An Overview of Limited Feedback in Wireless
梵文Communication Systems
David J.Love,Member,IEEE,Robert W.Heath Jr,Senior Member,IEEE,Vincent K.N.Lau,Senior Member,IEEE,David Gesbert,Senior Member,IEEE,Bhaskar D.Rao,Fellow,IEEE,and Matthew Andrews,
Member,IEEE
Abstract—It is now well known that employing channel adap-tive signaling in wireless communication systems can yield large improvements in almost any performance metric.Unfortunately, many kinds of channel adaptive techniques have been deemed impractical in the past becau of the problem of obtaining channel knowledge at the transmitter.The transmitter in many systems(such as tho using frequency division duplexing)can not leverage techniques such as training to obtain channel state information.Over the last few years,rearch has repeatedly shown that allowing the receiver to nd a small number of information bits about the channel conditions to the transmitter can allow near optimal channel adaptation.The practical systems,which are commonly referred to as limited orfinite-rate feedback systems,supply benefits ne
arly identical to unrealizable perfect transmitter channel knowledge systems when they are judiciously designed.In this tutorial,we provide a broad look at thefield of limited feedback wireless communications.We review work in systems using various combinations of single antenna,multiple antenna,narrowband,broadband,single-ur, and multiur technology.We also provide a synopsis of the role of limited feedback in the standardization of next generation wireless systems.
Index Terms—Wireless communications,Limited feedback, MIMO systems,Quantized precoding,Multiur MIMO systems.
I.I NTRODUCTION
土字
T HE INCREASES in wireless data rates over the years have been accompanied by large steps in communication system design.Past improvements in coding,modulation,and scheduling have led to the current systems deployed today. Manuscript received June23,2008;revid August26,2008.This material is bad in part upon work supported by the National Science Founda-tion under grants CCF-0513916,CCF-514194,and CNS-626797;Samsung Electronics;the AT&T Foundation;UC Discovery Grant,com07-10241;the DARPA IT-MANET program,Grant W911NF-07-1-0028;RGC fund615105; and by the U.S.Army Rearch Office under the Multi-University Rearch Initiative(MURI)grant-W911NF-04-1-0224.
D.J.Love is with the School of Electrical and Computer Engi-neering,Purdue University,West Lafayette,IN47907USA(e-mail: djlove@ecn.purdue.edu).
clear是什么意思R.W.Heath,Jr.is with the Wireless Networking and Communications Group,Department of Electrical and Computer Engineering,The University of Texas at Austin,Austin,TX78712USA(e-mail:rheath@ece.utexas.edu). V.K.N.Lau is with the Dept of Electrical&Electronic Engineering, Hong Kong University of Science and Technology,Hong Kong(e-mail: eeknlau@ee.ust.hk).
D.Gesbert is with the Depart.Mobile Communications,Eurecom Institute, Sophia Antipolis,France(e-mail:gesbert@eurecom.fr).
B.D.Rao is with the Electrical and Computer Engineering Department, University of California,San Diego,La Jolla,CA92093USA(e-mail: brao@ece.ucsd.edu).
M.Andrews is with Alcatel-Lucent Bell Labs,Murray Hill,NJ07974USA (e-mail:andrews@).
Digital Object Identifier10.1109/JSAC.2008.081002.Next generation systems are poid to make u of a variety of channel adaptive techniques.The sorts of signaling ap-proaches allow the transmitte
r to adapt to the propagation con-ditions.This implies that the transmitter requires some form of knowledge of the wireless channel conditions,often referred to as channel state information(CSI)at the transmitter(CSIT). Employing most kinds of channel adaptive techniques has been impossible in the past becau two-way communication is accomplished using frequency division duplexing(FDD). The forward and rever links in FDD generally have highly uncorrelated channels becau they are parated in frequency. One way of overcoming this problem is by using other forms of ,statistical reciprocity).The sorts of systems u the fact that the forward and rever links often share the same fading distribution.Statistical approaches can perform very well in situations where the channel exhibits some form of(slowly varying)structure,such as having a large mean ,a large Rician K-factor)or strong correlation(either in space,time,or frequency).Generally, however,statistical adaptation comes with a non-negligible performance loss compared with adaptation techniques that u the instantaneous channel realization.
The big innovation that has overcome the challenge of making instantaneous channel adaptation practical is the u of feedback.A system employing feedback us a low rate data stream on the rever side of the link to provide information to the transmitter of the forward side of the link.This information conveys some notion of the forward link , channel state,received power,inte
rference level,etc.),and the transmitter us the information to adapt forward link transmission.The value of feedback varies with the system scenario.However,generally speaking,the value is greater when the channel introduces some form of disturbance(such as spatial interference,intersymbol interference,multiur interference,etc.)that cannot be handled by the receiver alone. The feedback information itlf can be digital or analog. In this tutorial,we concentrate on digital feedback,which is commonly referred to as limited feedback orfinite-rate feedback.
The history of feedback in communication systems traces back to Shannon[289],[290]and other early work such as [106],[285],[286],[326],[327].Interest has continued to grow in the us of feedback.Feedback has had broad impact in areas such as control systems,source coding,informa-tion theory,and communication theory.We concentrate and summarize the prent state of rearch into applications of
0733-8716/08/$25.00c 2008IEEE
自行车包
limited feedback in wireless communication systems,where its interest has recently en much revival,particularly in relation with multiple-input multiple-output (MIMO)systems.Our goal is to examine what has been accomplished and make some comments on the direction of this area of res
earch.We will divide the work into two main areas:single-ur (e Section II)and multiur communication (e Section III).Becau the true measure of the impact of rearch is in the applications it generates,we look at the role of limited feedback in current and future standardized wireless systems in Section IV.We provide some concluding remarks in Section V.
Throughout the paper we u some common notation.The complex numbers are denoted by C .The transpo of a vector is denoted by a superscript T,and the conjugate transpo by a superscript ∗.A diagonal matrix is created from a vector with the function diag (·).The two-norm of a vector (or matrix)is reprented by · 2,and the Frobenius norm of a matrix is reprented by · F .The ceiling function is written as · ,and the floor function is similarly written as · .The ba two logarithm is written log 2(·).The determinant of a matrix is evaluated with det(·).
II.F EEDBACK IN S INGLE -U SER W IRELESS S YSTEMS The design of single-ur wireless systems has a long and storied history.We address the role of limited feedback in single and multiple antenna systems.A.Single Antenna Systems
Single antenna wireless links are the most commonly found wireless links.Single-ur wireless systems are often split into the categories of narrow and broadband depending on the relationship between the bandwidth and delay spread
of the propagation channel.For this reason,the bene fits of channel adaptation using limited feedback will be divided into narrowband and broadband systems.
1)Narrowband Systems:The k th channel u of a narrow-band system is mathematically modeled as
y [k ]=h [k ]x [k ]+n [k ].
(1)
where y [k ]is the complex received symbol,h [k ]is the complex channel respon,x [k ]is the transmitted symbol,and n [k ]is noi distributed according to CN (0,1)(assuming the noi is normalized to unit variance).The transmitted signal x [k ]is subject to a long term power constraint where E h,x [|x [k ]|2]≤ρ.To allow the receiver to perform co-herent detection,channel estimation techniques are usually performed.Most of the work on limited feedback assumes that the receiver has perfect knowledge of the h [k ]for all k.We will note when discussing work that makes other assumptions.Additionally,various ergodicity and stationarity assumptions must hold for the process {h [k ]},but the are beyond the scope of this paper.
Becau our focus is on adapting the transmitted signal to the channel conditions,modeling how the channel varies across a codeword block is critical.We primarily focus on a block-fading channel model,where the channel is constant for
veral channel us before changing independently.There-fore,the t th channel block satis fies h [tK ch ]=h [tK ch +1]=···=h [(t +1)K ch −1]=h (t )where K ch is the length of the fading block.The transmitted data will also have a block structure.Let K bl denote the codeword block length.We refer to the vector [x [0]x [1]···x [K bl −1]]as the transmitted codeword.The relationship between the channel block length K ch and the codeword block length K bl is important.In this tutorial,we will refer to the ca when K ch =K bl as the slow-fading scenario and the ca when K ch
K bl →0when K bl →∞as the fast-fading scenario.More discussion on the relation between codeword block length and time variation of the as E x |x [k ]|2
|h [k ]=h (t )≤ρt where the expectation is over all possible codewords.To satisfy the long-term power constraint,we have to require that E h [ρt ]≤ρ.If the transmitter has knowledge of the channel conditions for each channel block,ρt could be adaptively chon to maximize performance.Variable rate encoding is also very common.In this kind of framework,the rate is varied according to the instantaneous channel conditions.
Assuming perfect knowledge of the channel at the transmit-ter and receiver,the ergodic capacity is [51],[98]
R =E h log 2 1+ρ(h )|h |2
(2)where ρ(h )is a function that allocates power subject to water-filling.Interestingly,this rate can be achieved asymptotically
with fixed rate encoding [49].For the fast-fading ca,we can construct the codewords as
x [k ]=
ρ(h [k ])s [k ](3)
where {s [k ]}K bl
−1
冬至吃啥k =0is a codeword designed independently of the channel conditions (but who rate is determined using distribution information)such that E s |s [k ]|
2
≤1and ρ(h [k ])is chon according to the water filling algorithm.The problem with capacity achieving power allocation frameworks is that they require the transmitter to perfectly know h [k ](or at least its magnitude).As mentioned earlier,in systems such as tho using FDD,this knowledge is not available.For this reason,the solution is for the receiver to utilize the rever link as a feedback channel,nd channel state information on this channel,and give the transmitter some kind of side information u [k ]about the current channel realization h [k ].A general scheme is shown in Figure 1.The receiver can obtain some level of channel information using techniques such as training.Using this knowledge,the receiver can design feedback to be nt as overhead on the rever link.The problem of codeword design with side information was brought up in [49].This paper considers more general channel models than just (1),without restriction to block fading.In addition,[49]does not require the receiver to perfectly know h [k ]but instead assumes the receiver has access to some side information w [k ].Thus,the problem becomes one of encoding and decoding using this side information along with knowledge of the joint probability density function p (h,u,w ).
LOVE et al.:AN OVERVIEW OF LIMITED FEEDBACK IN WIRELESS COMMUNICATION SYSTEMS1343
Channel Side
Channel Fig. 1.Block diagram of a single antenna limited feedback system. The receiver obtains information about the wireless channel(either perfect or imperfect)through techniques such as training.This receiver channel information is then fed into a quantizer that returns a small number of feedback bits to be nt as overhead on the rever link.The transmitter can u the received feedback bits to adapt the transmitted signal to the forward channel.
This work was later extended to the fast-fading ca
(through a block-fading construction)in[172]adding the additional requirement of a cardinality constraint on the side
information u[k].The problem of properly designing the side information u[k]is shown to be one of scalar quantization
that can be solved using the Lloyd algorithm.The fast-fading
assumption employed in this paper allows the codeword rate to befixed becau a codeword block spans a large number
of channel realizations.
Extensive analysis of theflat-fading single antenna system has been conducted in[155]when the transmitter is provided
with a quantized version of the magnitude of h[k].This quantized version is taken by dividing up the non-negative
part of the real line into quantization regions.This quantization
approach is similar to techniques ud in the temporal water-filling proof in[98],which took the limit as the quantization
noi goes to zero.In[155],the power allocation strategy then
us the quantized channel realization subject to either a short-term power constraint(whereρt≤ρfor any channel block t) or a long-term power constraint(where the power allocated to the t th channel blockρt is restricted in expected value to be
bounded byρ).An overview of the possible power constraints
is available in[48].
A model other than block fading was discusd in[267].
This work assumed periodic feedback,where feedback is nt everyfixed number of channel us.The channel model considered was afinite-statefirst-order Markov model. From a practical perspective,another approach to the prob-lem of adapting to the channel conditions is to concentrate
on lecting from afixed t of per channel u constellations and varying the density(or equivalently the average energy) of the constellations.On-off rate adaptation was propod in[37],where the transmission was turned on and off subject to the channel conditions.A more general system where the rate of the transmitter is adjusted bad on the channel is addresd in[52].Here the effect on the probability of error subject to an average rate constraint is analyzed.The ideas were later extended to take into account queue length[53]. Various other works have looked at the application of rate variation[18],[48],[147],[237],[299],[300],[311],some using specific constellation families and some combining the rate variation with adaptive power allocation.Analysis of adaptive modulation with feedback imperfections has been studied in[84],[238].Discussion can also be found in the overview paper[85].A diversity-bad approach is given in [293].
Work taking practical code designs into account has been relatively limited.Adaptive M-ary orthogonal coding for high bandwidth expansion systems(such as CDMA)has been propod in[171],and adaptive trellis coded modulation for high bandwidth efficiency has been studed in[17],[95],[169], [170],[232].The works consider joint optimization of the coding rate and modulation level coding bad on maintaining a target average error rate or average throughput requirement. Outdated knowledge of channel state information has been considered.
In addition to the performance benefit associated with adaptive coded modulation systems,there is another important benefit of channel state knowledge at the transmitter.In[229], the authors studied the concept of incorporating knowledge of channel side information at the transmitter on the LDPC code design.It is shown that substantial reduction of LDPC decod-ing complexity can be obtained utilizing the side information. Another approach to feedback is the u of repeat requests when channel conditions cau codeword errors.In fact, regardless of the availability of explicit CSIT,there is always ACK/NAK signaling exchange in the upper layers in most communication systems.Such ACK/NAK exchange is ud for automatic repeat request(ARQ)in the upper layers so that an error-free logical channel can be prented to the application layers.In fact,the ACK/NAK signaling exchange can also be utilized at the physical layer of the transmitter to learn about the actual channel conditions.This information is particularly uful when the CSIT(through explicit feedback [FDD]or implicit feedback[TDD])is not perfect. Consider the ca when the channel state information obtained by limited feedback(orfinite-rate feedback)may be outdated or suffering from feedback errors.Becau of the errors,the transmitter must adapt the transmit power and/or data rate according to this imperfect CSIT.In order to effectively exploit the imperfect channel information at the transmitter,it is important to take into account the error statistics of the CSIT in the adaptation.However,it is very difficult for the transmitter to obtain and keep track of the error statistics
becau they usually depend on the channel envi-ronment and Doppler spectrum.In such cas,the ACK/NAK signaling from the upper layer ARQ is very uful to provide a truly clod-loop adaptation.For example,if the transmitter is overly aggressive in the ,in adjusting the data rate),the packet will be corrupted at the receiver and a NAK will result.Bad on the NAK information,the transmitter can reduce the data rate and/or increa the transmit power until an ACK is received.Such an approach is very robust to CSIT errors and does not require explicit knowledge of CSIT error statistics at the transmitter.In fact,this clod-loop adaptation framework has been commercially deployed in IS95in outer-loop power control.
Selective repeat ARQ is studied in[27].ARQ schemes with reliable and unreliable feedback are studied in[26].Power and rate adaptation utilizing ACK/NAK feedback has appeared in
1344IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS,VOL.26,NO.8,OCTOBER 2008
[102],[125],[358].In [141],the authors considered a two level stochastic scheduling bad on learning automata.In [338],the authors modeled the power,rate adaptation (as well as ur lection)using Markov Decision Process (MDP)and obtained an optimal as well as low complexity control policy.Fro
m the works,it is found that robust performance can be obtained by jointly considering both limited CSIT feedback as well as ACK/NAK signaling in the design of transmitter adaptation policy.
2)Broadband and Wideband Systems:A single antenna broadband model is complicated by the fact that previously transmitted symbols interfere with the current symbols.A discrete-time model for this kind of t-up is
y [k ]=
L =0
h [k, ]x [k − ]+n [k ].(4)
where the channel is now frequency lective and repre-nted by an (L +1)-tap finite impul respon filter
[h [k,0]···h [k,L ]]at the k th channel u.
Becau of the dif ficulty in dealing with the intersymbol interference resulting from frequency lective channels,es-pecially for recently standardized wideband systems (UMTS-LTE,WiMAX,WiFi)
,industry and academia have turned toward the u of orthogonal frequency division multiplex-ing (OFDM).In OFDM,the signal x [k ]is jointly designed over K sc +L channel us assuming that the channel is constant during a block of K ch channel us with K ch ≥K sc +L.The transmitter constructs a K sc collection of parallel subchannels in the frequency domain.The k th trans-mission across the parallel subchannels can be written ˜x
= ˜x 0 ˜k ···˜x K sc
−1 ˜k
T .This vector is then multplied by an inver discrete Fourier transform (DFT)matrix,and the last L entries of the transformed signal are appended to the beginning of the vector (termed a cyclic pre fix).
After reception,the receiver removes this cyclic pre fix and multiplies the signal by a DFT matrix.This then gives a post-processing input-output relation in the frequency domain of
˜y ˜k =diag ˜h ˜k ˜x [˜k ]+˜n ˜k (5)at OFDM channel u ˜k.
Here vector notation has been ud where the v th entry of each vector corresponds to the input-out
put relation for the v th subcarrier.
Adapting the subcarrier powers with limited feedback has been the focus of veral works.Using a one bit per sub-carrier (or per block of subcarriers)design that simply turns subchannels off and on was propod by [177].Later work on quantized feedback in OFDM to activate or deactivate subchannels was the focus of [306],[307].More general schemes for jointly quantizing the per subcarrier power al-locations have been discusd in [61],[201],[207],[262].A sub-channel grouping approach was developed in [3],[4].Techniques ud to address the problem of adaptation with unquantized (but stale or imperfect)CSIT studied in [348]can also be employed.The ca of using feedback for bit interleaved coded OFDM was addresd in [310].An overview of adaptive modulation with OFDM is available in [270].With the emergence of systems such as ultra-wideband (UWB)there has been an incread interest in adaptive周记格式
signaling over very large bandwidths (often on the order
of 109Hertz).One possible approach to signaling in the systems is to nd a narrowband signal over an adaptively chon frequency band.When a narrowband channel is chon by probing over a wideband channel,feedback allows the transmitter to choo a frequency band with good performan
ce (generally de fined as having a large signal-to-interference plus noi ratio (SINR)).The low signal-to-noi ratio (SNR)scaling of the maximum achievable rate is the focus of [44].Training a wideband channel with feedback to optimize rate is discusd in [5].Extending feedback analysis to wideband channels that are spar in the delay and Doppler domains is considered in [103].B.Multiple Antenna Systems
The application of limited feedback to multiple antenna wireless systems has received much attention in the recent past.The spatial degree-of-freedom and the potentially sizable bene fits available by adapting over it make limited feedback a very attractive option.
The degrees of freedom with multiple antenna systems can be exploited to offer rate and diversity bene fits as well as beamforming and interference canceling capabilities.While the diversity gain can be typically extracted without the need of CSIT feedback (e.g.,space time codes),CSIT plays a crucial role for beamforming and interference mitigation at the transmitter side,as will be clari fied below.
1)Narrowband Systems:A single-ur narrowband multi-ple antenna system can be reprented by an expression of the form
y [k ]=H [k ]x [k ]+n [k ](6)at the k th channel u.Assuming M t transmit antennas and
M r receive antennas,y [k ]is an M r -dimensional receive vector,H [k ]is an M r ×M t channel respon matrix,x [k ]is an M t -dimensional transmit vector,and n [k ]is M r -dimensional noi.We assume the noi to have i.alized entries distributed according to CN (0,1).The transmitter power
constraint requires that E H ,x x [k ] 2
肝火旺吃什么药
2 ≤ρ.As in the single antenna ca,we concentrate on the scenario where the receiver has access to H [k ].Given this,there are a variety of ways to design x [k ]if the transmitter is given access to some quantized information relating to H [k ].
Again,this analysis will depend on the time evolution model of the channel.If we u our previous notation of block-fading,the t th channel block satis fies H [tK ch ]=H [tK ch +1]=···=H [(t +1)K ch −1]=H (t )where K ch is the length of the fading block.For power constraint reasons,
E x x [k ] 2
2|H [k ]=H (t ) ≤ρt for the t th block.Varying ρt to perform temporal water-filling provides capacity ben
e fits,but unless otherwi noted,our discussion assumes ρt =ρfor all channel blocks.
内部审核报告1a)Covariance Quantization
When the transmitter and receiver both perfectly know the channel,the ergodic capacity is [96],[320]
R =E H
max Q :tr (Q )≤1,Q ∗
=Q ,Q 0
log 2det (I +ρHQH ∗) .(7)
1345
This codeword t is chon according to some spatial power constraint criteria such that E s s [k ](s [k ])∗
=I and such that the encoding rate per channel block approaches the achievable rate of the instantaneous channel.For fast-fading,a fixed rate codeword t can be ud satisfying similar conditions to tho above but with a fixed encoding rate.
One of the first looks at trying to design the covariance matrix using imperfect channel information was the covariance design for multiple-input single-output (MISO)systems using statistical information published in [331].For a limited rate feedback approach,the general idea is to u the fact that the receiver knows H [k ]through procedures such as training.Using this channel knowledge,the receiver can quantize some function of H [k ]using vector quantization (VQ)techniques.Naturally,the aspects of the channel that the transmitter cares about are tho that allow the design of the covariance for the t th channel block [294].Using this line of reasoning,the receiver can determine a rate maximizing covariance and feed this back to the transmitter.Employing a codebook of possible covariance matrices Q ={Q 1,...,Q 2B }that is known to both the transmitter and receiver,the receiver can arch for the codebook index that solves
n opt [k ]=argmax 1≤n ≤2B
log 2det (I +ρH [k ]Q n H ∗[k ])
and nd the B bit binary label corresponding to covariance Q n opt [k ]to the transmitter.This gives a maximum achievable rate in bits per channel u of
R Q =E H
max Q ∈Q
log 2det (I +ρHQH ∗) (8)
using a codebook Q known to both the transmitter and
receiver.
The covariance codebook can be either fixed or randomly generated (using a ed known to both the transmitter and receiver).Designing a fixed covariance codebook to maximize the average rate is a challenging problem that depends on the stationary distribution of the channel [40],[168].Vector qu
antization approaches using the Lloyd algorithm have been shown to ef ficiently generate codebooks that achieve a large rate [168].Random approaches for covariance design have also been propod [69]using ideas pioneered in [278].In fact,it was shown in [69]that the rate loss with B bits of feedback decreas exponentially with the number of feedback bits.
While the codebook approach is optimal for a block-to-block independently fading channel,temporal correlation
Feed-to u or code-characterizations of the feedback side information can be further leveraged [356].
As a final remark,many of the above works considered block-fading channels and optimize the ergodic capacity in the covariance optimization problem under limited feedback.However,ergodic capacity may not be an appropriate perfor-mance measure in non-ergodic channels (such as the slow-fading ca).In slow-fading channels,there are systematic packet errors due to channel outages despite the u of powerful channel coding.This happens becau given limited CSIT there is still uncertainty about the actual CSI,and the transmitted packet will be corrupted whenever the data rate exceeds the instantaneous mutual information.In addition to limited CSIT feedback,there m
ight be feedback error due to noisy feedback links.This will also contribute to packet errors due to channel outage.When there is a noisy feedback link,the index mapping is also an important design parameter that will affect the robustness of the CSIT feedback.As a result,joint adaptation between the data rate,covariance matrix,and feedback index mapping is important to control the packet errors to a reasonable target.In order to account for the potential penalty of packet errors,it is important to consider system goodput (b/s/Hz successfully delivered to the receiver)instead of ergodic capacity as the system performance measure in the optimization framework.The design of robust limited feedback schemes and the joint rate,covariance,and feedback index mapping optimization for system goodput is a relatively unexplored topic.In [342],the authors extend the VQ op-timization framework to consider joint rate and covariance adaptation using Lloyd’s algorithm for slow-fading MIMO channels.
1b)Beamforming
While optimal covariance quantization is of interest to analyze how clo to perfect transmitter channel knowledge a limited feedback system can perform,limited feedback can have immediate impact enhancing existing clod-loop signaling approaches.Beamforming is characterized by the u of a rank one covariance matrix.Note that using a rank one Q matrix is optimal whenever the sin
gle-ur channel is itlf rank one.This notably occurs when the ur terminal is equipped with a single antenna.In this situation the availability of CSIT is critical.
In beamforming,the single-ur MIMO expression in (6)is
restricted so that x [k ]=√
ρf [k ]s [k ]where f [k ]is a channel dependent vector referred to as a beamforming vector and s [k ]is a single-dimensional complex symbol chon independently of the instantaneous channel conditions.For power constraint
