A time-variant channel prediction and feedback framework for interference alignment

1.1 MB·

by  Zhinan Xu

IEEETRANSACTIONSONVEHICULARTECHNOLOGY,ACCEPTEDFORPUBLICATION. 1
A Time-Variant Channel Prediction and Feedback
Framework for Interference Alignment
Zhinan Xu, Markus Hofer, Thomas Zemen
Abstract—In interference channels, channel state information IA under CSI mismatch for interference channels is studied
(CSI) can be exploited to reduce the interference signal dimen- in [4] showing that full DoF are achievable if the variance
sions and thus achieve the optimal capacity scaling, i.e. degrees
of the CSI measurement error is proportional to the inverse
of freedom, promised by the interference alignment technique.
of the SNR. Similar results are found in [5] for interference
However,imperfectCSI,duetochannelestimationerror,imper-
fect CSI feedback and time selectivity of the channel, lead to broadcast channels.
7 a performance loss. In this work, we propose a novel limited Limited feedback via quantization is a promising approach
1 feedback algorithm for single-input single-output interference totransferCSItothetransmittersideinfrequencydivisiondu-
alignment in time-variant channels. The feedback algorithm
0 plex (FDD) systems. Several approaches address the problem
encodes the channel evolution in a small number of subspace
2
of limited feedback for IA [6]–[10] assuming perfect channel
coefficients, which allow for reduced-rank channel prediction

n to compensate for the channel estimation error due to time estimation. In [6], channel coefficients are quantized using a
a selectivity of the fading process and feedback delay. An upper Grassmannian codebook for frequency-selective single-input
J bound for the rate loss caused by feedback quantization and single-output (SISO) channels. The work in [7] extends the
2 channelpredictionisderived.Basedonthisbound,wedevelopa
results to multiple-input multiple-output (MIMO) channels.
dimensionswitchingalgorithmforthereduced-rankpredictorto
Both [6] and [7] show that the full DoF are achievable as
] findthebesttradeoffbetweenquantization-andprediction-error.
T Besides, we characterize the scaling of the required number of long as the feedback rate is high enough (which scales with
I feedback bits in order to decouple the rate loss due to channel the transmit power). This result aligns with the one that is
.
s quantization from the transmit power. Simulation results show found for MIMO broadcast channels in [11]. The work in
c that a rate gain over the traditional non-predictive feedback
[ strategy can be secured and a 60% higher rate is achieved at [8] addresses the problem of improving the sum rate under
20dB signal-to-noise ratio with moderate mobility. limitedfeedbackbyinvolvingadditionaliterativecomputation
1
ofpre-quantizationfiltersatthereceivers.Tofurtherreducethe
v Index Terms—Interference alignment, IA, channel state in-
feedback overhead, [9] considers differential limited feedback
6 formation, time-variant fading channels, time-variant channel
7 prediction, basis expansion model, limited feedback. on the Grassmannian manifold by exploiting temporal corre-
3 lation of the time-selective fading channels. In the context of
0 opportunistic transmission for IA, [10] shows that multi-user
0 I. INTRODUCTION
diversity can be exploited based on 1-bit feedback from each
. Interferenceisacruciallimitationinnextgenerationcellular
1 user, while preserving the full DoF. Instead of quantizing the
0 systems. To address this problem, interference alignment (IA)
CSI, [12] considers analog feedback and shows that the DoF
7 has attracted much attention and has been extensively studied
ofIAcanbepreservedaslongastheforwardandreverselink
1 lately. IA is able to achieve the optimal capacity scaling,
: SNRs scale together.
v i.e. degrees of freedom (DoF), at a high signal-to-noise ratio
However, for a practical system, the imperfection of CSI is
i (SNR) and a rate of K/2·log(SNR)+o(log(SNR)) for the
X caused by various aspects:
K userinterferencechannelwithtime-variantcoefficients[1].
r
a However, this result is based on the assumption that global (a) For time-variant channels, CSI is acquired with the aid
channelstateinformation(CSI)isperfectlyknownatallnodes. of pilot symbols. The channel varies over time due to
This is extremely hard to achieve due to the large amount of the mobility of the users. If the channel changes after
required feedback information. the transmission of the pilot symbols, the receiver cannot
CSI imperfection degrades IA performance. Assuming the detect the channel variation, which leads to a reduction in
imperfectchannelmatrixisthesummationofthetruechannel sum rate due to the use of outdated channel estimates.
matrix and an independent error matrix, the impact of imper- (b) For FDD, CSI is fed back through limited capacity
fect CSI on IA has been investigated in [2]–[5]. The work of feedback channels. The error due to quantized feedback
[2] derives both upper and lower bounds on the achievable degrades the IA performance.
rates assuming noisy CSI. The error performance of IA is (c) The feedback information arrives at the transmitter with a
studied in [3] and adaptive schemes are proposed to introduce delay which causes a further performance degradation.
robustness against CSI imperfection. The performance loss of
A related body of research tackling the above mentioned
problems exists for single-cell multiuser MIMO systems [13],
Copyright (c) 2015 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be [14]. For interference channels, [15] studies (a) and (b) for
[email protected]. MIMO IA using a minimum mean square error (MMSE)
TheauthorsarewithAITAustrianInstituteofTechnology,Vienna,Austria
estimator.Thestudiesin[13]–[15]considerblockfadingchan-
(e-mail:{zhinan.xu,markus.hofer,thomas.zemen}@ait.ac.at).
ManuscriptreceivedXXX,XX,2015;revisedXXX,XX,2015. nels.Theworkof[16]extends[15]consideringtime-selective
IEEETRANSACTIONSONVEHICULARTECHNOLOGY,ACCEPTEDFORPUBLICATION. 2
continuousfadinginthepayloadpart,whileassumingconstant feedbackbitstodecoupletheratelossduetoquantization
fading for the training part. from the transmit power.
In this paper, we jointly consider (a)-(c) for interference The following notation is used throughout this paper: We
alignment in wideband SISO systems with symbol extension denote a scalar by a, a column vector by a and a matrix
overfrequency[1],i.e.precodingacrossorthogonalfrequency by A. The superscript ∗, T and H stand for conjugate,
dimensions. We address channel estimation, feedback and transpose and Hermitian transpose, respectively. The notation
prediction jointly and analyze the impact of imperfect CSI (cid:107)·(cid:107), (cid:107)·(cid:107) , tr(·), vec(·), det(·), (cid:100)·(cid:101) and E[·] denotes vector
F
in terms of sum rate loss. Previous works solely tackle the 2-norm, Frobenius norm, trace, vectorization, determinant,
theoreticalaspectofimperfectCSI[2]–[5],orlimitedfeedback ceiling operation and the expectation operation, respectively.
assuming perfect channel estimation [6]–[10]. Besides, the Wedenoteby[A]k,(cid:96) the(k,(cid:96))-thelementofmatrixAandby
prediction aspect for IA is studied in time division duplex [a]k the k-th element of vector a.
(TDD) systems [17] or assuming perfect feedback for FDD The rest of the paper is organized as follows: In Section
systems [18]. However, it is almost impossible to feed back II, we introduce the system model of IA. Section III presents
allchannelimpulseresponsesofthepayloadforFDDsystems thereduced-rankchannelestimationandpredictionalgorithm.
duetolimitedcapacityfeedbackchannels.Ourapproachtakes Section IV describes the proposed training and feedback
advantageoftheband-limitednatureoftime-variantchannels, algorithm. The performance is quantified in terms of sum rate
which allows us to describe the channel variation of the loss in Section V, where a subspace switching algorithm is
payload using a few subspace coefficients. The feedback of given. The numerical results are provided in Section VI. We
subspace coefficients using vector quantization for channel conclude the paper in Section VII.
predictionisnotconsideredinanyofthepreviousworkstothe
best of our knowledge. We exploit this concept in the context II. SYSTEMMODEL
of IA. Although we present our results for SISO interference Let us consider a K user time- and frequency-selective
channels, the complete strategy can be generalized to MIMO SISO interference channel, which consists of K transmit-
interference channels using a method similar to [7] by vector- ter and receiver pairs. We denote by h (t,τ) the time-
k,(cid:96)
ization of channel matrices. The scientific contributions of the variant impulse response between transmitter (cid:96) and receiver
paper: k, where t is time and τ is delay. Orthogonal frequency
division multiplexing (OFDM) is used to convert the time-
• Wetackletheproblems(a)and(c)byreduced-rankchan- and frequency- selective channel into N parallel time-
nel prediction using discrete prolate spheroidal (DPS)
selective and frequency-flat channels. The sampled impulse
sequences[19].Thankstotheenergyconcentrationofthe
response is defined as h [m,s] = h (mT ,sT ), where
sequencesintheDopplerdomain,weareabletodescribe k,(cid:96) k,(cid:96) s c
1/T is the bandwidth and T = (N + G)T denotes
thechannelevolutionbyonlyafewsubspacecoefficients. c s c
the OFDM symbol duration with a cyclic prefix length G.
• To address problem (b), we show that the subspace The S-tap time-variant sampled impulse response between
coefficients can be quantized and fed back using vector
transmitter (cid:96) and receiver k is denoted by h [m] =
quantization, which greatly reduces the redundancy of k,(cid:96)
[h [m,1],...,h [m,S]]T, ∀k,(cid:96) ∈ {1,...,K}. Every el-
the codebook by exploiting the rotation invariance. In k,(cid:96) k,(cid:96)
ement of the channel impulse response vector h [m] is
addition, we highlight the importance to feed back the k,(cid:96)
independent identically distributed (i.i.d.) with a power delay
subspace coefficients in delay domain, resulting in a (cid:104) (cid:105)
profile (PDP) E{h [m]h [m]H} = diag( p1 ,...,pS ).
reduction of noise. With the subspace coefficients, the k,(cid:96) k,(cid:96) k,(cid:96) k,(cid:96)
transmitter is able to perform channel prediction to com- We assume the channel gain (cid:80)S ps =N.
s=1 k,(cid:96)
bat the time selectivity of the channel. We generalize the The variation of a wireless channel for the duration of
results from [20] allowing for more channel extensions the transmission of a data packet is caused by user mobility
than channel delay taps. and multipath propagation. We define the normalized Doppler
• Thesubspacevectortobequantizedhascorrelatedentries frequency of the time-selective fading process {hk,(cid:96)[m,s]} as
insomecases.Wecharacterizethesecondorderstatistics
ν =f T (1)
of the subspace vector, which is used for whitening D D s
the vector to match the statistics of the quantization where f denotes the Doppler frequency in Hertz (Hz).
D
codebook, improving the results in [20]. The temporal covariance function over consecutive OFDM
• An upper bound of the rate loss due to the channel symbols becomes
prediction- and quantization-error is derived, which is
R [m]=E{h [a]Hh [a+m]}. (2)
usedtofacilitateanadaptivesubspacedimensionswitch- hk,(cid:96) k,(cid:96) k,(cid:96)
ing algorithm. The temporal covariance matrix is defined as [R ] =
hk,(cid:96) a,m
• Weshowthatthereexistsatradeoffbetweenquantization R [a−m] for a,m∈[0,...,M −1].
hk,(cid:96)
error and prediction error at a given feedback rate. The Using OFDM, The observed frequency selective channel
subspace dimension switching algorithm is efficient to can be converted into N narrowband frequency-flat channels
capture the tradeoff and find the subspace dimension as
associated with a higher rate.
w [m]=D h [m] (3)
• We characterize the scaling of the required number of k,(cid:96) N×S k,(cid:96)
IEEETRANSACTIONSONVEHICULARTECHNOLOGY,ACCEPTEDFORPUBLICATION. 3
where D is the N ×S submatrix of the N ×N DFT symbol i can be expressed as
N×S
matrix D . The DFT matrix D is defined as [D ] =
N N N i,j ui[m]Hy [m]=ui[m]HW [m]vi[m]si[m]+
√1 e−j2π(i−1)(j−1)/N, ∀i,j ∈{1,...,N}. The diagonal ma- k k k k,k k k
N (cid:124) (cid:123)(cid:122) (cid:125)
trix containing the channel frequency response can be written desiredsignal
as Wk,(cid:96)[m]=diag(wk,(cid:96)[m]). uik[m]H(cid:88)Wk,k[m]vkj[m]sjk[m]+
We consider two different communication phases: (i) the j(cid:54)=i
CSIacquisitionviapilotsand(ii)thetransmissionofpayload. (cid:124) (cid:123)(cid:122) (cid:125)
inter-streaminterference
In the CSI acquisition phase, the pilot symbols from different
transmitters are orthogonalized in time. During the transmis- ui[m]H(cid:88)(cid:88)d(cid:96) W [m]vj[m]sj[m]+ui[m]Hn [m] (5)
k k,(cid:96) (cid:96) (cid:96) k k
sion of payload, all transmitters will send simultaneously.
(cid:96)(cid:54)=kj=1
However, for a given transmitter, its signal is only intended to (cid:124) (cid:123)(cid:122) (cid:125)
bereceivedbyasingleuserforagivensignalinginterval.The inter-userinterference
signal received is the superposition of the signals transmitted for i ∈ {1,...,dk} and k ∈ {1,...,K}. Considering i.i.d
by all transmitters. The received signal at receiver k in these Gaussian input of sik[m], the achievable sum rate is given by
two phases can thus be modeled by
R [m]
sum
 
Wk,k[m]xk[m]+nk[m], m∈Pk =(cid:88) 1 log 1+ NdkP (cid:12)(cid:12)uik[m]HWk,k[m]vki[m](cid:12)(cid:12)2.
yk[m]=Wk,k(cid:80)[m]xkW[m]+[m]x [m]+n [m], elsewhere k,i N 2 Ik1,i[m]+Ik2,i[m]+1 
k(cid:54)=(cid:96) k,(cid:96) (cid:96) k
(6)
where P denotes the pilot position indices of user k. The where
k
vector xk[m] ∈ CN×1 denotes the transmitted symbol for I1 [m]=(cid:88)NP (cid:12)(cid:12)ui[m]HW [m]vj[m](cid:12)(cid:12)2, and (7)
userk withpowerconstraintE{x [m]Hx [m]}=PN,where k,i d (cid:12) k k,k k (cid:12)
k k k
j(cid:54)=i
P is the transmit power per subcarrier. Additive complex
symmetricGaussiannoiseatreceiverkisdenotedbynk[m]∼ I2 [m]=(cid:88)(cid:88)d(cid:96) NP (cid:12)(cid:12)ui[m]HW [m]vj[m](cid:12)(cid:12)2, (8)
CN(0,I ). The SNR is defined as SNR=P. k,i d (cid:12) k k,(cid:96) (cid:96) (cid:12)
N (cid:96)
(cid:96)(cid:54)=kj=1
In this work we consider a user velocity and carrier fre-
denote inter-stream interference and inter-user interference,
quencysuchthattheDopplerbandwidthofthefadingprocess
respectively.
f is much smaller than the subcarrier spacing f = 1 .
D sc TcN The precoding and decoding vectors can be designed
Hence, the inter-carrier interference resulting from Doppler
according to [1]. Each transmitter computes the precoding
shift is small enough to be neglected for the processing at
vectors vi[m] such that the interference signals from the
the receiver side, see the discussion in [21, Sec. II] and [22, k
undesiredK−1transmittersarealignedatallreceiversleaving
Sec. II].
the interference free subspace for the intended signal. With
Throughout the paper, we adopt a widely used assumption,
perfect CSI, the following IA conditions should be satisfied
where all channels h ∀k,(cid:96) have the same gain and time-
k,(cid:96)
selective fading statistics. Transmit power P is assumed for uik[m]HWk,k[m]vkj[m]=0, ∀k, ∀i(cid:54)=j (9)
all transmitters. ui[m]HW [m]vj[m]=0, ∀k (cid:54)=(cid:96), ∀i, j (10)
k k,(cid:96) (cid:96)
(cid:12)(cid:12)uik[m]HWk,k[m]vki[m](cid:12)(cid:12)≥c>0, ∀k, i (11)
where c is a constant. Accordingly, the interference terms can
A. SISO Interference Alignment with Perfect CSI be perfectly canceled satisfying I1 [m]=I2 [m]=0.
k,i k,i
IA can achieve optimal DoF when infinite channel exten- III. REDUCED-RANKCHANNELESTIMATIONAND
sions exist [1]. Using IA over N orthogonal subcarriers, each PREDICTION
transmitter k sends a linear combination of d <N symbols
k In this section, we introduce the idea of channel prediction.
si[m], along the linear precoding vectors vi ∈ CN×1,
k k First, a well-known minimum mean square error (MMSE)
yielding
solution is given. In subsection III-A, we present the reduced-
rank predictor and its relation to the MMSE solution. To
(cid:88)dk
x [m]= vi[m]si[m], (4) simplify notations, we drop the indices of transmitters and
k k k
receivers and focus on the prediction problem for a specific
i=1
subcarrier. Let us denote by w[m,n], n[m,n] y[m,n] and
where si[m] ∈ C denotes the transmitted symbol and x[m,n] the n-th element of the vector w[m] n[m] y[m] and
E{(cid:12)(cid:12)sik[mk](cid:12)(cid:12)2} = PN/dk. The precoding vector vki[m] fulfills xov[mer],tirmesepceacntivbeelyw.Trihtteenchaasnnelsamplesofthen-thsubcarrier
(cid:13)(cid:13)vi[m](cid:13)(cid:13)2 =1. Defining the decoding vector ui[m]∈CN×1
subkject to (cid:13)(cid:13)ui[m](cid:13)(cid:13)2 =1, the received signal atkreceiver k for gn =[w[0,n],...,w[M −1,n]]T, (12)
k
IEEETRANSACTIONSONVEHICULARTECHNOLOGY,ACCEPTEDFORPUBLICATION. 4
where M is the length of a single data block. B. The Choice of Subspace Dimension - An Upper Bound
A number of N pilot symbols x[m,n] ∈
√ √ P In wireless communication systems, detailed second-order
{ P,− P},∀m ∈ P known at the receivers allow us
statistics are difficult to obtain due to the short time-interval
to acquire channel knowledge. With the pilot symbols, we
over which the channel can be assumed to be stationary [25].
obtain the noisy channel observations at m ∈ P according
For this reason, we assume incomplete second-order statistics
to w(cid:48)[m,n] = 1y[m,n]x[m,n] = w[m,n] + √1 n(cid:48)[m,n],
P P in this work, where only the support W = (−νD,νD) of the
where n(cid:48)[m,n] = √1 n[m,n]x[m,n] has the same statistical Doppler spectrum is known to the transmitters and receivers
P
properties as n[m,n]. The noisy observation vector of the withν (cid:28)1/2.Forthecaseofunknownsupport,pleaserefer
D
n-th subcarrier over time toadaptivechannelestimationusinghypothesestest[22],[26].
The shape of the Doppler spectrum is assumed to be flat with
(cid:20) (cid:21)T
g(cid:48)n = w(cid:48)[0,n],...,w(cid:48)[M −1,n] (13) support W, which is given by
(cid:40)
1 , ν ∈W
is used for channel prediction. Defining the M × 1 vector S (ν,W)= |W| (19)
flat
0, otherwise.
r [m] = [R [m],R [m − 1],...,R [m − M + 1]], the
h h h h
estimator minimizing the MSE can be derived as [23]
The covariance function of such a fading process becomes
1
w˜MMSE[m,n]=r(hP)[m]H(R(hP)+ PINP)−1g(cid:48)n(P) (14) Rflat[m,W]= sinπ(2mπ|mWν|D). (20)
where the covariance matrix R(hP) ∈CNP×NP of the channel The corresponding covariance matrix is [Rflat[W]]l,m =
at pilot positions is obtained as a sub-matrix of Rh ∈ by R [l−m,W] for l,m ∈ [0,...,M −1]. The eigenvector
extracting K-spaced rows and/or columns, i.e. [R(hPk,)(cid:96)]i,m = offltahte covariance matrix Rflat[m,W] are also known as DPS
[R ] . The vectors g(cid:48)n(P) contains the sequences [27], [28], which are utilized as the basis functions
hk,(cid:96) K(i−1)+i,K(m−1)+m
respective elements for m ∈ P in the same order as in (12). u [W] in this paper. The band-limiting region of the DPS
p
The N ×1 vector r(P)[m] contains the respective elements sequencesu [W]ischosenaccordingtothesupportW ofthe
P h p
of R [m−m ] for m ∈P in the same order as r [m]. Dopplerspectrumofthetime-selectivefadingprocess.Toease
h P P h
the notation, we drop W in the rest of the paper. Given u ,
p
[19,Sec.III.D]showsthattheDPSsequencescanbeextended
A. Reduced-Rank Channel Predictor over Z in the minimum-energy band-limited sense, enabling
channel prediction in (18). The energy of the DPS sequences
The channel gn can be approximated by a reduced rank
is most concentrated in the interval of block length M. This
representation[19],[24],whichexpandsgn byDorthonormal
energy concentration is defined as
basisfunctionsu =[u [0],...,u [M−1]]T,p∈{0,...,D−
p p p
1} M−1
D(cid:88)−1 (cid:88) |up[m]|2
gn ≈Uφn = φnpup, (15) κ = m=0 . (21)
p ∞
p=0 (cid:88) |u [m]|2
p
where U = [u0,...,uD−1] collects D basis vectors of a m=−∞
temporal covariance matrix R and φn = [φn,...,φn ]
h 0 D−1 Thevaluesκ areclusterednear1forp≤(cid:100)2ν M(cid:101)anddecay
contains the subspace coefficients for the channel gn. p D
rapidly for p > (cid:100)2ν M(cid:101). The optimal subspace dimension
Let us define f[m] = [u [m],...,u [m]]T, which col- D
0 D−1 thatminimizesthemeansquareerror(MSE)foragivennoise
lects the values of the basis functions at time m. The estimate
level is found to be [19]
of φn can be calculated according to
 
φ˜n =G−1 (cid:88) w(cid:48)[m,n]f[m]∗, (16) Dub = argmin |W1|M M(cid:88)−1κp+ MDP. (22)
m∈P D∈{1,...,M} p=D
=G−1U(P)Hg(cid:48)n(P) (17)
Later in Sec. V we will see that D is the upper bound of
ub
the subspace dimension when quantized feedback is used.
where G = (cid:80) f[m]f[m]H = U(P)HU(P) and U(P) =
m∈P Remark 1. The reduced-rank channel prediction is a close
[u(P),...,u(P) ]. The vector u(P) contains the respective
0 D−1 p approximate of the MMSE predictor, especially at high SNR.
elements for m ∈ P in the same order as in (12). Thus, the
At high SNR (P → ∞), the MMSE predictor converges
estimated (predicted) n-th subchannel at time instant m ∈ Z
to a maximum-likelihood (ML) predictor i.e., w˜ [m,n] =
is given by ML
r(P)[m]HR−1(P)g(cid:48)n(P). For the reduced-rank predictor, more
h h
D−1 basis functions tend to be taken as P → ∞ according to
w˜[m,n]= (cid:88)φ˜nu [m]=f[m]Tφ˜n. (18) (22). Therefore, it also converges to a ML predictor due to
p p
p=0 the relationship shown in [19, Eq.38].
IEEETRANSACTIONSONVEHICULARTECHNOLOGY,ACCEPTEDFORPUBLICATION. 5
The delay domain coefficients can also be decomposed into
Pilots Payload
Downlink two parts as
M T
t
[γ˜1,...,γ˜N]=[γ1,...,γS,0 ]+[γ¨1,...,γ¨N]
D×(N−S)
1 … K 1 … K …… 1 … K (27)
where
Feedback delay
Uplink [γ1,...,γS,0 ]T =DH[φ1,...,φN]T, (28)
TD t D×(N−S) N
and
Fig.1. Signalingmodel,whereM denotesthelengthofthepilotsequence [γ¨1,...,γ¨N]T =DH[φ¨1,...,φ¨N]T. (29)
andT thepayloadlength. N
Eachvector γs containsthe subspacecoefficients correspond-
ing to the s-th channel tap hs[m]. The entries of φ¨n are a set
IV. TRAININGANDFEEDBACKFORIA of random Gaussian variables with the covariance matrix
(cid:32) (cid:33)
In this section, we consider a limited feedback scheme I
for the subspace coefficients φ˜n estimated at the receiver E{φ¨nφ¨nH}=G−1U(P)H MK U(P)G−1 (30)
P
side. Figure 1 shows the working principle of the feedback
1
system. The subspace coefficients are estimated using the = G−1. (31)
pilot symbols. Each receiver estimates the channels to all K P
transmitters separately. To this end, the pilot symbols from Due to uncorrelated noise over N subchannels, {φ¨1,...,φ¨N}
d d
different transmitters are orthogonalized in time. The number is a set of uncorrelated random Gaussian variables
ofpilotsymbolsM foreachtransmittersatisfiesM =KM . with the same distribution. The same is also true for
p p
The pilot placement for the k-th transmitter is defined as {γ¨1,...,γ¨N} according to Wiener-Khinchine theorem. Since
d d
only {γ˜1,...,γ˜S} are relevant to the actual channel realiza-
(cid:8) (cid:9)
P = k+(i−1)K, where i∈{1,...,M } . (23)
k p tions and thus have to be fed back to the transmitters. By
neglecting {γ˜S+1,...,γ˜N}, the SNR can be increase by a
Error-free dedicated broadcast channels with delay T are
D
factor of N/S.
assumed from each receiver to all the other nodes, i.e. all the
Toestablishatractableanalysis,weassumethatthenumber
transmittersandallotherreceivers.Duringthefeedbackphase,
of delay taps S is known at the receiver side. For practical
each receiver broadcasts the estimated subspace coefficients
wireless channels, the number of taps can be estimated via
using N bits. Upon reception of the quantized feedback, the
d
most significant tap detection [29], [30].
transmitters and receivers can calculate the IA precoders and
decoders, respectively.
B. Reformulation of Subspace Representation for the SISO
Interference Channels
A. Noise Reduction
With the basis coefficients {γ˜1,...,γ˜S} obtained from
Assuming a wide-sense-stationary fading process, the N (24),thepredictedchannelimpulseresponsecanbecalculated
narrowband channels from the same transmitter receiver pair as
havethesameDopplerbandwidth,thusallN fadingprocesses h˜[m]=[γ˜1,...,γ˜S]Tf[m]
(32)
share the same set of basis functions. Due to the fact that =F[m]η˜
N ≥ S, the impulse response h[m] contains less coefficients
where F∈CS×DS and η˜∈CDS×1 are defined as follows:
thanthefrequencyresponsew[m].Thus,h[m]isbettersuited
forCSIfeedback.Foranoisychannel,theequivalentsubspace  f[t]T 0T ... 0T 
D×1 D×1
coefficients in the delay domain can be expressed as  0T f[t]T ... 0T 
[γ˜1,...,γ˜N]T =DH[φ˜1,...,φ˜N]T. (24) F[m]= D...×1 ... ... D...×1  (33)
N
0T 0T ... f[t]T
D×1 D×1
Among all N taps of the estimated channel, only the first
and
S columns {γ˜1,...,γ˜S} are from the contribution of the
 γ˜1 
channel, while the rest N − S columns are due to noise
.
impairment. Hence, the elimination of these channel taps can η˜= . . (34)
 . 
improve the SNR. This can be seen by rewriting (16) as γ˜S
φ˜n =φn+φ¨n (25) Note that the vector η˜ is not unique since the achievable rate
(cid:88) (cid:88) for IA is invariant to a norm change and phase rotation of η˜
=G−1 w[m,n]f[m]∗+G−1 n(cid:48)[m,n]f[m]∗
(this will be shown in Sec. V). Therefore, it is equivalent to
m∈P m∈P know η˜ or αη˜ at the transmitter side, where α ∈ C. Thus,
(26)
the CSI feedback problem becomes feeding back a point on
where the subspace coefficients φ˜n are decomposed into two the Grassmannian manifold G , which can be realized by
DS,1
parts, corresponding to the channel and noise, respectively. vector quantization.
IEEETRANSACTIONSONVEHICULARTECHNOLOGY,ACCEPTEDFORPUBLICATION. 6
C. Vector Quantization and Feedback The corresponding RVQ codebook Crnd contains 2Nd unit-
After the subspace coefficient vector η˜ ∈ CDS×1 is ob- norm vectors, i.e. Crnd = {ηˆ1,...,ηˆ2Nd}. Using codebook
tained at the receiver side, the receiver quantizes the vector Crnd, the quantized version of η˜ can be obtained as
according to its codebook and broadcasts the index to the ηˆ=argmax|ηˆHη˘|. (41)
transmitter side through a feedback channel using N bits. i
d ηˆi∈Crnd
If the vector η˜ has correlated entries, the design of the
After receiving the feedback information, the transmitters can
optimal codebook is difficult. In the special case with i.i.d en-
reconstructthequantizedvectorbyaddingthecorrelationΣηˆ.
tries, the optimal codebook for quantization can be generated
numerically using the Grassmannian line-packing approach
[31], [32]. However, it is still challenging to find the optimal V. RATELOSSANALYSIS
codewordswhichachievethequantizationboundpromisedby In this section, we analyze the rate loss of our pro-
[33], except for some specific cases (e.g. lower dimensional posed scheme. We decouple the channel prediction- and
packing problems, such as vector quantization problems, with quantization-error and derive an upper bound of the rate loss.
small codebook size [34]). To overcome this, random vector We show that a smaller subspace dimension is favorable for
quantization (RVQ) codebooks are proposed. The codewords quantization due to less coefficients. On the other hand, more
of Crnd are independent unit-norm vectors from the isotropic subspace coefficients will reduce the prediction error for the
distribution on the complex unit sphere [11], [35]. RVQ is case of D > 1. Therefore, there exists a tradeoff between
ub
commonly used to analyze the effects of quantization because quantization error and prediction error when selecting the
it is mathematically tractable and asymptotically optimal with subspace dimension at a given feedback rate. We develop
a distortion on the order of 2−DNSd−1. a subspace dimension switching algorithm to find the best
When the vector to be quantized has correlated entires, tradeoff such that the rate loss upper bound is minimized.
a better codebook can be designed by skewing the RVQ We also characterize the scaling of the required number of
codebook to match the correlation structure [36], [37]. The feedbackbitsinordertodecoupletheratelossduetoimperfect
skewed codebook yields performance gain compared to the quantization from the transmit power.
RVQ codebook. However, the exact characterization of the
quantization error has remained an open question. A recent
A. Leakage Interference Due to Imperfect CSI
study in [38] tries to derive the SNR loss for single user
Imperfect CSI results in residual interference, thus, IA
MIMO beamforming system using skewed codebooks, but it
conditions(9)and(10)cannotbesatisfied.Uponreceptionof
still remains in terms of expectations of eigenvalues.
the quantized subspace vector ηˆ, the quantized channel hˆ[m]
Toovercomethedifficultyinobtaininganalyticalresults,we
can be constructed in the same way as (32). The precoding
utilize a RVQ codebook, which will facilitate our analytical
vector vˆi[m] and decoding vector uˆi[m] are calculated using
performance analysis in Sec. V. Considering a factorization k k
of the covariance matrix Rη˜ = ΣΣH, the vector η˜ can be wˆk,(cid:96)[m] = DN×Shˆk,(cid:96)[m] as the true channel, which results
whiten as in
η˘=Σ−1η˜. (35) uˆik[m]HWˆ k,k[m]vˆkj[m]=0, ∀k, ∀i(cid:54)=j (42)
The covariancematrix Rη˜ =Bdiag(λ1,...,λS) contains the uˆik[m]HWˆ k,(cid:96)[m]vˆ(cid:96)j[m]=0, ∀k (cid:54)=(cid:96), ∀i, j (43)
(cid:12) (cid:12)
covariance matrix of the subspace coefficients for each tap s, (cid:12)uˆi[m]HWˆ [m]vˆi[m](cid:12)≥c>0, ∀k, i. (44)
(cid:12) k k,k k (cid:12)
i.e.
We modify the upper bound for the average loss in sum rate
λs =E{γ˜sγ˜sH} (36)
[12] for a time-variant channel as
=E{[φ˜1,...,φ˜N]d∗dT[φ˜1,...,φ˜N]H} (37)
=G−1U(P)HE{[g(cid:48)1s,.s..,g(cid:48)N]d∗ ∆R< N1T(cid:88) (cid:88) log2(cid:0)1+E(cid:2)Ik1,i[m]+Ik2,i[m](cid:3)(cid:1). (45)
s
k,i m∈T
dT[g(cid:48)1,...,g(cid:48)N]H}U(P)G−1 (38)
s (cid:18) (cid:19) We define bˆi,j[m] = uˆi[m]∗ ◦ vˆj[m] as the Hadamard
=G−1U(P)H psR(hP)+ P1IMK U(P)G−1. (39) vpˆrjo[mdu]c.tTohfetlheeakkda,eg(cid:96)ceoidnitnegrfevreeckntcoeriunˆik(7[m)(cid:96)a]nadn(d8)pcreacnobdeinrgewvreicttteonr
where d is the s-th column of the DFT matrix D . The (cid:96)
s N×S as
equations (37), (38) and (39) are due to (24), (17) and (3),
respectively. I1 [m]=(cid:88)NP (cid:12)(cid:12)w [m]Tbˆi,j [m](cid:12)(cid:12)2, and (46)
k,i d (cid:12) k,k k,k (cid:12)
WedonotassumeexactknowledgeofthePDPinourwork, k
i(cid:54)=j
insteaEd{ahfl[mat]hP[DmP]Ha}ss=umdpitaiogn((cid:2)p1,...,pS(cid:3))= NSIS (40) Ik2,i[m]=(cid:88)k(cid:54)=(cid:96)(cid:88)jd=(cid:96)1 NdP(cid:96) (cid:12)(cid:12)(cid:12)wk,(cid:96)[m]Tbˆik,,j(cid:96)[m](cid:12)(cid:12)(cid:12)2. (47)
is used. In case the assumed PDP matches the PDP of the We define the predicted channel frequency response as
truechannel,thevectorη˘willbeisotropicallydistributedwith
w˜ [m]=[w˜1 [m],...,w˜n [m]]T (48)
uncorrelated entries. k,(cid:96) k,(cid:96) k,(cid:96)
IEEETRANSACTIONSONVEHICULARTECHNOLOGY,ACCEPTEDFORPUBLICATION. 7
and the prediction error as z˜ [m]=w [m]−w˜ [m]. The
k,(cid:96) k,(cid:96) k,(cid:96)
average power leakage of the inter-stream interference in (46)
NP (cid:20)(cid:12) (cid:12)2(cid:21)
can be upper bounded by I˜i,j[m]= E (cid:12)z˜ [m]Tbˆi,j [m](cid:12)
k,k d (cid:12) k,k k,k (cid:12)
E=(cid:2)I(cid:88)k1,i[NmP](cid:3)E(cid:20)(cid:12)(cid:12)w [m]Tbˆi,j [m](cid:12)(cid:12)2(cid:21) (49) =NdkkPE(cid:20)(cid:12)(cid:12)(cid:12)e˜k,k[m]Hqˆik,,jk[m](cid:12)(cid:12)(cid:12)2(cid:21) (54)
i(cid:54)=j dk (cid:12) k,k k,k (cid:12) ≈NPE(cid:20)qˆi,j [m]HE(cid:2)e˜ [m]e˜ [m]H(cid:3)qˆi,j [m](cid:21)
=(cid:88)NdPE(cid:20)(cid:12)(cid:12)(cid:12)(cid:0)w˜k,k[m]T+z˜k,k[m]T(cid:1)bˆik,,jk[m](cid:12)(cid:12)(cid:12)2(cid:21) (50) dk k,k k,k k,k k,k (55)
i(cid:54)=j k N2P (cid:13) (cid:13)2 (cid:20) NP(cid:21)
=(cid:88)NP(cid:32)E(cid:20)(cid:12)(cid:12)w˜ [m]Tbˆi,j [m](cid:12)(cid:12)2+(cid:12)(cid:12)z˜ [m]Tbˆi,j [m](cid:12)(cid:12)2 = Sdk E(cid:13)(cid:13)qˆik,,jk[m](cid:13)(cid:13) ·MSE m,D, S (56)
dk (cid:12) k,k k,k (cid:12) (cid:12) k,k k,k (cid:12) =J˜i,j[m]
i(cid:54)=j k,k
(cid:16) (cid:17)(cid:21)(cid:33)
+2Re w˜ [m]Tbˆi,j [m]bˆi,j [m]Hz˜ [m]∗ (51) where (54) is due to Parseval’s theorem. In order for tractable
k,k k,k k,k k,k
results, we arrive at (55) by assuming the independence of
≈(cid:88)(cid:32)NdkPE(cid:20)(cid:12)(cid:12)(cid:12)z˜k,k[m]Tbˆik,,jk[m](cid:12)(cid:12)(cid:12)2(cid:21) q(ˆqˆik,ik,j,(cid:96),j(cid:96)[m[m]]anisdae˜kf,u(cid:96)n[mcti]o,nwhoifchhˆkis,(cid:96)t[hme]caasnedfothretrheefoMreMh˜SkE,(cid:96)[pmre]d.iTcthoer
i(cid:54)=j (cid:124) (cid:123)(cid:122) (cid:125) prediction error e˜ [m] is independent of h˜ [m]). For our
k,(cid:96) k,(cid:96)
I˜ki,,jk[m] reduced-rank predictor, it still provides a good approximation
NP (cid:20)(cid:12) (cid:12)2(cid:21)(cid:33) duetothecloserelationbetweenareduced-rankpredictorand
+ E (cid:12)w˜ [m]Tbˆi,j [m](cid:12) (52)
d (cid:12) k,k k,k (cid:12) the MMSE predictor (see discussion in Remark 1). Equation
(cid:124) k (cid:123)(cid:122) (cid:125) (56) follows from E(cid:2)e˜ [m]e˜ [m]H(cid:3) = E(cid:107)e˜k,k[m](cid:107)2I ,
Iˆki,,jk[m] where (cid:107)e˜k,k[m](cid:107)2 = (cid:107)z˜kk,k,k[m](cid:107)k2,k. The MSE per SsubchanS-
nel E(cid:2)|z˜ [m,n]|2(cid:3), ∀n ∈ {1,...,N}, is the sum of a
where (52) is obtained by ignoring the term k,k
E(cid:104)Re(cid:16)w˜ [m]Hbˆi,j[m]bˆi,j[m]Hz˜ [m](cid:17)(cid:105). An equality square bias and a variance term [19] MSE[m,D,SNR] =
k,(cid:96) k,(cid:96) k,(cid:96) k,(cid:96) bias2[m,D] + var[m,D,SNR] where the variance can be
holds from (51) to (52) with the MMSE predictor (39),
approximated by
where z˜ [m] is zero-mean Gaussian and independent
k,(cid:96)
of w˜ [m]. However, for our reduced-rank predictor,
k,(cid:96) f[m]TG−1f[m]
we notice that an exact characterization of this var[m,D,SNR]= . (57)
(cid:104) (cid:16) (cid:17)(cid:105) SNR
term E Re w˜ [m]Hbˆi,j[m]bˆi,j[m]Hz˜ [m] is
k,(cid:96) k,(cid:96) k,(cid:96) k,(cid:96)
mathematically intractable. As discussed in Remark 1, The square bias term is calculated as [19]
the reduced-rank predictor is closely related to the MMSE
predictor.Wealsofoundviasimulationthatthistermisrather bias2[m,D]
small. Similar to (52), the inter-user interference term in (47)
can be upper bounded by (cid:90)12 (cid:12)(cid:12) (cid:88) (cid:12)(cid:12)2
= (cid:12)1−f[m]TG−1 f[(cid:96)]e−j2πν(m−(cid:96))(cid:12) S (ν)dν (58)
(cid:12) (cid:12) h
E(cid:2)I2 [m](cid:3)≈(cid:88)(cid:88)d(cid:96) (cid:32)NPE(cid:20)(cid:12)(cid:12)z˜ [m]Tbˆi,j[m](cid:12)(cid:12)2(cid:21) −12 (cid:12) (cid:96)∈P (cid:12)
k,i d (cid:12) k,(cid:96) k,(cid:96) (cid:12)
(cid:96)
k(cid:54)=(cid:96)j=1
(cid:124) (cid:123)(cid:122) (cid:125) where S (ν) denotes the actual power spectral density of the
h
I˜ki,,j(cid:96)[m] fading process. Due to the removal of the noise terms in (24),
NP (cid:20)(cid:12) (cid:12)2(cid:21)(cid:33) the noise variance is reduced by a factor of N/S, therefore
+ d E (cid:12)(cid:12)w˜k,(cid:96)[m]Tbˆik,,j(cid:96)[m](cid:12)(cid:12) . (53) resulting in an SNR of NP in equation (56).
(cid:96) S
(cid:124) (cid:123)(cid:122) (cid:125) A similar result can be obtained for the first term of the
Iˆki,,j(cid:96)[m] inter-user interference in (53), i.e.
In the following Sections V-B and V-C, we will show N2P (cid:13) (cid:13)2 (cid:20) NP(cid:21)
that the first and second terms in (52) and (53) are caused I˜i,j[m]≈J˜i,j[m]= E(cid:13)qˆi,j[m](cid:13) MSE m,D, .
k,(cid:96) k,(cid:96) Sd (cid:13) k,(cid:96) (cid:13) S
by the channel prediction error and the quantization error, (cid:96)
(59)
respectively.
B. Leakage Interference Due to Channel Prediction Error C. Leakage Interference Due to Channel Quantization Error
Defining [qˆi,j [m]T,qi,j [m]T]T = DHbˆi,j [m]∗, where
k,k k,k N k,k To obtain tractable expressions, we restrict the subsequent
qˆi,j [m] ∈ CS×1 and qi,j [m] ∈ C(N−S)×1, and analysis to a flat PDP, such that the vector to be quantized
k,k k,k
[e˜ [m]T,0 ]T = DHz˜ [m], the first term of the η˘ is isotropically distributed with uncorrelated entires. The
xk,k 1×(N−S) N k,k k,k
inter-stream interference I˜i,j[m] in (52) can be written as interference leakage caused by the quantization error Iˆi,j[m]
k,k k,k
IEEETRANSACTIONSONVEHICULARTECHNOLOGY,ACCEPTEDFORPUBLICATION. 8
(cid:104) (cid:16) (cid:17)(cid:105)
in (52) can be rewritten as E d2 η˘k,(cid:96) ,ηˆ in (67) is the expectation of the quan-
c (cid:107)η˘k,(cid:96)(cid:107) k,(cid:96)
NP (cid:20)(cid:12) (cid:12)2(cid:21) tization error. As shown in [33], for quantizing a vector
Iˆi,j[m]= E (cid:12)w˜ [m]Tbˆi,j [m](cid:12) arbitrarily distributed on the Grassmannian manifold G
k,k d (cid:12) k,k k,k (cid:12) DS,1
k using RVQ, the second moment of the chordal distance using
NP (cid:20)(cid:12) (cid:12)2(cid:21)
= E (cid:12)h˜ [m]Hqˆi,j [m](cid:12) (60) Nd quantization bits can be bounded as
d (cid:12) k,k k,k (cid:12)
k Γ( 1 )
=NdPE(cid:20)(cid:12)(cid:12)(cid:12)qˆik,,jk[m]HFk,k[m]Σk,kη˘k,k(cid:12)(cid:12)(cid:12)2(cid:21). (61) Q(Nd)≤ DDSS−−11 (c2Nd)−DS1−1 , (68)
k
where Γ(·) denotes the Gamma function.
Since ηˆk,k is the quantized version of η˘k,k and Furthermore, we have
(cid:107)ηˆ (cid:107) = 1, from Parseval’s theorem we have
k,k
qˆik,,jk[m]HFk,k[m]Σk,kηˆk,k = 0. We can define an (cid:13)(cid:13)qˆi,j [m]HF [m]Σ (cid:13)(cid:13)2 =(cid:88)S (cid:13)(cid:13)qˆi,j [m,s]λ f [m](cid:13)(cid:13)2
orthonormal basis in CDS as (cid:13) k,k k,k k,k(cid:13) (cid:13) k,k k,k k,k (cid:13)
s=1
  (cid:13) (cid:13)2
ηˆk,k,(cid:13)(cid:13)ΣΣk,kFFk,k[[mm]]HHqaˆˆiik,,,jjk[[mm]](cid:13)(cid:13),a1,a2,...,aDS−2 , =(cid:13)(cid:13)qˆik,,jk[m](cid:13)(cid:13) (cid:107)λk,kfk,k[m](cid:107)2(69)
(cid:13) k,k k,k k,k (cid:13)
(62) and
where [a ,a ,...,a ] is an orthonormal basis of
(cid:32)(cid:20) 1 2 DS−2 (cid:21)H(cid:33) E(cid:107)η˜k,k(cid:107)2 =DS, (70)
null ηˆ , Σk,kFk,k[m]Hqˆik,,jk[m] . We can decompose
k,k (cid:107)Σk,kFk,k[m]Hqˆik,,jk[m](cid:107) accordingtoAppendixA.Pluggingtheaboveresultsinto(67),
η˘ into the above orthonormal basis, i.e. the inter-stream interference leakage caused by quantization
k,k error Iˆi,j[m] can be finally bounded by
k,k
(cid:107)η˘k,k(cid:107)2 =(cid:12)(cid:12)ηˆkH,kη˘k,k(cid:12)(cid:12)2+ Iˆi,j[m]≤Jˆi,j[m]
(cid:12) (cid:12)2 k,k k,k
(cid:12)(cid:12)(cid:12)(cid:12)(cid:13)(cid:13)qqˆˆiki,,,jjk[[mm]]HHFFk,k[[mm]]ΣΣk,k(cid:13)(cid:13)η˘k,k(cid:12)(cid:12)(cid:12)(cid:12) +D(cid:88)S−2(cid:12)(cid:12)aHmη˘k,k(cid:12)(cid:12)2. (63) =dkN(DPSD−S1)E(cid:13)(cid:13)(cid:13)qˆik,,jk[m](cid:13)(cid:13)(cid:13)2(cid:107)λk,kfk,k[m](cid:107)2Q(Nd).
(cid:12)(cid:13) k,k k,k k,k(cid:13) (cid:12) m=1 (71)
Inserting (63) into (61) yields Accordingly, the inter-user interference can be bounded as
NP (cid:20)(cid:12) (cid:12)2(cid:21) Iˆi,j[m]≤Jˆi,j[m]
E (cid:12)qˆi,j [m]HF [m]Σ η˘ (cid:12) (64) k,(cid:96) k,(cid:96)
d (cid:12) k,k k,k k,k k,k(cid:12) NPDS (cid:13) (cid:13)2
k = E(cid:13)qˆi,j[m](cid:13) (cid:107)λ f [m](cid:107)2Q(N ).
NP (cid:34)(cid:13) (cid:13)2 d(cid:96)(DS−1) (cid:13) k,(cid:96) (cid:13) k,(cid:96) k,(cid:96) d
= E (cid:13)qˆi,j [m]HF [m]Σ (cid:13) (72)
d (cid:13) k,k k,k k,k(cid:13)
k
(cid:13) (cid:13)2
(cid:32) DS−2 (cid:33)(cid:35) The only stochastic part in the equation is (cid:13)qˆi,j[m](cid:13) , whose
(cid:107)η˘k,k(cid:107)2−|ηˆkH,kη˘k,k|2− (cid:88) (cid:12)(cid:12)dHmη˘k,k(cid:12)(cid:12)2 (65) value relies on the applied IA algorithm. (cid:13) k,(cid:96) (cid:13)
m=1
NP (cid:13) (cid:13)2 Theorem 1. When the proposed prediction and limited feed-
= d (DS−1)E(cid:13)(cid:13)qˆik,,jk[m]HFk,k[m]Σk,k(cid:13)(cid:13) back strategy is used for IA CSI feedback, the average rate
k loss due to channel prediction and quantization can be upper
(cid:104) (cid:105)
×E (cid:107)η˘k,k(cid:107)2−|ηˆkH,kη˘k,k|2 (66) bound by
= dk(DNSP−1)E(cid:13)(cid:13)(cid:13)qˆik,,jk[m]HFk,k[m]Σk,k(cid:13)(cid:13)(cid:13)2E(cid:107)η˘k,k(cid:107)2 ∆R(cid:46)∆Rub = N1T (cid:88) (cid:88) dklog2(cid:32)1+NP (cid:18)K− d1 (cid:19)·
(cid:20) (cid:18) η˘ (cid:19)(cid:21) k m∈T k
×E d2c (cid:107)η˘kk,,kk(cid:107),ηˆk,k (67) (cid:18)NMSE(cid:20)m,D,NP(cid:21)+ DSζ[m]Q(Nd)(cid:19)(cid:33),
S S DS−1
(cid:112)
where d (x ,x ) = (1−|xHx |2) is the chordal dis-
c 1 2 1 2 (73)
tance between two unit norm vectors x and x . Equa-
1 2
tion (66) follows from the fact that the quantization er- where ζ[m]=(cid:107)λ f [m](cid:107)2.
k,(cid:96) k,(cid:96)
ror is isotropic in the nullspace of ηˆk,k and there- Proof: Equation (73) is obtained by inserting (56),
fore the average power of η˘k,k in each dimension of (59), (71) and (72) into (45) and using the fact
(cid:26) (cid:27)
taino(cid:107)gnΣΣle(kk6,,okk7fFF)kkη˘f,,okk[[lmmlo.]]wHHqqˆˆsikik,,f,,jjkkro[[mmm]](cid:107)t,hde1i,ndd2e,p.e.n.d,ednDceS−of2theisneoqrumala.nEdquthae- (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)qqˆˆikik,,,,jj(cid:96)(cid:96)[[mm]](cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)22 ≤<(cid:13)(cid:13)(cid:13)bˆik,1,j(cid:96),[m∀]((cid:13)(cid:13)(cid:13)i2,k=,j(cid:80),(cid:96)Nn).=1T(cid:12)(cid:12)uhˆikis[mc,ann](cid:12)(cid:12)2b(cid:12)(cid:12)evˆj(cid:96)[smho,wn]n(cid:12)(cid:12)2a<s
Equationk,k(67) shows that the leakage interference can (cid:80)Nn=1(cid:12)(cid:12)uˆki[m,n](cid:12)(cid:12)2(cid:80)Nn=1(cid:12)(cid:12)vˆj(cid:96)[m,n](cid:12)(cid:12)2 =1.
be bounded by the chordal distance between the true and Remark 2. We notice that the rate loss upper bound (73)
the quantized subspace coefficients. The term Q(N ) = derivedusingthemethod[12]isknowntobelooseespecially
d
IEEETRANSACTIONSONVEHICULARTECHNOLOGY,ACCEPTEDFORPUBLICATION. 9
when CSI quality is poor, mainly due to the use of Jensen’s quantizationerrorusingrandomperturbations[45,Sec.VI.B],
(cid:13) (cid:13)2
inequality. Besides, we use the fact (cid:13)qˆi,j[m](cid:13) < 1, which which has been shown to be a good approximation of the
(cid:13) k,(cid:96) (cid:13)
quantization error using RVQ.
(cid:13) (cid:13)2
further loosens the bound. However, the term (cid:13)qˆi,j[m](cid:13)
(cid:13) k,(cid:96) (cid:13)
existsinboththepredictionandquantizationerrors,thususing A. Validation of The Rate Analysis
this inequality is not critical for the purpose of deriving a
First,weexaminetheeffectofimperfectchannelprediction
subspace switching algorithm in Section V-D, especially at
and quantization. Fig. 2 shows the power of leakage inter-
high SNRs.
ference (for a specific (k,(cid:96)) and (i,j)) versus the evolution
Theorem2. Thesumratelossduetothequantizationerrorcan of time for ν = 0.001 (6.05 km/h). The leakage powers
D
be bounded by a finite value when P −→ ∞, if the number due to prediction error and quantization error are shown
of feedback bits per receiver grows as respectively for MC simulations of I˜i,j[m] and Iˆi,j[m], and
k,(cid:96) k,(cid:96)
for the analytical upper bound J˜i,j[m] and Jˆi,j[m]. Note that
N =(DS−1)log P. (74) k,(cid:96) k,(cid:96)
d 2 (cid:13) (cid:13)2
there still exists a stochastic part E(cid:13)qˆi,j[m](cid:13) in Jˆi,j[m]
Proof: See Appendix B. (cid:13) k,(cid:96) (cid:13) k,(cid:96)
Note that practical implementation of large codebooks in and J˜ki,,j(cid:96)[m]. However, as explained in Remark 2, using the
(cid:13) (cid:13)2
order to achieve a low quantization error remains a long- upper bound (cid:13)qˆi,j[m](cid:13) < 1 has only a minor impact on
(cid:13) k,(cid:96) (cid:13)
standing problem and is still under investigation. Ongoing re-
the subspace switching algorithm, and thus the deterministic
searchtopicsthatarehelpfulincomplexityreductionincludes part of Jˆi,j[m] and J˜i,j[m] are more relevant. Therefore, we
progressive refinement [39] and hierarchical codebooks [40]. k,(cid:96) k,(cid:96)
(cid:13) (cid:13)2
take an empirical value of E(cid:13)qˆi,j[m](cid:13) from the simulation
(cid:13) k,(cid:96) (cid:13)
in order to make the comparison with the true leakage power.
D. Adaptive Subspace Dimension Switching Algorithm
We can observe that the leakage due to the prediction error
Whenthesubspacecoefficientsareunquantized,theoptimal
increases over time due to increased MSE. The leakage due
subspace dimension that minimizes the prediction error is
to the quantization error is almost a constant throughout the
given by (22). However, for D > 1, a subspace dimension
ub frame.Inaddition,theresultscorrespondingtoMCsimulation
higher than one is favorable for channel prediction, while
and the analytical upper bound are quite close. The sum of
resulting in a higher quantization error. Hence, a limited
bothleakagetermsisslightlyhigherthanthetrueinterference
feedback system exhibits a tradeoff between the quality of
leakage power due to the ignorance of the last term in (51).
channel prediction and quantization. The selection of the
For comparison, we include also the prediction error without
subspace dimension to find the best tradeoff becomes more
noise reduction as described in Sec. IV-A. It can be seen that
relevantandthus,aselectionmetricisneededforthispurpose.
thepredictionerrorislargerduetothehighernoiselevel.Note
The rate loss upper bound developed in (73) is suitable. We
that the interference leakage with non-flat PDPs (not shown)
propose an adaptive subspace dimension switching algorithm,
is similar to the one with flat PDP and matches well with the
which finds the subspace dimension minimizing (73), i.e.
analytical bound as well.
D = argmin ∆R . (75)
ub
D∈{1,...,Dub} B. Choice of Subspace Dimension
Inthissection,wedemonstratethesubspacechoiceandthe
VI. SIMULATIONRESULTS accuracy of the subspace dimension switching algorithm with
In this section, the sum rate of the proposed scheme is various SNRs, number of feedback bits and number of delay
evaluated through Monte-Carlo (MC) simulations. For the IA taps.
designinthissection,weusetheclosed-formIAalgorithm[1] Fig. 3 shows contour lines of the subspace dimension
over N = 5 channel extensions with an additional precoding obtained according to (75) as a function of SNR and the
subspace optimization [41], since it has been shown that the number of feedback bits. In order to demonstrate the sub-
original closed-from IA solution [1] yields low rate if no space dimension switching algorithm, we select ν = 0.004
D
further optimization is performed [41], [42]. We consider a (24.2km/h) for the simulation, which represents the velocity
K = 3 user interference channel, where each channel has of a slowly moving car. It can be seen that a higher subspace
S delay taps and a flat PDP E{h [m]h [m]H} = NI . dimension is suggested when both SNR and the number of
k,(cid:96) k,(cid:96) S S
Eachdelaytaph [m,s]istemporallycorrelatedaccordingto feedback bits are high. This is because higher SNR allows
k,(cid:96)
Clarke’s model [43] with R [m] = J (2πν m), where J forhighsubspacedimensionforchannelpredictionduetothe
hk,(cid:96) 0 D 0
is the 0-th order Bessel function of the first kind. The OFDM relativelysmallvarianceofareduced-rankpredictor.Thiswill
symbol rate 1/T = 1.4×104Hz is chosen according to the alsoresultinmoresubspacecoefficients,whichinturnrequire
s
3GPPLTEstandard[44].Thecarrierfrequencyf =2.5GHz. morebitsforfeedbacktomaintainalowquantizationerror.In
c
The normalized Doppler frequency is obtained as ν = caseofalowfeedbackrate,alowersubspacedimensionisstill
D
vf T /c , where c is the speed of light, and v is the relative favorable in order for a low quantization error, and therefore
c s 0 0
velocity between transmitter and receiver. In order to enable the best tradeoff between prediction and quantization.
the performance analysis with exponentially large codebooks, Fig. 4 illustrates the sum rate using the same setup as in
we replace the RVQ process by the statistical model of the Fig. 3, with subspace dimension D ∈ {1,2,3}, respectively.
IEEETRANSACTIONSONVEHICULARTECHNOLOGY,ACCEPTEDFORPUBLICATION. 10
0.2
I˜ik,j,(cid:2)[t]
0.18 Iˆik,j,(cid:2)[t]
Normalized by P000...111246 IIJJJT˜˜˜ˆ˜iikkiiiukkk,,,,,jj,,jjj,,,(cid:2)(cid:2)r(cid:2)(cid:2)(cid:2)[[[[[ettttt]]]]]L+w+eiaItˆJkˆihk,ikaj,,o(cid:2)j,g(cid:2)[u[tet]t]noisereduction e[bits/s/Hz] 1150 DD==32
e at
ag 0.1 m r
Leak0.08 Su 5
ce D=1
n0.06
e
er 0
erf0.04 0
Int0.02 20 40 60 80
60 40
0 SNR[dB] 20
0 5 10 15 20 25 30 35 40 45 80
OFDM Symbol Index Number of feedback bits N
d
Fig. 2. Evolution of Interference leakage with time at SNR=25dB and Fig. 4. Sum rate with subspace dimension D ∈ {1,2,3}, as a function
normalizedDopplerfrequencyνD=0.001.Thelengthofthepilotsequence of SNR and the number of feedback bits at normalized Doppler frequency
M = 15. The length of the payload T = 45. The number of channel taps νD =0.004. The length of the pilot sequence M =15. The length of the
S =3.ThenumberofsymbolextensionsN =5.Thenumberoffeedback payloadT =45.ThenumberofchanneltapsS=2.Thenumberofsymbol
bitsNd=15. extensionsN =5.
80
6
70 D=2
4
D=3
60
2
z] D=1
50 s/H 0 2
B] s/
SNR[d3400 D=2 um rate[bit --42 01..155 Sswuibtcshpiancge dimension D=2
S 5 10 15
D=1
20 -6
quantizedCSIη˜
10 D=1 -8 alodwpetr.SbDoSund(76)
unquantizedfeedback
0 -10
10 20 30 40 50 60 70 80 5 10 15 20 25 30 35
Number of feedback bits Number of feedback bits N
d
Fig. 3. Subspace dimension obtained according to (75), as a function of Fig.5. SumrateversusthenumberoffeedbackbitsatSNR=30dBandthe
SNR and the number of feedback bits at normalized Doppler frequency normalizedDopplerfrequencyνD=0.004.Thelengthofthepilotsequence
νD = 0.004. The length of the pilot sequence M = 15. The length of M = 15. The length of the payload T = 45. The number of channel taps
the payload T = 45. The number of channel taps S = 2. The number of S=3.ThenumberofsymbolextensionsN =5.
symbolextensionsN =5.
of the rate loss upper bound ∆R , as discussed in Remark
ub
It can be seen that the dimension suggested in Fig. 3 matches
2. For such a setting, (22) suggests that the optimal subspace
well with the dimension that achieves a higher rate.
dimension D is 2 for unquantized feedback. However, as
ub
Fig.5showsthesumrateversusthenumberoffeedbackbits
discussed earlier, higher dimension D will lead to a larger
atanSNR=30dBandthenormalizedDopplerfrequencyν =
D quantization error due to more subspace coefficients. To find
0.004 (24.2km/h). The lower bound of the average achievable
thebestsubspacedimension,wepresenttheachievedrateand
rate is defined as
the corresponding lower bound at both D ={1,2}. It can be
R =E(cid:2)Rperfect(cid:3)−∆R . (76) observedthattheachievedsumrateincreaseswiththenumber
lb sum ub
of feedback bits. For D =1, it achieves an initial higher rate
Due to the fact that the average sum rate given perfect CSI due to smaller quantization error. The achieved rate becomes
is a constant, we can use this lower bound to examine the a constant with the increase of N due to the dominance of
d
effectiveness of the subspace switching algorithm (75). Note the prediction error. When more than 15 bits are used, the
thattheratelowerboundcanbenegativeduetothelooseness two dimensional subspace outperforms the one dimensional