 1,2, ..., NT, (2d)
where Q is the input covariance matrix, which can be a function of the instantaneous channel matrix, and qkk is the
kth diagonal entry of the matrixQ. The expectations in (2a)
and (2b) are with respect to the distribution ofH. The power constraints are described in (2b), (2c), and (2d);P¯ represents the long-term power constraint, Pmax represents the
short-term power constraint, andPˆrepresents the per-antenna power constraint. For simplicity, we drop the time index 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 prob-lem subject to simultaneous power constraints (2b) and (2c), or simultaneous power constraints (2b) and (2d). Interested readers are referred to [16] for a suboptimal power allocation subject to all three constraints.
III. GENERALSTRUCTURE OFOPTIMALPOWER
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 (1) with NT =
NR = 1. Let γ :=
¯
P|h|2
N0 denote the instantaneous received
coefficient, and letf(γ)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 isP(γ)γ/P, and the¯ ergodic capacity is [2] C=Eγ log 1 + P(γ¯)γ P . (3)
Based upon the coding theorem in [14], the ergodic capacity is the solution to the following optimization problem:
max P(γ) Z log 1 +P(γ¯)γ P f(γ)dγ, (4) subject to Eγ[P(γ)]≤P ,¯ ∀γ:P(γ)≤Pmax,
which corresponds to a scalar version of (2). To determine the general power policy, letα:=Pmax/P¯ 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 αand a certain thresholdαth (which is obtained at
the end of this section), and can be separated into the following three cases:
Case 1, α≤1: 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) =Pmax,∀γ. (5)
Case 2, αth≤α: In this case, the long-term power
con-straint dominates and the power policy is water-filling in time [2] P(γ) = ( 0, γ < γ00 ¯ P γ0 0 −P¯ γ, γ≥γ 0 0 , (6)
where the thresholdγ00 is determined from substituting (6) into
Eγ[P(γ)] = ¯P . (7)
Case 3, 1< α≤αth: In this case both long- and short-term
power constraints play a role, and the power allocation in [14] provides the solution to the optimization problem. The power allocation in this case becomes
P(γ) = 0, γ < γ0 ¯ P γ0 − ¯ P γ, γ0≤γ < γ0 1−αγ0 Pmax, 1−γαγ0 0 ≤γ , (8)
where the threshold γ0 is determined by substituting (8)
into (7), i.e., the solution to equation
Z 1−γαγ0 0 γ0 P¯ γ0 −P¯ γ f(γ)dγ+ Z ∞ γ0 1−αγ0 Pmaxf(γ)dγ= ¯P . (9) Optimal power allocation (8) comes from the Karush-Kuhn-Tucker (KKT) conditions [17]. As we mentioned previ-ously, [14] determines a similar solution, which is valid only for Case 3, since the threshold γ0 in (8) has a valid solution
only in this regime. In Section IV, for the Rayleigh fading channel, we show that the thresholdγ0is uniquely determined
from (7) if1< α≤αth(Case 3). Note that the thresholdγ0
in (8) andγ00 in (6) are, in general, different.
Now, we obtain the value of αth. In order to eliminate
the short-term power constraint, the power policy in (6) should always satisfy the short-term power constraintP(γ)≤
Pmax,∀γ, but the maximum value of P(γ) in (6) occurs as
γ→ ∞and is equal toP /γ¯ 00. Therefore,
¯ P γ0 0 ≤Pmax ⇒ 1 γ0 0 ≤α ⇒ αth= 1 γ0 0 . (10)
In general, the values ofγ00 andαthdepend on the
distribu-tion ofγ 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 ofαthis important in practical wireless communication
systems, since it might be the case that the allowedαis larger than αth. 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 := max (NR, NT) andm := min (NR, NT). The fading matrix
H can be represented using singular value decomposition (SVD) as
H=UΛVH, (11) where U and V are unitary matrices and Λ is a diagonal matrix with entries equal to the square roots of the eigenvalues of the matrix W= ( HHH, if NR≤NT HHH, if N R> NT .
Denote the eigenvalues of W by λk,1 ≤ k ≤ m. The
equivalent channel model is [4]
˜
y=Λ˜x+˜n, (12) where we use the transformation ˜y=UHy, ˜x=VHx, and
˜
n = UHn. The equivalent channel consists of m parallel
channels. Note that the trace of the input covariance matrix is invariant with respect to this transformation, e.g., tr(Qx) =
tr(Qx˜), whereQxandQx˜are the covariance matrices of the
random vectorsxand˜x, respectively. LetΛ0:=
q ¯
P
mN0Λbe the normalized channel with
diago-nal entries equal to square root ofλ0k= mNP¯
0λk,1≤k≤m.
In the SVD equivalent channel model, the power allocation policy is a function ofΛ0. Therefore, the covariance matrix can be given asQ(Λ0)or Q(λ0), whereλ0:= [λ01, λ02, ..., λ0m]. To maximize the ergodic capacity, the covariance matrix should be diagonal [4]. Let Pk(λ0),1 ≤ k ≤ m denote the kth
diagonal entry of the covariance matrix. Note that each Pk
is a function of the vectorλ0, or in other words, a function of allλ0
1) Long-Term and Short-Term Power Constraints: If the input power is subject to long- and short-term power con-straints, we remove the constraint in (2d). Then, using the SVD equivalent channel model and the above definitions, the optimization problem in (2) becomes
max Pk(λ0),k=1,2,...,mC=Eλ 0 "m X k=1 log 1 + Pk(λ 0)λ0 k ¯ P /m # , (13) subject to Eλ0 " m X k=1 Pk(λ0) # ≤P ,¯ ∀λ0: m X k=1 Pk(λ0)≤Pmax.
The optimal power allocation structure can be found by examining the KKT conditions. Formally, we state and prove the following theorem.
Theorem 1. The solution to the optimization problem(13)for
¯ P ≤Pmaxis Pk(λ0) = 1 v − ¯ P mλ0 k + , ifPm k=1 1 v− ¯ P mλ0 k + ≤Pmax 1 β+v− ¯ P mλ0 k + ,otherwise , (14) where(x)+ := max (0, x). The Lagrange multipliersv andβ are obtaining by solving
m X k=1 1 β+v − ¯ P mλ0k + =Pmax, (15) Eλ0 " m X k=1 Pk(λ0) # = ¯P . (16) Proof: See Appendix 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 fixed for all fading coefficients (all λ0) and is obtained such that (16) 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 coefficientsλ0as we can see from (15). 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 1.
Algorithm 1. 1) Computing the constant v: Let β(v, λ0)
denote the solution of (15). Substituting the power allo-cation (14)into (16)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.
2) 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 thresholdv, that is Pk =
1 v − ¯ P mλ0 k + .
Fig. 1. Illustration of the optimal power allocation for a2×2MIMO system subject to long- and short-term power constraints
3) Check the summation: If the summation of these powers (Pm
k=1Pk) is not larger than Pmax, 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 constantv, findβ according to(15).
5) Penalizing the powers: Repeat step 2, but this time with the threshold β + v instead of v, i.e., Pk = 1 β+v − ¯ P mλ0 k +
. Then, we have the optimal power al-location and we are done. Note that this step penalizes the previously computed powers by addingβ into the de-nominator of 1v such that the short-term power constraint is satisfied with equality.
An illustration of the optimal power policy for a2×2MIMO system in Figure 1 gives some insight about the structure of the power allocation and Algorithm 1. The horizontal axis corresponds to time. For every time slot the value of mλP¯0
k
for k = 1,2 (for the two singular values of the channel) is sketched. In fact, the vertical axis corresponds to how weak 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 1 of Algorithm 1. 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 exceed the short-term power constraintPmax
in any time slot (step 3). As it can be seen, at time slots 1, 2, 4, 5, and 7, the power policy is water-filling across time and space (step 2). However, for time slots 3 and 6, we have to adjust the water level, according to step 5 of Algorithm 1, 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 ofβdepends on the channel fading coefficients, and can be obtained from step 4 of Algorithm 1.
In Section III-A, finding the optimal power allocation was separated into three cases depending on the value of α and a constant αth. An analogous situation arises for MIMO
systems, as we now briefly discuss.
dominates and the long-term power constraint can be removed. The optimal power allocation is the well-known water-filling across antennas, but not across time, as obtained in [4].
Case 2, αth≤α: In this case, the power allocation in
Theorem 1 is valid, but it simplifies to Pk = 1 v0 − ¯ P mλ0 k + , (17)
where the constant v0 is determined by substituting (17) into (16). 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 examined in [3].
Case 3, 1< α≤αth: Both power constraints play a role
and the optimal power allocation is given by Theorem 1, which was discussed earlier.
The only remaining task is to obtain the value ofαth. Note
that for Case 2, we must ensure that the power allocation in (17) does not violate the short-term power constraint for all channel fading coefficients. As a result,
m X k=1 1 v0 − ¯ P mλ0 k + ≤Pmax,∀λ0 ⇒ m X k=1 1 v0 ≤Pmax ⇒ m v0P¯ ≤ Pmax ¯ P . Therefore, we have αth= m v0P¯. (18)
2) 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 (2c). First, note that for a Hermitian matrix Qwe have [18]
max
1≤k≤NTqkk ≤1≤maxk≤NTeigenvaluek(Q), (19)
where qkk is the kth diagonal entry of the matrix Q, and
eigenvaluek(Q) is the kth eigenvalue of the matrix Q. If
we consider the SVD method discussed earlier, then under the transformation ˜x = VHx, the eigenvalues of the input
covariance matrix do not change. On the other hand, the eigenvalues of the matrix tr(Qx˜) are equal to the diagonal
entries Pk(λ0),1≤k≤m. Therefore, the constraint
Pk(λ0)≤P , kˆ = 1, ..., m, (20)
is sufficient to satisfy (2d). Because of inequality (19), this condition is only sufficient and not necessary (It is possible that (2d) is satisfied, but (20) is not). Therefore, we can find a suboptimal solution by assuming a more stringent constraint. Specifically, we consider the optimization problem
max Pk(λ0),k=1,2,...,m C=Eλ0 "m X k=1 log 1 + Pk(λ 0)λ0 k ¯ P /m # , (21) subject to Eλ0 " m X k=1 Pk(λ0) # ≤P ,¯ ∀λ0:Pk(λ0)≤P , kˆ = 1, ..., m.
Theorem 2. The solution to the optimization problem(21)for
¯ P ≤mPˆ is Pk(λ0) = 0, if mλP¯0 k ≥ 1 v 1 v− ¯ P mλ0 k , if 1v > mλP¯0 k ≥ 1 v−Pˆ ˆ P , otherwise . (22)
Proof: The proof is similar to the proof of Theorem 1 and is omitted due to space considerations.
In (22),vis a constant that is fixed for all fading coefficients, and is determined by substituting the power allocation (22) into (16). Let α:= mP /ˆ P¯. Note that the definition of α is slightly different from Sections III-A and III-B1, since there arem per-antenna power constraints here. Again, the general power allocation can be specialized in three cases.
Case 1, α≤1: In this case, the long-term power constraint can be removed and the power policy is given byPk(λ0) = ˆP.
Case 2, αth≤α: In this case, the per-antenna power
con-straint can be removed. The power allocation in Theorem 2 is valid but simplifies to (17), where v0 is determined from (16). By the same procedure as before, we can find that αth=m/(v0P¯).
Case 3, 1< α≤αth: In this case, both power constraints
play a role and the power policy is given by (22). IV. RAYLEIGHFADINGCHANNELS
In Section III, we obtained the optimal power allocation policies for general channel models subject to various com-binations of power constraints. In this section, we consider the IID Rayleigh fading channel model, simplify the power policies to the extent possible, and provide some numerical results.
A. Analysis
Many of the constants and thresholds for the power alloca-tions developed in Section III are funcalloca-tions of the distribution of the fading process. In this section, we simplify the calcu-lations needed to obtain these constants and thresholds in the IID Rayleigh fading case.
1) SISO and MISO: Consider the parameter γ defined in Section III-A. For SISO Rayleigh fading γ is an exponential random variable with expected value P /N¯ 0. Let γ¯ be the
average SNR of the system, which in the case of SISO and MISO systems is ¯γ = ¯P /N0. We denote the probability
density function of γ by f(γ). Consider the thresholds γ00, αth, andγ0in the optimal power allocation from Section III-A.
In [19], the value ofγ00 is derived for Rayleigh fading channel
as the solution to the equation e− γ00 ¯ γ γ0 0 ¯ γ −E1 γ0 0 ¯ γ = ¯γ, (23) whereE1(x)is the exponential integral defined by
E1(x) :=
Z +∞
1
t−1e−xtdt, x≥0.
Note that the value of αth is only a function of¯γ. For the
Theorem 3. If 1< α≤αth (Case 3), then the threshold γ0
in(8)for the Rayleigh fading SISO channel can be determined from e−x x − e−1−xαx¯γ x −E1(x)+E1 x 1−αxγ¯ +α¯γe−1−xαxγ¯= ¯γ, (24) where x:=γ0/γ. Furthermore,¯ γ0is unique and γ0∈[0,1α].
Proof: See Appendix B.
Note that a MISO system with NT =n can be analyzed
just like a SISO system with different distributions for the equivalent channel gainγ, which depends onnand the multi-antenna 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 [16] for analytical results on the thresholds αth and γ0 in MISO systems with Rayleigh
fading. Numerical results are provided in Section IV-B for both SISO and MISO systems.
2) MIMO: In the MIMO case, because of the more intricate structure of the power policies, finding a simplified equation for expressing some of the thresholds is challenging. In partic-ular, when the power allocation contains a Lagrange multiplier, which depends on the instantaneous fading coefficients (like β in Sections III-B1), there is no closed form.
However, we can find αth in the Rayleigh fading channel
model. In Sections III-B1 and III-B2, we have αth =
m/(v0P¯), where v0 is the constant in the optimal power policy (17). From (16), αthbecomes the solution to
Z ∞ 1 αth αth− 1 x fλ0(x)dx= 1, (25)
where fλ0(x) is the probability distribution function of a
normalized unordered eigenvalue of the Wishart matrix [4]. Let λ¯ = mNP¯
0 denote the average SNR per parallel channel.
From the calculation of [3] and [20], (25) reduces to
m X k=1 k−1 X p=0 k−1 X q=0 (k−1)!(−1)p+q (n−m+k−1)!p!q! n −m+k−1 k−1−p n −m+k−1 k−1−q ·Gn−m+p+q( 1 ¯ λαth ) =m¯λ, (26) whereGr(s)is defined as Gr(s) := Γ(r+ 1, s) s −Γ(r, s)∀integers r≥0. B. Numerical Results and Discussion
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. 1) SISO and MISO: In Figure 2, we plot αth versus γ¯
for SISO systems, and MISO systems with SVD and antenna selection, each with n= 2and n= 8. Note that in practical wireless communication systems, it is advantageous to have αthsmall, since this means that we can ignore the short-term
power constraint for a wider range ofα. According to Figure 2, as we increase the average SNR, the value ofαthgets smaller
−100 −5 0 5 10 15 20 1 2 3 4 5 6 7 8 9 Average SNR (dB) αth SISO MISO−Selection(n=2) MISO−SVD(n=2) MISO−Selection(n=8) MISO−SVD(n=8)
Fig. 2. The value ofαth, the threshold separating Cases 2 and 3, versus
average SNRγ¯for SISO and MISO systems with Rayleigh fading.
and finally approaches 1 for large average SNRs. It can be easily seen that (23) has a unique solution andγ00always lies in the interval[0,1][19]. Therefore,αth= 1/γ00is unique and is
always larger than one. The behavior ofαthfor large average
SNR can be intuitively explained 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 long-term powers, and correspondingly αth 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 ofαthis always
smaller than for the SISO system. As we can see in Figure 2, the more antennas at the transmitter, the smallerαth.
Next, we examine the effect ofαon the capacity. Figure 3 shows the capacity versus average SNR for each of the two schemes in the MISO channel with the two extreme possibilities forα:α=∞andα= 1(We are not interested in the caseα <1for 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 αth ≤ α,∀γ),¯
and ifα= 1, the problem reduces to constant power allocation since the power policy is P(γ) = ¯P. 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 1 < α < ∞ between the two extremes, the capacity versus average SNR curve lies between the two curves forα= 1andα=∞. As we can see from Figure 3, the difference between the capacities for the two extreme values of αbecomes negligible at high SNR for both SVD and antenna selection. We observe that the value of the short-term power constraint (Pmax) 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≥1.
−15 −10 −5 0 5 10 15 0.5 1 1.5 2 2.5 3 Average SNR (dB) Capacity (Bits) SVD, α=∞ SVD, α=1 Antenna Selection, α=∞ Antenna Selection, α=1
Fig. 3. Ergodic capacity versus average SNR for MISO systems with Rayleigh fading,n= 2andα=∞andα= 1.
−100 −5 0 5 10 15 20 1 2 3 4 5 6 Average SNR (dB) αth 2 × 2 MIMO system 4 × 2 MIMO system 8 × 2 MIMO system
Fig. 4. The value ofαth, the threshold separating Cases 2 and 3, versus Average SNR per parallel channel (mNP¯
0) for MIMO systems with Rayleigh
fading.
2) MIMO: Figure 4 shows the value of αth versus the
average SNR per parallel channel (λ¯ = mNP¯
0) for MIMO
systems according to (26). This plot suggests similar obser-vations as for MISO systems: Water-filling has more short-term fluctuations at low SNR, which makesαthlarger. Again,
the short-term power constraint is more relevant at low SNR compared to high SNR. Also, note that Figures 2 and 4 show the effect of the distribution of the channel singular values on αth. The observation is that, as we increase the number of
antennas at the transmitter, the value ofαthdecreases. In fact,
we can conclude that for a well-conditioned Wishart matrix in which the number of transmit antennas is much larger than the number of receive antennas [21], the value of αthis smaller
compared to the case of an ill-conditioned Wishart matrix. 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 αthbecomes smaller.
Figure 5 examines the effect of Pmax on the ergodic
−150 −10 −5 0 5 10 15 0.5 1 1.5 2 2.5 3 3.5 4 Ratio of Capacities Average SNR (dB) Cα=∞/CNo CSIT (4 × 2 System) Cα=∞/CNo CSIT (2 × 2 System) Cα=∞/Cα=1 (2 × 2 System) Cα=∞/Cα=1 (4 × 2 System)
Fig. 5. Ratio of ergodic capacities versus average SNR per parallel channel (mNP¯
0) for MIMO systems with Rayleigh fading subject to long- and
short-term power constraints.
capacity of2×2and4×2MIMO systems with IID Rayleigh fading subject to long- and short-term power constraints. In the two lower curves, we compare the capacity of two extreme cases:Pmax = ¯P andPmax → ∞ (or α= 1 andα→ ∞).
These curves suggest that ifPmaxis larger than or equal toP¯,
the value of the short-term power level (Pmax) 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 [22], in which the curves for the two cases of space-time water-filling and spatial water-water-filling are equivalent to the case with α=∞andα= 1, respectively. Therefore, for a fixed 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 3, Pmax
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 αmax becomes
smaller for a larger number of transmit antennas.
Figure 5 also shows the ratio of the ergodic capacity forα=
∞to the ergodic capacity if there is no CSI at the transmitter (two upper curves). For the 2×2 MIMO system, the loss in the ergodic capacity is negligible at high SNR. This always happens for anm×nIID Rayleigh fading MIMO system with m ≤ n as discussed in [23], since at high SNR, the water-filling 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 the4×2MIMO system, because if there are more transmit antennas than receive antennas, there is a power boost of NT/NR from having CSIT [23].
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 2×2 and 4×2 MIMO
−15 −10 −5 0 5 10 15 0.5 1 1.5 2 2.5 3 Average SNR (dB) Ratio of Capacities
CPower Allocation/CNo CSIT (4 × 2 System) C
Power Allocation/CNo CSIT (2 × 2 System)
CPower Allocation/CNo Per−antenna (4 × 2 System) CPower Allocation/CNo Per−antenna (2 × 2 System)
Fig. 6. Ratio of ergodic capacities versus average SNR for MIMO systems with Rayleigh fading subject to long-term and per-antenna power constraints withα= 2.
systems with IID Rayleigh fading in Figure 6. Consider the problem in Section III-B2 withα= 2(orPˆ= ¯P). In Figure 6, CPower Allocation is the ergodic capacity obtained from power
allocation (22),CNo CSIT is the ergodic capacity if there is no
CSIT, andCNo Per-Antennais 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 CPower Allocation/CNo CSIT, which shows how much
gain one can get from the suboptimal power allocation (22) compared to the naive equal power allocation across time and space. This ratio is considerably larger than 1 for low average SNR, and decreases to 1 for large average SNR in the case of 2×2MIMO 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
CPower Allocation/CNo Per-Antenna, which shows the impact of a
per-antenna power constraint on the ergodic capacity. Note that this ratio is exactly equal to1for average SNR larger than
−4.1 dBand0.45 dBfor the4×2and2×2MIMO systems, respectively. The reason is that the value ofαthis equal to 2
at average SNR equal to −4.1 dB and0.45 dBfor the 4×2
and 2×2 MIMO systems, respectively, as can be seen from Figure 4, which means that for average SNR larger than those, we are in Case 2 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 1 is consistent with the observation in [13] 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 log-normal shadowing. We anticipate such a result based upon our observations in conjunction with results in [22], in which the authors demonstrate that with log-normal shadowing, space-time water-filling achieves significantly higher ergodic capac-ity 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 maximize 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 short-term power constraint that is larger than a long-short-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.
APPENDIXA PROOF OFTHEOREM1
Since (13) is a convex optimization problem, we prove the theorem using the KKT conditions [17]. For simplicity, we drop λ0 from Pk(λ0) and simply denote it by Pk in the
proof of Theorems 1, 2 and 3. To maximize the capacity, the long-term power constraint should be satisfied with equality, which is possible since α ≥ 1. Let θ1, θ2, ..., θm denote
the Lagrange multipliers corresponding to the constraints that force the powers to be positive (P1≥0, P2≥0, ..., Pm≥0,
respectively), θ0 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
θkPk = 0, k= 1,2, ..., m, (27a) θk ≥0, k= 1,2, ..., m, (27b) Pk≥0, k = 1,2, ..., m, (27c) θ0( m X k=1 Pk−Pmax) = 0, (27d) θ0≥0, (27e) m X k=1 Pk ≤Pmax, (27f) − mf(λ 0 ) mPk+ ¯ P λ0 k −θk+θ0+vf(λ0) = 0, k= 1,2, ..., m, (27g) Eλ0 " m X k=1 Pk # = ¯P , (27h)
where (27g) is obtained by setting the derivative of the Lagrangian function with respect to Pk to zero. Now, based
on the above conditions, we obtain some restrictions on the solution:
Restriction 1: if Pk 6= 0, then from (27a), θk = 0, so θ0=
(P 1
k+ ¯P /(mλ0k)
−v)f(λ0)≥0 (from (27g) and (27e)). Restriction 2: if mλP¯0
k
≥ 1
v, then Pk = 0 (from Restriction
1).
Now, consider two different situations: Situation 1, Pm k=1 1 v− ¯ P mλ0 k +
≤Pmax: In this case, the
power allocation Pk = 1 v− ¯ P mλ0 k + ,1 ≤k≤m is a valid solution and satisfies all the KKT conditions (note that in this case θ0 = 0, and from Restriction 1 and 2, above power
allocation results). Situation 2, Pm k=1 1 v− ¯ P mλ0 k +
> Pmax: In this case,
θ06= 0and from (27d), we have Pm
k=1Pk−Pmax= 0. Now,
consider two cases: 2.1: If mλP¯0
k
≥ 1
θ0/f(λ0)+v, thenPk= 0 (because ifPk>0,
then from (27a) θk = 0, and from (27g)Pk = θ0/f(1λ0)+v −
¯ P mλ0 k >0, which is a contradiction). 2.2: If mλP¯0 k < θ0/f(1λ0)+v, then θk= 0 (because ifθk >0,
then from (27a) Pk = 0, and from (27g)θk =θ0+vf(λ0)− f(λ0) ¯ P /(mλ0 k) >0, so mλP¯0 k > 1 θ0/f(λ0)+v, which is a contradiction).
With the explanation in 2.1 and 2.2 cases, and defining β := θ0/f(λ0), we now can determine the power allocation in situation 2. That is Pk = 1 β+v − ¯ P mλ0 k + ,1 ≤ k ≤ m, whereβ is the answer toPm
k=1 1 β+v− ¯ P mλ0 k + =Pmax.
The power allocation described in Situation 1 and Situa-tion 2 completes the proof.
APPENDIXB PROOF OFTHEOREM3
Putting the exponential distribution f(γ) into (9), (24) results. Note that in the power allocation (8), the inequality γ0≤ 1−γαγ0
0 should always be satisfied for the power allocation
to be valid. Therefore, the thresholdγ0should lie in the region
[0,1/α] (0 ≤ γ0 ≤ 1/α, therefore, 0 ≤ x ≤ 1/γα). Now,¯
define the function g(x)as
g(x) :=e −x x − e−1−xαx¯γ x −E1(x) +E1 x 1−αx¯γ +αγe¯ −1−xαx¯γ −¯γ. Then, we have: ∂g(x) ∂x = e−1−xαx¯γ −e−x x2 ≤0,∀x∈[0, 1 ¯ γα], (28) lim x→0+g(x) = (α−1)¯γ >0 (since1< α), (29) lim x→(1 ¯ γα) −g(x) = e−x1 x1 −E1(x1)−¯γ, x1= 1 ¯ γα. (30)
Note that 1 < α ≤ αth for the power allocation (8), and
from (10) αth = γ10 0
, where γ00 is given by (23). Therefore,
x1= ¯γα1 ≥
γ00
¯
γ, and from [19], we have e−x1 x1 −E1(x1)−¯γ≤0 for x1= ¯γα1 . Therefore, lim x→( 1 ¯ γα) −g(x)≤0. (31)
From (28), we observe that the derivate of g(x) with respect to x is not positive, and therefore it is non-increasing; From (29) and (31), we observe that the function g(x) is positive at0and is non-positive at γα¯1 . Therefor,xis uniquely determined and x ∈[0,γα¯1 ]. Finally, because x= γ0/¯γ, we
observe that γ0 is uniquely determined and γ0 ∈[0,1α], and
the proof is complete.
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