Capacity Limits and Multiplexing Gains of MIMO Channels with Transceiver Impairments

0.48 MB·English

by  Emil Björnson

IEEECOMMUNICATIONSLETTERS 1
Capacity Limits and Multiplexing Gains of
MIMO Channels with Transceiver Impairments
Emil Bjo¨rnson, Per Zetterberg, Mats Bengtsson, and Bjo¨rn Ottersten
Transmitter-Distortion Noise
Abstract—The capacity of ideal MIMO channels has an high- η n
t MIMO Channel
SNR slope that equals the minimum of the number of transmit
and receive antennas. This letter shows that physical MIMO Intended x √PH y Received
Signal Signal
channelsbehavefundamentallydifferent,duetodistortionsfrom
transceiver impairments. The capacity has a finite upper limit Fig. 1. Block diagram of the generalized MIMO channel considered in
2 that holds for any channel distribution and SNR. The high-SNR thisletter.Unlinktheclassicalchannelmodelin[1],thetransmitter-distortion
1 slope is thus zero, but the relative capacity gain of employing generatedbyphysicaltransceiverimplementationsisincludedinthemodel.
0 multiple antennas is at least as large as for ideal transceivers.
2 II. GENERALIZEDCHANNELMODEL
Index Terms—Channel capacity, high-SNR analysis, multi-
p antenna communication, transceiver impairments. Consider a flat-fading MIMO channel with Nt transmit an-
e tennas and Nr receive antennas.The received signal y CNr
S ∈
I. INTRODUCTION in the classical affine baseband channel model of [1] is

8 In the past decade, a vast number of papers have stud- y=√PHx+n, (1)
1
ied multiple-input multiple-output (MIMO) communications where P is the SNR, x CNt is the intended signal,
] motivated by the impressive capacity scaling in the high- and n (0,I) is circul∈ar-symmetric complex Gaussian
T SNR regime. The seminal article [1] by E. Telatar shows noise. T∼heCNchannel matrix H CNr×Nt is assumed to be
.I that the MIMO capacity with channel knowledge at the a random variable H having any∈multi-variate distribution fH
s
[c arencdeiNverrabreehtahveetsotaaslmnuimn(bNert,oNfrtr)alnosgm2(itPa)n+d rOec(e1i)v,ewanhteernenNast, wreiatlhiztahteionnosrm(ia.eli.,zerdangkai(nHE){=tr(HmiHnH(N)t},N=rN))tNalrmaonsdt fsuulrle-lrya—nk
1 respectively, and P is the signal-to-noise ratio (SNR). The this basically covers all physical channel distributions.
v factormin(Nt,Nr)istheasymptoticgainoversingle-antenna The intended signal in (1) is only affected by a multi-
3 channelsandiscalleddegreesoffreedomormultiplexinggain. plicative channel transformation and additive thermal noise,
9 Some skepticism concerning the applicability of these re- thusidealtransceiverhardwareisimplicitlyassumed.Physical
0 sults in cellular networks has recently appeared; modest gains
transceivers suffer from a variety of impairments that are
4
of network MIMO over conventional schemes have been
not properly described by (1) [5]–[10]. The influence of
.
9 observed and the throughput might even decrease due to the
impairments is reduced by compensation schemes, leaving a
0 extra overhead [2], [3]. One explanation is the finite channel
residual distortion that is well-modeled as additive Gaussian
2
coherencetimethatlimitstheresourcesforchannelacquisition
1 noise on the baseband with a variance that scales with P [7].
[4] and coordination between nodes [3], thus creating a finite
: Building on the analysis and measurements in [6]–[8], the
v fundamental ceiling for the network spectral efficiency—
combinedinfluenceofimpairmentsinthetransmitterhardware
i
X irrespectively of the power and the number of antennas. is modeled by the generalized MIMO channel
While these results concern large network MIMO systems,
ar there is another non-ideality that also affects performance and y=√PH(x+ηt)+n, (2)
manifestsitselfforMIMOsystemsofanysize:transceiverim- wherethetransmitter-distortionη CNt describesthe(resid-
t ∈
pairments[5]–[10].Physicalradio-frequency(RF)transceivers ual) impairments. This term describes the mismatch between
suffer from amplifier non-linearities, IQ-imbalance, phase the intended signal x and the signal actually generated by the
noise, quantization noise, carrier-frequency and sampling-rate transmitter; see the block diagram in Fig. 1.
jitter/offsets, etc. These impairments are conventionally over- Under the normalized power constraint1 tr(Q) = 1 with
looked in information theoretic studies, but this letter shows Q=E xxH (similar to [1]), the transmitter-distortion is
{ }
thattheyhaveanon-negligibleandfundamentalimpactonthe (cid:0) (cid:1)
η 0,Υ (Q) with Υ =diag(υ (Q),...,υ (Q)).
spectral efficiency in modern deployments with high SNR. t ∼CN t t 1 Nt
This letter analyzes the capacity of a generalized MIMO The distortion depends on the intended signal x in the sense
channel with transceiver impairments. We generalize results thatthevarianceυ (Q)isanincreasingfunctionofthesignal
n
from [5] and [7] and emphasize the high-SNR behavior, since power q at the nth transmit antenna (i.e., the nth diago-
n
the capacity has a finite limit and thus is fundamentally nal element of Q). We neglect any cross-correlation in Υ
t
different from the ideal case in [1]. The main conclusion
is that the classic multiplexing gain is zero, but the relative 1The power constraint is only defined on the intended signal, although
distortionsalsocontributeasmallamountofpower.However,thisextrapower
improvement over single-antenna channels can be even larger
is fully characterized by the SNR, P, and we therefore assume that P is
forthesephysicalMIMOchannelsthanwithidealtransceivers. selectedtomakethetotalpowerusagefulfillallexternalsystemconstraints.
IEEECOMMUNICATIONSLETTERS 2
between antennas.2 In multi-carrier (e.g., OFDM) scenarios, N > N . Informally speaking, the lower and upper bounds
t r
the channel model (2) can describe each individual subcarrier. aretightwhenthehigh-SNRcapacity-achievingQisisotropic
However,thereissomedistortion-leakagebetweensubcarriers in a subspace of size N and size min(N ,N ), respectively.
t r t
and for simplicity we model this on a subcarrier basis as The following corollaries exemplify these extremes.
leakage between the antennas (the impact of what is done on Corollary 1. Suppose the channel distribution is right-
the different antenna at one subcarrier is likely to average out rotationally invariant (e.g., H HU for any unitary matrix
when having many subcarriers). To capture a range of cases U). The capacity is achieved b∼y Q = 1 I for any P and α.
we propose (cid:16) α (cid:17) The lower bound in (6) is asymptoticalNlyttight for any Nt.
υ (Q)=κ2 (1 α)q + (3)
n − n N Proof: The right-rotational invariance implies that the
t
where the parameter α [0,1] enables transition from one N dimensions of HHH are isotropically distributed, thus
∈ t
(α=0)tomany(α=1)subcarriers.Theparameterκ>0isthe the concavity of E logdet() makes an isotropic covariance
levelofimpairments.3 Thismodelisagoodcharacterizationof matrixoptimal.The{lowerb·ou}ndin(6)isasymptoticallytight
phase noise and IQ-imbalance, while amplifier non-linearities as it is constructed using this isotropic covariance matrix.
(that make υn(Q) increase with P) are neglected [6]—the This corollary covers Rayleigh fading channels that are
dynamic range is assumed to always match the output power. uncorrelated at the transmit side, but also other channel
distributionswithisotropicspatialdirectivityatthetransmitter.
III. ANALYSISOFCHANNELCAPACITY The special case of a deterministic channel matrix enables
The transmitter knows the channel distribution fH, while stronger adaptivity of Q and achieves the upper bound in (6).
the receiver knows the realization H. The capacity of (2) is Corollary2. Supposeα=1andthechannelHisdeterminis-
C (P)= sup (x;y,H) (4) ticandfullrank.LetHHH=UMΛMUHM denoteacompact
Nt,Nr fX:tr(E{xxH})=tr(Q)=1I eigendecomposition where ΛM =diag(λ1,...,λM) contains
where fX is the PDF of x and (, ; ) is conditional mutual the non-zero eigenvalues and the semi-unitary UM ∈CNt×M
information. Note that (x;y,HI)=· ·E· (x;yH=H) . contains the corresponding eigenvectors. The capacity is
I H{I | } M (cid:32) (cid:33)
Lemsmupa 1E.HT(cid:110)helocga2pdaectit(cid:0)yI+CPNtH,NQr(HPH) (cPanHbΥeteHxpHre+ssIe)d−1a(cid:1)s(cid:111) (5) ford =(cid:2)µCNt,1N(cid:3)r(Pw)h=ere(cid:88)i=µ1ilosgs2elec1te+dtλoimNλκ2itadk+ie1(cid:80)M d =(71).
Q:tr(Q)=1 i −λi + i=1 i
The capacity is achieved by Q = U diag(d ,...,d )UH.
and is achieved by x (0,Q) for some feasible Q 0. M 1 M M
Proof: For any ∼reaCliNzation H = H and fixed P(cid:23), (2) The upper bound in (6) is asymptotically tight for any Nt.
Proof: The capacity-achieving Q is derived as in [1],
is a classical MIMO channel but with the noise covariance
using the Hadamard inequality. The capacity limit follows as
(PHΥ HH +I). Eq. (5) and the sufficiency of using a
t Q= 1 U UH achieves the upper bound.
Gaussian distribution on x then follows from [1]. M M M
Although the capacity behaves differently under impair-
Although the capacity expression in (5) appears similar to
ments,theoptimalwaterfillingpowerallocationinCorollary2
that of the classical MIMO channel in (1) and [1], it behaves
is the same as for ideal transceivers (also noted in [7]). When
very differently—particularly in the high-SNR regime.
N N , the capacity limit Mlog (1+ Nt ) is improved
Theorem 1. The asymptotic capacity limit C ( ) = t ≥ r 2 Mκ2
Nt,Nr ∞ by increasing Nt, because a deterministic H enables selective
lim C (P) is finite and bounded as
P→∞ Nt,Nr transmissionintheNr non-zerochanneldimensionswhilethe
Mlog (cid:18)1+ 1(cid:19) C ( ) Mlog (cid:18)1+ Nt (cid:19) (6) transmitter-distortion is isotropic over all Nt dimensions.
2 κ2 ≤ Nt,Nr ∞ ≤ 2 Mκ2 Weconcludetheanalysisbyelaboratingonthefactthatthe
lower bound in (6) is always asymptotically achievable.
where M=min(N ,N ). The lower bound is asymptotically
t r
achieved by Q= 1 I. The two bounds coincide if N N . Corollary 3. If the channel distribution fH is unknown
Nt t ≤ r at the transmitter, the worst-case mutual information
Proof: The proof is given in the appendix. min (x;y,H) is maximized by Q= 1 I (for any α) and
This theorem shows that physical MIMO systems have a apprfoHaIches Mlog (cid:0)1+ 1(cid:1) as P . Nt
finite capacity limit in the high-SNR regime—this is funda- 2 κ2 →∞
mentally different from the unbounded asymptotic capacity
for ideal transceivers [1]. Furthermore, the bounds in (6) hold A. Numerical Illustrations
for any channel distribution and are only characterized by the Consider a channel with N = N = 4 and varying SNR.
t r
number of antennas and the level of impairments κ. Fig. 2 shows the average capacity over different determinis-
The bounds in (6) coincide for N N , while only the tic channels, either generated synthetically with independent
t r
≤
upperboundgrowswiththenumberoftransmitantennaswhen (0,1)-entries or taken from the channel measurements in
CN
[11]. The level of impairments is varied as κ 0.05, 0.1 .
2Thepaper[8]predictssuchacorrelation,butitistypicallysmall. ∈{ }
Ideal and physical transceivers behave similarly at low and
3The error vector magnitude, EVM = EE{{(cid:107)(cid:107)ηxt(cid:107)(cid:107)22}}, is a common measure medium SNRs in Fig. 2, but fundamentally different at high
for quantifying RF transceiver impairments. Observe that the EVM equals
SNRs.Whiletheidealcapacitygrowsunboundedly,thecapac-
κ2 for the considered υn(Q) in (3). EVM requirements in the range κ ∈
[0.08,0.175]occurinLongTermEvolution(LTE)[9,Section14.3.4]. itywithimpairmentsapproachesthecapacitylimitC4,4( )=
∞
IEEECOMMUNICATIONSLETTERS 3
70 Low SNR DoF Regime Saturation Regime
Ideal Transceiver Hardware
60 Transceiver Impairments: κ=0.05
z] Transceiver Impairments: κ=0.1 40
H
Capacity [bits/s/345000 C a p afocrit yD iLffiemrietsntC κ4,4(∞) acity [bits/s/Hz]2300 DLCififmaeprietasnc tfi ot Nyrt SIdloepaelNt =12 Nt =4
erage 20 Cap Deterministic, α=1 (Average)
Av Synthetic 10 Deterministic, α=0 (Average)
10 Channels Measured
Uncorr Rayleigh Fading, any α
Channels
0 0
−10 0 10 20 30 40 50 60 70 −10 0 10 20 30 40 50
SNR [dB] SNR [dB]
Fig.2. Averagecapacityofa4x4MIMOchanneloverdifferentdeterministic Fig. 3. Capacity of a MIMO channel with Nr =4 and impairments with
channelrealizationsanddifferentlevelsoftransceiverimpairments. κ=0.05.WeconsiderdifferentNt,channeldistributions,andα-values.
4log2(1 + κ12) in Theorem 1. The difference between the sfiinnigteleP-in)pouftasninNgtle×-oNurtpMutIM(SOISOch)acnhnaenlnoevle.rthecorresponding
uncorrelatedsyntheticchannelsandtherealisticallycorrelated
measured channels vanishes asymptotically. Therefore, only Definition 1. The finite-SNR multiplexing gain, (P), is the
M
the level of impairments, κ, decides the capacity limit. ratio of MIMO to SISO capacity at a given P. For (2) we get
Next,weillustratethecaseN N anddifferentα.Fig.3 C (P)
t ≥ r (P)= Nt,Nr . (9)
considers Nt ∈{4, 12}, while having Nr =4, κ=0.05, and M C1,1(P)
two different channel distributions: deterministic (we average ThisratiobetweentheMIMOandSISOcapacityquantifies
overindependent (0,1)-entries)anduncorrelatedRayleigh the exact gain of multiplexing. The concept of a finite-SNR
CN
fading. We show α 0, 1 in the deterministic case, while multiplexinggainwasintroducedin[12]foridealtransceivers,
therandomcasegive∈sQ{ =}1 Iandsamecapacityforanyα. while the refined Definition 1 can be applied to any channel
Nt
Thesechannelsperformsimilarlyandhavethesamecapac- model. The asymptotic behavior of (P) is as follows.
ity limit when N =4. The convergence to the capacity limit M
t Theorem 2. Let h denote the SISO channel. The finite-SNR
is improved for the random distribution when N increases,
t multiplexing gain, (P), for (2) and any α satisfies
but the value of the limit is unchanged. Contrary, the capacity M
limits in the deterministic cases increase with N (and with E H 2 E H 2
t {(cid:107) (cid:107)F} lim (P) {(cid:107) (cid:107)2}, (10)
α since it makes the distortion more isotropic). Fig. 3 shows NtE h2 ≤ P→0M ≤ E h2
{| | } {| | }
thatthereisamediumSNRrangewherethecapacityexhibits log (1+ Nt )
roughly the same M-slope as achieved asymptotically for M lim (P) M 2 Mκ2 , (11)
idealtransceivers.Followingtheterminologyof[3],thisisthe ≤ P→∞M ≤ log2(1+ κ12)
where and denotetheFrobeniusandspectralnorm,
degrees-of-freedom (DoF) regime while the high-SNR regime (cid:107)·(cid:107)F (cid:107)·(cid:107)2
respectively. The upper bounds are achieved for deterministic
is the saturation regime; see Fig. 3. This behavior appeared
channels (with full rank and α = 1). The lower bounds are
in [3] for large cellular networks due to limited coherence
achieved for right-rotationally invariant channel distributions.
time,butwedemonstrateitsexistenceforanyphysicalMIMO
Proof: The low-SNR behavior is achieved by Taylor
channel (regardless of size) due to transceiver impairments.
approximation: Q = 1 I gives the lower bound, while the
Nt
per-realization-optimal Q=uuH (where u is the dominating
IV. GAINOFMULTIPLEXING
eigenvector of HHH) gives the upper bound. The high-SNR
The MIMO capacity with ideal transceivers behaves as
behavior follows from Theorem 1 and its corollaries.
Mlog (P)+ (1)[1],thusitgrowsunboundedlyinthehigh-
2 O This theorem indicates that transceiver impairments have
SNRregimeandscaleslinearlywiththeso-calledmultiplexing
little impact on the relative MIMO gain, which is a very
gain M = min(N ,N ). On the contrary, Theorem 1 shows
t r positive result for practical applications. The low-SNR be-
thatthecapacityofphysicalMIMOchannelshasafiniteupper
havior in (10) is the same as for ideal transceivers (since
bound, giving a very different multiplexing gain:
PHΥ HH +I I), while (11) shows that physical MIMO
t
≈
Mc∞lassic =Pl→im∞ClNotg,N2(rP(P)) =0. (8) c(ahlathnonuelgshciadneaalcthraienvseceMive(rPs )on>lyMcaninacthhieevheigMh-(SPN)R=reMgi)m.e
In view of (8), one might think that the existence of a non-
A. Numerical Illustrations
zero multiplexing gain is merely an artifact of ignoring the
transceiver impairments that always appear in practice. How- The finite-SNR multiplexing gain is shown in Figs. 4 and 5
ever, the problem lies in the classical definition, because also for uncorrelated Rayleigh fading and deterministic channels,
physicalsystemscangainincapacityfromemployingmultiple respectively, with N 4, 8, 12 , N =4, κ=0.05, α=1.
t r
∈{ }
antennasandutilizingspatialmultiplexing.Apracticallymore The limits in Theorem 2 are confirmed by the simulations.
relevant measure is the relative capacity improvement (at a Although the capacity behavior is fundamentally different for
IEEECOMMUNICATIONSLETTERS 4
4.5
ain Nt =12 APPENDIX:PROOFOFTHEOREM1
G As a preliminary, consider any full-rank channel realization
g
exin Nt =8 H. Let HHH=UMΛMUHM denote a compact eigendecom-
ultipl 4 High-SNR Limit position (with UM ∈ CNt×M, ΛM ∈ CM×M; see Corollary
M 2). The mutual information is increasing in P and satisfies
R
SN Low-SNR Nt =4 log det(cid:0)I+PHQHH(PHΥ HH+I)−1(cid:1)
− 2 t
Finite Limit ITdreaanls Tceraivnesrc Iemivpear irHmaerdnwtsa: κre=0.05 =log2det(cid:0)I+PUHM(Q+Υt)UMΛM(cid:1)
3−.52 0 −10 0 10 20 30 40 50 −log2det(cid:0)I+PUHMΥtUMΛM(cid:1) →
SNR [dB] log det(cid:0)UH(Q+Υ )U Λ (cid:1) log det(cid:0)UHΥ U Λ (cid:1)
Fig. 4. Finite-SNR multiplexing gain for an uncorrelated Rayleigh fading 2 M t M M − 2 M t M M
channelwithNr =4andNt≥4. =log2det(cid:0)I+UHMQUM(UHMΥtUM)−1(cid:1) (12)
SNR Multiplexing Gain678 HNigLt him=-Sit4NsR Nt =8 Nt IT=drea1an2ls Tceraivnesrc Iemivpear irHmaerdnwtsa: κre=0.05 aristhPm==→an(cid:88)lido∞M=g1u2.slToidnghege2t(cid:16)(cid:16)tfihI1res++truµeΥlqie(u−tΥda1el/i−ttt2(y1QI/f+2oΥQlAl−toΥBwH−t/s)2Hf=Πr/o2ΥdmΠeHtteΥ/(x2IHtpU+/aM2nBUd(cid:17)MAin)g)(cid:17).tThehilso(1egn3a-)-
−5
Finite athbeleismtapkaicntgotfheΛlimitcaPnc→els∞ouatn.dWaechtiheevneaidneenxtpifryestshieonpwrohjeecre-
e 4 M
Averag−2 0 −10 0 10SNR [dB2]0 30 40 50 otinotnomUaHMtriΥxΠ1t/2Υ.HtT/h2UeMith=stΥroHtng/2eUstMeig(UenHMvaΥluteUisMd)e−n1oUtedHMµΥi(1t·/)2.
Fig. 5. Average finite-SNR multiplexing gain of deterministic channels TheconvergenceasP isuniform,thuswecanachieve
(generatedwithindependentCN(0,1)-entries)withNr =4andNt≥4. bounds by showing that→all∞realizations has the same asymp-
totic bound. A lower bound is given by any feasible Q; we
physical and ideal transceivers, the finite-SNR multiplexing selectQ= 1 IasitgivesΥ = κ2Iandmakes(12)indepen-
gain is remarkably similar—not unexpected since the asymp- Nt t Nt
dentofH.Since(13)isaSchur-concavefunctionintheeigen-
totic limits in Theorem 2 are almost the same for any level
values, an upper bound is achieved by replacing µ () with
ohfightr-aSnNscReirveegrimime,pwaihrmereent(sa.) Tthheeremisaianfadsifteferrecnocneverisgeinncethtoe the average eigenvalue M1 tr(Υ−t 1/2QΥ−t H/2ΠΥHt/i2U·M) ≤
the limits under impairments and (b) deterministic channels M1 tr(Υ−t 1/2QΥ−t H/2) = NMtκ2, where the inequality follows
achieve an asymptotic gain higher than M when Nt >Nr. from removing the projection matrix (since ΠΥHt/2UM (cid:22) I).
NotethattheupperandlowerboundscoincidewhenN N ,
t r
V. CONCLUDINGIMPLICATIONS thus Q= M1 I is asymptotically optimal in this case. ≤
Unlikeconventionalcapacityanalysis,thecapacityofphysi-
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