Robust Receiver Design for MIMO Single-Carrier
Block Transmission over Time-Varying Dispersive
Channels Against Imperfect Channel Knowledge
Chih-Yuan Lin, Member, IEEE, Jwo-Yuh Wu, Member, IEEE, and Ta-Sung Lee, Senior Member, IEEE
Abstract—We consider MIMO single-carrier block transmis-sion over time-varying multipath channels, under the assumption that the channel parameters are not exactly known but are estimated via the least-squares training technique. While the channel temporal variation is known to negate the tone-by-tone frequency-domain equalization facility, it is otherwise shown that in the time domain the signal signatures can be arranged into groups of orthogonal components, leading to a very natural yet efficient group-by-group symbol recovery scheme. To realize this figure of merit we propose a constrained-optimization based receiver which also takes into account the mitigation of channel mismatch effects caused by time variation and imperfect estima-tion. The optimization problem is formulated in an equivalent unconstrained generalized-sidelobe-canceller setup. This enables us to directly model the channel mismatch effect into the system equations through the perturbation technique and, in turn, to further exploit the statistical assumptions on channel temporal variation and estimation errors for deriving a closed-form solution. Within the considered framework the proposed robust equalizer can be combined with the successive interference cancellation mechanism for further performance enhancement. Flop count evaluation and numerical simulation are used to evidence the advantages of the proposed scheme.
Index Terms—MIMO, single carrier block transmission, time-varying multipath channels, channel estimation, constrained op-timization, generalized sidelobe canceller, perturbation analysis.
I. INTRODUCTION
M
ULTI-INPUT multi-output (MIMO) single-carrier (SC) block transmission with cyclic prefix (CP) has been identified as one key technique for supporting high data rates over frequency selective fading channels [1]; such a system configuration can be found in the uplink mode of the next-generation wireless communication standards like 3GPP-LTE [2], [3]. One particularly attractive feature unique to MIMO-SC systems is the low-complexity per-tone frequency-domain equalization (FDE) scheme, which facilitates the developmentManuscript received April 2, 2007; revised September 18, 2007 and February 3, 2008; accepted June 24, 2008. The associate editor coordinating the review of this paper and approving it for publication was S. Aissa. This work is sponsored jointly by the National Science Council under grants NSC-97-2221-E-009-101-MY3, NSC-97-2221-E-009-056-MY2 and NSC-96-2628-E-009-003-MY3, by the Ministry of Education of Taiwan under the MoE ATU program, and by the MediaTek research center at National Chiao Tung University, Taiwan.
The authors are with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: cylin.cm90g@nctu.edu.tw; jywu@cc.nctu.edu.tw; tslee@mail.nctu.edu.tw).
Digital Object Identifier 10.1109/T-WC.2008.070348
of several efficient (batch or adaptive) receiver implementa-tions [1], [4]. Such a figure of merit, however, hinges crucially on the time-invariant channel assumption. When the channel is otherwise subject to fast temporal variation, orthogonality among the signal components in the frequency domain will no longer be preserved, and tone-by-tone signal recovery is then rendered impossible. Moreover, in a time-varying environment channel parameter mismatch due to the mere availability of the out-dated channel estimate will become another detrimen-tal factor dominating the system performance. For MIMO-SC block transmission over time-varying dispersive channels, robust receiver designs which can efficiently tackle the joint impacts due to channel time variation and estimation errors, to the best of our knowledge, are hardly found in the literature. This paper proposes a robust receiver design scheme for MIMO-SC systems when the multipath channels undergo time selectivity, and are not exactly known but instead estimated via the least-squares (LS) training technique [5], [6]. In lieu of performing signal recovery in the frequency domain, the proposed approach relies on received data processing in the time domain. Specifically, by exploiting the cyclic shifting property of the time-domain channel matrix it is shown that the signal signatures can be arranged into groups of orthogonal columns. In case that the inter-group interference can be effectively mitigated, the orthogonality structure in conjunction with space-time matched filtering will lead to a low-complexity intra-group symbol recovery scheme. Toward realizing such a time-domain processing facility, a linear weighting matrix is designed for each group based on the constrained optimization formulation [7], [8]. To further mit-igate channel mismatch due to time variation and estima-tion errors, we leverage the generalized side-lobe canceller (GSC) principle [9], [10], [11] to transform the constrained optimization problem into an equivalent unconstrained setup. The unconstrained GSC formulation enables us to leverage the perturbation analysis technique [12], [13] to explicitly model the channel mismatch effect into the system equations, in turn providing a unique two-fold advantage. First, this allows a very natural cost function for weighting matrix design against channel mismatch. Second, we can then exploit the underlying statistical assumptions on the channel errors, due to both temporal variation and imperfect estimation, to derive an analytic solution.
We note that constrained optimization based designs re-sistant to signal/channel parameter uncertainty has been
dressed in other contexts such as beamforming [14], [15], [16], and multiuser detection [17], [18]. The adopted approaches therein are however significantly different from the proposed method in this paper. In [14], [15], and [16], the parameter mismatch is considered instead to be deterministic, the design criteria are of a min-max type, and the solutions are obtained via the convex optimization techniques. In [17] and [18] the parameter errors are modeled as random variables with a Gaussian distribution; the problem is then solved by exploiting the Gaussinality assumption and an associated linear or nonlin-ear programming setup. The proposed GSC based formulation combined with perturbation analysis in this paper relies on a stochastic error modeling strategy similar to that in [17] and [18]. However, as we will show it merely calls for the knowledge of the first- and second-order error statistics but, unlike [17] and [18], is free from any priori assumptions on the error distributions. For high-rate MIMO-OFDM transmission over channels with long delay spreads, such an approach has been adopted in [19] for robust receiver design against channel estimation error. The rest of this paper is organized as follows. Section II presents some preliminary results. Section III intro-duces the motivation behind the proposed scheme, proposes a constrained-optimization based solution framework, and briefly reviews the equivalent unconstrained GSC formulation, all under the perfect channel knowledge assumption. Section IV shows the proposed robust equalization scheme. Section V compares the algorithm complexity. Section VI contains the simulation results. Finally, Section VII concludes this paper.
Notation List: Let Cm and Cm×n be the sets of
m-dimensional complex vectors and m× n complex matrices. Denote by (·)T, (·)∗, and (·)H, respectively the transpose,
the complex conjugate, and the complex conjugate transpose. Im and 0m denote the m× m identity and zero matrices;
0m×n is the m× n zero matrix. The symbols ⊗ and ,
respectively, stand for the Kronecker product [20, p-242] and Hadamard product [20, p-298]. Forx ∈ CmandX ∈ Cm×m,
let diag{x1, ..., xm} be an m × m diagonal matrix with xn,
1 ≤ n ≤ m, on the main diagonal, and let Diag {X} be an
m× m diagonal matrix with the diagonal entries of X on
the main diagonal. The notation E{y} stands for the expected value of the random variable y. Forx ∈ CmandX ∈ Cm×n,
denote by x the vector two-norm and X the matrix Frobenius norm.
II. PRELIMINARY
A. Signal Model
Consider the discrete-time baseband model of a MIMO-SC system with N transmit antennas, M receive antennas, CP length G, and symbol block size Q. Let hm,n(k, l), 0 ≤ l ≤
L, be the lth tap of the channel between the nth and mth transmit-receive antenna pair at time instant k, where L denotes the delay spread assumed common to all MN subchannels. Assuming G≥ L, after CP removal the received symbol block at the mth receive antenna can be expressed as
rm(t) = N n=1 Hm,n(t)sn(t) + vm(t), (1) training data
data training data
"
data"
"
TSC symbols
training data
data training data
"
data"
"
TSC symbols
Fig. 1. Adopted frame structure.
where sn(t) := [sn,1(t), ..., sn,Q(t)]T ∈ CQ is the tth symbol
block sent from the nth transmit antenna,vm(t) ∈ CQ is the
channel noise vector, and Hm,n(t) ∈ CQ×Q is the channel
matrix whose ith column, denoted byc(i)m,n(t), is
c(i) m,n(t) = Ji−1 h(i)T m,n(t) 0 · · · 0 T , 1 ≤ i ≤ Q, (2) in which h(i)m,n(t) := [hm,n(˜t + (i − 1)Q,0) hm,n(˜t +
(i)Q,1) · · · hm,n(˜t+ (L + i − 1)Q, L) ]T with(·)Q denoting
the modulo-Q operation and ˜t:= (t − 1)(Q + G) + G, and J := 01×(Q−1) 1 IQ−1 0(Q−1)×1 . (3)
In case that the channel is time-invariant, i.e., hm,n(k, l) = hm,n(l) for some hm,n(l), Hm,n(t) is a circulant matrix and symbol recovery can be done via the tone-by-tone FDE technique [1]. In the considered time-varying channel environ-ment,Hm,n(t) is no longer circulant and the FDE facility is
negated. Since there are no specific advantages of processing the data in the frequency domain, in this paper we will instead focus on the time-domain signal model (1) for receiver design; as will be shown next this can lead to an effective framework for addressing the robust signal recovery problem against imperfect channel knowledge. The following assumptions are made throughout the paper.
1) The number of receive antennas is equal to or greater than the number of transmit antennas, i.e., M ≥ N. 2) The source symbols of each transmit antenna
sn(t) is zero mean, unit-variance, and
E{sn1,q1(t1)sn2,q2(t2)∗} = δ(n1− n2)δ(t1− t2)δ(q1− q2), where δ(·) is the Kronecker delta.
3) The elements of vm(t)’s are i.i.d. complex circular
Gaussian with zero mean and variance σ2 v.
B. Time-Varying Channel Estimation & Equalization This paper considers the burst-by-burst transmission such that 1) each data burst consists of T symbol blocks, 2) the leading block per burst serves as the training symbol for channel estimation, and 3) the resultant channel estimate is used for receiver design to recover the subsequent T-1 source symbol blocks. A schematic description of the frame structure is shown in Figure 1. Since channel estimation and equal-ization is done on a burst-wise basis, in the sequel we shall focus on the initial burst. The time-varying channel estimation scheme adopted in this paper is briefly reviewed as below; the robust equalizer design which exploits the channel error characteristics will be discussed in Section IV. We assume that the MIMO channel is estimated by using the LS training technique [5], [6], which, in a time-varying environment, is known to yield the optimal estimate of the “averaged” channel impulse response within one symbol duration [21], namely,
h(av)m,n:= 1 Q Q−1 q=0 hm,n(G + q, 0) · · ·Q1 Q−1 q=0 hm,n(G + q, L) T . (4)
Associated with each 1 ≤ m ≤ M let us define h(av)m =
[h(av)Tm,1 · · · h(av)Tm,N ]T, the resultant channel estimate is given
by [21] h(av) m = h(av)m + A +(im + vm ) :=Δhm ∈ CN (L+1)×1,1 ≤ m ≤ M, (5) where A = [diag{t1}FL, ..., diag{tN}FL] ∈ CQ×N (L+1)
withFL/
√
Qbeing the first L+1 columns of the Q× Q FFT matrix, A+ denotes the pseudoinverse of A, t
n ∈ CQ is
the frequency-domain training sequence for the nth transmit antenna, and im=
N
n=1im,n∈ CQ with the qth entry [21]
im,nq = Q−1 q1=1 L+1 l=0 1 Q Q−1 q2=0 hm,n(G + q2, l)e−j2π(q−q1)lQ × [tn](q−q1) Qe −j2πq1q2 Q ,1 ≤ q ≤ Q. (6) Assuming that the channel variation is piecewise linear in time, h(av)m,n can also be treated as the optimal estimate of
the channel parameters in the middle instant of the training period, i.e., hm,n(G + Q/2, l), 0 ≤ l ≤ L [21]. For fixed
1 ≤ m ≤ M and 1 ≤ q ≤ Q we assume that hm,n(q, l), ∀n, l,
are independent circular complex Gaussian variables with zero-mean and variance σ2
l. Then the channel estimation error
Δhm in (5) is zero mean with covariance matrix RΔh,m:=
A+E{i
miHm} + σ2vIN (L+1)
(A+)H ∈ CN (L+1)×N (L+1), in
which the (q, q)th entry of E{i
miHm}, 1 ≤ q, q≤ Q, can be
directly computed from (6) as (7) (shown in the bottom of this page), where Rh(q, l) = E{hm,n(k, l)h∗m,n(k + q, l)} =
σ2lJ0(2πfdqTs), with J0, fd, and Tsrespectively denoting the zeroth-order Bessel function, Doppler frequency, and sampling period. Moreover, assuming i) the taps among subchannels are mutually independent, and ii) the noise is spatially uncorre-lated, we have
RΔh,m1,m2 := E{Δhm1ΔhHm2} = 0N (L+1). (8)
The results (7) and (8) will be used for robust equalizer design in Section IV.
III. GROUP-WISESYMBOLDETECTION: PERFECT
CHANNELKNOWLEDGE
A. Motivation
This subsection highlights the motivation behind the pro-posed approach. Particularly, we will show that the
time-domain channel matrix Hm,n(t) in (2) is imbedded with
certain column-wise orthogonality structure; such an appeal-ing feature will naturally lead to an inter-group interference cancellation framework followed by a low-complexity intra-group symbol recovery scheme. We shall first focus on the ideal case that the channel is perfectly known at the receiver, and will discuss more realistic situations in Section IV. To proceed, we first observe from (2) that, for2 ≤ i ≤ Q, the ith column c(i)m,n(t) of the channel matrix Hm,n(t) is simply an
(i − 1)-step down-shifted version of the zero-padded channel impulse response vector[h(i)Tm,n(t) 0 · · · 0]T. As a result, if the
symbol block size Q is chosen to be an integer multiple of G, i.e., Q= P G for some positive integer P, then for each fixed 1 ≤ i ≤ G we have
c(i+p1G)
m,n (t)Hc(i+pm,n2G)(t) = 0 for 0 ≤ p1= p2≤ P −1, (9)
since the locations of the respective nonzero entries never overlap; a schematic description of such an orthogonality relation is depicted in Figure 2. Equation (9) suggests that the Q columns ofHm,n(t) can be divided into G groups of
orthogonal vectors as follows H(i) m,n(t) = c(i)m,n(t) c(i+G)m,n (t) · · · c(i+(P −1)G)m,n (t) ∈ CQ×P , 1 ≤ i ≤ G. (10) To further exploit the benefit from the orthogonality condition (9), for each fixed 1 ≤ i ≤ G let us stack Hm,n(i) (t) for all
1 ≤ m ≤ M and 1 ≤ n ≤ N to form H(i) n (t) := H(i)1,n(t)T H(i) 2,n(t)T · · · H(i)M,n(t)T T ∈ CM Q×P , and H(i)(t) := H(i) 1 (t) H(i)2 (t) · · · H(i)N(t) ∈ CM Q×N P , 1 ≤ i ≤ G, (11) Collecting the M received signal blocks rm’s in (1) into a
vector, the overall input-output relation can be rearranged as r(t) :=r1(t)T · · · rM(t)T T =G i=1 H(i)(t)s(i)(t) + [v 1(t)T· · · vM(t)T]T :=v(t) , (12)
where s(i)(t) :=s(i)
1 (t)T· · · s(i)N (t)T
T
∈ CN P, s(i) n (t) :=
sn,i(t) sn,i+G(t) · · · sn,i+G(P −1)(t)T ∈ CP. Toward
sym-bol extraction based on (12) we propose to first design
E{imiHm} q,q = 1 Q2 N n=1 Q−1 q1= 0, q2= 0 q1= q, q2= q L l=0 Q−1 q3,q4=0 Rh(q3−q4, l)e −j2π[(q−q1)−(q −q2)]l Q [tn] (q−q1)Q[tn] ∗ (q−q2) Qe −j2π(q1−q2)l Q . (7)
( ) , t m n H ⎧ ⎪ ⎪ ⎪⎪ ⎨ ⎪ ⎪ ⎪ ⎪⎩ 1 L+ L (1) ( ) , t m n H Hm n(2), ( )t Hm n(3), ( )t H(4)m n, ( )t ( ) , t m n H ⎧ ⎪ ⎪ ⎪⎪ ⎨ ⎪ ⎪ ⎪ ⎪⎩ 1 L+ L (1) ( ) , t m n H Hm n(2), ( )t Hm n(3), ( )t H(4)m n, ( )t Fig. 2. A schematic description of the orthogonality condition (3.1) with G = 4.
an inter-group interference suppression matrix W(j)(t) ∈
CMQ×N Q such that r(j)(t) := W(j)(t)Hr(t) ≈ H(j)(t)HH(j)(t)s(j)(t) = ⎡ ⎢ ⎢ ⎣ H(j)1 (t)HH(j) 1 (t) · · · H(j)1 (t)HH(j)N (t) .. . ... ... H(j)N (t)HH(j) 1 (t) · · · H(j)N (t)HH (j) N (t) ⎤ ⎥ ⎥ ⎦ s(j)(t), (13) where the last equality in (13) follows directly from (11), and then recovers(j)(t) via the signal model (13). Benefiting
from the orthogonality condition (9), the matched-filtered channel matrixH(j)(t)HH(j)(t) in (13) exhibits an appealing
structure. Indeed, since the nonzero entries among the P columns of H(i)m,n(t) in (10) do not overlap and, for each
1 ≤ i ≤ G, H(i)n (t) is obtained by stacking H(i)m,n(t) over
all 1 ≤ m ≤ M, it is easy to check that H(i)
n1(t) HH(i)
n2(t) = D(i)n1n2(t), 1 ≤ n1, n2≤ N, (14) where D(i)n1n2(t) ∈ CP ×P is diagonal. Equation (14) implies that, for subsequent intra-group symbol recovery through separating the NP coupled streams in (13), the problem is reduced to solving a set of P independent linear equations, each with dimension N×N. Such a decoupled nature reduces computations, especially when the block length Q (and hence P = Q/G) is large; it can also limit the error propagation effect in the symbol recovery stage. The key challenge of the proposed two-stage equalization scheme is the design of the interference suppression matrixW(j)(t) for fulfilling (13); the
underlying mathematical formulation is discussed next. B. Solution Based on Constrained Optimization
Based on (13), the linear weighting matrix W(j)(t) for
re-covering the jth symbol group should be designed to minimize the inter-group interference power, and then extract the desired signal component through space-time matched filtering. A typical technique toward fulfilling such a two-fold task is through constrained optimization [7], [8]; more precisely, we shall designW(j)(t) by solving the following problem
min W(j)(t) E ⎧ ⎪ ⎨ ⎪ ⎩ W(j)(t)H ⎛ ⎝ G i=1,i=j H(i)(t)s(i)(t) + v(t) ⎞ ⎠ 2⎫⎪ ⎬ ⎪ ⎭, s.t.W(j)(t)HH(j)(t) = H(j)(t)HH(j)(t); (15)
the recovery of the entire symbol vector s(t) is then done group-wise via repeatedly solving (15) for 1 ≤ j ≤ G. By using the standard Lagrangian multiplier technique [22, p-215], the solution to (15) is given by
W(j)(t) =R(j) I (t)−1H(j)(t) % H(j)(t)HR(j) I (t)−1H(j)(t) &−1 · H(j)(t)HH(j)(t), (16) where R(j)I (t) := G
i=1,i=jH(i)(t)H(i)(t)H + σ2nIMQ. An
alternative approach to solving (15) lies in transforming the constrained optimization formulation into an equivalent un-constrained setup via the GSC principle [9]. This relies on the following decomposition of W(j)(t):
W(j)(t) = H(j)(t) − B(j)(t)U(j)(t), (17)
whereH(j)(t) ∈ CMQ×N P represents the non-adaptive
por-tion which verifies the desired space-time matched filtering constraint, B(j)(t) ∈ CMQ×(MQ−N P ) is the signal
block-ing matrix with B(j)(t)HH(j)(t) = 0
(MQ−NP )×NP, and
U(j)(t) ∈ C(MQ−NP )×NP is the adaptive component which
forms the remaining free parameters to be determined. With (10), the equalized output becomes
z(j)(t) := W(j)(t)Hr(t) = z(j) d (t) − U(j)(t)Hz (j) b (t), (18) where z(j)d (t) :=H(j)(t)HH(j)(t)s(j)(t) + H(j)(t)H ⎛ ⎝ G i=1,i=j H(i)(t)s(i)(t) + v(t) ⎞ ⎠ :=i(j)(t) , (19) and z(j)b (t) := B(j)(t)H ⎛ ⎝ G i=1,i=j H(i)(t)s(i)(t) + v(t) ⎞ ⎠ . (20)
Since the desired signal component, namely,
H(j)(t)HH(j)(t)s(j)(t), in the matched filtered branch
z(j)d (t) is contaminated by i(j)(t), toward effective interference
suppression equation (18) suggests that the matrix U(j)(t)
should be chosen so thatU(j)(t)Hz(j)
b (t) is as close to i(j)(t)
as possible; we can thus determineU(j)(t) via
min U(j)(t)E ' i(j)(t) − U(j)(t)Hz(j) b (t) 2 ( . (21)
By following the standard procedure [10], the solution to (21) is given by U(j)opt(t) =%B(j)(t)HR(j) I (t)B(j)(t) &−1 · B(j)(t)HR(j) I (t)H(j)(t); (22)
the resultant optimal GSC weight is thus1 W(j)opt(t) =H(j)(t) − B(j)(t)U(j) opt(t) =H(j)(t)− B(j)(t)%B(j)(t)HR(j) I (t)B(j)(t) &−1 · B(j)(t)R(j) I (t)H(j)(t). (23)
We note that i) the GSC scheme solves a constrained op-timization problem via a simple unconstrained formulation, ii) solutions (16) and (23) are obtained based on the crucial perfect channel knowledge assumption; when channel param-eter mismatch occurs due to imperfect estimation and time variation, they are only suboptimal because the formulations do not take into account channel error mitigation, iii) as will be seen in the next section, the GSC framework is advantageous in that it allows us to directly model the channel mismatch effect into the system equations for facilitating robust equalizer design.
IV. PROPOSEDROBUSTGSC EQUALIZER
Given only the channel estimate H(i)av, 1 ≤ i ≤ G, which is
acquired through training at t= 1, one may simply choose to modify solution (16) via replacing H(i)(t) by H(i)
av to get the time-invariant equalizer ) W(j)= ( R(j) I )−1H (j)av % H(j)H av ( R(j)I )−1H (j)av &−1 H(j)H av H (j)av, (24) where R(j)I = G i=1,i=jH (i)
avH (i)Hav + σn2IMQ, and use (24)
once and for all toward subsequent symbol recovery. Such a strategy, even though quite simple, fails to combat channel mismatch and could incur very poor equalization performance. The GSC principle, on the contrary, provides a very natural and effective framework for robust time-varying equalizer design, as is shown in this section. We will first introduce the problem formulation in Section IV-A, and then derive the solution in Section IV-B. Some discussions regarding the proposed robust scheme are given in Section IV-C.
A. Problem Formulation
Recall that the mechanism of GSC filter (17) basically involves space-time signal matched filtering and signal block-ing followed by interference suppression through (21). While the signal combining and nulling components (H(j)(t) and
B(j)(t), respectively) are immediately fixed upon the (possibly
imperfect) knowledge ofH(j)(t), the adaptive portion U(j)(t)
can nonetheless be allowed to be time-varying, and is then designed to tackle the channel parameter mismatch effects caused by time variation and estimation error. It is such inherent channel tracking capability that makes GSC principle a promising approach in the considered scenario.
Specifically, when only a channel estimate H(j)av is
avail-able, exact signal matched filtering over 2 ≤ t ≤ T is
1The resultant GSC based solution will coincide with the optimal one
derived under the original constrained-optimization based formulation if rankB(j)(t)= MQ − NP [23]; this condition is fulfilled if the columns
ofB(j)(t) form an orthonormal basis for the left null-space associated with
H(j)(t) .
impossible; the best one can do, however, is to linearly combineH(j)(t)s(j)(t) just with H(j)
av to get the
approxima-tion H(j)Hav H(j)(t)s(j)(t). This implies that the non-adaptive
portion of the GSC weight should be set as H(j)av throughout
2 ≤ t ≤ T and, in turn, the blocking matrix is likewise fixed according to the relation B(j)Hav H (j)av = 0(MQ−NP )×NP.
Hence, given the channel knowledge H(j)av only, the signal
matching and blocking matrices are restricted to be time-invariant. Nevertheless, to reliably recover the desired signal against background time-varying interference, the adaptive component must account for the temporal variation. This thus suggests the following modified GSC decomposition
)
W(j)(t) = H(j)
av − B(j)avU
(j)(t). (25)
With (25), the equalized output instead reads z(j)(t) = )W(j)(t)Hr(t) = z(j) d (t) − U (j)(t)Hz(j) b (t), (26) where z(j)d (t) := H(j)H av H(j)(t)s(j)(t) + H(j)H av G i=1,i=j H(i)(t)s(i)(t) + H(j)H av v(t) :=i(j)d (t) , (27) is the (approximate) space-time matched filtered signal, and
z(j)b (t) := B(j)H av H(j)(t)s(j)(t) + B(j)H av G i=1,i=j H(i)(t)s(i)(t) + B(j)H av v(t), (28)
is the corresponding blocking component. Due to the out-of-date channel knowledge, the signal of interest in thez(j)d (t) is non-coherently combined and is corrupted by the interference i(j)d (t); also, since the blocking matrix is determined via
B(j)Hav H (j)av = 0(MQ−NP )×NP, and hence B(j)Hav H(j)(t) =
0(MQ−NP )×NP in general, there is a signal leakage term
B(j)Hav H(j)(t)s(j)(t) into the blocking branch z(j)b (t). Toward
signal recovery against interference, a natural strategy as suggested by the GSC principle is to treat z(j)b (t) in (28) as an aggregate interference and to design U(j)(t) such that U(j)(t)Hz(j)
b (t) is best close to i (j) d (t).
For this we shall first note that in (27) and (28) only the channel estimate H(i)av’s (acquired through training at t= 1)
are available but the true channel matrices H(i)(t)’s are
unknown: the mismatches between H(i)av’s and H(i)(t)’s are
due to time variation as well as channel estimation errors. To facilitate subsequent analysis we must seek for explicit rules linking the unknownH(i)(t) to the channel estimate H(i)
av. A
commonly used model which specifies such channel parameter deviation can be found in [24], [25], and in terms of the current matrix formulation it reads
H(i)(t) = D(i)
c (t) H(i)av + δH(i)(t), 1 ≤ i ≤ G, (29)
whereD(i)c (t) is an MQ×MQ diagonal matrix depending on
the temporal channel variation whose entries are assumed to be zero-mean Gaussian random variables (explicit formulae for D(i)c (t) and the covariance of δH(i)(t) are given in Appendix
A). By using (29) we can rewrite (27) and (28) as (30) (shown in the bottom of this page), and
z(j)b (t) = B(j)Hav D(j)c (t) H(j)avs(j)(t) + B(j)H av G i=1,i=j D(i) c (t) H(i)avs(i)(t) + B(j)H av G i=1 δH(i)(t)s(i)(t) + B(j)Hav v(t). (31) Based on (30) and (31), we specifically propose to design U(j)(t) by minimizing the following cost function
J(t) = E ' i(j)(t) − U(j)(t)Hz(j) b (t) 2 ( , (32)
where the expectation is taken with respect to (w.r.t.) the source symbol, background noise, channel estimation error, and channel temporal variation (assuming all are mutually independent).
B. Optimal Solution Let us expand (32) into J(t) =U(j)(t)HE * z(j)b (t)z(j)b (t)H+U(j)(t) − U(j)(t)HE*z(j) b (t)i (j)(t)H+ − E*i(j)(t)z(j)b (t)H+U(j)(t) + E*i(j)(t)i(j)(t)H+. (33) Since for a given matrix A we have
∂T r(U(j)(t)HAU(j)(t))/∂U(j)(t) = %
U(j)(t)HA&T,
∂T r(AU(j)(t))/∂U(j)(t) = AT,
and ∂T r(U(j)(t)HA)/∂U(j)(t) = 0 [26], taking the
first-order partial derivative of J(t) w.r.t. U(j)(t) yields E * z(j)b (t)z(j)b (t)H+U(j)(t) = E*z(j) b (t)i (j) (t)H+. (34)
The optimal U(j)(t) can then be obtained by solving the matrix equation (34), provided that the covariance matrices E
*
z(j)b (t)z(j)b (t)H+ and E*z(j) b (t)i
(j)(t)H+ are available.
With (30), (31), and by taking expectation w.r.t. source sym-bols, channel noise, and channel temporal variation charac-terized via (29), we can reach the following intermediate
expressions (see Appendix A for detailed proof): E * z(j)b (t)z(j)b (t)H+= Ee *
B(j)Hav %Gi=1D(i)c (t) H(i)avH (i)Hav D(i)c (t)H
+NIM ⊗ DQ(t) + σv2IMQ B(j)av + , (35) E * z(j)b (t)i(j)(t)H+= Ee * B(j)Hav %G
i=1,i=jD(i)c (t) H(i)avH (i)Hav D(i)c (t)H
+NIM⊗ DQ(t) + σv2IMQ H(j)av + , (36) where DQ(t) :=diag * L l=0σl2 % 1 −,,ρ(t, l)),,2 & , ...,Ll=0σl2 % 1 −,,ρ(t + Q − 1, l)),,2 & + , (37) with ρ(k, l) := Rh(k, l)(σl σl)−1, σl2 = σl2 + [RΔh,m](n−1)(L+1)+l+1,(n−1)(L+1)+l+1, t := (t − 1)(G +
Q) − Q/2, and Ee{·} denotes the expectation involving channel estimation error yet to be carried out. To explicitly determine the expectations in (35) and (36) we must further seek for tractable relations linking the channel estimates ( H(i)av for1 ≤ i ≤ G and B(j)av) and the background estimation
errors. While H(i)av can be directly modeled as the actual
channel parameter corrupted by the errors, namely,
H(i)
av = H(i)av+ ΔH(i)av,1 ≤ i ≤ G, (38)
an exact expression of the blocking component B(j)av in
terms of ΔH(j)av remains formidable to characterize, since it
is determined through B(j)Hav H (i)av = 0(MQ−NP )×NP and,
thus, is obtained as an orthonormal basis of the left null space of H(i)av. To resolve this difficulty, we will leverage the
perturbation technique [12], [13] to get an approximate, but analytic, relation among B(j)av and the channel estimation error
ΔH(j)av. The result is shown in the next lemma (the proof can
be found in [12]). Lemma 4.1: Let H(j)av = U(j)h Σ (j) h V (j)H h be a singular
value decomposition of the actual channel matrix H(j)av. The
blocking matrix B(j)av can be approximated by
B(j) av ≈ B(j)av − U (j)h Σ(j)−1h V(j)Hh ΔH(j)Hav B(j)av :=ΔB(j) av . (39)
By substituting B(j)av in (39) into (35) and (36), we have
the following main results (see Appendix B for detailed derivations). z(j)d (t) = H(j)H av D(j)c (t) H(j)avs(j)(t) + H(j)Hav G i=1,i=j D(i) c (t) H(i)avs(i)(t) + H(j)Hav G i=1 δH(i)(t)s(i)(t) + H(j)Hav v(t) :=¯i(j)(t) . (30)
Proposition 4.1: The covariance matrices involved in (34) can be expressed as E * z(j)b (t)z(j)b (t)H+=B(j)H av % R(j)L (t) + R(j)I (t) +R(j)e,1(t) + NIM⊗ DQ(t) & B(j) av, (40) E * z(j)b (t)i(j)(t)H+=B(j)H av % R(j)I (t) + R(j)e,2(t) +NIM ⊗ DQ(t) H(j) av, (41) where R(j)L (t) := D(j) c (t)H(j)avH(j)Hav D(j)c (t)H, (42) R(j)I (t) := G i=1,i=j D(i)
c (t)H(i)avH(i)Hav Dc(i)(t)H+ σ2vIMQ,
(43) and R(j)e,1(t) :=4i=1Xi(t) and R(j)e,2(t) :=
3
k=1Yk(t), in
which the component matricesXi(t)’s and Yk(t)’s are defined
in Table I.
Based on (34), (40), and (41), the robust solution of U(j)(t), which is on average optimal for mitigating channel temporal variation and estimation error, is thus
U(j)opt(t) = B(j)Hav % R(j)L (t) + R(j)I (t) + R(j)e,1(t) +NIM ⊗ DQ(t)) B(j)av −1 × B(j)H av % R(j)I (t) + R(j)e,2(t) + NIM⊗ DQ(t) & H(j) av. (44) In the practical situation when only a channel estimate is available, the sampled-version of the overall robust GSC weighting matrix is accordingly given by
) W(j) r (t) = H(j)av − B(j)avU ¯ (j) opt(t). (45) C. Discussions
1) The proposed design formulation aims for joint mit-igation of channel temporal variation and estimation error. Compared with (22) obtained under exact channel knowledge, the main distinctive feature of the proposed robust scheme (44) lies in replacing the two R(j)I (t)’s in (22) respectively by the “composite” interference covariance matricesR(j)L (t)+R(j)I (t)+R(j)e,1(t)+NIM⊗
DQ(t) and R(j)I (t) + R(j)e,2(t) + NIM ⊗ DQ(t). We
should note that (i) the common term R(j)I (t) can be
regarded as the analogue of R(j)I (t) with the avail-ability of H(i)av’s only (rather than the exact H(j)(t)),
(ii) the term R(j)L (t) arises from signal leakage into the blocking branch due to channel mismatch, (iii) the matricesR(j)e,1(t) and R(j)e,2(t) are due to the aggregate impacts caused by channel estimation errors, and (iv)
the quantity NIM⊗DQ(t) accounts for the background
channel temporal variation.
2) Another scheme for estimating time-varying channels is to treat the channel estimates for two consecutive data bursts as the end points, and further leverage linear inter-polation for acquiring the channel information within the entire time frame [21]; this approach applies whenever the channel varies linearly w.r.t. time. In such a scenario, the impacts due to channel temporal variation would be largely reduced and channel estimation error, instead, becomes the dominant adverse factor to be combated. The proposed design strategy can be used for construct-ing an associated error-resistant GSC equalizer. Indeed, the cost function (32) instead involves only the averages w.r.t. source, noise, and channel estimation errors; by following the procedures shown in Section IV-B, the sampled-version of the resultant robust GSC weight can be obtained by (the derivations are highlighted in Appendix C) ) W(j) a (t) = H(j)(t)− % B(j)(t)H%R (j) I (t) + Re(t) & B(j)(t)&−1 · B(j)(t)HR (j) I (t) H(j)(t), (46) where H(j)(t) and R(j)
I (t) respectively denote the
esti-mates of H(j)(t) and R(j)
I (t) (through linear
interpo-lation), and Re(t) is the estimated version of Re(t)
defined in Table I.
3) The proposed group-wise symbol recovery scheme can be directly combined with the successive interference cancellation (SIC) mechanism for further performance improvement; the resultant solutions in each processing layer is essentially of the form (45), except that in R(j)I (t), R(j)e,1(t), and R(j)e,2(t) the signature matrices corresponding to the previously detected signal com-ponents are removed. Detailed description of the SIC based implementation is listed in Table II.
4) We finally note that the analytic expression of B(j)av in
Lemma 4.1 involves only the first-order error perturba-tion. Even though more accurate approximation to B(j)av
can be obtained by incoPersonNamerporating higher order terms [27], the analysis would however become intractable.
V. ALGORITHMCOMPLEXITY
The computation of the proposed GSC weighting matrix basically involves solving B(j)Hav Hˆ(j)av = 0(MQ−NP )×NP for
the blocking matrix B(j)av followed by inverting an (MQ −
N P) × (MQ − NP ) matrix. Due to the sparse structure of ˆH(j)av (see (2) and (11)), the computation of B(j)av only
requires(4MN(ML − 2N)(L + 1)/G) Q flops (i.e., O (Q)). The complexity for matrix inversion can be further reduced by leveraging the conjugate gradient [28] based reduced-rank (RR) implementation [29] (listed in Table III). Table IV shows the flop counts (measured in terms of the number of complex-valued additions and multiplications) of the GSC method, GSC with RR implementation as well as two comparative
TABLE I
FORMULAE OFX(j)(t), X(i,j)(t), X1(t)˜X4(t), Y1(t)˜Y3(t),ANDRe(t).
diagonal block submatrix of the
stacks a matrix into a vector columnwise and denotes the first th block submatrix of
diagonal block submatrix of the
stacks a matrix into a vector columnwise and denotes the first th block submatrix of
diagonal block submatrix of the
stacks a matrix into a vector columnwise and denotes the first th block submatrix of
diagonal block submatrix of the
stacks a matrix into a vector columnwise and denotes the first th block submatrix of
receivers, namely, the layered space-frequency equalization (LSFE) scheme [1] and the group-wise V-BLAST detector [30]. From the table we observe that the computation complex-ity of the three methods is all about O(Q3), while the adoption
of the RR implementation can reduce the complexity of the GSC scheme to O(Q2). Specifically, for the system parameters
(N, M, Q, L, G, Lb, Nb, I) = (2, 2, 64, 7, 8, 1, 2, 10) adopted
in the simulation section the flop counts are, respectively, F CGSC ≈ 1.46 + ×107, F CGSC(RR) ≈ 3.2 × 106, F CLSF E ≈ 2 × 106, and F C
GB ≈ 1.2 × 107 (Lb and Nb respectively
denote the tap order of feedback filter and the number of decision stages for the LSFE detector, and I the number of iterations involved in the RR implementation).
VI. SIMULATIONRESULTS
This section illustrates the simulated performance of the proposed scheme. We consider a MIMO-SC system with carrier frequency 5 GHz, transmission bandwidth 20 MHz, N = 2 transmit antennas, M = 2 receive antennas, symbol block size Q = 64, and CP length G = 8. The velocity of the moving transmitter is set to be 120 Km/h. The source symbols are drawn from the QPSK constellation. The chan-nels are characterized by the Jakes’s model [31] with order L = 7 and the impulse response is normalized such that 2 n=1 7 l=0E *,,h(m,n) (k, l),,2+ = 1 for each 1 ≤ m ≤ 2 and k ≥ 0. The input SNR at the mth receive antenna is defined as SNR := σ−2
v , and the data burst length is set
to be T = 15. The number of iterations involved in the RR implementation is set to be I = 10. After inter-group interference suppression the MMSE V-BLAST detector [32] is used for symbol recovery.
0 5 10 15 20 25 10-4 10-3 10-2 10-1 SNR (dB) BE R GSC (without SIC) GSC (with SIC)
Fig. 3. BER performances of the proposed method with and without the SIC mechanism (perfect channel knowledge).
A. Impacts of SIC Based Implementation
We first compare the performances of the proposed GSC equalizers with and without the SIC implementation. Figures 3 and 4, respectively, show the simulated bit-error-rate (BER) with perfect and imperfect channel knowledge (for the latter the channels are estimated via the LS training technique [5] with training power Pt= 32). We note that, when channel is
perfectly known, the robust GSC solution (45) reduces to (23). The results show that the proposed method combined with the SIC mechanism yields about a 2˜3 dB gain with exact channel knowledge, and a 4˜5 dB gain with LS channel estimate. This would benefit from the increased receive diversity attained by the SIC mechanism.
TABLE II
ALGORITHM SUMMARY OF THESICMECHANISM.
Initialization: l l l l l (0) (0) (0) 1 3 1 3 ( ) 1 (0) ( ) ( ) 2 1 , ( ) ( ) 1 (0) ( ) ( ) 2 1 ( ) ( ); ( ) ( ); ( ) ( ); ( ) ( ( ) ); ( ) ( ) ; i L i i H av c i j c i i i H L i i H av av I i c c v MQ t t t t t t t t t t σ + = + = = = = = = +
∑
∑
r r X X X X Y D H K D R D H H D I Recursion: for if j =1 (1) (0) (1) (0) 1 ( )t = 1 ( );t 3( )t = 3 ( )t X X X X else(
)
( 1) ( 1) ( ) ( 1) ( 1) 1 1 ( ) ( 1) ( 1) ( 1) ( 1, ) 3 3 1, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) j j j j j H c H H j j j j j j c j j t t t t t t t t − − − − − − − − − = − = − X X D D D X X D D X end end l( ) ( ) ( ) ( 1) ( ) ( ) ( 1) ( ) ( ) ,2 1 2 , ( 1) ( ) ( , ) 3 , ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ) ( ) ( ) ( ( ) ) ; j j j j j j H j j j H av e c H c c j j c H j H j j j H c j j t t t t t t t t t t − − − ⎡ ⎤ ⎡ ⎤ =⎢⎣ − ⎥⎦+⎢⎣ − ⎥⎦ ⎡ ⎤ +⎢⎣ − ⎥⎦ R X D D D Y D H K D X D D X 1. 2. ( ) ( 1) ( ) l l( ) ( ) ( ) ( ) ( ) ( ) j j H ( ) ; j j j j H av av I t = I− t − c t c t R R D H H D 4. ( ) ( ) ( ) ( ) ( ) ,1( ) 1 ( ) ,( ( )) 3 ( ) 3 ( ) ; j j j j j H e t = t + j j t + t + t R X D X X X 5. 6. Compute using (4.21) ( )j r W 7. ( )j( )(
( )j H ( )j( ))
r t =dec t s W r 8. (j 1)( ) ( )j( ) ( )j( )l( ) ( )j j( ) av c t t t t + = − r r D H s 3. 1≤ ≤j G+1 l l( ) ( ) ( )j( ) ( )j( ) j j H ( )j( ) ;H av av c c L t = t t R D H H D Initialization: l l l l l (0) (0) (0) 1 3 1 3 ( ) 1 (0) ( ) ( ) 2 1 , ( ) ( ) 1 (0) ( ) ( ) 2 1 ( ) ( ); ( ) ( ); ( ) ( ); ( ) ( ( ) ); ( ) ( ) ; i L i i H av c i j c i i i H L i i H av av I i c c v MQ t t t t t t t t t t σ + = + = = = = = = +∑
∑
r r X X X X Y D H K D R D H H D I Recursion: for if j =1 (1) (0) (1) (0) 1 ( )t = 1 ( );t 3( )t = 3 ( )t X X X X else(
)
( 1) ( 1) ( ) ( 1) ( 1) 1 1 ( ) ( 1) ( 1) ( 1) ( 1, ) 3 3 1, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) j j j j j H c H H j j j j j j c j j t t t t t t t t − − − − − − − − − = − = − X X D D D X X D D X end end l( ) ( ) ( ) ( 1) ( ) ( ) ( 1) ( ) ( ) ,2 1 2 , ( 1) ( ) ( , ) 3 , ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ) ( ) ( ) ( ( ) ) ; j j j j j j H j j j H av e c H c c j j c H j H j j j H c j j t t t t t t t t t t − − − ⎡ ⎤ ⎡ ⎤ =⎢⎣ − ⎥⎦+⎢ − ⎥ ⎣ ⎦ ⎡ ⎤ +⎢⎣ − ⎥⎦ R X D D D Y D H K D X D D X 1. 2. ( ) ( 1) ( ) l l( ) ( ) ( ) ( ) ( ) ( ) j j H ( ) ; j j j j H av av I t = I− t − c t c t R R D H H D 4. ( ) ( ) ( ) ( ) ( ) ,1( ) 1 ( ) ,( ( )) 3 ( ) 3 ( ) ; j j j j j H e t = t + j j t + t + t R X D X X X 5. 6. Compute using (4.21) ( )j r W 7. ( )j( )(
( )j H ( )j( ))
r t =dec t s W r 8. (j 1)( ) ( )j( ) ( )j( )l( ) ( )j j( ) av c t t t t + = − r r D H s 3. 1≤ ≤j G+1 l l( ) ( ) ( )j( ) ( )j( ) j j H ( )j( ) ;H av av c c L t = t t R D H H D 0 5 10 15 20 25 30 35 10-2 10-1 SNR (dB) BE RRobust GSC (without SIC) Robust GSC (with SIC)
Fig. 4. BER performances of the proposed method with and without the SIC mechanism (LS channel estimate).
B. Comparison with Existing Works
We compare the BER performance of the proposed method (combined with SIC) with the two alternative solutions group-wise V-BLAST [30] and LSFE [1]. The group size of the group-wise V-BLAST is set to be16 (this is the same as the group size in our scheme NP = 2 × 8 = 16). For the LSFE detector, the number of decision stages and the tap order of the feedback filter are respectively Nb = 2 and Lb = 1 (Nb= 2
is the suggested optimal choice [1] for two transmit antennas,
0 5 10 15 20 25 10-4 10-3 10-2 10-1 SNR (dB) BE R LSFE [32] Group-wise VBLAST [27] GSC [1] [30] 0 5 10 15 20 25 10-4 10-3 10-2 10-1 SNR (dB) BE R LSFE [32] Group-wise VBLAST [27] GSC [1] [30]
Fig. 5. BER performances of the three methods (perfect channel knowledge).
and through simulation further increasing Lb does not seem to improve performance). Figures 5 and 6, respectively, show the results with perfect and imperfect channel knowledges. As we can see, even in the ideal case the proposed approach can outperform the two comparative choices. When only a chan-nel estimate is available, the performances of all equalizers degrade, but the robust solution (45) does yield the lowest BER. With a fixed SNR level (25 dB) Figure 7 illustrates the BER of all methods when the burst duration T increases from 13 to 25. The results show that the proposed robust equalizer
0 5 10 15 20 25 30 35 10-2 10-1 SNR (dB) BE R LSFE [32] Group-wise VBLAST [27] RR-GSC GSC Robust RR-GSC Robust GSC [1] [30] 0 5 10 15 20 25 30 35 10-2 10-1 SNR (dB) BE R LSFE [32] Group-wise VBLAST [27] RR-GSC GSC Robust RR-GSC Robust GSC [1] [30]
Fig. 6. BER performances of the three methods (LS channel estimate).
14 16 18 20 22 24 10-3 10-2 T BER LSFE [32] Group-wise VBLAST [27] RR-GSC GSC Robust RR-GSC Robust GSC [1] [30] 14 16 18 20 22 24 10-3 10-2 T BER LSFE [32] Group-wise VBLAST [27] RR-GSC GSC Robust RR-GSC Robust GSC [1] [30]
Fig. 7. BER performances of the three method w.r.t. various burst length (LS channel estimate).
(45) can significantly limit the BER penalty and is thus quite resistant to the increase of T. Due to space limitation, more performance comparisons regarding different mobile velocities and antenna configurations are relegated to a supplementary file available at [33].
C. Channel Estimation with Linear Interpolation
Figure 8 further depicts the simulated BER when the chan-nel information at each time instant is acquired through the linear interpolation technique [21]; in this case the proposed robust scheme is given by (46). Compared with Figure 6, for each receiver the incurred performance loss w.r.t. the exact channel knowledge case is less severe due to the availability of more timely channel information. The proposed robust solution (46), as expected, achieves the best performance since it is capable of combating channel estimation errors.
VII. CONCLUSIONS
We study the robust receiver design problem for MIMO-SC systems when the multipath channels are time-varying and are estimated through the LS training technique. By exploit-ing certain group-wise orthogonality structure imbedded in the time-domain channel matrix, the proposed receiver aims
0 5 10 15 20 25 30 35 10-3 10-2 10-1 100 SNR (dB) BER LSFE [32] Group-wise VBLAST [27] RR-GSC GSC Robust RR-GSC Robust GSC [1] [30] 0 5 10 15 20 25 30 35 10-3 10-2 10-1 100 SNR (dB) BER LSFE [32] Group-wise VBLAST [27] RR-GSC GSC Robust RR-GSC Robust GSC [1] [30]
Fig. 8. BER performances of the three methods (interpolation-based channel estimate).
for inter-group interference suppression followed by a low-complexity intra-group symbol recovery scheme.
The design of the interference rejection matrix is mathe-matically formulated via constrained optimization technique. To further tackle the adverse effects due to imperfect channel knowledge we leverage the GSC principle to reformulate the problem into an equivalent unconstrained setup. The constraint-free GSC formulation resorts to proper weighting matrix decomposition and has several unique advantages: 1) it provides a simple yet efficient channel tracking mechanism by setting the adaptive component to be time-varying, 2) it allows for a very natural cost function for weighting matrix design against channel uncertainty due to time variation and estimation errors, 3) it enables us to directly model the channel mismatch effect into the filtered signal model via the perturbation analysis and, accordingly, we can then exploit the channel error statistics to derive a closed-form solution. The proposed approach can be further implemented in an SIC fashion for performance enhancement; it can also be used for estimation-error resistant receiver design when the channel estimate at each time instant is acquired via linear interpolation. Compared with the existing works the proposed scheme yields significantly improved simulated BER, with comparable algorithm complexity.
APPENDIXA
DERIVATIONS OF(35)AND(36)
With (30), (31) and by averaging over source symbols and channel noises, we have
E * z(j)b (t)z(j)b (t)H+= Ee *
B(j)Hav %Gi=1D(i)c (t) H(i)avH (i)Hav D(i)c (t)H
+σ2
vIMQ+ Eh{RδH(t)} B(j)av
+
and E * z(j)b (t)i(j)(t)H+= Ee *
B(j)Hav %Gi=1,i=jD(i)c (t) H(i)avH (i)Hav D(i)c (t)H
+σ2 vIMQ+ Eh * R(j)δH(t)+&H (j)av + , (48)
where Ee{·} and Eh{·} denote the expectations taken with
respect to the channel estimation error and channel temporal variation, respectively,
RδH(t) = G
i=1
δH(i)(t) H(i)Hav D(i)c (t)H
+G i=1 D(i) c (t) H(i)avδH(i)(t)H +G i=1 δH(i)(t)δH(i)(t)H, (49) and R(j)δH(t) = G i=1,i=j
δH(i)(t) H(i)Hav D(i)c (t)H
+ G i=1,i=j D(i) c (t) H(i)avδH(i)(t)H +G i=1 δH(i)(t)δH(i)(t)H, (50) in which D(i) c (t) ∈ CMQ×MQ D(i) c (t) := diag -Ji−1[a 1(t) · · · aP(t)]T . ⊗ IM, (51) with ap(t) = ρ % t+ ((p − 1)(L + 1) + i − 1)Q,0 & σ0 σ0, ..., ρ % t+ ((p − 1)(L + 1) + L + i − 1)Q, L & σL σL , ρ(t, l) = Rh(t, l)(σl σl)−1, σ2 l = σ2l + [RΔh,m](n−1)(L+1)+l+1,(n−1)(L+1)+l+1,
where ap(t) ∈ R1×(L+1) and δH(i)(t) ∈ CMQ×N P has
the (m, n)th Q × P block submatrix whose pth column
δc(i,p)m,n(t) ∈ CQ is given by (52) (shown in the bottom
of this page), where, for a fixed t, δhm,n(t, l), 0 ≤ l ≤ L, are zero-mean Gaussian random variables with variance σ2l
%
1 − |ρ(t, l)|2&. We claim that Eh{RδH(t)} = Eh
*
R(j)δH(t)+= NIM⊗ DQ(t), (53)
the result then follows from (47), (48), and (53).
Proof of (53): Since the nonzero entries of
δH(i)(t) are zero-mean Gaussian random variables, we have Eh{δH(i)(t)} = 0MQ×N P, for 1 ≤ i ≤ G, and from (49) and (50) we have Eh{RδH(t)} =Eh * R(j)δH(t)+ =Eh / G i=1 δH(i)(t)δH(i)(t)H 0 . (54)
Let Diag{C1, ...,Cm} ∈ Cmn×mn be the block
diag-onal matrix with the diagdiag-onal block submatrices Cp ∈
Cn×n, 1 ≤ p ≤ m. By the definition of δH(i)(t) ∈
CMQ×N P in (52), the (m
1, m2)th Q × Q block submatrix
ofG
i=1δH(i)(t)δH(i)(t)H∈ CMQ×MQ,1 ≤ m1, m2≤ M,
is given by (55) (shown in the bottom of next page). Since the elements of δh(i,p)m,n(t), ∀m, n, are independent (see (52)),
we have Eh * δh(i,p)m 1,n(t)δh(i,p)m2,n(t)H + = ' D(i)p (t), m1= m2 0L+1, m1= m2 , (56) D(i)p (t) ∈ C(L+1)×(L+1) is a diagonal matrix with
the lth diagonal element D(i)p (t)
l,l =
σl−12 (1 − ,,ρ(t + (i + (p − 1)(G + 1) + l − 2)Q, l− 1),,2) for 1 ≤
l ≤ L + 1. Equation (56) directly implies that
Eh*Gi=1δH(i)(t)δH(i)(t)H+is a diagonal matrix with the
mth Q×Q diagonal block given by (57) (shown in the bottom
of the next page). By some direct rearrangements and since Q=PG, (57) becomes Eh ⎧ ⎨ ⎩ G i=1 δH(i)(t)δH(i)(t)H (m,m)⎫⎬ ⎭= NDQ(t), (58) where DQ(t) := diag* L l=0σ2l % 1 −,,ρ(t, l)),,2 & , ..., L l=0σl2 % 1 −,,ρ(t + Q − 1, l)),,2 & + ∈ CMQ×MQ.
Equation (53) directly follows from (58). APPENDIXB
DERIVATIONS OF(40)AND(41)
To derive (40) and (41), we have to determine the expec-tation quantities associated with the channel estimation error in (35) and (36). Based on (2) the(m, n)th block submatrices ofH(i)av ∈ CMQ×N P andΔH(i)av ∈ CMQ×N P are respectively
defined as H(i) av (m,n) = Ji 1 h(av)T m,n ,0, ..., 0 T ⊗ IP 2 ∈ CQ×P, (59) δc(i,p)m,n(t) = Ji+(p−1)G−1[δh m,n(t + ((p − 1)(L + 1) + i − 1)Q,0), ..., δhm,n(t + ((p − 1)(L + 1) + i + L − 1)Q, L) :=δh(i,p)T m,n ,0, ..., 0]T. (52)
