Constant-Envelope Precoding with Time-Variation Constraint on the Transmitted Phase Angles
aa r X i v : . [ c s . I T ] N ov Constant-Envelope Precoding with Time-VariationConstraint on the Transmitted Phase Angles
Sudarshan Mukherjee and Saif Khan Mohammed
Abstract —We consider downlink precoding in a frequency-selective multi-user massive MIMO system with highly efficientbut non-linear power amplifiers at the base station (BS). A low-complexity precoding algorithm is proposed, which generatesconstant-envelope (CE) transmit signals for each BS antenna.To avoid large variations in the phase angle transmitted fromeach antenna, the difference of the phase angles transmitted inconsecutive channel uses is limited to [ − απ , απ ] for a fixed < α ≤ . To achieve a desired per-user information rate,the extra total transmit power required under the time variationconstraint when compared to the special case of no time variationconstraint (i.e., α = 1 ), is small for many practical values of α . In a i.i.d. Rayleigh fading channel with BS antennas, single-antenna users and a desired per-user information rate of bit-per-channel-use, the extra total transmit power required isless than . dB when α = 1 / . Index Terms —Massive MIMO, constant envelope.
I. I
NTRODUCTION
In massive MIMO systems a base station (BS) with a largenumber of antennas ( N , a few hundreds) communicates withseveral user terminals ( M , a few tens) on the same time-frequency resource [1], [2]. There has been recent interest inmassive MIMO systems due to their ability to increase spectraland energy efficiency even with very low-complexity multi-user detection and precoding [3], [4], [5]. However, physicallybuilding cost-effective and energy-efficient large arrays is achallenge. Specifically in the downlink, the power amplifiers(PAs) used in the BS should be highly power-efficient. Dueto the trade-off between the efficiency and linearity of thePA [6], highly efficient but non-linear PAs must be used.The efficiency of the PA is related to the amount of backoffnecessitated (to reduce non-linear distortion) by the peak toaverage ratio of the input waveform. For minimum backoffand hence maximum efficiency, the input waveform shouldhave a constant or nearly constant envelope (CE). With this motivation, in [7] we had proposed a CE precodingalgorithm for the frequency-flat multi-user MIMO broadcastchannel, which was then extended to frequency-selective chan-nels in [8]. With N sufficiently larger than M and i.i.d.Rayleigh fading, numerical studies done in both these papersrevealed that in order to achieve a desired per-user ergodicinformation rate, the proposed CE precoding algorithm needed Sudarshan Mukherjee and Saif Khan Mohammed are with the Dept. ofElectrical Engineering, Indian Institute of Technology (I.I.T.), Delhi, India.S. K. Mohammed is also associated with the Bharti School of Telecommu-nication Technology and Management (BSTTM), I.I.T. Delhi. This work wassupported by EMR funding from the Science and Engineering Research Board(SERB), Department of Science and Technology (DST), Government of India. That is, the discrete-time complex baseband signal transmitted from eachBS antenna has a constant magnitude irrespective of the channel gains andthe information symbols to be communicated. only about . − . dB extra total transmit power comparedto that required under the less stringent and commonly usedtotal average transmit power constraint. It was also observedthat even under a stringent per-antenna CE constraint, an O ( N ) array gain is achievable, i.e., with every doubling inthe number of BS antennas the total transmit power can bereduced by dB while maintaining a fixed information rateto each user (assuming that the number of users is fixed).However, in the CE precoding algorithm proposed in [7],[8] the phase angle of the complex baseband signal transmittedfrom each BS antenna is unconstrained (i.e., it’s principal valuelies in ( − π , π ] ), and therefore it is possible that the phasecould vary very fast between consecutive channel uses. Aphase variation of ◦ (or more) between consecutive channeluses will result in zero crossings in the baseband signal, whichwith practical PAs could lead to distortion in the transmittedsignal. In this paper, we address this problem by proposinga CE precoding algorithm with an additional constraint thatthe difference of the phase angle transmitted in consecutivechannel uses be limited to the interval [ − απ , απ ] for a fixed < α ≤ (the special case of α = 1 was considered in [8]).It is shown that the complexity of the proposed CE algorithmis independent of α and is the same as the algorithm proposedin [8]. Numerical studies on the i.i.d. Rayleigh fading channelsuggest that an O ( N ) array gain is achieved even underthe additional phase angle variation constraint. To achieve adesired per-user information rate, the extra total transmit powerrequired under the time variation constraint when compared tothe special case of no time variation constraint (i.e., α = 1 ),is small when α is close to and N ≫ M . For example, with α = 1 / , N = 80 , M = 5 single-antenna users and a desiredper-user rate of bit-per-channel-use (bpcu), the magnitudeof the phase variation is limited to απ = 90 ◦ and the extratransmit power required is less than dB.II. S YSTEM M ODEL AND
CE P
RECODING
In the previous works and also in this paper, without loss ofgenerality we assume single-antenna users. It is assumed thatthe BS has knowledge of the channel vector to each user. Thecomplex baseband constant envelope signal transmitted fromthe i -th BS antenna at time t is of the form When a user terminal (UT) has multiple antennas, the proposed algorithmcan still be applied by treating each antenna at the UT as a separate user. In a massive MIMO system ( N ≫ M ), the amount of time-bandwidthresource required for channel estimation at the BS in the uplink is proportionalto M , while in the downlink it is proportional to N . Since N ≫ M , itis suggested that massive MIMO systems would operate in a time divisionduplexed (TDD) mode, so that downlink CSI can be estimated from the CSIacquired in the uplink through uplink training [1], [2], [3]. x i [ t ] = r P T N e jθ i [ t ] , i = 1 , , · · · , N, (1)where j ∆ = √− , P T is the total power transmitted from the N BS antennas and θ i [ t ] ∈ [ − π , π ) is the phase of the CEsignal transmitted from the i -th BS antenna at time t . Theequivalent discrete-time complex baseband channel betweenthe i -th BS antenna and the k -th user (having a single-antenna)has a finite impulse response of length L samples, denoted by ( h k,i [0] , h k,i [1] , · · · , h k,i [ L − . The signal received at the k -th user ( k = 1 , , · · · , M ) at time t is given by y k [ t ] = r P T N N X i =1 L − X l =0 h k,i [ l ] e jθ i [ t − l ] + w k [ t ] , (2)where w k [ t ] ∼ CN (0 , σ ) is the AWGN at the k -thuser at time t (AWGN is i.i.d. across time and across theusers). For the sake of brevity let us denote the vectorof phase angles transmitted at time instance t by Θ[ t ] =( θ [ t ] , · · · , θ N [ t ]) . In the following we briefly summarize theCE precoding algorithm proposed in [8]. Suppose that, at timeinstances t = 1 , , ..., T we are interested in communicatingthe information symbol √ E k u k [ t ] ∈ U k ⊂ C to the k -th user. Let E [ | u k [ t ] | ] = 1 , k = 1 , · · · , M . Also, let u [ t ] = ( √ E u [ t ] , · · · , √ E M u M [ t ]) ∈ U × · · · × U M bethe vector of information symbols to be communicated attime t . In [8], we had proposed an algorithm for finding thetransmit phase angles θ i ( t ) , i = 1 , , . . . , N , t = 1 , , . . . , T in such a way that the received noise-free signal at eachuser is almost the same as the information symbol intendedfor that user, i.e., p P T /N P Ni =1 P L − l =0 h k,i [ l ] e jθ i [ t − l ] ≈√ P T √ E k u k [ t ] , ∀ k = 1 , , . . . , M , t = 1 , , . . . , T .In [8] we find the transmit phase angles as a solu-tion to the optimization problem in (3), where Θ u [ t ] =( θ u [ t ] , · · · , θ uN [ t ]) , t = 1 , . . . , T denotes the vectors of trans-mit phase angles, for the given information symbol vectors u [ t ] , t = 1 , . . . , T . The main idea in (3) is to choose thetransmit phase angles in a way so as to minimize the energyof the difference between the received noise-free signal andthe intended information symbol for all users. Note that theobjective function f ( · · · ) in (3) is a function of N T variables( N phase angles transmitted at T time instances). Finding anexact solution to the problem in (3) is prohibitively complex,and therefore in [8] we had proposed a low-complexity near-optimal solution to (3). The CE precoding idea is primarilybased on our previous work in [7] (for frequency-flat channels)where we had analytically shown that for a broad class offrequency-flat channels (including i.i.d. fading), for a fixed M and fixed symbol energy levels ( E , · · · , E M ), by having asufficiently large N ≫ M it is always possible to choose thetransmit phase angles in such a way that the received signalsat the users are arbitrarily close to the desired informationsymbols.III. CE P RECODING WITH C ONSTRAINED T IME V ARIATION OF T RANSMIT P HASE A NGLES
Note that for the CE precoding method, the transmit phaseangles can take any value in the interval [ − π , π ) (see (3)). Therefore it is possible that between consecutive time in-stances, the phase angle transmitted from a BS antenna couldchange by a large magnitude, which will distort the transmitsignal at the output of the PA. To address this issue, in this pa-per we propose a CE precoder where for each BS antenna thedifference between the phase angles transmitted in consecutivetime instances is constrained to lie in the interval [ − απ , απ ] for a given < α ≤ , i.e., | θ i [ t ] − θ i [ t − | ≤ απ forall t = 1 , . . . , T , i = 1 , . . . , N . This constraint ensures thatthe maximum variation in the transmitted phase angle betweenconsecutive time instances is at most απ (e.g., with α = 1 / the maximum phase angle variation is only ◦ ). In this paper,under the time-variation constraint we propose an optimizationproblem to find the transmit phase angles for given informationsymbols u k [ t ] , k = 1 , . . . , M , t = 1 , . . . , T , as given by (4).Exactly solving (4) has prohibitive complexity, and thereforein the following we propose a low-complexity near-optimalsolution to (4). The essential idea of this low complexitysolution is to iteratively optimize f ( · · · ) as a function ofone variable at a time while fixing the other variables totheir previous values. In one iteration of this low-complexityalgorithm, we have N T sub-iterations. In the first sub-iterationwe start with θ [1] and minimize f ( · · · ) as a function of θ [1] while keeping the other ( N T − variables fixed totheir previous values. We then update θ [1] with its optimumvalue and then move onto the second sub-iteration where weminimize f ( · · · ) as a function of θ [1] while keeping the othervariables fixed. In general, in the ( N ( q − r ) -th sub-iterationwe minimize f ( · · · ) as a function of θ r [ q ] (i.e., the phaseangle transmitted from the r -th BS antenna in the q -th timeinstance) while keeping the other variables fixed. Since thechannel is causal and has a memory of L time instances, itfollows that in the summation on the right hand side of thedefinition of f ( · · · ) in (3), only the terms corresponding to t = q, ( q + 1) , · · · , min( T, q + L − depend on θ r [ q ] . Giventhis fact, the minimization of f ( · · · ) only w.r.t. θ r [ q ] is givenby (5). In (5), for any complex number z , ARG ( z ) ∆ = { φ ∈ ( − π , π ] | e jφ = z/ | z |} is the principal value of the phaseangle of z and z ∗ denotes the conjugate of z . From (5) it isclear that the new value of θ r [ q ] depends on S r,q ( k, t ) . Notethat, for every different ( r, q ) we need not recalculate S r,q ( k, t ) explicitly using the sum in the R.H.S. of its definition in (5).Instead, S r,q ( k, t ) can be calculated by subtracting the currentvalue of h k,r [ t − q ] e jθ r [ q ] / √ N (i.e., value at the start of the ( N ( q −
1) + r ) -th sub-iteration) from the current value of S ( k, t ) ∆ = P Ni =1 P L − l =0 h k,i [ l ] e jθi [ t − l ] √ N − √ E k u k [ t ] , i.e. S r,q ( k, t ) = S ( k, t ) − h k,r [ t − q ] e jθ r [ q ] √ N . (6)Note that with change in θ r [ q ] , we also need to change S ( k, t ) for all k = 1 , . . . , M , t = q, . . . , min( T, q + L − . Themodified value of S ( k, t ) after the ( N ( q −
1) + r ) -th sub- (Θ u [1] , · · · , Θ u [ T ]) = arg min Θ[ t ] ∈ [ − π,π ) N t =1 ,...,T f ( θ [1] , · · · , θ N [1] , · · · , θ [ T ] , · · · , θ N [ T ]) where f ( θ [1] , · · · , θ N [1] , · · · , θ [ T ] , · · · , θ N [ T ]) ∆ = T X t =1 M X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P Ni =1 P L − l =0 h k,i [ l ] e jθ i [ t − l ] √ N − √ E k u k [ t ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3) (Θ u [1] , · · · , Θ u [ T ]) = arg min (Θ[1] , Θ[2] , ··· , Θ[ T ]) | θ i [ t ] − θ i [ t − | ≤ απi =1 ,...,N , t =1 ,...,T f ( θ [1] , · · · , θ N [1] , · · · , θ [ T ] , · · · , θ N [ T ]) . (4) θ ′ r [ q ] = arg min θ r [ q ] | θ r [ q ] − θ r [ q − | ≤ απ min( T, ( q + L − X t = q M X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S r,q ( k, t ) + h k,r [ t − q ] e jθ r [ q ] √ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where S r,q ( k, t ) ∆ = (cid:16) N X i =1 L − X l =0 , ( i,l ) =( r, ( t − q )) h k,i [ l ] e jθ i [ t − l ] √ N (cid:17) − √ E k u k [ t ]= arg min θ r [ q ]( θ r [ q ] − θ r [ q − ∈ [ − απ , απ ] min( T, ( q + L − X t = q M X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S r,q ( k, t ) + h k,r [ t − q ] e jθ r [ q − e j ( θ r [ q ] − θ r [ q − √ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = θ r [ q −
1] + arg min ω ∈ [ − απ , απ ] min( T, ( q + L − X t = q M X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S r,q ( k, t ) + h k,r [ t − q ] e jθ r [ q − e j ω √ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = θ r [ q −
1] + arg max ω ∈ [ − απ , απ ] ℜ e jω n − min( T, ( q + L − X t = q M X k =1 h k,r [ t − q ] e jθ r [ q − S ∗ r,q ( k, t ) o! = θ r [ q −
1] + απ , − π ≤ c < − απ − c , − απ ≤ c < απ − απ , απ ≤ c ≤ π , where c ∆ = ARG (cid:16) − min( T, ( q + L − X t = q M X k =1 h k,r [ t − q ] e jθ r [ q − S ∗ r,q ( k, t ) (cid:17) . (5) iteration is given by S ′ ( k, t ) = P Ni =1 P L − l =0 , ( i,l ) =( r, ( t − q )) h k,i [ l ] e jθ i [ t − l ] √ N − p E k u k [ t ] + h k,r [ t − q ] e jθ ′ r [ q ] √ N = S ( k, t ) + h k,r [ t − q ] √ N (cid:16) e jθ ′ r [ q ] − e jθ r [ q ] (cid:17) (7)where θ ′ r [ q ] is the new updated value of the phase angle to betransmitted from the r -th BS antenna at time instance q , andis given by (5). After the last sub-iteration of an iteration (i.e.,where we update θ N [ T ] ), we start with the first sub-iteration(where we update θ [1] ) of the next iteration. It is clear that thevalue of the objective function f ( · · · ) reduces monotonicallyfrom one sub-iteration to the next. Numerically, it has beenobserved that the value of the objective function convergesin a few iterations ( ≤ ) and further iterations lead to littlereduction in the value of f ( · · · ) . Further, the value that theobjective function converges to, is observed to be small when N ≫ M . The complexity of each sub-iteration is O ( M L ) and is independent of α (see (5)). Since we update N T phaseangles in each iteration, the total complexity of each iterationis O ( N M LT ) . With a fixed number of iterations, the overallcomplexity of the proposed algorithm is O ( N M LT ) , i.e., aper-channel-use complexity of O ( N M L ) , which is the sameas that of the algorithm proposed in [8] to solve (3).IV. I NFORMATION T HEORETIC P ERFORMANCE A NALYSIS
For a given set of information symbol vectors u [ t ] , t =1 , · · · , T , let b Θ u [1] , b Θ u [2] , · · · , b Θ u [ T ] denote the output phaseangles of the proposed iterative CE precoding algorithm (see Section III). Let b θ ui [ t ] be the phase angle transmitted from the i -th antenna at time t . The signal received at the k -th user isthen given by y k [ t ] = p P T p E k u k [ t ] + p P T I uk [ t ] + w k [ t ] I uk [ t ] ∆ = (cid:16) N X i =1 L − X l =0 h k,i [ l ] √ N e j b θ ui [ t − l ] − p E k u k [ t ] (cid:17) (8)Note that I uk [ t ] behaves like multi-user interference(MUI). Also, let y k ∆ = ( y k [1] , · · · , y k [ T ]) T , u k ∆ =( √ E k u k [1] , · · · , √ E k u k [ T ]) T , I uk ∆ = ( I uk [1] , · · · , I uk [ T ]) T and w k ∆ = ( w k [1] , · · · , w k [ T ]) T . Let H = { h k,i [ l ] } denote theimpulse responses of the channels between the N BS antennasand the M users. For a given H , an achievable rate for the k -thuser is given by the mutual information I ( y k ; u k | H ) /T [9].For any arbitrary distribution on u k , it is difficult to compute I ( y k ; u k | H ) . A lower bound on I ( y k ; u k | H ) /T is anachievable information rate for the k -th user. Therefore, inthe following we derive a lower bound to I ( y k ; u k | H ) /T assuming u k [ t ] , t = 1 , · · · , T to be i.i.d. CN (0 , i.e., propercomplex Gaussian having zero mean and unit variance. I ( y k ; u k | H ) T = 1 T (cid:16) h ( u k ) − h ( u k − y k / p P T | y k , H ) (cid:17) = log ( πeE k ) − (cid:16) h ( v k | y k , H ) /T (cid:17) ( a ) ≥ log ( πeE k ) − (cid:16) h ( v k | H ) / T (cid:17) ( b ) ≥ log ( E k ) − (cid:16) log ( | R v | ) / T (cid:17) v k ∆ = u k − (cid:0) y k / p P T (cid:1) (9)where h ( · ) denotes the differential entropy operator, and theinequality in step (a) is due to the fact that conditioning reduces entropy [9]. The inequality in step (b) follows fromthe fact that the proper complex Gaussian distribution is theentropy maximizer, i.e., h ( v k | H ) ≤ log (( π e ) T | R v | ) , where R v ∆ = E [ v k v Hk ] is the autocorrelation matrix of v k and | R v | denotes its determinant [10]. From (8) and the definition of v k in (9) we get v k = − I uk − (cid:0) w k / √ P T (cid:1) . Since I uk and w k areindependent, it follows that R v = E [ I uk I u H k | H ] + ( σ /P T ) I ,where the expectation is over u , · · · , u M . Substituting thisexpression for R v in (9), we get I ( y k ; u k | H ) T ≥ R k ( H , E , P T σ ) , where R k ( H , E , P T σ ) ∆ = " log ( E k ) − log (cid:12)(cid:12)(cid:12) E [ I uk I u H k | H ] + σ P T I (cid:12)(cid:12)(cid:12) T + (10) Here [ x ] + ∆ = max(0 , x ) and E ∆ = ( E , E , · · · , E M ) T isthe vector of the average information symbol energies of the M users. The ergodic information rate lower bound for the k -th user is then given by E [ R k ( H , E , P T /σ )] (expectationis over H ). V. N
UMERICAL RESULTS AND DISCUSSION
We consider a frequency selective channel with a uni-form power delay profile, i.e., E [ | h k,i [ l ] | ] = 1 /L , l =0 , , · · · , ( L − . The channel gains h k,i [ l ] are i.i.d. Rayleighfaded, i.e., proper complex Gaussian (mean , variance /L ).The ergodic sum rate P Mk =1 E [ R k ( H , E , P T /σ )] can bemaximized as a function of ( E , · · · , E M ) . This is howeverdifficult. Nevertheless, since the users have identical channelstatistics, it is likely that the optimal E vector has equalcomponents, i.e., E k = E ′ , k = 1 , . . . , M . Using nu-merical methods, for a given P T /σ we therefore find theoptimal E ′ which results in the largest ergodic sum rate. With E k = E ′ , k = 1 , . . . , M , we observe that all users havethe same ergodic rate, i.e. E [ R ( H , E , P T /σ )] = · · · = E [ R M ( H , E , P T /σ )] . Subsequently, we refer to this rateachieved by each user as the per-user ergodic information rate.In Fig. 1 we plot the minimum P T /σ required by theproposed CE precoder to achieve a per-user information rateof bpcu as a function of increasing N with fixed M = 5 users and L = 4 . The special case of α = 1 corresponds to anunconstrained (time-variation) CE precoder and therefore hasthe best performance. We see that for a given N , more transmitpower is required for a smaller α . This is expected sincea smaller α places a more stringent constraint on the time-variation of the transmitted phase angles, which reduces theinformation rate. However, even with α = 1 / (i.e., limitingthe magnitude of the time variation between consecutive timeinstances to be less than ◦ ), the extra transmit powerrequired when compared to α = 1 is less than dB when N is sufficiently larger than M (in this case N > M ). In this paper, E is fixed and does not vary with H . The achievable sumrate could be improved by adapting E with H , but then this would be difficultto realize in practice since we do not know the exact analytical dependence ofthe optimal E (which maximizes the ergodic sum rate) on the instantaneous H . Since E does not vary with each channel realization H , the optimal E depends on the multi-user channel only through its statistics.
20 40 60 80 100 120 140 160−15−10−50510
Number of BS antennas N M i n . r e q . P T σ t oa c h i e v e - bp c u / u s e r Proposed CE Prec. α = 1Proposed CE Prec. α = Proposed CE Prec. α = Proposed CE Prec. α = Coop. bound (TAPC)ZF (TAPC)
Fig. 1. Minimum required P T /σ to achieve a per-user ergodic rate of bpcu, plotted as a function of increasing N . Fixed M = 5 users and L = 4 . Also, for a fixed α the extra transmit power required whencompared to α = 1 , decreases with increasing N . From thefigure, it is also observed that irrespective of the value of α ,for sufficiently large N ≫ M the required P T /σ reducesby roughly dB with every doubling in N (i.e., an O ( N ) array gain with N BS antennas). For the sake of completeness,we have also considered the sum rate achieved under onlyan average total transmit power constraint (TAPC) which isclearly less stringent than the per-antenna CE constraint. UnderTAPC, we have plotted an achievable sum rate (ZF - Zero-Forcing precoder) and an upper bound on the sum capacity(cooperative users). It can be observed that even with α = 1 / ,the extra total transmit power required by the CE precoderwhen compared to the sum capacity achieving precoder underTAPC, is roughly dB when N ≫ M . CE precoding withnon-linear PAs is beneficial, since this dB loss is less thanthe gain in power efficiency that one can achieve by using anon-linear power-efficient PA instead of using a highly linearinefficient PA [6]. R EFERENCES[1] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, O. Edfors, F. Tufvessonand T. L. Marzetta, “Scaling up MIMO: opportunities and challenges withvery large arrays,”
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