The Capacity of Gaussian MIMO Channels Under Total and Per-Antenna Power Constraints
aa r X i v : . [ c s . I T ] D ec The Capacity of Gaussian MIMO ChannelsUnder Total and Per-Antenna PowerConstraints
Sergey Loyka
Abstract
The capacity of a fixed Gaussian multiple-input multiple-output (MIMO) channel and the optimaltransmission strategy under the total power (TP) constraint and full channel state information arewell-known. This problem remains open in the general case under individual per-antenna (PA) powerconstraints, while some special cases have been solved. These include a full-rank solution for the MIMOchannel and a general solution for the multiple-input single-output (MISO) channel. In this paper, thefixed Gaussian MISO channel is considered and its capacity as well as optimal transmission strategies aredetermined in a closed form under the joint total and per-antenna power constraints in the general case.In particular, the optimal strategy is hybrid and includes two parts: first is equal-gain transmission andsecond is maximum-ratio transmission, which are responsible for the PA and TP constraints respectively.The optimal beamforming vector is given in a closed-form and an accurate yet simple approximation tothe capacity is proposed. Finally, the above results are extended to the MIMO case by establishing theergodic capacity of fading MIMO channels under the joint power constraints when the fading distributionis right unitary-invariant (of which i.i.d. and semi-correlated Rayleigh fading are special cases). Unlikethe fixed MISO case, the optimal signaling is shown to be isotropic in this case.
Index Terms
MIMO, channel capacity, power constraint.
The material in this paper was presented in part at the IEEE International Symposium on Information Theory, Barcelona,Spain, July 2016.S. Loyka is with the School of Electrical Engineering and Computer Science, University of Ottawa, Ontario, Canada, e-mail:[email protected]
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I. I
NTRODUCTION
The capacity of a fixed multiple-input multiple-output (MIMO) Gaussian channel under thetotal power (TP) constraint and full channel state information (CSI) at both ends is well-knownas well as the optimal transmission strategy to achieve it [1]-[4]: the optimal strategy is Gaussiansignaling over the channel eigenmodes with power allocation given by the water-filling (WF)algorithm. In the special case of multiple-input single-output (MISO) channel, this reducesto the rank-1 signalling, i.e. beamforming, where the beamforming vector is proportional tothe channel vector (i.e. stronger channels get more power), which mimics the maximum ratiocombining (MRC) in diversity reception systems [2][3], which we term here ”maximum ratiotransmission” (MRT). Recently, this problem was considered under individual per-antenna (PA)power constraints [6]-[8], which is motivated by the distributed design of active antenna arrayswhere each antenna has its own RF amplifier with limited power (as opposed to a commonamplifier and a passive beamforming network in the case of TP constraint ), so that powersof different antennas cannot be traded off with each other. The optimal transmission strategyfor a fixed channel was established in [8], which corresponds to beamforming (i.e. rank-1transmission) with uniform amplitude distribution across antennas and where the beamformingvector compensates for channel phase differences so that all transmitted signals are coherentlycombined at the receiver. This mimics the well-known equal gain combining (EGC) in a diversity-reception system. Hence, we term this strategy ”equal gain transmission” (EGT) here. A fixedmultiple-input multiple-output (MIMO) Gaussian channel under PA constraints was consideredin [9] and [13], where a numerical algorithm to evaluate an optimal Tx covariance was developedbased on a partial analytical solution [9] and a closed-form full-rank solution was obtained [13],while the general solution remains illusive. This is in stark contrast to the capacity under the TPconstraint, for which the general solution is well-known for this channel. The capacity of theergodic-fading MISO channel under the long-term average PA constraint and full CSI at bothends was established in [12]. The following further considerations make the TP constraint important: (i) for battery-operated devices, the TP determines thebattery life; (ii) the TP constraint is important when a power/energy supply is significantly limited; (iii) the growing importanceof ”green” communications makes the TP important since it is the TP rather than the PA power that determines the carbonfootprint of the system.
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Single-user PA-constrained results were extended to multi-user scenarios in [7] and [11], wherea precoder was developed that achieves a 2-user MISO Gaussian broadcast channel (BC) capacity[7] and an iterative numerical algorithm was developed to obtain optimal covariance matrices tomaximize the sum-capacity of Gaussian MIMO multiple-access (MAC) channel [11], for whichno closed-form solution is known.One may further consider a hybrid design of a Tx antenna array where each antenna has itsown power amplifier and yet some power can be traded-off between antennas (correspondingto a common beamforming network) under the limited total power (e.g. due to the limitationof a power supply unit). This implies individual (PA) as well as total (TP) power constraints.Ergodic-fading MIMO channels were considered in [10] under long-term TP and short-termPA constraints and a sub-optimal signalling transmission strategy was proposed. An optimalstrategy to achieve the ergodic capacity under the above constraints remains unknown. A fixed(non-fading) MISO channel was considered in [14] under full CSI at both ends and joint TPand PA constrains. It was shown that beamforming is still an optimal strategy. A closed-formsolution was established in the case of 2 Tx antennas only and the general case remains an openproblem.The present paper provides a closed-form solution to this open problem, which is based onKarush-Kuhn-Tucker (KKT) optimality conditions for the respective optimization problem. Inparticular, we show that the optimal strategy is hybrid and consists of 2 parts: 1st part, whichincludes antennas with stronger channel gains and for which PA constraints are active, performsEGT (when PA constraints are the same for all antennas) while 2nd part, which includes antennaswith weaker channel gains and for which PA constraints are inactive, performs MRT. This mimicsthe classical equal gain and maximum ratio combining (EGC and MRC) strategies of diversityreception. Amplitude distribution across antennas as well as the number of active PA constraintsare explicitly determined. Sufficient and necessary conditions for the optimality of the MRTand the EGT are given. In particular, the MRT is optimal when channel gain variation amongantennas is not too large and the EGT is optimal for sufficiently large total power constraint.Based on the fact that the capacity under the joint (PA+TP) constraints is upper boundedby the capacities under the individual (either PA or TP) constraints, a compact yet accurateapproximation to the capacity is proposed.While closed-form solutions for the optimal signaling and the capacity of the fixed Gaussian
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MISO channel under the joint power constraints are established in sections III and IV, one maywonder whether they can be extended to the MIMO case and whether fading can be included aswell, which is important from the practical perspective for modern wireless systems. Section Vpartially addresses this question by considering a class of fading MIMO channels and establishingits ergodic capacity under the joint power constraints when the fading distribution is right unitary-invariant (see section V for details), of which i.i.d. and semi-correlated Rayleigh fading are specialcases. Unlike the fixed MISO case, the optimal signaling is shown to be isotropic in this case.This extends the respective result in [19] established under the TP constraint and i.i.d. Rayleighfading to the joint PA and TP constraints as well as to the class of right unitary-invariant fadingdistributions.
Notations : bold lower-case letters denote column vectors, h = [ h , h , .., h m ] T , where T isthe transposition, while bold capital denote matrices; R + is the Hermitian conjugation of R ; r ii denotes the i -th diagonal entry of R ; ⌊ x ⌋ is the integer part while ( x ) + = max[0 , x ] is thepositive part of x ; ∇ R is the derivative with respect to R ; R ≥ means that R is positivesemi-definite; | h | p = ( P i | h i | p ) /p is the l p -norm of vector h and | h | = | h | is the l norm.II. C HANNEL M ODEL AND C APACITY
Discrete-time model of a fixed Gaussian MISO channel can be put into the following form: y = h + x + ξ (1)where y, x , ξ and h are the received and transmitted signals, noise and channel respectively; h ∗ i is i -th channel gain (between i -th Tx antenna and the Rx). Without loss of generality, we orderthe channel gains, unless indicated otherwise, as follows: | h | ≥ | h | ≥ .. | h m | > , and m isthe number of transmit antennas. The noise is assumed to be Gaussian with zero mean and unitvariance, so that the SNR equals to the signal power. Complex-valued channel model is assumedthroughout the paper, with full channel state information available both at the transmitter andthe receiver. Gaussian signaling is known to be optimal in this setting [1]-[4] so that finding thechannel capacity C amounts to finding an optimal transmit covariance matrix R : C = max R ∈ S R ln(1 + h + Rh ) (2)where S R is the constraint set. In the case of the TP constraint, it takes the form S R = { R : R ≥ , tr R ≤ P T } , (3) December 15, 2016 DRAFT where P T is the maximum total Tx power, and the MRT is optimal [3] so that the optimalcovariance R ∗ is R ∗ = P T hh + / | h | (4)and the capacity is C MRT = ln(1 + P T | h | ) (5)Under the PA constraints, S R = { R : R ≥ , r ii ≤ P } , (6)where r ii is i -th diagonal entry of R (the Tx power of i -th antenna), P is the maximum PApower, and the EGT is optimal [8] so that the optimal covariance R ∗ is R ∗ = P uu + , (7)where the entries of the beamforming vector u are u i = e jφ i , φ i is the phase of h i , and thecapacity is C EGT = ln(1 + P | h | ) (8)Note from (5) and (8) that it is the l norm of the channel h that determines the capacity underthe PA constraint while the l norm does so under the total power constraint. In the next section,we will see how this observation extends to the case of the joint PA and TP constraints.III. T HE C APACITY U NDER THE J OINT C ONSTRAINTS
Following the same line of argument as for the total power constraint [1]-[4], the channelcapacity C under the joint PA and TP constraints is as in (2) where S R is as follows: S R = { R : R ≥ , tr R ≤ P T , r ii ≤ P } (9)and P T , P are the maximum total and per-antenna powers. This is equivalent to maximizing theRx SNR: max R h + Rh s.t. R ∈ S R (10)The following Theorem gives a closed-form solution to this open problem. December 15, 2016 DRAFT
Theorem 1.
The MISO channel capacity in (2) under the per-antenna and total power constraintsin (9) is achieved by the beamforming with the following input covariance matrix R ∗ = P ∗ uu + (11) where P ∗ = min( P T , mP ) and u is a unitary (beamforming) vector of the form: u i = a i e jφ i (12) where φ i is the phase of h i and a i represents amplitude distribution across antennas: a i = c , i = 1 ..kc | h i | , i = k + 1 ..m (13) and c = 1 √ m ∗ , c = p − k/m ∗ | h mk +1 | (14) m ∗ = P ∗ /P , h mk +1 = [ h k +1 ...h m ] T is the truncated channel vector, and k is the number of activeper-antenna power constraints, ≤ k ≤ ⌊ m ∗ ⌋ , determined as the least solution of the followinginequality | h k +1 | ≤ h th = | h mk +1 | √ m ∗ − k (15) if P T < mP and k = m otherwise. The capacity is C = ln(1 + γ ∗ ) (16) where γ ∗ = h + R ∗ h is the maximum Rx SNR under the TP and PA constraints, γ ∗ = P ∗ ( c | h k | + c | h mk +1 | ) (17) where the 2nd term is absent if k = m .Proof: see Appendix.Note from (12) that the beamforming vector always compensates for channel phases so thatthe transmitted signals are combined coherently at the receiver, while the amplitude distributionacross Tx antennas depends on the number of active PA constraints: amplitudes are always thesame for those antennas for which PA constraints are active (which represent stronger channels)and they are proportional to channel gain when for inactive PA constraints (weaker channels). December 15, 2016 DRAFT
In accordance with this, (17) has two terms: 1st term c | h k | represents the gain due to theequal gain transmission (EGT, | u i | = c ) for active PA constraints while 2nd one c | h mk +1 | -due to the maximum ratio transmission (MRT, | u i | = c | h i | ) for inactive PA constraints, whichmimic the equal gain combining (EGC) and maximum ratio combining (MRC) in the case ofdiversity reception systems. These two terms are represented by l and l norms respectively,which mimic the respective observation for (8) and (5).Eq. (15) facilitates an algorithmic solution to find the number k of active PA constraints andhence the threshold h th : the inequality is verified for k in increasing order, starting from k = 0 ,and the algorithm stops when 1st solution is found (this will automatically be the least solution,as required).The following Corollary establishes conditions for the optimality of the MRT, which corre-sponds to k = 0 . Corollary 1.
All PA constraints are inactive and thus maximum ratio transmission is the optimalstrategy if and only if | h | ≤ | h | p P/P T (18) Proof:
Follows directly from Theorem 1 by using k = 0 . The necessary part is due to thenecessity of the KKT conditions for optimality.Note that this limits channel gain variance among antennas. In particular, it always holds ifall channel gains are the same. It also implies that at least 1 PA constraint is active if | h | > | h | p P/P T (19)In a similar way, one obtains a condition for the optimality of the EGT. Corollary 2.
All PA constraints are active and thus the equal gain transmission is the optimalstrategy if and only if P T ≥ mP (20)When the TP constraint is not active, i.e. P T ≥ mP and hence k = m , Theorem 1 reduces tothe respective result in [8] under the identical PA constraints. December 15, 2016 DRAFT
A. Examples
To illustrate the optimal solution, we consider the following representative example: h =[3 , , . , . T . Note that this example also applies to complex-valued channel gains since thebeamforming vector is always adjusted to compensate for the channel phases and hence theydo not affect the capacity or the amplitude distribution, which will stay the same for the moregeneral case of h = [3 e jφ , e jφ , . e jφ , . e jφ ] T (21)where φ ...φ are (arbitrary) phases, which affect the beamforming vector phases as in (12).Fig. 1 shows the capacity under the total and joint power constraints as the function of the totalpower P T when P = 1 . As the total power increases, more and more PA constraints becomeactive, starting with antennas corresponding to strongest channels. Note that the MRT is optimal( k = 0 ) if the total power is not too large: P T ≤ P | h | / | h | ≈ . (22)while the EGT is optimal if P T ≥ mP = 4 (23)Fig. 2 shows the amplitude distribution for the scenario in Fig. 1 under the joint PA+TPconstraints. While weak channels get less power at the beginning (when the MRT is optimal), itgradually increases as the strongest channels reach their individual power constrains until eventu-ally all channels have the same power (when the EGT is optimal). Note that while the amplitudes a and a of the strongest and weakest channels are monotonically decreasing/increasing, theamplitudes a , a of intermediate channels are not monotonic in P T , increasing first until theyreach the stronger level and then decreasing.In general, the capacity under the joint PA+TP constraints can be upper-bounded by the EGTand MRT capacities under the PA and TP constraints respectively: C ≤ min( C MRT , C
EGT ) (24)where C MRT , C
EGT are as in (5), (8), and the upper bound is tight everywhere except in thetransition region, so one can approximate the capacity C as C ≈ min( C MRT , C
EGT ) (25) December 15, 2016 DRAFT T P k PA TP PA+TP (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:3) (cid:1) (cid:2) (cid:3) (cid:2) (cid:4) (cid:5) (cid:6) (cid:7)
Fig. 1. The capacity of MISO channel under the PA, TP and joint PA+TP constraints and the number of active PA constraints k vs. total power P T ; P = 1 , h = [3 , , . , . T . T P a a a a (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:3) (cid:1) (cid:2) (cid:3) (cid:4)(cid:5) (cid:6) (cid:7)(cid:8) (cid:9) (cid:10) (cid:8) (cid:5) (cid:11) (cid:6) (cid:12) (cid:5) (cid:13)(cid:7) (cid:6) (cid:5) (cid:14) (cid:15) Fig. 2. The optimal amplitude distribution under the joint power constraints for the scenario in Fig. 1.
It is straightforward to show that (24) and (25) hold with strict equality under (18) or (20) forany h , or if | h | / | h m | = 1 for any P T and P . The approximation is sufficiently accurate if thevariance in the channel gains is not large, i.e. if | h | / | h m | is not too large, as the followingexample demonstrates in Fig. 3, where h = [4 , , . , T . Fig. 4 shows the respective amplitudedistribution. Notice that the variance of the amplitude distribution is smaller than that in Fig. 2,since the variance in the channel gains is smaller as well, and that the range of optimality ofthe MRT is larger while the range of optimality of the EGT is exactly the same as in Fig. 2. In December 15, 2016 DRAFT0 T P k PA TP PA+TP (cid:1) (cid:2) (cid:3) (cid:2) (cid:4) (cid:5) (cid:6) (cid:7)
Fig. 3. The capacity of MISO channel under the PA, TP and joint PA+TP constraints and the number of active PA constraints k vs. total power P T ; P = 1 , h = [4 , , . , T . Note that the approximation in (25) is accurate over the whole range of P T . T P a a a a (cid:1)(cid:2)(cid:3) (cid:4)(cid:5)(cid:3) (cid:1) (cid:2) (cid:3) (cid:4)(cid:5) (cid:6) (cid:7)(cid:8) (cid:9) (cid:10) (cid:8) (cid:5) (cid:11) (cid:6) (cid:12) (cid:5) (cid:13)(cid:7) (cid:6) (cid:5) (cid:14) (cid:15) Fig. 4. The optimal amplitude distribution under the joint power constraints for the scenario in Fig. 3. fact, it follows from (22) and (23) that while the range of optimality of the MRT depends onthe channel, that of the EGT does not.IV. D
IFFERENT
PA C
ONSTRAINTS
In a similar way, one may wish to consider a more general case where individual antennashave different power constraints, so that the constraint set is S R = { R : R ≥ , tr R ≤ P T , r ii ≤ P i } (26) December 15, 2016 DRAFT1
The channel capacity under these constraints is given in the following.
Theorem 2.
The MISO channel capacity in (2) under the per-antenna and total power constraintsin (26) is achieved by the beamforming with the input covariance matrix as in (11) and (12) where a i = c i , i = 1 ..kc | h i | , i = k + 1 ..m (27) c i = r P i P ∗ , c = p − k/m ∗ | h mk +1 | (28) and P ∗ = min( P T , P mi =1 P i ) , m ∗ = P ∗ /P , P = k P ki =1 P i is the average power of the activePA constraints, k is the number of active PA constraints, determined as the least solution of thefollowing inequality | h k +1 |√ P k +1 ≤ | h mk +1 | q P T − P ki =1 P i (29) if P T < P mi =1 P i and k = m otherwise, where channel gains { h i } are ordered in such a waythat {| h i | / √ P i } are in decreasing order. The capacity is as in (16) and the optimal SNR is γ ∗ = P ∗ k X i =1 c i | h i | + c | h mk +1 | ! (30) Proof:
Follows along the same lines as that of Theorem 1.Note that 1st term in (30) does not represent EGT anymore; rather, the amplitudes are adjustedto match the PA constraints. The conditions for optimality of the MRT can be similarly obtained.When the TP constraint is inactive, i.e. when P T ≥ P mi =1 P i , Theorem 2 reduces to the respectiveresult in [8], as it should be. The condition for the optimality of the MRT is as follows. Corollary 3.
All PA constraints are inactive and thus the MRT is optimal if and only if | h | ≤ | h | p P /P T (31) and at least 1 PA constraint is active otherwise. All PA constraints are active if and only if P T ≥ m X i =1 P i (32) December 15, 2016 DRAFT2
V. F
ADING
MIMO C
HANNELS
While the closed form solutions for the optimal signaling and the capacity of fixed MISOchannels under the joint power constraints have been obtained above, one may wonder whetherthey can be extended to the MIMO case and whether fading can be included as well, which isof particular importance for modern wireless systems.In this section, we partially answer this question by considering Gaussian fading MIMOchannels of the form y = Hx + ξ (33)where x , y are the transmitted and received (vector) signals, ξ is the Gaussian i.i.d. noise and H is the channel matrix. The entries of this matrix are random variables representing fadingchannel gains between each transmit and each receive antenna. We assume that the Tx has thechannel distribution information only (due to e.g. limitations of the feedback link and the channelestimation mechanism, see e.g. [16]). A class of ergodic fading distributions will be considered,of which i.i.d. Rayleigh fading is a special case. The following definition characterizes this class. Definition 1.
A fading distribution of H is right unitary-invariant if HU and H are equal indistribution for any unitary matrix U of appropriate size. To see a physical motivation behind this definition, observe that i.i.d. Rayleigh fading, whereeach entry of H is i.i.d. complex Gaussian with zero mean, satisfies this condition. A moregeneral class of distributions which fit into this definition can be obtained by considering thepopular Kronecker correlation model, see e.g. [17], where the overall channel correlation is aproduct of the independent Tx and Rx parts, which are induced by the respective sets of scatterers(e.g. around the base station and mobile unit), so that the channel matrix is H = R / r H R / t (34)where R r , R t are the Rx and Tx end correlations and the entries of H are i.i.d. complexGaussian with zero mean. While this model does not fit in general into Definition 1, its specialcase of no Tx correlation, R t = I , so that H = R / r H (35) December 15, 2016 DRAFT3 is indeed right unitary-invariant (since H and H U have the same distribution). Note that thismodel does allow an (arbitrary) Rx correlation. The uncorrelated Tx end may represent a basestation where the antennas are spaced sufficiently widely apart of each other thereby inducingindependence, see e.g. [18].The following Theorem establishes the ergodic capacity of a Gaussian MIMO channel undera right unitary-invariant fading distribution and the joint PA and TP constraints. Theorem 3.
Consider the ergodic-fading MIMO channel as in (33) for which the fading distri-bution is right unitary-invariant. Its channel capacity under the joint PA and TP constraints in (9) is as follows: C = E H { ln | I + P ∗ HH + |} (36) where E H is the expectation with respect to the fading distribution, P ∗ = min { P, P T /m } , andthe optimal Tx covariance matrix is R ∗ = P ∗ I , i.e. isotropic (independent) signaling is optimal.Proof: The proof consists of two parts. In Part 1, we establish the optimality of isotropicsignaling under the TP constraint only, while in Part 2, we extend this result to include the PAconstraints as well.
Part 1 : the ergodic capacity under the TP constraint can be presented in the following form: C = max R E H { ln | I + HRH + |} (37) = max R E H { ln | I + HU Λ U + H + |} (38) = max R E f H { ln | I + f H Λ f H + |} (39) = max Λ E H { ln | I + H Λ H + |} (40)where the maximization is subject to R ≥ , tr R ≤ P T . (37) is the standard expressionfor the ergodic MIMO channel capacity, see e.g. [19][16]; (38) follows from the eigenvaluedecomposition R = U Λ U + , where the columns of unitary matrix U are the eigenvectors of R and the diagonal matrix Λ collects the eigenvalues of R ; (39) follows from f H = HU ; (40)follows since f H and H have the same distribution and the constraint tr R = tr Λ ≤ P T dependsonly on the eigenvalues and hence the eigenvectors can be eliminated from the optimization. To December 15, 2016 DRAFT4 proceed further, let C ( Λ ) = E H { ln | I + H Λ H + |} (41)and observe that this is a concave function (since ln | · | is and E H preserves concavity, see e.g.[15]). Further observe the following chain inequality: C ( Λ ) = E H { ln | I + H Λ π H + |} (42) = 1 m ! X π E H { ln | I + H Λ π H + |} (43) ≤ E H ( ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I + 1 m ! X π H Λ π H + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)) (44) = E H { ln | I + P ∗ H H + |} (45)where Λ π is a diagonal matrix whose diagonal entries are a permutation π of those in Λ , and P ∗ = P T /m . (42) follows from the fact that a permutation can be represented by a unitary matrix(where each column and each row has all zero entries except for one) and hence C ( Λ ) = C ( Λ π ) ;(43) follows since (42) holds for any π and the total number of permutations is m ! ; the inequalityin (44) is due to the concavity of C ( Λ ) ; (45) follows from m ! P π Λ π = P ∗ I . Since the inequalityin (44) becomes equality when Λ = P ∗ I , the optimal signaling under the TP constraint is R ∗ = P ∗ I so that C = max Λ C ( Λ ) = E H { ln | I + P ∗ HH + |} , (46)which establishes Part 1. Part 2: consider first the case when P ≥ P T /m and observe that the capacity under the jointconstraints C cannot exceed that under the TP constraint only, which hence serves as an upperbound: C ≤ C . Since the TP optimal covariance R ∗ = P T I /m also satisfies the PA constraints(under the assumed condition), it is also optimal under the joint constraints and hence the upperbound is achieved: C = C . If, on the other hand, P < P T /m , observe that the TP constraintis redundant (since, due to the PA constraints, the total power does not exceed mP < P T ) andhence the jointly-constrained optimization with P T > mP is equivalent to the PA-constrainedoptimization only, which in turn is equivalent to the jointly-constrained optimization with newtotal power P ′ T = mP (since the new TP constraint is also redundant). However, the latter December 15, 2016 DRAFT5 problem is just a special case of P ≥ P T /m considered above, from which the optimality of R ∗ = P ′ T I /m = P I follows.Combining two parts, it follows that R ∗ = min { P, P T /m } I is optimal under the jointconstraints in general and hence C = C . This completes the proof.It follows from Theorem 3 and its proof that the same capacity expression holds under theTP constraint, the PA constraints and the joint PA and TP constraints (where P ∗ is definedaccordingly). This extends the earlier result in [19] established for i.i.d. Rayleigh fading and theTP constraint to the class of right unitary-invariant fading distributions (including, as a specialcase, the semi-correlated model in (35)) and to the PA as well as the joint PA and TP constraints.Note that the optimal signaling here is isotropic, so that the optimal covariance matrix is full-rank, unlike that in Theorem 1, which is of rank-1. The importance of isotropic signalling is dueto the fact that no channel state or distribution information is needed at the Tx end (and hencethe feedback requirements are minimal).Applying this Theorem to the semi-correlated channel in (35), one obtains its ergodic capacityunder the joint PA and TP constraints: C = E H { ln | I + P ∗ R r H H +0 |} (47)Unlike the fixed MISO case, the optimal signaling here is isotropic, R ∗ = P ∗ I , and henceindependent of the Rx correlation R r . However, the capacity does depend on R r but, as itfollows from (47), C depends on the eigenvalues of R r only, not on its eigenvectors: C = E H { ln | I + P ∗ U r Λ r U + r H H +0 |} = E ˜ H { ln | I + P ∗ Λ r ˜ H ˜ H +0 |} (48) = E H { ln | I + P ∗ Λ r H H +0 |} where R r = U r Λ r U + r is the eigenvalue decomposition, and ˜ H = U + r H . The last equality isdue to the fact that ˜ H and H are equal in distribution. Hence, different R r induce the samecapacity provided that they have the same eigenvalues.These properties are ultimately due to the right unitary invariance of the fading process. It canbe further shown (by examples) that Theorem 3 does not hold in general if fading distribution isnot right unitary-invariant: e.g. consider R t = diag { , , ..., } for which the optimal covariance December 15, 2016 DRAFT6 can be shown to be R ∗ = diag { min( P T , P ) , , ..., } (i.e. all the power is allocated to the onlynon-zero eigenmode of R t ). VI. C ONCLUSION
The Gaussian MISO channel has been considered under the joint total and per-antenna powerconstraints. Its capacity as well as the optimal transmission strategy have been established inclosed-form, thus extending earlier results established under individual constraints only or, inthe case of joint constraints, for 2 Tx antennas only. It is interesting to observe that the optimaltransmission strategy is hybrid, i.e. a combination of equal gain (for stronger antennas) andmaximum-ratio (for weaker antennas) transmission strategies. If the variance of channel gainsacross antennas is not too large, the maximum ratio transmission is optimal and individual powerconstraints are not active. Finally, the above results have been extended to the MIMO case byestablishing the ergodic capacity of fading MIMO channels under the joint power constraintswhen the fading distribution is right unitary invariant, which includes, as special cases, i.i.d.and semi-correlated Rayleigh fading. The optimal signaling in this case has been shown to beisotropic and hence the feedback requirements are minimal.VII. A
CKNOWLEDGEMENT
The author is grateful to R.F. Schaefer for insightful discussions, and to V.I. Mordachev andT. Griken for their support. VIII. A
PPENDIX
A. Proof of Theorem 1
The problem in (2) under the constraints in (9) is convex (since the objective is affine and theconstraints are affine and positive semi-definite). Since Slater’s condition holds, KKT conditionsare sufficient for optimality [15]. The Lagrangian for this problem is: L = − h + Rh + λ ( tr R − P T ) + X i λ i ( r ii − P ) − tr M R (49)where λ, λ i ≥ are Lagrange multipliers responsible for the total and per-antenna powerconstraints, and M ≥ is (matrix) Lagrange multiplier responsible for the positive semi-definite December 15, 2016 DRAFT7 constraint R ≥ . The KKT conditions are ∇ R L = − hh + + λ I − M + Λ = 0 (50) λ ( tr R − P T ) = 0 , λ i ( r ii − P ) = 0 , RM = 0 (51) tr R ≤ P T , r ii ≤ P, (52) M ≥ , λ i ≥ (53)where ∇ R is the derivative with respect to R and Λ = diag { λ ...λ m } is a diagonal matrixcollecting λ i ; (50) is the stationarity condition, (51) are complementary slackness conditions;(52) and (53) are primal and dual feasibility conditions.Combining both inequalities in (52), one obtains: tr R ≤ min( P T , mP ) = P ∗ (54)and from (50) hh + + M = λ I + Λ > (55)where the last inequality is due to the diagonal part of the equality: | h i | + m ii = λ + λ i > (56)since m ii ≥ and | h i | > . Therefore, hh + + M is full-rank, r ( hh + + M ) = m . Since r ( hh + ) = 1 and M ≥ , it follows that r ( M ) ≥ m − . Since r ( M ) = m implies M > andhence R = 0 - clearly not an optimal solution, one concludes that r ( M ) = m − and hence r ( R ) = 1 (this follows from complementary slackness M R = 0 ), i.e. beamforming is optimal: R ∗ = P ∗ uu + (57)where | u | = 1 . It remains to find the beamforming vector u . To this end, combining the lastequation with M R = 0 , one obtains:
M u = − h + uh + ( Λ + λ I ) u (58)from which it follows that u i = h + u h i / ( λ + λ i ) (59) December 15, 2016 DRAFT8 and hence φ ui = φ i + ϕ = φ i (60)where φ ui , ϕ are the phases of u i and h + u ; since the common phase ϕ does not affect R or theSNR, one can set ϕ = 0 without loss of generality to obtain u i = ah i / ( λ + λ i ) (61)where a = | h + u | .If λ i > (active i -th per-antenna constraint), then r ii = P ∗ | u i | = P from (51) and (11) sothat | u i | = c = 1 / √ m ∗ (62)Since λ i > , using (61), c = | u i | = a | h i | / ( λ + λ i ) < a | h i | /λ (63)so that | h i | > λc /a = h th (64)where h th is a threshold channel gain, i.e. PA constraints are active for all sufficiently strongchannels.When λ i = 0 (inactive i -th PA constraint) for at least one i , it follows from (56) that λ > ,i.e. the TP constraint is active: tr R = P T , which implies P T ≤ mP . One obtains from (61) inthis case u i = c h i , c = a/λ (65)which, when combined with the PA constraint r ii = P T | u i | ≤ P , requires | h i | ≤ h th (66)where c can be found from the TP constraint | u | = 1 : | u | = kc + c | h mk +1 | = 1 (67)and k < m is the number of active PA constraints, i.e. when (64) holds, which implies c = p − k/m ∗ | h mk +1 | (68) December 15, 2016 DRAFT9 so that k ≤ m ∗ and h th can be expressed as h th = λc a = c c = | h mk +1 | √ m ∗ − k (69)If k = m , i.e. all PA constraints are active, then one can take h th = 0 for consistency with (64).This implies P T ≥ mP so that m ∗ = m (note that (69) is not defined in this case).To find the number k of active PA constraints when P T < mP , so that m ∗ = P T /P < m andhence k ≤ m ∗ < m , observe that (64) and (65) imply | h k |√ m ∗ − k > | h mk +1 | (70)while (66) implies | h k +1 |√ m ∗ − k ≤ | h mk +1 | (71)both due to the ordering | h | ≥ | h | ≥ .. ≥ | h m | , so that k has to satisfy both inequalitiessimultaneously.The next step is to show that there exists unique k that satisfies both inequalities. First, weshow that there is at least one solution of (71). Lemma 1.
There exists at least one solution k , ≤ k ≤ m ∗ , of (71) .Proof: If m ∗ = m , then k = m clearly solves it, where we take h m +1 = 0 for consistency(recall that all channels with 0 gain do not affect the capacity). If m ∗ < m , then k = ⌊ m ∗ ⌋ solves it.The next Lemma shows that, in general, a solution is not unique. Lemma 2. If k ≤ ⌊ m ∗ ⌋ satisfies (71) , then all k ′ such that k ≤ k ′ ≤ ⌊ m ∗ ⌋ also satisfy it, i.e. asolution is not unique in general. Likewise, all k ′ ≤ k solve (70) if k solves it.Proof: Let (71) to hold for k < ⌊ m ∗ ⌋ , so that | h k +1 | ( m ∗ − k ) ≤ | h k +1 | + .. + | h m | (72)and hence | h k +2 | ( m ∗ − ( k + 1)) ≤ | h k +1 | ( m ∗ − ( k + 1)) ≤ | h k +2 | + .. + | h m | (73) December 15, 2016 DRAFT0 i.e. (71) also holds for k ′ = k + 1 . By induction, it holds for all k ≤ k ′ ≤ ⌊ m ∗ ⌋ . To prove 2ndclaim, note that it follows from (70) that | h k − | ( m ∗ − k + 1) ≥ | h k | ( m ∗ − k + 1) > | h k | + .. + | h m | (74)Finally, we show that a unique k satisfying both inequalities does exist. Proposition 1.
There exists a unique solution of (70) and (71) , which is also the least solutionof (71) .Proof:
Note, from Lemma 1, that a least solution k ′ of (71) exists, so that the followingholds | h k ′ +1 | ( m ∗ − k ′ ) ≤ | h k ′ +1 | + .. + | h m | (75) | h k ′ | ( m ∗ − k + 1) > | h k ′ | + .. + | h m | (76)where the last inequality is due to the fact that k ′ is the least solution; this inequality implies | h k ′ | ( m ∗ − k ) > | h k ′ +1 | + .. + | h m | (77)i.e. (70) holds for k = k ′ .It remains to show that M ≥ (dual feasibility). To this end, note that this is equivalent to x + M x ≥ ∀ x . It follows from (50), (61), (64), (65) and Caushy-Schwarz inequality that x + M x = −| h + x | + λ | x | + x + Λ x (78) ≥ −| h + x | + ac m X i =1 | h i || x i | (79) ≥ −| h + x | + | h | m X i =1 | h i || x i | (80) ≥ −| h + x | + m X i =1 | h i | | x i | ≥ (81)This completes the proof. December 15, 2016 DRAFT1 R EFERENCES [1] T.M. Cover, J.A. Thomas, Elements of Information Theory, Wiley, 2006.[2] J.R. Barry, E.A. Lee, D.G. Messerschmitt, Digital Coomunications (3rd Ed.), Kluwer, Boston, 2004[3] D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005.[4] E. Biglieri, J. Proakis, and S. Shamai, ”Fading Channels: Information-Theoretic and Communications Aspects,” IEEE Trans.Inform. Theory, vol. 44, No. 6, pp. 2619-2692, Oct. 1998.[5] G.L. Stuber, Principles of Mobile Communication, Springer, New York, 2011.[6] W. Yu and T. Lan, Transmitter optimization for the multi-antenna downlink with per-antenna power constraint, IEEE Trans.Signal Process., vol.55, no. 6, pp. 2646-2660, June 2007.[7] J. Park, W. Sung, and T. Duman, Precoder and capacity expressions for optimal two-user MIMO transmission with per-antenna power constraints, IEEE Comm. Letters, vol. 14, no. 11, pp. 996-998, Nov. 2010[8] M. Vu, MISO Capacity with Per-antenna power constraint, IEEE Trans. on Commun., vol. 59, no. 5, May 2011.[9] M. 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Cao et al, ”Optimal Transmission Rate for MISO Channels with Joint Sum and Per-antenna Power Constraints”, IEEEInternational Conference on Communications (ICC), London, June 08-12, 2015.[15] S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.[16] E. Biglieri et al, MIMO Wireless Communications, Cambridge University Press, New York, 2007.[17] J.P. Kermoal et al., A stochastic MIMO radio channel model with experimental validation, IEEE JSAC, v.20, N.6, pp.1211-1226, Aug. 2002.[18] S. Loyka, G. Tsoulos, Estimating MIMO System Performance Using the Correlation Matrix Approach, IEEE Communi-cation Letters, v. 6, N. 1, pp. 19-21, Jan. 2002.[19] I. E. Telatar, Capacity of Multi-Antenna Gaussian Channels, AT&T Bell Labs, Internal Tech. Memo, June 1995, (EuropeanTrans. Telecom., v.10, no. 6, Dec. 1999).[1] T.M. Cover, J.A. Thomas, Elements of Information Theory, Wiley, 2006.[2] J.R. Barry, E.A. Lee, D.G. Messerschmitt, Digital Coomunications (3rd Ed.), Kluwer, Boston, 2004[3] D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005.[4] E. Biglieri, J. Proakis, and S. Shamai, ”Fading Channels: Information-Theoretic and Communications Aspects,” IEEE Trans.Inform. Theory, vol. 44, No. 6, pp. 2619-2692, Oct. 1998.[5] G.L. Stuber, Principles of Mobile Communication, Springer, New York, 2011.[6] W. Yu and T. Lan, Transmitter optimization for the multi-antenna downlink with per-antenna power constraint, IEEE Trans.Signal Process., vol.55, no. 6, pp. 2646-2660, June 2007.[7] J. Park, W. Sung, and T. Duman, Precoder and capacity expressions for optimal two-user MIMO transmission with per-antenna power constraints, IEEE Comm. Letters, vol. 14, no. 11, pp. 996-998, Nov. 2010[8] M. Vu, MISO Capacity with Per-antenna power constraint, IEEE Trans. on Commun., vol. 59, no. 5, May 2011.[9] M. Vu, MIMO Capacity with Per-Antenna Power Constraint, IEEE Globecom, Houston, USA, 5-9 Dec., 2011.[10] M. Khoshnevisan, J.N. Laneman, Power Allocation in Multi-Antenna Wireless Systems Subject to Simultaneous PowerConstraints, IEEE Trans. Comm., v.60, No. 12, pp. 3855–3864, Dec. 2012.[11] Y. Zhu, M. Vu, Iterative Mode-Dropping for the Sum Capacity of MIMO-MAC with Per-Antenna Power Constraint, IEEETrans. Comm., v. 60, N. 9, pp. 2421–2426, Sep. 2012.[12] D. Maamari, N. Devroye, D. Tuninetti, The Capacity of the Ergodic MISO Channel with Per-antenna Power Constraintand an Application to the Fading Cognitive Interference Channel, IEEE Int. Symp. Information Theory, Hawaii, USA, July2014.[13] D. Tuninetti, ”On the capacity of the AWGN MIMO channel under per-antenna power constraints”, 2014 IEEE InternationalConference on Communications (ICC), Sydney, June 2014.[14] P. Cao et al, ”Optimal Transmission Rate for MISO Channels with Joint Sum and Per-antenna Power Constraints”, IEEEInternational Conference on Communications (ICC), London, June 08-12, 2015.[15] S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.[16] E. Biglieri et al, MIMO Wireless Communications, Cambridge University Press, New York, 2007.[17] J.P. Kermoal et al., A stochastic MIMO radio channel model with experimental validation, IEEE JSAC, v.20, N.6, pp.1211-1226, Aug. 2002.[18] S. Loyka, G. Tsoulos, Estimating MIMO System Performance Using the Correlation Matrix Approach, IEEE Communi-cation Letters, v. 6, N. 1, pp. 19-21, Jan. 2002.[19] I. E. Telatar, Capacity of Multi-Antenna Gaussian Channels, AT&T Bell Labs, Internal Tech. Memo, June 1995, (EuropeanTrans. Telecom., v.10, no. 6, Dec. 1999).