Waveforms for the Massive MIMO Downlink: Amplifier Efficiency, Distortion and Performance
11 Waveforms for the Massive MIMO Downlink:Amplifier Efficiency, Distortion and Performance
Christopher Moll´en, Erik G. Larsson and Thomas Eriksson
Abstract —In massive
MIMO , most precoders result in downlinksignals that suffer from high
PAR , independently of modulationorder and whether single-carrier or
OFDM transmission is used.The high
PAR lowers the power efficiency of the base stationamplifiers. To increase power efficiency, low-
PAR precoders havebeen proposed. In this article, we compare different transmissionschemes for massive
MIMO in terms of the power consumedby the amplifiers. It is found that (i)
OFDM and single-carriertransmission have the same performance over a hardened massive
MIMO channel and (ii) when the higher amplifier power efficiencyof low-
PAR precoding is taken into account, conventional andlow-
PAR precoders lead to approximately the same power con-sumption. Since downlink signals with low
PAR allow for simplerand cheaper hardware, than signals with high
PAR , therefore, theresults suggest that low-
PAR precoding with either single-carrieror
OFDM transmission should be used in a massive
MIMO basestation.
Index Terms —low-
PAR precoding, massive
MIMO , multiuserprecoding, out-of-band radiation, peak-to-average ratio, poweramplifier, power consumption.
I. I
NTRODUCTION W IRELESS massive
MIMO systems, initially conceivedin [2] and popularly described in [3], simultaneouslyserve tens of users with base stations equipped with tens orhundreds of antennas using multiuser precoding. Compared toclassical multiuser
MIMO , order-of-magnitude improvementsare obtained in spectral and energy efficiency [4], [5]. For thesereasons, massive
MIMO is expected to be a key component offuture wireless communications infrastructure [3], [6].This work compares different multiuser precoding schemesfor the massive
MIMO downlink. Under a total radiated powerconstraint, optimal multiuser
MIMO precoding is a rather well-understood topic, see, e.g., [7], [8], as is linear (and necessarilysuboptimal) precoding, see, e.g., [9] and the survey [10]. Itis also known that, for massive
MIMO specifically, linearprecoding is close to optimal under a total-radiated powerconstraint [5]. There are also numerous results on precodingunder per-antenna power constraints [11]–[13].In practice, massive
MIMO precoders optimized subject to atotal radiated power constraint yield transmit signals with highpeak-to-average ratio (
PAR ), regardless of whether single-carrier
C. Moll´en and E. Larsson are with the Communication Systems Division,Dept. of Electrical Eng. (
ISY ), Link¨oping University, Link¨oping, Sweden, andT. Eriksson is with the Dept. of Signals and Systems, Chalmers University ofTechnology, Gothenburg, Sweden.The research leading to these results has received funding from the EuropeanUnion Seventh Framework Programme under grant agreement number ICT-619086 (
MAMMOET ), the Swedish Research Council (Vetenskapsr˚adet) and
ELLIIT .Parts of this work were presented at the European Wireless conference 2014[1].
NormalizedFInstantaneousFPowerF | x ( t )| / E [| x ( t )|] [dB] -2 0 2 4 6 8 10 12 CCD F -8 -6 -4 -2 single-carrierFzero-forcing,F4-QAMOFDMFzero-forcing,FGausslow-PARFprecoding Fig. 1. The complimentary cumulative distribution of the amplitudes ofdifferent massive
MIMO downlink signals that have been pulse shaped by a root-raised cosine filter, roll-off 0.22. Single-carrier and
OFDM transmission havevery similar distributions for the linear precoders described in Section III-B3for any modulation order, cf. [15]. The low-
PAR precoding scheme is describedin [16] and in Section III-C. The system has 100 base station antennas, 10single-antenna users and the channel is i.i.d. Rayleigh fading with 4 taps. or OFDM transmission and whether a low or a high modulationorder is used, see Figure 1. To avoid heavy signal distortionand out-of-band radiation, transmission of such signals requiresthat the power amplifiers are backed off and operated at a point,where their transfer characteristics are sufficiently linear [14].The higher the signal
PAR is, the more backoff is needed; andthe higher the backoff is, the lower the power efficiency ofthe amplifier will be. Against this background, precoders thatyield signals with low
PAR would be desirable.The possibility to perform low-
PAR precoding is a uniqueopportunity offered by the massive
MIMO channel—signalpeaks can be reduced because the massive
MIMO downlinkchannel has a large nullspace and any additional signaltransmitted in the nullspace does not affect what the usersreceive. In particular,
PAR -reducing signals from the channelnullspace can be added to the downlink signal so that theemitted signals have low
PAR [3], [17]. A few low-
PAR precoders for massive
MIMO have been proposed in theliterature [16]–[18]. In [17], the discrete-time downlink signalswere constrained to have constant envelopes. There it wasestimated that, in typical massive
MIMO scenarios, 1–2 dBextra radiated power is required to achieve the same sum-rateas without an envelope constraint. While some extra radiatedpower is required by low-
PAR precoders, it was argued in [17]that the overall power consumption still should decrease dueto the increased amplifier efficiency.Another unique feature of the massive
MIMO downlink Note that the precoder in [17] can transmit symbols from any general inputconstellation with single-carrier transmission and
OFDM —the received signalsdo not have to have constant envelopes, only the downlink signal emitted fromeach base station antenna has constant envelope. a r X i v : . [ c s . I T ] O c t P r ec od e r u [ n ] p ( τ ) Pu M [ n ] cont. channel { H [ ℓ ]} P User ky k [ n ] p * ( − τ ) s [ n ] u ( t ) u M ( t ) x ( t ) x M ( t ) p ( τ ) β k h k ( τ ) β k h k M ( τ ) y k ( t ) (a) The continuous-time model of the downlink. P r ec od e r u [ n ] u M [ n ] disc. channel { H [ ℓ ]} User k y k [ n ] s [ n ] PP d k [ n ]+ z k [ n ] β k h k [ n ] β k h k M [ n ] (b) The equivalent discrete-time model of the downlink.Fig. 2. The downlink of a massive MIMO system. channel is that certain types of hardware-induced distortiontend to average out when observed at the receivers [19]. Ourstudy confirms that the variance of the in-band distortion causedby nonlinear base station amplifiers does decrease with thenumber of base station antennas.The objective of the paper at hand is to more accuratelyquantify the benefits of low-
PAR precoding for the massive
MIMO downlink, taking into account in-band distortion andout-of-band radiation stemming from amplifier nonlinearitiesand imperfect channel state information due to pilot-basedchannel estimation. The difference between
OFDM and single-carrier transmission is also investigated. The main technicalcontribution of the paper is a comprehensive end-to-endmodelling of massive
MIMO downlink transmission, whichis treated in continuous time in order to capture the effects ofnonlinear amplification, the associated capacity bound, and theestimations of the power consumption for relevant amplifiermodels. All conclusions are summarized in Section V.We stress that the effect of amplifier nonlinearities onwireless signals have also been studied by others [14], [20],and for
MIMO specifically in [21]. In relation to this literature,the novel aspects of our work include: (i) a specific focus onthe massive
MIMO downlink channel, which facilitates low-
PAR precoding; (ii) a classification and comparison of precoderscommonly considered for massive
MIMO ; (iii) an estimate ofthe amplifier power consumption of low-
PAR precoding incomparison to that of other standard precoders.II. S
YSTEM M ODEL
The downlink shown in Figure 2(a) is studied. The basestation is equipped with M antennas and it serves K single-antenna users over a frequency-selective channel. All signalsare modeled in complex baseband.We let s k [ n ] be the n -th symbol that is to be transmitted touser k and collectively denote all the n -th symbols by s [ n ] (cid:44) ( s [ n ] , . . . , s K [ n ]) . The base station precodes the symbols toproduce the discrete-time signals { u m [ n ] } , where u m [ n ] is theprecoded signal of antenna m . These signals are scaled such that M (cid:88) m =1 E (cid:2) | u m [ n ] | (cid:3) = 1 , ∀ n (1)and pulse shaped by a filter with impulse response p ( τ ) intothe continuous-time transmit signals u m ( t ) (cid:44) (cid:88) n u m [ n ] p ( t − nT ) , (2)where T is the symbol period. After pulse shape filtering, thetransmit signal u m ( t ) has a bandwidth smaller or equal to thebandwidth B of the pulse p ( τ ) . The bandwidth B is the widthof the interval, over which the spectrum of p ( τ ) is non-zero.For example, if a root-raised cosine filter of period T withroll-off σ were used, then BT = 1 + σ .The continuous-time signal u m ( t ) is then amplified totransmit power by an amplifier that, in general, is nonlinear.The amplified signal is given by x m ( t ) = g (cid:0) | u m ( t ) |√ b (cid:1) e j (arg u m ( t )+Φ( | u m ( t ) | / √ b )) , (3)where g ( | u m ( t ) | ) is the AM - AM conversion and Φ( | u m ( t ) | ) the AM - PM conversion, see for example [15]. For now, theconversions g ( u ) and Φ( u ) are generic functions. Later in ouranalysis however, appropriate assumptions will be made tospecify them. The factor b is the backoff that has to be done toavoid nonlinear amplification and distortion. By backing off thesignal power to a suitable operating point, the signal amplitudewill stay in a region with sufficiently linear amplification mostof the time, see [15]. In this article, all backoffs are given indB relative to the backoff of the 1-dB compression point—thepoint, where the output signal is 1 dB weaker than what itwould have been if the amplification were perfectly linear. Thesignals are amplified so that lim t →∞ M (cid:88) m =1 E (cid:34) t t / (cid:90) − t / | x m ( t ) | d t (cid:35) = P, (4)where P is the transmitted power of the base station.The nonlinear relation in (3) generally widens the spectrumof the amplified signal, i.e. its signal energy is no longerconfined to the bandwidth B of the pulse p ( τ ) . The energyoutside the ideal bandwidth is called out-of-band radiation andis quantified by the Adjacent Channel Leakage Ratio ( ACLR ),which is defined in terms of the power P [ − B/ ,B/ of x m ( t ) in the useful band and the powers P [ − B/ , − B/ , P [ B/ , B/ in the immediately adjacent bands: ACLR (cid:44) max (cid:18) P [ − B/ , − B/ P [ − B/ ,B/ , P [ B/ , B/ P [ − B/ ,B/ (cid:19) , (5)where P B (cid:44) (cid:90) f ∈B S x ( f )d f. (6)and S x ( f ) is the power spectral density of x m ( t ) . In Figure 3,four power spectral densities of different amplified signals areshown to illustrate the out-of-band radiation. Half the in-bandspectrum is shown together with the whole right band. − − − − − − − P o w e r q S pe c t r a l q D en s i t y q[ d B ] Frequency fT PAOoperationOatO1dBOcompression10 dBOback-offInOband RightOband low-PAROprecodedOsingle-carrierMaximum-ratioOprecodedOOFDM
Fig. 3. The power spectral densities after amplification of two signal typeswith the PA operating at the 1 dB compression point (upper two curves) andwell below saturation (lower two curves). The signals are from the system laterdescribed in Table I, where the bandwidth of the pulse p ( τ ) is BT = 1 . . This signal is broadcast over the channel, whose small-scalefading impulse response from antenna m to user k is h km ( τ ) and large-scale fading coefficient to user k is β k . Specifically,user k receives the signal y k ( t ) = (cid:112) β k M (cid:88) m =1 (cid:0) h km ( τ ) (cid:63) x m ( τ ) (cid:1) ( t ) + z k ( t ) , (7)where z k ( t ) is a stationary white Gaussian stochastic processwith spectral height N that models the thermal noise of theuser equipment. The received signal is passed through a filtermatched to the pulse p ( τ ) and sampled to produce the discrete-time received signal y k [ n ] (cid:44) (cid:0) p ∗ ( − τ ) (cid:63) y k ( τ ) (cid:1) ( nT ) . (8)In analyzing this system, we will look into an equivalentdiscrete-time system, see Figure 2(b). In order to do that, thedistortion produced by the nonlinear amplifier has to be treatedseparately, since the nonlinearity widens the spectrum and isnot accurately described by symbol-rate sampling. The small-scale fading coefficients of the discrete-time impulse responseof the channel, including the pulse-shaping and matched filter,between antenna m and user k are denoted h km [ (cid:96) ] (cid:44) T (cid:0) p ( τ ) (cid:63) h km ( τ ) (cid:63) p ∗ ( − τ ) (cid:1) ( (cid:96)T ) . (9)For these channel coefficients, we assume that E [ h ∗ km [ (cid:96) ] h km [ (cid:96) (cid:48) ] ] = 0 , ∀ (cid:96) (cid:54) = (cid:96) (cid:48) , (10) L − (cid:88) (cid:96) =0 E (cid:2) | h km [ (cid:96) ] | (cid:3) = 1 , ∀ k, m, (11)and that h km [ (cid:96) ] is zero for integers (cid:96) / ∈ [0 , L − , where L isthe number of channel taps.The n -th received sample at user k is then given by y k [ n ] = (cid:112) P β k (cid:16) M (cid:88) m =1 L − (cid:88) (cid:96) =0 h km [ (cid:96) ] u m [ n − (cid:96) ] + d k [ n ] (cid:17) + z k [ n ] , (12)where the noise sample z k [ n ] (cid:44) (cid:0) p ∗ ( − τ ) (cid:63) z k ( τ ) (cid:1) ( nT ) . Tomake z k [ n ] ∼ CN (0 , N /T ) i.i.d., it is assumed that p ( τ ) isa root-Nyqvist pulse of period T and signal energy /T . Theterm d k [ n ] describes the in-band distortion —the part of the distortion that can be seen in the received samples y k [ n ] —caused by the nonlinear amplification of the transmit signal. Itis given by d k [ n ] (cid:44) √ P M (cid:88) m =1 (cid:16)(cid:0) x m ( t ) −√ P u m ( t ) (cid:1) (cid:63) h km ( τ ) (cid:63) p ∗ ( − τ ) (cid:17) ( nT ) . (13)By introducing the following vectors u [ n ] (cid:44) ( u [ n ] , . . . , u M [ n ]) T y [ n ] (cid:44) ( y [ n ] , . . . , y K [ n ]) T d [ n ] (cid:44) ( d [ n ] , . . . , d K [ n ]) T z [ n ] (cid:44) ( z [ n ] , . . . , z K [ n ]) T and matrices H [ (cid:96) ] (cid:44) h [ (cid:96) ] · · · h M [ (cid:96) ] ... . . . ... h K [ (cid:96) ] · · · h KM [ (cid:96) ] B (cid:44) diag( β , . . . , β K ) , (14)the received signals can be written as y [ n ] = √ P B (cid:16) L − (cid:88) (cid:96) =0 H [ (cid:96) ] u [ n − (cid:96) ] + d [ n ] (cid:17) + z [ n ] . (15)If the transmission were done in a block of N symbols peruser, and a cyclic prefix were used in front of the blocks, i.e. u [ n ] = u [ N + n ] , for n = − L, . . . , − , (16)where n = 0 is the time instant when the first symbol is receivedat the users, then the received signal in (15) is easily given inthe frequency domain. If the discrete Fourier transforms of thetransmit signals, received signals and channel are denoted by ˜u [ ν ] (cid:44) N − (cid:88) n =0 e − j πnν/N u [ n ] , (17) ˜y [ ν ] (cid:44) N − (cid:88) n =0 e − j πnν/N y [ n ] , (18) ˜H [ ν ] (cid:44) L − (cid:88) (cid:96) =0 e − j π(cid:96)ν/N H [ (cid:96) ] , (19)then the received signal at frequency index ν is given by ˜y [ ν ] = √ P B (cid:0) ˜H [ ν ] ˜u [ ν ] + ˜d [ ν ] (cid:1) + ˜z [ ν ] , (20)where ˜d [ ν ] describes the in-band distortion caused by thenonlinear amplification, ˜z [ ν ] ∼ CN (0 , N T I K ) and I K is the K -dimensional identity matrix. The frequency-domain notationin (17)–(19) will be useful when we later consider OFDM -basedtransmission methods.In this paper, we limit ourselves to look at block transmissionwith a cyclic prefix (16). To require a cyclic prefix simplifiesour exposition and does not limit its generality much. A prefixis present in almost all modern digital transmission schemes, asa guard interval or as a delimiter between blocks. A prefix thatis correlated with the symbols is arguably a waste of spectralresources. However, by letting N be much greater than L , thiswaste can be made arbitrarily small.III. D OWNLINK T RANSMISSION
In the downlink, a precoder chooses, based on the channelstate information available at the base station, transmit signals such that the users receive the symbols intended for them. Thesymbols to be transmitted fulfil E (cid:2) | s k [ n ] | (cid:3) = ξ k , ∀ n, k, (21)where { ξ k } are power allocation coefficients that are normalizedsuch that K (cid:88) k =1 ξ k = 1 . (22)We assume that the uplink and downlink are separated intime, using so called time-division duplexing, and that eachuser sends an N p -symbol long pilot sequence in the uplinkthat is orthogonal to the pilots of all other users. The pilotsare used by the base station to estimate the small-scale fadingcoefficients { h km [ (cid:96) ] } . The large scale fading coefficients { β k } are assumed to be known. Note that, to achieve orthogonalitybetween pilots, N p ≥ KL . Further, it is assumed that thechannel estimates ˆ h km [ (cid:96) ] = h km [ (cid:96) ] − (cid:15) km [ (cid:96) ] , ∀ k, m, (cid:96), (23)where (cid:15) km [ (cid:96) ] is the estimation error, are obtained through linearminimum-mean-square estimation, so that ˆ h km [ (cid:96) ] and (cid:15) km [ (cid:96) ] are uncorrelated. In analogy with (19), we will denote theFourier transforms of the channel estimates and the estimationerror { ˆ˜ h km [ ν ] } and { ˜ (cid:15) km [ ν ] } respectively. Their variances are δ k (cid:44) L − (cid:88) (cid:96) =0 E (cid:104) | ˆ h km [ (cid:96) ] | (cid:105) = E (cid:104) | ˆ˜ h km [ ν ] | (cid:105) , (24) E k (cid:44) L − (cid:88) (cid:96) =0 E (cid:2) | (cid:15) km [ (cid:96) ] | (cid:3) = E (cid:2) | ˜ (cid:15) km [ ν ] | (cid:3) . (25)Note that if { h km [ (cid:96) ] } are i.i.d. across k and m and if theuplink is perfectly linear, then δ k = N p ρ p β k N p ρ p β k , E k = 11 + N p ρ p β k , (26)where ρ p is the ratio between the power used to transmit thepilots and the thermal noise variance of a base station antenna. A. Achievable Data Rates
To treat single-carrier and
OFDM transmission together, let ¯ y k [ n ] (cid:44) (cid:40) y k [ n ] , if single-carrier transmission ˜ y k [ n ] , if OFDM transmission (27)be the n -th received sample at user k . A lower bound on thecapacity of the downlink channel to user k is given by [19] R k (cid:44) log (1 + SINR k ) , (28)where the signal-to-interference-and-noise ratio is given by SINR k = | E [ ¯ y ∗ k [ n ] s k [ n ] ] | /ξ k E [ | ¯ y k [ n ] | ] − | E [ ¯ y ∗ k [ n ] s k [ n ] ] | /ξ k (29)To evaluate (28), we consider the following signal: r k [ n ] = (cid:40)(cid:80) Mm =1 (cid:80) L − (cid:96) =0 ˆ h km [ (cid:96) ] u m [ n − (cid:96) ] , if SC (cid:80) Mm =1 ˆ˜ h km [ n ]˜ u m [ n ] , if OFDM . (30) and define the deterministic constant g k (cid:44) √ δ k ξ k E (cid:2) s ∗ k [ n ] r k [ n ] (cid:3) . (31)Here g k is normalized by √ δ k so that it does not depend onthe estimation error. This normalization will later allow us tosee the impact of the channel estimation error on the SINR .The fact that g k does not depend on δ k is seen by expanding g k as is done in (73) in the Appendix.Now the received signal can be written ¯ y k [ n ] = (cid:112) P β k (cid:0) g k (cid:112) δ k s k [ n ]+ i k [ n ]+ e k [ n ]+ ¯ d k [ n ] (cid:1) + ¯ z k [ n ] . (32)In this sum, the first term is equal to the signal of interest,scaled by g k √ δ k . The second term i k [ n ] (cid:44) r k [ n ] − g k (cid:112) δ k s k [ n ] (33)is a term comprising interference and downlink channel gainuncertainty (as in [5]). The third term in (32) e k [ n ] (cid:44) (cid:40)(cid:80) Mm =1 (cid:80) L − (cid:96) =0 (cid:15) km [ (cid:96) ] u m [ n − (cid:96) ] , if SC (cid:80) Mm =1 ˜ (cid:15) km [ n ]˜ u m [ n ] , if OFDM (34)is the error due to imperfect channel state knowledge at the basestation. The last two terms in (32) are the in-band distortion thatthe users see because of nonlinear amplification and thermalnoise respectively: ¯ d k [ n ] (cid:44) (cid:40) d k [ n ]˜ d k [ n ] ¯ z k [ n ] (cid:44) (cid:40) z k [ n ] , if SC ˜ z k [ n ] , if OFDM . (35)The interference i k [ n ] is uncorrelated with s k [ n ] in (32),because E [ s ∗ k [ n ] i k [ n ] ] = E (cid:2) s ∗ k [ n ]( r k [ n ] − g k δ k s k [ n ]) (cid:3) = E (cid:2) s ∗ k [ n ] r k [ n ] (cid:3) − g k δ k ξ k = 0 . (36)The in-band distortion ¯ d k [ n ] , on the other hand, is correlatedwith s k [ n ] and i k [ n ]+ e k [ n ] . These two correlations, we denote,similarly to (31), c k (cid:44) √ δ k ξ k E (cid:2) s ∗ k [ n ] ¯ d k [ n ] (cid:3) , (37) ρ k (cid:44) I k + E k E (cid:2) ( i ∗ k [ n ] + e ∗ k [ n ]) ¯ d k [ n ] (cid:3) , (38)where E k (cid:44) E (cid:2) | e k [ n ] | (cid:3) , (39) I k (cid:44) E (cid:2) | i k [ n ] | (cid:3) , (40)are the channel error and interference variances. We note that,when (26) holds, E k = E k . The in-band distortion can now bedivided into three parts: ¯ d k [ n ] = c k (cid:112) δ k s k [ n ] + ρ k ( i k [ n ] + e k [ n ]) + d (cid:48) k [ n ] , (41)The first part of the in-band distortion: d (cid:48) k [ n ] is uncorrelatedto s k [ n ] and i k [ n ] + e k [ n ] , for the same reason s k [ n ] and i k [ n ] are uncorrelated in (36). The factor c k is thus the amount ofamplitude that the nonlinear amplification “contributes” to theamplitude of the desired signal. Usually, in a real-world system, this is a negative contribution in the sense that | g k + c k | < | g k | .It should therefore be seen as the amount of amplitude lost (inwhat is usually called clipping ). Similarly, the other correlation ρ k is the amount of interference that is clipped by the nonlinearamplification. Finally, we denote the variance of the in-banddistortion D k (cid:44) E (cid:2) | d (cid:48) k [ n ] | (cid:3) . (42)With this new notation, the two expectations in (29) can bewritten as follows. | E [ ¯ y ∗ k [ n ] s k [ n ] ] | /ξ k = δ k ξ k P β k | g k + c k | (43) E (cid:2) | ¯ y k [ n ] | (cid:3) = P β k ( δ k ξ k | g k + c k | +( I k + E k ) | ρ k | + D k )+ N T (44)This simplifies (29), which becomes SINR k = δ k ξ k P β k | g k + c k | Pβ k (cid:0) ( I k + E k ) | ρ k | + D k (cid:1) + N T . (45)From (45), the two consequences of nonlinear amplificationcan be seen: (i) in-band distortion with variance D k and (ii)signal clipping by c k , a reduction of the signal amplitude thatresults in a power-loss.We also see that the variance δ k is the fraction between thepower that would have been received if the channel estimateswere perfect and the actually received power. It can thus beseen as a measure of how much power that is lost due toimperfect channel state information at the base station.The bound (28) is an achievable rate of a system thatuses a given precoder and where the detector uses (31) as achannel estimate and treats the error terms in (32) as additionaluncorrelated Gaussian noise. This detector has proven to beclose to the optimal detector in environments, where themassive MIMO channel hardens.
B. Linear Precoding Schemes
With knowledge of the channel, the base station can precode the symbols in such a way that the gain g k is big and theinterference I k small.
1) OFDM-Transmission: In OFDM transmission, the pre-coder is defined in the frequency domain. The time domaintransmit signals are obtained from the inverse Fourier transform u [ n ] (cid:44) N − (cid:88) ν =0 e j πnν/N ˜u [ ν ] (46)of the precoded signals ˜u [ ν ] = ˜W [ ν ] s [ ν ] , ν = 1 , . . . , N − (47)where ˜W [ ν ] is a precoding matrix for frequency ν . Theprecoding is linear, because the precoding matrix does notdepend on the symbols, only on the channel.To ensure that (1) is fulfilled, it is required that E (cid:104) (cid:107) ˜W [ ν ] (cid:107) F (cid:105) = K, ∀ ν. (48)
2) Single-Carrier Transmission:
The transmit signals ofsingle-carrier transmission are given by the cyclic convolution u [ n ] = N − (cid:88) (cid:96) =0 W [ (cid:96) ] s [ n − (cid:96) ] , (49)where the indices are taken modulo N . The impulse responseof the precoder is given in terms of its frequency domaincounterpart: W [ (cid:96) ] (cid:44) N − (cid:88) ν =0 e j πν(cid:96)/N ˜W [ ν ] . (50)
3) Conventional Precoders:
In this paper, three conventionalprecoders are studied. They will be given as functions of thechannel estimates { ˆH [ (cid:96) ] } (a sequence of matrices defined interms of { ˆ h km [ (cid:96) ] } in the same way as H [ (cid:96) ] is defined in (14)in terms of { h km [ (cid:96) ] } ) and its Fourier transform ˆ˜H [ ν ] (cid:44) N − (cid:88) (cid:96) =0 ˆH [ (cid:96) ] e − j πν(cid:96)/N . (51)The factors α x used in the definitions below are normalizationconstants, chosen such that (48) holds. a) Maximum-Ratio Precoding: Maximum-ratio precodingis the precoder that maximizes the gain g k and the receivedpower of the desired signal. It is given by ˜W [ ν ] = α MR ˆ˜H H [ ν ] , for MR . (52)While it maximizes the received power of the transmission,interference I k (cid:54) = 0 is still present in the received signal. Intypical scenarios with favorable propagation, maximum-ratioprecoding suppresses this interference increasingly well withhigher number of base station antennas and in the limit ofinfinitely many antennas, the interference becomes negligiblein comparison to the received power [2]. For maximum-ratioprecoding and an i.i.d. Rayleigh fading channel, both withsingle-carrier and OFDM transmission, the array gain andinterference terms are [5] g k = √ M , I k = δ k , for MR . Because the precoding weights of antenna m only depend onthe channel coefficients { ˆ h km [ (cid:96) ] } of that antenna, maximum-ratio precoding can be implemented in a distributed fashion,where the precoding is done locally at each antenna.Note that this definition of maximum-ratio precoding makesit equivalent to time-reversal precoding for single-carriertransmission, see for example [22]. b) Zero-Forcing Precoding: The zero-forcing precoder isgiven by ˜W [ ν ] = α ZF ˆ˜H H [ ν ] (cid:0) ˆ˜H [ ν ] ˆ˜H H [ ν ] (cid:1) − , for ZF . (53)It nulls the interference I k at the cost of a lower gain g k compared to maximum-ratio precoding. For zero-forcingprecoding and an i.i.d. Rayleigh fading channel, both withsingle-carrier and OFDM transmission, the gain and interferenceterms are [5] g k = √ M − K, I k = 0 , for ZF . c) Regularized Zero-Forcing Precoding: Regularized zero-forcing precoding aims at maximizing the received
SINR (45).In the limit of an infinite number of antennas, the optimallinear precoder is given by [23] ˜W [ ν ] = α RZF ˆ˜H H [ ν ] (cid:0) ˆ˜H [ ν ] ˆ˜H H [ ν ]+ ρ I K (cid:1) − , for RZF , (54)where ρ ∈ R + is a system parameter, which depends on theratio P T /N and on the path losses { β k } of the users. Theregularized zero-forcing precoder balances the interferencesuppression of zero-forcing and array gain of maximum-ratioprecoding [10] by changing the parameter ρ . How to findthe optimal parameter ρ is described in [23] and later inSection IV-B.The interference I k and gain g k of regularized zero-forcingdepend on the parameter ρ and no closed-form expressionfor them is known. However, when the transmit power P islow compared to the noise variance N /T , then a big ρ isoptimal and the interference and array gain are close to theones of maximum-ratio precoding. And when the transmitpower relative the noise variance is high, a small ρ is optimaland the interference and array gain are close to the ones ofzero-forcing. C. Discrete-Time Constant-Envelope Precoding
The low-
PAR precoding scheme originally proposed in [24]and extended in [16], [17], here called discrete-time constant-envelope precoding, is briefly described in two sections, firstfor single-carrier transmission, then for
OFDM .
1) Single-Carrier Transmission:
The discrete-time constant-envelope precoder produces transmit signals that have constant-envelope when viewed in discrete time, i.e. | u m [ n ] | = 1 √ M , ∀ n, m. (55)It does so by minimizing the difference between the receivednoise-free signal and the desired symbols under a fixed modulusconstraint, { u m [ n ] } =arg min {| u m [ n ] | = M − / } N − (cid:88) n =0 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L − (cid:88) (cid:96) =0 ˆH [ (cid:96) ] u [ n − (cid:96) ] − √ γ s [ n ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (56)where γ ∈ R + is a system parameter that is chosen to maximizethe system performance. Intuitively, a small γ makes theinterference I k small, but the gain g k small too. On the otherhand, a large γ makes the array gain big but also makes ithard to produce the desired symbol at each user, which resultsin an excessive amount of interference. In Section IV-B, it willbe shown how the parameter γ is chosen such that the datarate is maximized.The optimization problem in (56) can be approximatelysolved at a low computational complexity by cyclic optimiza-tion: minimizing the norm with respect to one u m [ n ] at a time,while keeping the other variables fixed. Such a solver is notmuch heavier in terms of computations than the zero-forcingprecoder [16].
2) OFDM Transmission: O FDM transmission in connectionwith discrete-time constant-envelope precoding can be done byusing the same algorithm as for the single-carrier transmission. Instead of precoding the symbols { s [ n ] } directly, the basestation would precode their inverse Fourier transform ˜s [ n ] (cid:44) √ N N − (cid:88) ν =0 e j πνn/N s [ ν ] . (57)The convolution in (56) should then be seen as a cyclic, i.e.the indices should be taken modulo N . D. Power Allocation among Users
The power allocation { ξ k } between users has to be decidedaccording to a chosen criterion, for example that all users shallbe served with the same data rate. This “egalitarian” criterionis used in this paper and is given by the max-min problem: { ξ k } = arg max { ξ k } : eq.(22) min k SINR k , (58)where SINR k is given in (45). Note that out of all the termsin (45), apart from ξ k itself, only the clipping c k , in-banddistortion D k and the correlation ρ k might depend on ξ k . That g k and I k do not depend on { ξ k } , can be seen from (75) and(80) in the Appendix. Extensive simulations over Rayleighfading channels indicate that only D k depends on the powerallocation ξ k and that this dependence is linear.To solve (58), a first-order approximation of the dependenceon ξ k is made. The in-band distortion is assumed to be: D k = D (cid:48) + δ k ξ k D (cid:48)(cid:48) , (59)where D (cid:48) and D (cid:48)(cid:48) are non-negative constants.For the { ξ k } that solve (58), there is a common SINR suchthat
SINR k = SINR , for all k , because (45) is an increasingfunction in ξ k . Rearranging (45) gives the power allocation ξ k = SINR
P β k (( I k + E k ) | ρ k | + D (cid:48) ) + N T δ k P β k ( | g k + c k | − D (cid:48)(cid:48) SINR ) . (60)Because the power allocations sum to one (22), SINR K (cid:88) k =1 P β k (( I k + E k ) | ρ k | + D (cid:48) ) + N T δ k P β k ( | g k + c k | − D (cid:48)(cid:48) SINR ) = 1 , (61)the common SINR can be found by solving this equation. Theoptimal power allocations are thus given by (60), where
SINR is the largest solution to (61).Note that, if D (cid:48)(cid:48) = 0 , (61) can be solved explicitly, whichgives an expression for SINR and the optimal power allocation ξ k = δ k β k | g k + c k | ( P β k (( I k + E k ) | ρ k | + D (cid:48) ) + N /T ) (cid:80) Kk (cid:48) =1 P β k (cid:48) (( I k (cid:48) + E k (cid:48) ) | ρ k (cid:48) | + D (cid:48) )+ N /Tδ k (cid:48) β k (cid:48) | g k (cid:48) + c k (cid:48) | . (62)Specifically for maximum-ratio precoding when (26) holds,this power allocation becomes ξ MR k = P β k ( | ρ k | + D (cid:48) ) + N /Tβ k δ k (cid:80) Kk (cid:48) =1 P β k (cid:48) ( | ρ k (cid:48) | + D (cid:48) )+ N /Tβ k (cid:48) δ k (cid:48) , ∀ k, (63)and, for zero-forcing precoding, it becomes ξ ZF k = P β k ( E k | ρ k | + D (cid:48) ) + N /Tβ k δ k (cid:80) Kk (cid:48) =1 P β k (cid:48) ( E k (cid:48) | ρ k (cid:48) | + D (cid:48) )+ N /Tβ k (cid:48) δ k (cid:48) , ∀ k. (64) − −
10 0 10 20 − − − − − − − −
200 Sample Index ℓ T ap E ne r g y E [| w m k [ ℓ ]| ] [ d B ] M = M = M = M = M = M = Fig. 4. The normalized energy of the taps of the impulse response of thezero-forcing filter for single-carrier transmission over a frequency-selective4-tap channel. The base station serves K = 10 users. These two expressions (63) and (64) are equivalent to thecorresponding formulas in [25] in the special case there is noamplifier distortion. It should be noted that when the numberof users is large, K (cid:38) , the term D (cid:48)(cid:48) is close to zero. E. Single-Carrier vs. OFDM Transmission
In terms of the achievable data rate (28), which has beenproven tight when the channel hardens, single-carrier and
OFDM transmission are equivalent in massive
MIMO . Due to channelhardening, all tones of the
OFDM transmission have equallygood channels { ˜H [ n ] } , therefore the advantage of OFDM —thepossibility to do waterfilling across frequency—results in littlegain. This is summarized in the following Proposition andproven in the Appendix.
Proposition 1:
If the same precoding scheme f : ˆ˜H [ n ] (cid:55)→ ˜W [ n ] is used for all tones n , the rate in (28) is equal forsingle-carrier transmission (49) and for OFDM (46).With regards to implementation, the two transmission meth-ods differ. While
OFDM requires a Fourier transform to be doneby the users, single-carrier transmission does not. While
OFDM causes a delay of at least N symbols, since precoding anddetection are done block by block, single-carrier transmissioncan be implemented for frequency-selective channels withshort filters with much smaller delay. Channel inversion withfilters with few taps is only possible in massive MIMO —in
SISO systems or small
MIMO systems, pre-equalization of afrequency-selective channel requires filters with a huge numberof taps. This can be seen in Figure 4, which shows the powerprofile { E (cid:2) | w mk [ (cid:96) ] | (cid:3) , (cid:96) = . . . , − , , , . . . } of the impulseresponse of the zero-forcing precoder for different numbersof base station antennas. With few antennas, zero-forcingrequires many filter taps, while, with massive MIMO , it requiresapproximately the same number of filter taps as the numberof channel taps.Since the symbol period of
OFDM is longer than that ofsingle-carrier transmission,
N T compared to T , OFDM is lesssensitive to synchronization errors in the sampling in (8).While a small time synchronization error, in the order of T ,leads to a simple phase rotation in OFDM , it would lead todifficult intersymbol interference in single-carrier transmission.For small frequency synchronization errors, in the order NT ,however, OFDM suffers from intersymbol interference, while
TABLE IS
YSTEM P ARAMETERS number of tx-antennas M = 100 number of users K = 10 and channel model L = 4 -tap Rayleigh fadingpulse shape filter root-raised cosine, roll-off 0.22amplifier type class B, see (69) and (66)path loss exponent α = 3.8 (typical urban scenario)single-carrier transmission only experiences a simple phaserotation.In some implementational aspects, the two transmissionmethods are similar though. The computational complexitiesare roughly the same—the Fourier transform that single-carriertransmission spares the users from, has to be done by thebase station instead, see (50). We stress that the signals ofsingle-carrier and OFDM transmission practically have the same
PAR in massive
MIMO , at least in an i.i.d. Rayleigh fadingenvironment. In Figure 1, there was only a small gap betweenthe
PAR of single-carrier and
OFDM transmission and, whenthe number of channel taps is greater than
L > , the gap ispractically closed.The operational differences between single-carrier and OFDM transmission are summarized in Table II.IV. N
UMERICAL E VALUATIONS OF R ATE
To estimate the power consumed by the amplifiers at differentsum-rates and to compare different precoders to each other, theexpectations in (28) that lack closed-form expressions werenumerically evaluated for the system specified in Table I. Allcontinuous signals were modelled by κ = 7 times oversampleddiscrete-time signals. Specifically the channel from antenna m to user k was assumed to be Rayleigh fading h km ( (cid:96)T /κ ) ∼CN (0 , / ( κL )) , for (cid:96) = 0 , . . . , κ ( L − and i.i.d. across k , m and (cid:96) . The users were assumed to be uniformly spread outover an annulus-shaped area, with inner radius and outerradius . The path loss of user k was then assumed to be β k = ( / k ) α , (65)where k is the distance between user k and the base station,which is located in the middle of the annulus, and where α isthe path loss exponent.Further, it was assumed that the pilots used for channelestimation were N p = KL symbols long and sent with thesame power ρ p from all users. The power was chosen suchthat a signal sent from the cell edge, where the path loss is β min = 1 / α , would be received at the base station withpower 0 dB above the noise variance, i.e. ρ p β min = N /T and ρ p = 100 α N /T . A. Effects of Nonlinear Power Amplifiers
The power amplifiers of the simulated system have beenmodeled by the Rapp model [15], where the phase distortionis neglected, so Φ( u ) = 0 , ∀ u , and the AM - AM conversion is Power1Efficiency1 η [a] C li pp i ng P o w e r- Lo ss c k d B ] -0.4-0.3-0.2-0.10 DTCE1PrecodingMaximum-Ratio1PrecodingZero-Forcing1Precoding (a) The fraction of power lost due to clipping
Power5Efficiency5 η [%]0 20 40 60 80 PA n - B and D i s t o r t i on D k [ d B ] -80-60-40-20 205dB175dB145dB105dB75dB5.25dB35dB05dB-1.8dB (b) The variance of the received in-band distortion Power1Efficiency1 η [%] A C L R d B ] -90-80-70-60-50-40-30-20-10 (c) The amount of out-of-band radiationFig. 5. Measurements on a Rapp-modeled ( p = 2 ) class B amplifier for three signal types. The signals have been pulse shape filtered with a root-raisedcosine, roll-off factor 0.22. The encircled points correspond to some selected operating points of the amplifier, which are specified by the backoff from the1-dB compression point. It is assumed that all users are at the same distance to the base station, i.e. ξ k = 1 /K , for all k . given by g ( u ) = A max u/u max (1 + ( u/u max ) p ) p , (66)where the parameter p = 2 approximates a typical moderate-cost solid-state power amplifier [26]. The parameter A max isthe highest possible output amplitude and u max = A max /g (cid:48) (0) determines the slope of the asymptote that g ( u ) approachesfor small u .To ensure that the total radiated power is P , as is requiredby (4), the parameters are chosen as follows: u max = M − / (67) A max = u max √ Pλ , (68)where λ is a correction factor to compensate for the powerlost due to clipping, which is chosen such that (4) holds. Notethat the correction factor is different for different signal typesand backoffs.Massive MIMO will require simple, inexpensive and powerefficient amplifiers [3]. The most basic class B amplifiers havethese properties [27], and could therefore potentially be suitedfor massive MIMO . The power efficiency of such an amplifieris given by [15] η = π E (cid:2) g ( | u m ( t ) | ) (cid:3) A max E [ g ( | u m ( t ) | ) ] , (69)Note that η ≤ π/ , with equality only if the continuous-timeinput signal u m ( t ) has perfectly constant envelope and theamplifier operates in saturation.The two phenomena of nonlinear amplification—in-banddistortion and amplitude clipping—are studied by looking atthe variance of the in-band distortion σ k (cid:44) D k ξ k | g k | and theclipped power a k (cid:44) | g k + c k | | g k | relative to the ideal amplitude.The clipped power was computed together with the powerefficiency of the amplifiers for several backoffs and averagedover many channel realizations for the system specified inTable I, in which all users are at the same distance to the basestation and ξ k = 1 /K , for all k . By treating the backoff asan intermediate variable, the clipped power can be given as a function of the efficiency, see Figure 5(a). It is noted thatthe power lost due to clipping is small (smaller than −0.4 dB)even when the amplifiers are operated close to saturation.Similarly, the variance of the in-band distortion, Figure 5(b),and the ACLR , Figure 5(c), were computed for several backoffsand averaged over different channel realizations. It can be seenthat the amount of energy radiated out-of-band is monotonicallydecreasing with the backoff. A constraint on the
ACLR willtherefore constrain the maximum efficiency that the amplifiercan operate at. Further, it is noted that the efficiency is not asimple function of the backoff, but it depends on the signal type.We also note that, whereas the clipping power-loss is smallat operating points with high efficiency, the in-band distortion(at least for the conventional precoders) and the out-of-bandradiation are not. The latter two phenomena will thus be themain factors to determine the operating point of the amplifiers.Because of their similar amplitude distributions, all the linearprecoding schemes ( MR , ZF , RZF precoding) result in similarcurves in Figures 5(a), 5(b) and 5(c). Therefore, only the resultsof single-carrier maximum-ratio and zero-forcing precodingare shown. The curves are identical to the ones of
OFDM transmission.In Figure 6, it can be seen how the effects of the nonlinearamplifiers change when the number of antennas, users andchannel taps are changed in a single-carrier
MIMO system. Tomake comparisons easy, all users in the system of Figure 6have the same path loss and the amplifiers are backed off by1 dB, enough to see distinct clusters around each symbol pointin all cases.When the number of users and channel taps are small,the distribution of the in-band distortion is different arounddifferent symbol points and the phase tends to be moreaccurately reproduced than the amplitude, resulting in oblongclouds around the outer symbol points. It is a well-knownphenomenon in
OFDM with a great number of subcarriers thatthe distribution of the in-band distortion is almost circularlysymmetric Gaussian and identically distributed around eachsymbol point, which means that the distortion can be regardedas uncorrelated additive noise [20]. In multiuser
MIMO , a similareffect is observed—when the number of users and channel taps -2 -1 0 1 2-2-1012 M = 4, K = 1, L = 4 σ = -18 dB, a = -0.2 dB -2 -1 0 1 2-2-1012 M = 4, K = 1, L = 15 σ = -18 dB, a = -0.2 dB-10 -5 0 5 10 Q uad r a t u r e A m p li t ude M = 100, K = 1, L = 4 σ = -23 dB, a = -0.2 dB -10 -5 0 5 10-10-50510 M = 100, K = 1, L = 15 σ = -27 dB, a = -0.2 dB-3 -2 -1 0 1 2 3-3-2-10123 M = 100, K = 10, L = 4 σ = -24 dB, a = -0.2 dB Inphase Amplitude -2 0 2-3-2-10123 -3 -1 1 3 M = 100, K = 10, L = 15 σ = -23 dB, a = -0.2 dB k k kkk k k kk k k k Fig. 6. Received signal points without thermal noise when broadcasting16-
QAM symbols with nonlinear amplification (1 dB backoff) over a
MIMO channel by single-carrier zero-forcing precoding for different number of users K , antennas M and channel taps L . are great, the noise is almost circularly symmetric Gaussianand identically distributed around each symbol point for single-carrier transmission too. This is intuitive, since the precodedtransmit signals are the sums of many independent symbolsand the receive signals are the sums of many different transmitsignals. The in-band distortion therefore gets mixed up andits distribution becomes symmetric and identical around allsymbol points, just as is the case in OFDM .In Figure 6, the variance of the in-band distortion seemsto roughly follow the scaling law predicted by [28], whichsays that the distortion variance relative to the signal powerscales as O ( (cid:112) K/M ) . Although the in-band distortion seemsto disappear with increasing number of antennas, the amplitudereduction due to clipping does not (in Figure 6, it remainsat −0.2 dB for all system setups), which was not observed in[28]. However, it can be argued that the clipping power-loss issmall and only needs to be considered when the amplifiers areoperated close to saturation. B. Data Rate and Power Consumption
In this section the power P cons that the base station amplifiersconsume is estimated. Even if the discrete-time constant-envelope precoder outperformed the other precoders in thecomparisons in Figure 5, in the end, it is in terms of consumedpower the precoders should be compared.The rate R k ( P, θ ) in (28) is a function of the transmit power P = ηP cons , and therefore a function of the operating point ofthe power amplifiers, which are parametrized by their efficiency η . In the case of discrete-time constant-envelope precoding, therate is also a function of the parameter θ = γ . And in the caseof regularized zero-forcing, it is a function of the parameter θ = ρ . For a given out-of-band radiation requirement, specifiedby a maximum ACLR level
ACLR max , the sum-rate of the systemis thus given by R ( P cons ) = max η,θ K (cid:88) k =1 R k ( ηP cons , θ ) , (70)where the maximization is over all θ ∈ R + and over alloperating points η ∈ [0 , η max ] , where η max is the highestoperating point that still results in an ACLR below
ACLR max . Ifthe
ACLR is not constrained, η max is taken to be the maximumpossible efficiency of the given amplifiers and signal type.The relation between consumed power and the average sum-rate of the system that is shown in Figure 7 has been obtainedby computing (70) for many different user distributions { β k } and taking an average. Both the cases (i) when the out-of-bandradiation is constrained by requiring the ACLR to be below−45 dB, which is the
ACLR requirement in
LTE [29], and (ii)when it is not constrained are considered.It can be seen that maximum-ratio precoding works well forlow rate requirements but is limited by interference to belowa certain maximum rate. Because the
SINR in (45) containsdistortion that scales with the radiated power, all precoders havea vertical asymptote, above which the rate cannot be increased.Except for discrete-time constant-envelope precoding, whosecurve starts to bend away upwards in the plot for 50 users,this vertical asymptote lies outside the scale and cannot beseen for the other precoders however. Since the array gain | g k | is smaller for discrete-time constant-envelope precodingthan for zero-forcing and regularized zero-forcing, its verticalasymptote is located at a lower rate than the asymptote ofzero-forcing and regularized zero-forcing precoding.Further, it can be seen that regularized zero-forcing andzero-forcing perform equally well when the number of usersis small. Regularized zero-forcing has an advantage over zero-forcing when the number of users is big though, because ofits ability to balance the resulting array gain and the amountof interuser interference received by the users.The low- PAR precoding scheme, discrete-time constant-envelope precoding, seems to consume roughly the sameamount of power as the conventional precoding schemes, bothwhen the
ACLR is constrained and when it is not, at leastfor low to medium rate requirements. At very high rates, theoptimal linear precoder has an advantage over discrete-timeconstant-envelope precoding—because the vertical asymptoteof discrete-time constant-envelope precoding is at a lower rate Ergodic Sum-Rate [bpcu]0 20 40 60 80 C on s u m ed P o w e r P c on s / N [ d B ] m a x A C L R = − d B no ACLR constraint K = 10 Ergodic Sum-Rate [bpcu]0 50 100 150 200 250 C on s u m ed P o w e r P c on s / N [ d B ] Zero-forcing precodingMaximum-ratio precodingRegularized zero-forcingDTCE precoding no ACLR constraint m a x A C L R = − d B K = 50 Fig. 7. The estimated consumed power of a base station with M = 100 antennas required to serve K = 10 (above) and K = 50 (below) users witha certain rate over a frequency-selective channel with L = 4 taps with andwithout a constraint on the ACLR . Ergodic Sum-Rate [bpcu]0 20 40 60 80 PA E ff i c i en cy η [ % ] no ACLR constraintmax ACLR = − K = 10 no ACLR constraint K = 50 max ACLR = − Fig. 8. The power efficiency of the amplifiers at the optimal operating pointfor different sum-rate requirements. The legend in Figure 7 also applies here. than it is for the optimal linear precoder.The value of η that corresponds to the optimal operatingpoint of the amplifiers is shown in Figure 8. When there is noconstraint on the out-of-band radiation, it is optimal to operatethe amplifiers in saturation, where the power efficiency is high,for low rate requirements. For higher rate requirements, theamplifiers should be backed off to lower the in-band distortionfor the conventional precoding schemes. The amplifiers of thelow- PAR precoding scheme, however, continue to operate closeto saturation also at high rates. When the
ACLR is constrained tobelow −45 dB, the optimal efficiency of the amplifiers coincideswith η max (the highest permissible operating point), i.e. 34 %for discrete-time constant-envelope precoding and 27 % formaximum-ratio and zero-forcing precoding, over the wholerange of rates investigated, both when serving 10 and 50 users.This corresponds to a backoff of 8 dB and 11 dB respectively. V. C ONCLUSIONS
We have compared four different multiuser
MIMO precoders:maximum-ratio, zero-forcing, regularized zero-forcing anddiscrete-time constant-envelope precoding. They can be usedfor single-carrier transmission and for
OFDM transmission. Thedifferent precoders and transmission methods are summarizedin Table II.In massive
MIMO , there is little operational differencebetween
OFDM and single-carrier transmission in terms ofcomplexity, and in terms of
PAR . It also turns out that single-carrier and
OFDM transmission have the same performancein terms of data rate. Additionally, massive
MIMO allowsfor time-domain channel inversion to be done with a shortfilter, for which the number of taps is of the same order ofmagnitude as the number of channel taps. This makes single-carrier transmission easy to implement and allows for runningprecoding with little delays. Since
OFDM requires the users tobe equipped with an additional
FFT , this would suggest thatsingle-carrier transmission should be considered in massive
MIMO systems.A massive
MIMO system with centralized baseband process-ing also allows for low-
PAR precoding, which increases thepower efficiency of the amplifiers but requires more radiatedpower to compensate for the lowered array gain comparedto conventional precoders. For the simplistic power amplifiermodel used, simulations have shown that the amplifiers of thebase station consume the same amount of power when usinglow-
PAR precoding as when using the optimal conventionalprecoder. Since low-
PAR transmit signals are more hardware-friendly and could enable cheaper and simpler base stationdesigns, this would suggest that a low-
PAR precoding schemethat also pre-equalizes the channel and suppresses interference(such as discrete-time constant-envelope precoding) should beused in massive
MIMO base stations with centralized basebandprocessing.Furthermore, in massive
MIMO , simulations have shown thatthe power efficiency of the amplifiers at the optimal operatingpoint is limited by the out-of-band radiation requirement andthat in-band distortion caused by nonlinear amplifiers has twoparts: one clipping part that decreases the amplitude of thereceived signal and one part that can be seen as additiveuncorrelated noise, which drowns in the thermal noise inrepresentative cases. The amplitude lost due to clipping is smalleven when the amplifiers are operated close to saturation.A
PPENDIX P ROOF OF P ROPOSITION
OFDM transmission, bothwith a cyclic prefix added in front of each transmission block,result in the same achievable rate (28). To do that, the effectsof the amplifiers are neglected. However, it is reasonable toassume that the in-band distortion caused by the amplifiersaffects the two transmission schemes in the same way given thatthe amplitude distributions and
PAR of the two transmissionschemes are the same. The data rate of single-carrier and
OFDM are the same if the array gains | g SC k | = | g OFDM k | andinterference variances I SC k = I OFDM k are the same. T A B LE II P R E C OD I NG S C H E M E S AND T R AN S M I SS I ON M ET HOD SF O R M A SS I V E M I M O S i ng l e - C a rr i e r O F D M g e n e r a l S i m p l e m a t c h e d - fi lt e r d e t ec ti on R unn i ngp r ec od i ng a ndd e t ec ti on , littl e d e l a y S e n s iti v e t o ti m e s yn c h r on i za ti on e rr o r s M a t c h e d - fi lt e r d e t ec ti on i n fr e qu e n c ydo m a i n (r e qu i r e s FF T a tt h e u s e r s ) B l o c k - w i s e p r ec od i ng a ndd e t ec ti on S e n s iti v e t o fr e qu e n c y s yn c h r on i za ti on e rr o r s LinearPrecoders–high
PAR M R M a x i m i ze s a rr a yg a i n E n a b l e s l o ca l p r ec od i ng ; d i s t r i bu t e d a rr a y s L o w c o m p l e x it y P e rf o r m s w e ll a tl o w d a t a r a t e s H a s r e s i du a li n t e rf e r e n ce L - t a p F I R fi lt e r S I N R a s i n ( ) , w it h | g k | = M , I k = δ k H a s r e s i du a li n t e rf e r e n ce S I N R a s i n ( ) , w it h | g k | = M , I k = δ k Z F N u ll s i n t e r u s e r a nd i n t e r s y m bo l i n t e rf e r e n ce R e qu i r e s i nv e r s i ono f K × K - m a t r i ce s ∼ L - t a p F I R fi lt e r S I N R a s i n ( ) , w it h | g k | = M − K , I k = S I N R a s i n ( ) , w it h | g k | = M − K , I k = R - Z F L i n ea r p r ec od e r t h a t m a x i m i ze s S I N R O p ti m i za ti onov e r p a r a m e t e r ρ n ee d e d P e rf o r m a n ce - w i s e s i m il a r t o Z F f o r s m a ll K , bu t b e tt e rf o r b i g K R e qu i r e s i nv e r s i ono f K × K - m a t r i ce s ∼ L - t a p F I R fi lt e r S I N R a s i n ( ) , w it h | g k | ∈ [ M − K , M ] , I k ∈ [ , δ k ] S I N R a s i n ( ) , w it h | g k | ∈ [ M − K , M ] , I k ∈ [ , δ k ] D T C E L o w P A R N on li n ea r p r ec od e r O p ti m i za ti onov e r p a r a m e t e r γ n ee d e d F ilt e r w it hd e l a y ∼ L e qu i v a l e n t w h e n w a t e r fi lli ng a nd j o i n t s e qu e n ce d e t ec ti on i s no t don e . We start by expanding the array gain (31) for single-carriertransmission: | g SC k | = 1 ξ k δ k (cid:12)(cid:12)(cid:12) E (cid:104) s ∗ k [ n ] (cid:88) m (cid:88) (cid:96) ˆ h km [ (cid:96) ] u m [ n − (cid:96) ] (cid:105)(cid:12)(cid:12)(cid:12) (71) = 1 ξ k δ k (cid:12)(cid:12)(cid:12) E (cid:104) s ∗ k [ n ] (cid:88) m,(cid:96) ˆ h km [ (cid:96) ] (cid:88) k (cid:48) ,(cid:96) (cid:48) w mk [ (cid:96) (cid:48) ] s k (cid:48) [ n − (cid:96) − (cid:96) (cid:48) ] (cid:105)(cid:12)(cid:12)(cid:12) (72)Since different symbols are uncorrelated and since they haveenergy ξ k , only terms for which k (cid:48) = k and (cid:96) = − (cid:96) (cid:48) willremain, so | g SC k | = 1 δ k (cid:12)(cid:12)(cid:12) E (cid:104) (cid:88) m (cid:88) (cid:96) ˆ h km [ (cid:96) ] w mk [ − (cid:96) ] (cid:105)(cid:12)(cid:12)(cid:12) . (73)This is a cyclic convolution evaluated in , which can becomputed in the frequency domain followed by an inversetransform: | g SC k | = 1 δ k (cid:12)(cid:12)(cid:12) (cid:88) m N (cid:88) n E (cid:104) ˆ˜ h km [ n ] ˜ w mk [ n ] (cid:105)(cid:12)(cid:12)(cid:12) . (74)If the same precoding scheme f has been used for allfrequencies, i.e. f : ˆ˜H [ n ] (cid:55)→ ˜W [ n ] , ∀ n , then all the termsin the inner sum are the same: | g SC k | = 1 δ k (cid:12)(cid:12)(cid:12) (cid:88) m E (cid:104) ˆ˜ h km [0] ˜ w mk [0] (cid:105)(cid:12)(cid:12)(cid:12) , (75)which is also the array gain | g OFDM k | of OFDM .We now study the interference (40) for single-carrier trans-mission, by using (73): I SC k = E (cid:104)(cid:12)(cid:12)(cid:12) (cid:88) m (cid:88) (cid:96) ˆ h km [ (cid:96) ] u m [ n − (cid:96) ] − g k s k [ n ] (cid:12)(cid:12)(cid:12) (cid:105) (76) = E (cid:20)(cid:12)(cid:12)(cid:12) (cid:88) m (cid:88) (cid:96) ˆ h km [ (cid:96) ] (cid:88) k (cid:48) (cid:88) (cid:96) (cid:48) w mk (cid:48) [ (cid:96) (cid:48) ] s k (cid:48) [ n − (cid:96) − (cid:96) (cid:48) ] − s k [ n ] E (cid:104) (cid:88) m (cid:88) (cid:96) ˆ h km [ (cid:96) ] w mk [ − (cid:96) ] (cid:105)(cid:12)(cid:12)(cid:12) (cid:21) (77) = E (cid:20)(cid:12)(cid:12)(cid:12) s k [ n ] (cid:16)(cid:88) m,(cid:96) ˆ h km [ (cid:96) ] w mk [ − (cid:96) ] − E (cid:104)(cid:88) m,(cid:96) ˆ h km [ (cid:96) ] w mk [ − (cid:96) ] (cid:105)(cid:17) + (cid:88) (cid:88) ( k (cid:48) ,n (cid:48) ) (cid:54) =( k, s k (cid:48) [ n (cid:48) ] (cid:88) m (cid:88) (cid:96) ˆ h km [ (cid:96) ] w mk (cid:48) [ n (cid:48) − (cid:96) ] (cid:12)(cid:12)(cid:12) (cid:21) (78)Since different symbols are uncorrelated and since they haveenergy ξ k , the square is expanded into the following. I SC k = (cid:88) k (cid:48) (cid:88) n (cid:48) ξ k (cid:48) E (cid:104)(cid:12)(cid:12)(cid:12) (cid:88) m (cid:88) (cid:96) ˆ h km [ (cid:96) ] w mk (cid:48) [ n (cid:48) − (cid:96) ] (cid:12)(cid:12)(cid:12) (cid:105) − ξ k (cid:12)(cid:12)(cid:12) E (cid:104) (cid:88) m (cid:88) (cid:96) ˆ h km [ (cid:96) ] w mk [ − (cid:96) ] (cid:105)(cid:12)(cid:12)(cid:12) (79)The two terms are cyclic convolutions in n (cid:48) and respectivelyand can be computed in the frequency domain. Along the sameline of reasoning as in (75), the interference variance is givenby I SC k = (cid:88) k (cid:48) ξ k (cid:48) E (cid:104)(cid:12)(cid:12)(cid:88) m ˆ˜ h km [0] ˜ w mk (cid:48) [0] (cid:12)(cid:12) (cid:105) − ξ k (cid:12)(cid:12)(cid:12) E (cid:104)(cid:88) m ˆ˜ h km [0] ˜ w mk [0] (cid:105)(cid:12)(cid:12)(cid:12) , (80) which is precisely the interference variance I OFDM k of OFDM at tone , or at any other tone.That the rate (28) is the same for single-carrier and OFDM transmission was proven, in a different way, for the specialcase maximum-ratio precoding in [22].R
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