IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 2, FEBRUARY 2010 681

Linearly Time-Varying Channel Estimation forMIMO/OFDM Systems Using

Superimposed TrainingXianhua Dai, Han Zhang, and Dong Li

Abstract—Channel estimation for multiple-inputmultiple-output/orthogonal frequency-division multiplexing(MIMO/ OFDM) systems in linearly time-varying (LTV)wireless channels using superimposed training (ST) isconsidered. The LTV channel is modeled by truncateddiscrete Fourier bases. Based on this model, a two-stepapproach is adopted to estimate the LTV channel overmultiple OFDM symbols. We also present a performanceanalysis of the channel estimation and derive a closed-formexpression for the channel estimation variances. It isshown that the estimation variances, unlike that of theconventional ST-based schemes, approach to a fixed lower-bound as the training length increases, which is directlyproportional to information-pilot power ratios. To furtherenhance the channel estimation performance with a limitedpilot power, an interference cancellation procedure isintroduced to iteratively mitigate the information sequenceinterference to channel estimation. Simulation results showthat the proposed algorithm outperforms frequency-divisionmultiplexed trainings schemes.

Index Terms—MIMO/OFDM, superimposed training, lin-early time-varying channel estimation.

I. Introduction

THE combination of multiple-input multiple-output(MIMO) antennas and orthogonal frequency-

division multiplexing (OFDM) can achieve a lowererror rate and/or enable high-capacity wirelesscommunication systems by flexibly exploiting diversitygain and/or the spatial multiplexing gains. Suchsystems, however, rely upon the knowledge of channelstate information (CSI) which is often obtained throughchannel estimation.

In conventional pilot-aided channel estimation ap-proaches, single-input single-output (SISO) and MIMOchannels can be effectively estimated by utilizing thetime-division multiplexed (TDM) and (or) frequency-division multiplexed (FDM) training sequences [5]-[7]

Paper approved by S. N. Batalama, the Editor for Spread Spectrumand Estimation of the IEEE Communications Society. Manuscript re-ceived November 29, 2007; revised June 19, 2008 and June 23, 2009.

X. Dai and D. Li are with the School of Electrical and CommunicationEngineering, Sun Yat-Sen University, Guangzhou 510275, P. R. China(e-mail: issdxh@mail.sysu.edu.cn, lid2@mail2.sysu.edu.cn).

H. Zhang was with the School of Electrical and CommunicationEngineering, Sun Yat-Sen University, Guangzhou, China. He is nowwith the school of Physics and Telecommunication Engineering, SouthChina Normal University, Guangzhou 510630, P. R. China (e-mail:hanz928@hotmail.com).

Digital Object Identifier 10.1109/TCOMM.2010.01.070471

[19]-[21]. Although the channel estimates are in generalreliable, extra bandwidth or time slot is required fortransmitting known pilots. In recent years, an alternativeapproach, referred to as superimposed training (ST),has been widely studied in [8]-[17]. In the idea of ST,additional periodic training sequence are arithmeticallyadded to information sequence in time- or frequency-domain, and the channel transfer function can thus be es-timated by using the first-order statistics. The advantageof the scheme is that there is no loss in information rate,and thus enables higher bandwidth efficiency. In thisscheme, however, the information sequences are viewedas interference to channel estimation since pilot symbolsare superimposed at a low power to the informationsequences at the transmitter. To circumvent the problem,it was recommended in [8]-[13] [17] that a periodicimpulse train of the period larger than the channel orderis superimposed in time-domain, and the channel is thusestimated by averaging the estimations of multiple train-ing periods to reduce the information sequence interfer-ence. For a SISO/OFDM system, [10] suggested a similarscheme that superimposes the periodic impulse trainingsequences on time-domain modulated signals, while infrequency-domain, a novel block transmission methodis proposed in [15]-[16] where a information sequencedependent component is added to the superimposedtraining so as to remove the effect of the informationsequence on the channel estimation at receiver. Theseabovementioned schemes, however, are restricted to thecase that the channel is linearly time-invariant (LTI), andcannot be extended to the linearly time- varying (LTV)channel since the variation of channel coefficients maydegrade the simple average-based solution extensively.In addition, some useful power has to be allocatedto the superimposed training sequences to confrontthe information sequence interference during symboltransmission, thereby lowering the effective SNR. Thismay be a great disadvantage to wireless communicationsystems with a limited transmission power. On the otherhand, the interference to information sequence recoverydue to the embedded training sequences may degradethe symbol-error-rate (SER) performance severely at re-ceiver. Previous papers merely focus on the informa-tion sequence interference suppression; whereas fewresearches are contributed to the superimposed trainingeffect cancellation for information sequence recovery.

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682 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 2, FEBRUARY 2010

In this paper, a new superimposed training-based channel estimation scheme is proposed forMIMO/OFDM systems over linearly time-varyingchannels. The main contributions are twofold. First,we resort to the truncated Fourier bases to model theLTV channel, and then adopt a two-step approachto estimate the time-varying channel coefficientsover multiple OFDM symbols. Furthermore, aclosed-form expression of the estimation varianceis derived, which provides a guideline for designingthe superimposed pilot symbols. We demonstrate bysimulation that the estimation variance, unlike thatof conventional superimposed training-based schemesof LTI channel [8]-[17], approaches to a fixed lower-bound as the training length increases. Second, tofurther enhance the channel estimation performance ofwireless communication systems with a limited pilotpower, an iterative interference cancellation scheme isproposed. By the iterative mitigation procedure, thecontribution of the information sequence interferenceto channel estimation as well as the training effectson symbol recovery are effectively cancelled withinseveral iterations, respectively. In simulations presentedin this paper, we compare the results of our approacheswith that of the FDM training approaches [19]-[21]as latter serves as a ”benchmark” in related works.It is shown that the proposed algorithm outperformsFDM trainings, and the demodulator exhibits a nearlyindistinguishable SER performance from that of [21].

The rest of the paper is organized as follows. SectionII presents the channel and system models. In SectionIII, we estimate the LTV channel coefficients with theproposed two-step channel estimation approach. In Sec-tion IV, we derive the closed-form expression of thechannel estimation variances. In section V, we developan iterative interference cancellation scheme to enhancethe channel estimation performance (hence SER level)by mitigating the information sequence interference andthe residual pilot effects, respectively. Section VI reportson some simulation experiments carried out in order totest the validity of theoretic results, and we conclude thepaper with Section VII.

Notation: The letter t represents the time-domainvariable and k is the frequency-domain variable. Boldletters denote the matrices and column-vectors, and thesuperscripts [•](T) and [•](H) represent the transpose andconjugate transpose operations, respectively. denotes the(k, t) element of the specified matrix.

II. SystemModel

Consider an MIMO/OFDM system of N transmittersor mobile users and a receive array of M receive anten-nas with perfect synchronization. At transmit terminals,an inverse fast Fourier transform (IFFT) is used as amodulator. The modulated outputs are given by

Xn(i) = [xn(i, 0), · · · , xn(i, t), · · · , xn(i,B − 1)]T, n = 1, · · · ,N(1)

where B is OFDM symbol-size, Sn(i) =[sn(i, 0), · · · , sn(i, k), · · · , sn(i,B− 1)]T is the ith transmittedsymbol of the nth transmit antenna. F−1 is the IFFTmatrix with [F−1]k,t = e

j2πktB and j2 = −1. Then, Xn(i)

is concatenated by a cyclic-prefix (CP) of length L(L is larger than or at least equal to the maximumchannel delay L to cancel the inter-symbol interference),propagated through the respective channels. Atreceiver, the received signals of mth receive antenna,discarding CP and stacking the received signalsy(m)(i, t)t = 0, · · · ,B − 1, can be written in a vector-formas

Y(m)(x) = [y(m)(i, 0), · · · , y(m)(i, t), · · · , y(m)(i,B − 1)]T,m = 1, · · · ,M (2)

and the received signals y(m)(i, t) in (2) is given by

y(m)(i, t) =N∑

n=1

Xn(i) ⊗ h(m)n + v(m)(i, t)

=

N∑n=1

L−1∑l=0

h(m)n,l (t)xn(i, t − l) + v(m)(i, t)t = 0, · · · ,B − 1

(3)

where h(m)n (t) = [h(m)

n,0 (t), · · · , h(m)n,L−1(t), 01×B−L]T is the im-

pulse response vector of the propagating channel fromthe nth transmit to the mth receive antenna with thechannel coefficients h(m)

n,l (t), l = 0, · · · , L−1 being the func-tions of time variable t. The notation ⊗ represents thecyclic convolution and v(m)(i, t) is the additive Gaussiannoise.

As mentioned in [1], the coefficients of the time- andfrequency-selective channel can be modeled as Fourierbasis expansions. Thereafter, this model was intensivelyinvestigated and applied in block transmission, channelestimation and equalization (e.g. [2]-[5]). In this paper,we extend the block-by-block process [2]-[5] to the casewhere multiple OFDM symbols are utilized. Consider atime interval or segment t : (� − 1)Ω ≤ t ≤ �Ω, the chan-nel coefficients in (3) can be approximated by truncateddiscrete Fourier bases (DFB) within the segment as

h(m)n,l (t) ≈

Q−1∑q=0

h(m)n,l,qe

− j2π(q−Q/2)tΩ

t = (� − 1)Ω, · · · , �Ω, � = 1, 2, · · · (4)

where h(m)n,l,q is a constant coefficient, Q represents the

basis expansion order that is generally defined as Q ≥2 fdΩ/ fs [1]-[5], Ω > B is the segment length and � is thesegment index. Unlike [2]-[5], the approximation framecovers multiple OFDM symbols, denoted by i = 1, · · · , I,where I = Ω/B′ and B′ = B + L.

At receiver, an FFT operation is performed on thevector (2), and the demodulated outputs can be writtenas

U(m)(i) = [u(m)(i, 0), · · · , u(m)(i, k), · · · , u(m)(i,B − 1)]T

= FY(m)(i),m = 1, · · · ,M. (5)

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DAI et al.: LINEARLY TIME-VARYING CHANNEL ESTIMATION FOR MIMO/OFDM SYSTEMS USING SUPERIMPOSED TRAINING 683

From (3)-(4) and the duality of time and frequency, theFFT demodulated signals in (5) can be written as

u(m)(i, k) = FFT{N∑

n=1

L−1∑l=0

h(m)n,l (t)xn(i, t − l) + v(m)(i, t)}

=

N∑n=1

L−1∑l=0

FFT{h(m)n,l (t)} ⊗ FFT{xn(i, t)} + v(m)(i, k)

=

N∑n=1

L−1∑l=0

FFT{Q∑

q=0

h(m)n,l,qe

− j2π(q−Q/2)Ω } ⊗ Sn(i) + v(m)(i, k)

(6)

where FFT{•} represents the FFT vector of the speci-fied function, v(m)(i, k) is the frequency-domain noise.Compared with the FFT demodulated signals of OFDMsystems with LTI channels [19]-[21], the convolutionin (6) between the information sequences and the FFTvectors of time-varying channel coefficients may intro-duce inter-carrier interferences (ICI). Note that the vectorFFT{h(m)

n,l (t) in (6) depends crucially on the channel vari-ations during an OFDM symbol interval. If the channelchanges slowly compared to the duration of an OFDMsymbol, the channel variations and the resulting ICIcan be neglected. As mentioned in [22], if the Dopplerfrequency satisfies fdB/ fs ≤ 0.01 ∼ 0.02, the signal-to-interference ratio is greater than 30dB. Correspondingly,the FFT vector FFT{h(m)

n,l (t)} can be approximated as a δ-vector and, a LTI channel can thus be associated witheach OFDM symbol. We refer to such a channel as slowlytime-varying.

In this paper, we focus on the slowly time-varyingchannel estimation. Following the slowly time-varyingassumption where the time-varying channel coefficientscan be approximated as LTI during one OFDM symbolperiod but vary significantly across multiple OFDMsymbols [21], the corresponding LTV channel coefficientswithin an OFDM symbol can be approximated as

h(m)n,l (t) ≈

Q∑q=0

h(m)n,l,qe

− j2π(q−Q/2)tiΩ , t = (i − 1)B′, · · · , iB′ (7)

where ti = (�−1)Ω+(i−1)B+B/2 is the mid-sample of the�th OFDM symbol. In (7), the LTV channel coefficientsare in fact approximated by the mid-values of the LTVchannel model (4) at the ith symbol. Since the proposedchannel estimation will be performed within one singleframe Ω, we omit the frame index � and thus have ti =(i − 1)B + B/2 for simplification.

For the slowly time-varying scenario, the vectorsFFT{h(m)

n,l (t)} in (6) is approximated as a δ-sequences and,the FFT demodulated signals at the sub-carrier of the

i-th symbol can be thus rewritten as

u(m)(i, k) =N∑

n=1

L−1∑l=0

[Q∑

q=0

h(m)n,l,qe

− j2π(q−Q/2)tiΩ ]

× e− j2πkl

B sn(i, k) + v(m)(i, k)

=

N∑n=1

L−1∑l=0

�(m)n,l (i)e

− j2πklB sn(i, k) + v(m)(i, k)

=

N∑n=1

H(m)n (i, k)sn(i, k) + v(m)(i, k) (8)

where �(m)n,l (i) =

Q∑q=0

h(m)n,l,qe

− j2π(q−Q/2)tiΩ and

H(m)n (i, k) =

L−1∑l=0

Q∑q=0

h(m)n,l,qe

− j2π(q−Q/2)tiΩ e

j2πklB

=

L−1∑l=0

h(m)n,l (i)e

− j2πklB (9)

It should be emphasized that although the channelmodel is approximated to be LTI during one OFDM sym-bol interval, the use of conventional first-order-statisticsbased channel estimation [8]-[13] [17] is not allowedsince the channel coefficients’ variation during multipleOFDM symbols may degrade the simple average proce-dure extensively.

III. ST - Based Channel Estimation

In this section, we propose a two-step channel estima-tion approach to overcome the abovementioned short-coming of conventional superimposed training-basedschemes in estimating LTV channels.

A: ST-based Channel Estimation over One OFDM SymbolIn superimposed training-based approaches [8]-[17],

the pilot symbols are superimposed (arithmeticallyadded) to the information sequences as

sn(i, k) = bn(i, k) + pn(i, k), k = 0, · · · ,B − 1 (10)

where bn(i, k) and pn(i, k) are the information and pilotsequence, respectively. Compared with the FDM/TDMtraining aided methods [19]-[22], superimposed train-ing requires no additional bandwidth (or time-slot) fortransmitting the known pilots, and thus offers a higherdata rate. As in MIMO/OFDM system where the receivedsignals in (8) are overlapped or superimposed acrossdifferent users, the training signals of the desired usercannot be distinguished from others’ for estimating thechannel. To circumvent this problem, we adopt thetraining scheme as

pn(i, k) =√

Epe− j2πk(n−1)L

B , n = 1, · · · ,N, k = 0, · · · ,B − 1 (11)

where Ep is the fixed power of the pilot symbols.Note that the pilot symbols in (11) are complex ex-

ponential functions superimposed over the whole sub-carriers, the corresponding time-domain signals of vari-ous users are in fact a δ sequence as pn(i, t) =

√EpBδ(t−

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684 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 2, FEBRUARY 2010

(n − 1) − L)n = 1, · · · ,N that follows a disjoint set withan interval L. Therefore, using the specifically designedtraining sequence (11), the training signals of varioususers are decoupled. The sequence (11), however, pos-sibly leads to high signal peaks at the instant samplest = (n − 1)L, n = 1, · · · ,N. One of the simple ways tosuppress the above undesired signal peaks may referto the scrambling procedure [23] (details will not beaddressed here since it is beyond the scope of thispaper).

Substituting the specifically designed pilot sequencesinto (8), we have

u(m)(i, k) =N∑

n=1

L−1∑l=0

�(m)n,l (i)e

− j2πklB pn(i, k)

︸��������������������������︷︷��������������������������︸desired signal for channel estimation

+

N∑n=1

L−1∑l=0

�(m)n,l (i)e

− j2πklB bn(i, k)

︸�������������������������︷︷�������������������������︸information interference to channel estimation

+v(m)(i, k)

=√

Ep

N∑n=1

L−1∑l=0

�(m)n,l (i)e

− j2πklB e

− j2πk(n−1)LB + w(m)(i, k)

=√

Ep

NL∑κ=1

λ(m)κ e

− j2πkqB + w(m)(i, k) (12)

where w(m)(i, k) =∑N

n=1∑L−1

l=0 �(m)n,l (i)e

− j2πklB bn(i, k) + v(m)(i, k).

In (12), the channel transfer functions of MIMO/OFDMsystem are in fact incorporated into a single vector fol-lowing the relationship λ(m)

(n−1)L+l(i) = �(m)n,l (i), l = 0, · · · , L −

1, n = 1, · · · ,N, which allows us to use the single usersuperimposed training-based approach in estimating theMIMO channel coefficients.

Let H(m)(i) = [�(m)1,0 (i), · · · , �(m)

1,L−1(i), · · · , �(m)N,0(i), · · · , �(m)

N,L−1(i)]T

be the channel coefficient vector associated with theith OFDM symbol. Substituting (10)-(11) into (5), thedemodulated signal vectors U(m)(i) can be written by

U(m)(i) = AH(m)(i) + E(m)(i) +V(m)

(i) (13)

where V(m)

(i) is the noise vector in frequency-domain, E(m)(i) = [E(m)(i, 0), · · · ,E(m)(i,B − 1)]T

is the interference vector produced by theinformation sequences with the interference signalsE(m)(i, k) =

∑Nn=1 H(m)

n (i, k)bn(i, k). A = [A(1, 0), · · · ,A(1, L −1), · · · ,A(n, l), · · · ,A(N, 0), · · · ,A(N, L − 1)] is a B × NLmatrix with the column-vectors

A(n, l) =√

Ep[e− j2π0l

B , · · · , e − j2πklB , · · · , e − j2π(B−1)l

B ]

n = 1, · · · ,N, l = 0, · · · , L − 1 (14)

Since the matrix A is known, when B > NL, the matrix Ais of full column rank, and the channel coefficient vectorscan be thus estimated by

H(m)

(i) = A†U(m)(i)

= H(m)(i) +A†E(m)(i) +A†V(m)(i)m = 1, · · · ,M, i = 1, · · · , I (15)

where the superscript † is the pseudo-inverse operation,and the hat ∧ indicates the estimation. From (15), themainly computational effort is directly proportional tothe unknown parameter number NL.

By analogy, channel estimation can be performed intime-domain either. By (10)-(11), we have the IFFT de-modulated signals

xn(i, t) = [FFT−1Sn(i)]1,t

= x′n(i, t) +√

EpBδ(t − (n − 1) − L), n = 1, · · · ,N(16)

where x′n(i, t) is the IFFT modulated signals of the in-formation sequences bn(i, t). The received signals (3) intime- domain can be thus obtained as

y(m)(i, t) =N∑

n=1

L−1∑l=0

�(m)n,l (i)

√EpBδ(t − (n − 1) − L)

+

N∑n=1

L−1∑l=0

�(m)n,l (i)x′n(i, t) + v(m)(i, t)

= λ(m)(n−1)L+l(i)

√EpBδ(t − (n − 1) − L)

+

N∑n=1

L−1∑l=0

�(m)n,l (i)x′n(i, t) + v(m)(i, t). (17)

Consequently, the channel estimation can be performedequivalently to that of the first-order-statistical method[8]-[13] [17] in time-domain

λ(m)(n−1)L+l(i) = �

(m)n,l (i) =

y(m)(i, (n − 1)L + l)√EpB

= �(m)n,l (i) +

N∑g=1

L−1∑κ=0�

(m)g,κx′g(i, (n − 1)L + l − κ)√

EpB

+v(m)(i, (n − 1)L + l)√

EpB, i = 1, · · · , I. (18)

Using the specifically designed superimposed trainingsequences in (11), the problem of the indistinguishablepilots in MIMO channel estimation is solved acrossvarious users, which allows us to utilize the single-user superimposed training-based channel estimationapproach in either frequency- domain (13)-(15) or time-domain (16)-(18).

From (13) and (18), we note that the informationsequence interference vector (the second entry of (15)and (18)) can hardly be neglected unless using a largepilot power Ep. The conventional ST trainings statedin [8]-[13] [17] employ averaging the channel estimatesover multiple OFDM symbols (or training periods) tosuppress the information sequence interference in thecase that the channel is linearly time-invariant duringthe record length. This arithmetical average operation in[8]-[13] [17], however, is no longer feasible to the channelassumed in this paper wherein the channel coefficientsare time-varying over multiple OFDM symbols.

B. Channel Estimation over Multiple OFDM Symbols

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In this sub-section, we develop a weighted averageapproach to suppress the abovementioned informationsequence interference over multiple OFDM symbols, andthus overcoming the shortcoming of conventional ST-based schemes for linearly time-varying channel estima-tion.

We take the LTV channel coefficient estimation of eachOFDM symbol, i. e. �(m)

n,l (i), i = 1, · · · , I by equations (15)and (18) as a temporal result, and then form a vector ash

(m)n = [�(m)

n,l (1), · · · , �(m)n,l (I)]T, n = 1, · · · ,N, l = 0, · · · , L − 1.

Following the channel modeling in (7), we have

h(m)n = ηh

(m)n,l,q

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝e− j2π(0−Q/2)t1

Ω · · · e− j2π(Q−Q/2)t1

Ω

... · · · ...

e− j2π(0−Q/2)tI

Ω · · · e− j2π(Q−Q/2)tI

Ω

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (19)

where h(m)n,l,q = [�(m)

n,l,q, · · · , �(m)n,l,q, · · · , �(m)

n,l,q]T is estimation of

the complex exponential coefficients vector modeling theLTV channel (4), η is a I × (Q + 1) matrix with [η]q,i =e− j2π(q−Q/2)ti

Ω . Thus, when I ≥ Q + 1, the matrix η is of fullcolumn rank, and the basis expansion model coefficientscan be computed by

�(m)n,l,q = η

†�

(m)n,l , n = 1, · · · ,N, l = 0, · · · , L − 1. (20)

Substituting ti = (i − 1)B′ + B/2 into the matrix η, weobtain the pseudo-inverse matrix as

[η†]i,q =e− j2π(q−Q/2)((i−1)B′+B/2)/Ω

I(21)

By (19)-(21), the coefficients (4) can be computed by

�(m)n,l,q =

I∑i=1

e− j2pi(q−Q/2)((i−1)B′+B/2)/Ω�

(m)n,l (i)

I(22)

In fact, (22) is estimated over multiple OFDM symbolswith a weighted average function of ej2π(q−Q/2)ti/I.

Compared with the conventional ST that are gener-ally limited to the case of LTI channels, the proposedweighted average approach can be performed to esti-mate the LTV channels of MIMO/OFDM systems. Infact, the proposed channel estimation is composed oftwo steps: First, with specially designed training signalsin (11), we estimate the channel coefficients duringeach OFDM symbol as temporal results. Second, thetemporal channel estimates are further enhanced overmultiple OFDM symbols by using a weighted averageprocedure. That is, not only the target OFDM symbol,but also the OFDM symbols over the whole frame areinvoked for channel estimation. Similar to the averageprocedure of LTI case [8]-[13][17], it is thus anticipatedthat the weighted average estimation may also exhibita considerable performance improvement for the time-varying channels over a long frame Ω.

IV. Channel Estimation AnalysisIn this section, we analyze the performance of the

proposed channel estimator in Section III and derive aclosed-form expression of the channel estimation vari-ance which can be, in turn, used for superimposedtraining power allocation. Before going further, we makethe following assumptions:

(H1): The information sequence {bn(i, k)} is equi-powered, finite-alphabet, i.i.d., with zero-mean and vari-ance Eb.

(H2): The additive noise {v(m)(i, t)} is white, uncorre-lated with {bn(i, k)}, with E[v(m)(i, t)] = σ2

v.(H3): The LTV channel coefficients h(m)

n,l are complexGaussian variables, and statistically independent.

The interference vector caused by the informationsequence in (15) and (18) can be rewritten as

A†E(m)(i) =[ε(m)

1,0 (i), · · · , ε(m)1,L−1(i), · · · , ε(m)

N,0(i), · · · , ε(m)N,L−1(i)

]T=

1√EpB

[ N∑g=1

L−1∑κ=0

�(m)g,κx

′g(i,B − κ), · · · ,

N∑g=1

L−1∑κ=0

�(m)g,κx

′g(i, (N − 1)L + L − κ)

](23)

The additive noise vector is also given by

A†V(m)(i) = υ(m)(i)

=[υ(m)(i, 0), · · · , υ(m)(i,NL − 1)

]T=

1√EpB

[v(m)(i, 0), · · · ,

v(m)(i, (n − 1)L + l), · · · , v(m)(i,NL − 1)]

(24)

By (H2), v(m)(i, t) is also independent of ε(m)n,l (i). We first

calculate the variance of v(m)(i, t) in (24) by

var(υ(m)(i, t)

)=

1√EpB

E[∣∣∣v(m)(i, t)

∣∣∣2] = σ2v√

EpB(25)

Invoking the assumption (H1), the estimation errorε(m)

n,l (i) =∑N

g=1∑L−1κ=0 �

(m)g,κx′g(i, (N − 1)L + L − κ)/√EpB is ap-

proximately Gaussian distributed for large symbol-sizeB. According to the Parseval’s principal, the estimationvariance due to the information sequence interferencecan be obtained as

var(ε(m)n,l (i)) = E

[∣∣∣∣ε(m)n,l (i)

∣∣∣∣2]=

1√EpB2

N∑g=1

B−1∑k=0

∣∣∣H(m)g (i, k)

∣∣∣2

=1√EpB

N∑g=1

L−1∑κ=0

∣∣∣h(m)g,κ(i)

∣∣∣2 (26)

Since (26) depends upon the channel transfer functions(equivalently, the channel impulse response), we definethe normalized variance as

nvar(ε(m)n,l (i)) =

1∣∣∣�(m)(i)∣∣∣2 var(ε(m)

n,l (i)) (27)

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where �(m)(i) =∑N

n=1∑L−1

l=0 |�(m)(i)|2/NL. Following thedefinition of (27), we obtain the normalized variance as

nvar(ε(m)n,l (i)) =

Eb

N∑g=1

L−1∑κ=0

∣∣∣h(m)g,κ(i)

∣∣∣2BEp|�(m)(i)|2 =

NLB

Eb

Ep(28)

From (28), we can find that the estimation variance dueto the information interference is directly proportionalto the information-to-pilot power ratio Eb/Ep , therebyresulting in an inaccurate solution for the general casethat Ep Eb.

Then, we analyze the channel estimation performanceof the weighted average approach over multiple OFDMsymbols Ω (the whole frame ). Define

ε(m)n,l =

[ε(m)

n,l (1), · · · , ε(m)n,l (I)

]T(29)

υ(m) =[υ(m)(1), · · · , υ(m)(I)

](30)

By (H1)-(H3), the MSE of the weighted average channelestimator is given by

MSE(m)n,l =

{‖ h(m)

n,l − h(m)n,l ‖2

}= tr

{η†E

{ε(m)

n,l (ε(m)n,l )H

}(η†)H

}+ tr

{η†E

{υ(m)(υ(m))H

}(η†)H

}= tr

[ηHη

]−1 1I

I∑i=1

{var(υ(m)(i)) + var(ε(m)

n,l (i))}

(31)

where ‖ • ‖ is the Euclidean norm, tr(•) is the traceof matrix. Note that the column vectors of the matrixη in (21) are in fact the FFT vectors of a I × I matrix,we thus have ηHη = II(Q+1) and tr[ηHη]−1 = (Q + 1)/I.Substituting (25)-(26) into (31), we obtain the varianceof the weighted average estimation h(m)

n,l,q associated with

ε(m)n,l (i), i = 1, · · · , I as

ρ(m)n,l,q =

(Q + 1)Eb

BEpI2

I∑i=1

N∑g=1

L−1∑κ=1

∣∣∣�(m)g,κ(i)

∣∣∣2 . (32)

By analogy, the variance of the additive noise υ(m)(i), i =1, · · · , I can be also derived as

E[∣∣∣υ(m)

∣∣∣2] = σ2v(Q + 1)BIEp

=(Q + 1)σ2

v

ΩEp(33)

Combining the variances in (32) and (33), we have theweighted average estimation variances

MSE(m)n,l =

(Q + 1)Eb

ΩIEp

I∑i=1

N∑g=1

L−1∑κ=1

∣∣∣�(m)g,κ(i)

∣∣∣2 + (Q + 1)σ2v

ΩEp.

(34)

In (34), the last term is due to the additive noise(Q+ 1)/Ω 1. In general, since the LTV channel modelsatisfies , the additive noise is greatly suppressed bythe weighted average procedure. On the other hand,estimation variance due to the information sequenceinterference (32) may be the dominant component ofthe channel estimation error, especially for high SNR.

Similar to (26)-(27), we derive the normalized varianceof ρ(m)

n,l,q by removing the channel gain by

nvar(ρ(m)n,l,q) =

1∣∣∣h(m)∣∣∣2 ρ(m)

n,l,q (35)

where |h(m)| = ∑Ii=1

∑Nn=1

∑L−1l=1 |�(m)(i)

n,l |2/NLI. From (34) and(35), it follows that

nvar(ρ(m)n,l,q) =

(Q + 1)Eb

|h(m)|2BEpI2

I∑i=1

N∑g=1

L−1∑κ=1

∣∣∣�(m)g,κ(i)

∣∣∣2=

(Q + 1)NLEb

ΩEp

B′

B≈ (Q + 1)NL

Ω

Eb

Ep(36)

From (36), the normalized variance is directly pro-portional to the information-pilot power ratio (IPPR)Eb/Ep and the ratio of the unknown parameter numberNL(Q+1) over the frame length Ω. In particular, with thespecifically designed training sequence (11), the closed-form estimation variance (36) may provide a guidelinefor signal power allocation at transmitter, e.g. for a giventhreshold of the estimation variance φ (channel gainhas been normalized), the minimum training power Ep

should at least satisfy the approximated constraint asEp ≥ φΩEb/NL(Q + 1).

Compared with the variances of channel estimationover one OFDM symbol as in (26)-(28), the estimationvariances (34)-(36) of the weighted average estimator(19)-(22) is significantly reduced owing to the fact thatΩ/B(Q + 1) � 1. Theoretically, the weighted averageoperation can be considered as an effective approach inestimating LTV channel, where the information sequenceinterference can be effectively suppressed over multipleOFDM symbols. As stated in the conventional ST-basedschemes [8]-[17], channel estimation performance canbe improved along with the increment of the recordedframe length Ω, i.e. the estimation variance approachesto zero as Ω →∝. This can be easily comprehendedthat larger frame length Ω means more observationsamples, and hence lowers the MSE level. From theLTV channel model (4), however, we note that as theframe length is increased, the corresponding truncatedDFB requires a larger order Q to model the LTV channel(maintain a tight channel model), and the least ordershould be satisfied Q/2 ≥ fdΩ fs, where fd and fs are theDoppler frequency and sampling rate, respectively [1]-[5]. Consequently, as the frame length Ω increases, theLTV channel estimation variance (36) approaches to onlya fixed lower-bound associate with the system Dopplerfrequency as well as the information-pilot power ratio.This is quite different from the ST trainings in estimatingLTI channels [8]-[17].

Explicitly, the weighted average achieves a significantimprovement in estimating the LTV channel comparedwith the conventional ST [8]-[17]. As the frame lengthΩ is increased, however, the estimation variances ap-proach to a fixed lower-bound. The enhancement of thechannel estimation (19)-(22) should resort to increasingthe superimposed training power allocation. This will

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be a great disadvantage to the wireless communicationsystems that requires an accurate CSI but with a limitedpower.

V. Improved Channel Estimation

In this section, we provide an alternative to enhancethe channel estimation performance for the case wherethe training power is limited.

As aforementioned, the information sequences fromall users are the dominant interference to channel esti-mation. This fact motivates us to enhance the channelestimation performance via mitigating the informationsequence interference. To this end, we propose an it-erative cancellation approach, which can be consideredas a twofold process. First, the information sequencesare recovered by a hard detector based on the LTVchannel estimation in Section III. Second, the recovereddata symbols are removed from the received signals tocancel the information sequence interference and, thusto enhance the channel estimation performance (henceSER level).

To recover the information sequence, the contributionof the pilots is firstly removed from the received signalsby

u(m)(i, k) = u(m)(i, k) −N∑

n=1

H(m)n (i, k)pn(i, k)

=

N∑n=1

H(m)n (i, k)sn(i, k) + e(m)(i, k) + v(m)(i, k) (37)

where H(m)n (i, k) =

∑L−1l=0

∑Qq=0 h(m)

n,l,qej2π(q−Q/2)ti/Ω =∑L−1

l=0 �(m)n,l (i)ej2πkl/B is the estimated LTV channel

transfer function, e(m)(i, k) is the residual error ofthe superimposed pilots. Stacking the remaining signals(37) to form vectors as U(i, k) = [u(1)(i, k), · · · , u(M)(i, k)]T,we have

U(i, k) = H(i, k)b(i, k) +ω(i, k) (38)

where H(i, k) is a M × N matrix with [H(i, k)]m,n =H(m)

n (i, k), b(i, k) = [b1(i, k), · · · , bN(i, k)]T is the data vectorand ω(i, k) = [e(1)

n (i, k)+ v(1)(i, k), · · · , e(M)n (i, k) + v(M)(i, k)] is

the interference vector. When H(i, k)] is of full columnrank, we obtain

b(i, k) = H(i, k)†U(i, k). (39)

The data symbols, owing to the finite alphabet set prop-erty, can be recovered by a hard detector as

bn(i, k) = arg min︸︷︷︸bn(i,k)∈Θ

[‖ bn(i, k) − bn(i, k) ‖2

](40)

where Θ is the finite alphabet set from which the trans-mitted data takes, e.g. 4-PSK and 8-PSK signals etc.

Using the recovered data symbols (40), the informa-tion sequence interference can be mitigated by removing

the estimated information sequences from the receivedsignals as

u(m)(i, k) = u(m)(i, k) −N∑

n=1

L−1∑l=0

�(m)n,l (i)e

− j2πklB bn(i, k)

=

N∑n=1

L−1∑l=0

�(m)n,l (i)e

− j2πklB pn(i, k)

+

N∑n=1

L−1∑l=0

(�

(m)n,l (i) − �(m)

n,l (i))e− j2πkl

B bn(i, k)

+

N∑n=1

L−1∑l=0

�(m)n,l (i)e

− j2πklB

(bn(i, k) − bn(i, k)

)+ v(m)(i, k).

(41)

Obviously, the information sequence interference can beeffectively cancelled as the following inequality holds∣∣∣∣∣∣∣

N∑n=1

L−1∑l=0

�(m)n,l (i)e

− j2πklB

∣∣∣∣∣∣∣ >∣∣∣∣∣∣∣

N∑n=1

L−1∑l=0

(�

(m)n,l (i) − �(m)

n,l (i))e− j2πkl

B

∣∣∣∣∣∣∣+ 2β

∣∣∣∣∣∣∣N∑

n=1

L−1∑l=0

�(m)n,l (i)e

− j2πklB

∣∣∣∣∣∣∣ (42)

where β is the SER of the hard detector in (40). Notethat the above inequality depends on the SER level β. Itis shown in Appendix that the upper-bound of the initialis

β ≤ 12

⎧⎪⎪⎨⎪⎪⎩1 −√

NL(Q + 1)Ω

Eb

Ep

⎫⎪⎪⎬⎪⎪⎭ . (43)

From (43), the iterative approach allows a wide range ofsymbol-error rate β.

Replacing the received signals u(m)(i, k) in (12)-(16) and(19)-(22) by u(m)(i, k) in (41), we re-estimate the CSI withthe channel estimation approach in Section III. Owe tothe interference mitigation (41), the performance of there-estimation can be significantly improved. Obviously,the effectiveness of the interference mitigation dependscrucially on the SER of the hard detector (40). From (37),we notice that the SER performance of the hard detector(40), in addition to additive noise, is also affected by theresidual error of superimposed training. We rewrite theresidual error as

e(m)(i, k) =N∑

n=1

[H(m)

n (i, k) − H(m)n (i, k)

]pn(i, k),

m = 1, · · · ,M. (44)

Note that e(m)(i, k), k = 0, · · · ,B− 1 is distributed over thewhole frequency tones, whereas owing to the specifi-cally designed training signals in (11), the time-domainreceived signals affected by the residual error are con-centrated only during a sequence of sample periodst = (n − 1)L + κ, κ = 0, · · · , L − 1, n = 1, · · · ,N. Inorder to mitigate the residual error, a natural idea isto iteratively reconstruct the time-domain signals, i. e.y(m) = (i, (n − 1)L + t), n = 1, · · · ,N, t = 0, · · · ,B − 1. Using

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the estimated CSI, time-domain signal vectors associatedwith the recovered sequences by (40) can be obtained by

Y(m)

(i) =[y(m)(i, 0), · · · , y(m)(i, t), · · · , y(m)(i,B − 1)

]=

N∑n=1

F−1[H(m)

n (i, 0)bn(i, 0), · · · , H(m)n (i,B − 1)bn(i,B − 1)

].

(45)

Replaced the contaminated signals y(m)(i, (n−1)L+ t), n =1, · · · ,N, t = 0, · · · ,B − 1 by the reconstructed signals in(45), the received signal vector is then updated by

Y(m)

(i) =[y(m)(i, 0), y(m)(i, 1), · · · , y(m)(i, (n − 1)L + t), · · · ,

y(m)(i, (N − 1)L + L − 1), y(m)(i,NL), · · · , y(m)(i, (B − 1))].

(46)

The reconstructed signals in the current iterationy(m)(i, (n − 1)L + t) should be closer to the actual signalsy(m)(i, (n − 1)L + t) than that of the previous iteration asthe following inequality holds∣∣∣y(m)

cur (i, (n − 1)L + t) − y(m)(i, (n − 1)L + t)∣∣∣ <∣∣∣y(m)

pre (i, (n − 1)L + t) − y(m)(i, (n − 1)L + t)∣∣∣ (47)

where y(m)(i, (n− 1)L+ t) is the received signal associatedwith the data signal bn(i, k), y(m)

cur (i, (n − 1)L + t) andy(m)

pre (i, (n − 1)L + t) are the reconstructed signals in thecurrent and previous iteration, respectively.

By the procedure above, we can achieve a lower SERthan that of (40) by using the updated signals in (46). Thenewly recovered data sequence is then removed fromthe received signals to cancel the information sequenceinterference as (41), and thus to enhance the channelestimation performance in the next iteration. Meanwhile,the corresponding enhanced channel estimates are thenutilized to mitigate the residual error of pilots, sim-ilar to (45)-(46), thereby achieving an improved SERperformance. Repeat the process above until the in-crement changes of the improved channel estimationperformance over successive iterations is below a giventhreshold.

We resort to the channel estimation variance (as dis-cussed in (26)-(28), (34)-(36)) to evaluate the enhancedestimation performance of the proposed iterative ap-proach. Consider the following assumption:

(H4): The recovered information sequences bn(i, k) andthe symbol error vectors δ bn(i, k) = bn(i, k) − bn(i, k) areindependent of additive noise.

Invoking assumption (H4), the corresponding estima-tion variance can be derived from (32)-(36) as

MSE(m)n,l,iter ≤

4βsteady(Q + 1)Eb

IΩEp

I∑i=1

N∑n=1

L−1∑l=0

∣∣∣∣�(m)n,l (i)

∣∣∣∣2 + (Q + 1)σ2v

ΩEp(48)

where βsteady is the steady-state error probability ofdetected information symbols (SER). The normalized

variance due to the information interference, i.e. (the firstterm of (48)) is reduced into

nvar(ε(m)n,l,iter) ≤

4βsteady NL(Q + 1)Ω

Eb

Ep(49)

In many applications, the information sequences areusually pre-coded before transmission, e.g. convolu-tional coding and block interleaving over one OFDMsymbol, to confront the channel distortion and additivenoise. The demodulator may obtain a small steady-state SER, e.g. βsteady → 10−3 ∼ 10−4. With such asmall βsteady, the iterative approach enables a significantimprovement on channel estimation compared with theweighted average (34)-(36). In other words, the newiterative scheme provides an alternative to enhance theestimation performance for wireless communication sys-tems that requiring accurate CSI but with a limitedpower.

Obviously, the enhancement of the iterative schemeabove is at the cost of an increment in computationalcomplexity that is directly proportional to the iterationnumber. The iterative approach may offer a tradeoff be-tween computational complexity and pilot power. Thatis, for the case that the transmission power is not limited,the weighted average (12)-(16) and (19)-(22) is indeed afeasible solution to LTV channel estimations, whereas ifthe power is not enough to achieve a satisfactory esti-mation, the iterative method above provides a valuablealternative at the cost of an increment in computationalcomplexity.

VI. Simulations

We assume the MIMO/OFDM system with N = 4 andM = 6. The symbol-size is B = 512 and the transmitteddata Sn(i, k) is 8-PSK signals with symbol rate fs =107/second. Before transmission, the transmitted data arecoded by 1/2 convolutional coding and block interleav-ing over one OFDM symbol [19]. The channel is assumedto be L = 10 and, the coefficients h(m)

n (t) are generatedas low-pass, Gaussian and zero mean random processesand correlated in time with the correlation functionsaccording to Jakes’ mode rn(τ) = μ2

nJ0(2π fnτ), n = 1, · · · , 4where fn is the Doppler frequency associated with the nth user. The multi-path intensity profile is chosen to beφ(l) = el/10, l = 0, · · · , L − 1. The Doppler spectra are

ψ(l) =(π√

f 2d − f

)for f ≤ fd where fd is the Doppler frequency of theuser, otherwise, ψ(l) = 0. CP length is chosen to be 15 toavoid inter-symbol interferences. The additive noise is aGaussian and white random process with a zero mean.

Test Case 1. Channel estimation: We run simulationswith the Doppler frequency fn = 300Hz that correspondsto the maximum mobility speed of 162 km/h as theusers operate at carrier frequency of 2GHz. In orderto model the LTV channel, the frame is designed asΩ = B′ × 256 = (B + CPlength) × 256 = 136192, i. e.each frame consists of 256 OFDM symbols. During the

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frame, the channel variation is fnΩ/ fs = 4.1. On the otherhand, the channel variation during an OFDM symbol isfnB/ fs = 0.0154. Such a small variation can be neglected,and the LTV channel can be approximated as LTI duringeach OFDM symbol interval. Over the frame Ω, weutilize the truncated DFB of order Q = 10 > 2 fdΩ/ f s tomodel the LTV channel coefficients. In order to estimatethe MIMO/OFDM channels, the superimposed pilotsare designed according to (11) with the pilot powerEp = 0.05Eb. Fig.1 depicts the LTV channel coefficientestimation over the frame Ω. It is clearly observed thatalthough the channel coefficient is accurately estimatedduring the centre part of the frame, the outmost samplesover the whole frame still exhibit errors. A possibleexplanation is that as the Fourier basis expansions in(4) are truncated, and an effect similar to the Gibbsphenomenon, together with spectral leakages, will leadto some errors at the beginning and the end of the frame.This may be a common problem for the proceedingliteratures [1]-[5] that using basis expansions to modelthe LTV channels. To solve the problem, the frames aredesigned to be partially overlapped, e.g. the frames aredesigned as (� − 1)Ω− ΓB′ ≤ t ≤ �Ω, � = 2, 3, · · · , where Γis a positive integer. By the frame-overlap, the channel atthe beginning and the end of one frame can be modeledand estimated from the neighboring frames.

To further evaluate the new channel estimators, weuse the mean square errors to measure the channelestimation performance by

MSE(m)n =

Ω/B′∑i=1

MSEmn (i)

=B′

Ω

Ω/B′∑i=1

E

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

∣∣∣∣∣ B−1∑t=0

L−1∑l=0

h(m)n,l (i, t) −

Q∑q=0

h(m)n,l,qe

− j2π(q−Q/2)tΩ

∣∣∣∣∣2BL

∣∣∣∣h(m)n,l (i, t)

∣∣∣∣2⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

(50)

where h(m)n,l,q is the channel coefficient estimation and

MSEmn (i) denotes the MSE of the i th OFDM symbol.

We firstly test the two-step channel estimator (10)-(23)under the different pilot powers and different channelcoefficient numbers to verify the channel estimationvariance analysis (34)-(36). The LTV channel is the sameas that in Fig. 1. As shown in Fig. 2, the MSE of thechannel estimation approach are almost independentof the additive noises, especially as SNR > 5dB. Thisis consistent with the channel estimation analysis (34)where the additive noise has been greatly suppressed bythe weighted average procedure. Thus, the estimationerrors depend mainly on the information- pilot powerratio as well as the system unknowns NL. This is ratherdifferent from the FDM trainings [19]-[21].

We then compare the two-step channel estimationscheme with the conventional ST-based methods [8][9]under the different Doppler frequencies. In the conven-tional ST scheme, the LTV channel is firstly estimatedfrom the LTI assumption at each OFDM symbol, andthen all the estimations from the frame Ω are averaged

0 2 4 6 8 10 12 14−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Time (ms)

One

Tap

Cha

nnel

Coe

ffici

ent

True LTV channel: Jakes modelQ = 10 Estimated Coefficient

Fig. 1. One tap coefficient of the LTV channel and the estimation overthe frame Ω = 136192/102 = 13.62ms.

0 5 10 15 20 25 3010

−3

10−2

10−1

100

Mea

n S

quar

e E

rror

(M

SE

)

Signal−noise Ratio (dB)

E

p = 0.02E

b, NL = 40

Ep = 0.10E

b, NL = 40

Ep = 0.50E

b, NL = 40

Ep = 0.50E

b, NL = 20

Fig. 2. Mean square error of the weighted average estimation versusSNR for the LTV channel of fn = 300Hz, Ω = 13.62ms.

to confront the information sequence interferences. Itshows clearly in Fig. 3 that for the LTI channel offn = 0Hz, both the conventional ST and the weightedaverage estimator exhibit the similar performance. Inaddition, the estimation performance can be improvedwith the increment of the frame length. However, whenthe channel involved in simulations is linearly time-varying, the channel estimation performance of the con-ventional ST-based schemes is degraded extensively. Thesimulation reveals the shortcoming of the conventionalST in estimating the LTV channels. On the contrary, theMSE level is reduced by the weighted average processfor the LTV channels of fn = 100, 300Hz. We also observethat the MSE approaches to a constant as the incrementof the frame length, i. e. the lower-bound that associatedwith the given Doppler frequency.

The simulations in Figs.2 and 3 are consistent with theestimation analysis in section IV. That is, although theweighted average procedure can be used as an effectiveapproach in estimating the LTV channels, the estimationperformance can not be improved as that of the LTI case

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0 100 200 300 400 500 60010

−4

10−3

10−2

10−1

100

OFDM Symbol Number of the Frame

Mea

n S

quar

e E

rror

(M

SE

)

Weighted Avera. fn = 0HzWeighted Avera. fn = 100HzWeighted Avera. fn = 300HzConventional ST fn = 0HzConventional ST fn = 100HzConventional ST fn = 300Hz

Fig. 3. Mean square errors versus frame or average length underthe different Doppler frequencies of the LTV channel of Ω = 13.62ms,Ep = 0.05Eb and NL = 40.

by increasing the frame length Ω. To acquire a more ac-curate estimation result, more power should be assignedto the superimposed training signals which could haveotherwise been allocated to the information sequence.Thus, for the case that the training power is limited, weresort to the iterative interference cancellation approachin Section V to further enhance the channel estimationperformance.

Fig. 4 depicts the performance between the weightedaverage scheme and the iterative estimator in termsof channel MSE. As shown in Fig. 4, the new itera-tive estimation exhibits a more significant improvementthan that of weighted average estimation even with asmaller pilot power Ep = 0.05Eb, which conforms thatthe information sequence interferences can be effectivelycancelled by iterative procedure in Section V, and thusenables a significant improvement on channel estima-tion.

The channel MSE is also compared with that of theFDM training-based channel estimator [21] in Fig. 5.Since the channel is time-varying between OFDM sym-bols, FDM training-based channel estimation should beperformed under the LTI assumption within each OFDMsymbol. For estimating the MIMO/OFDM channels withthe unknowns of NL = 40, 64 known pilot symbolswhich are subject to optimal pilot specifications [21] areused in one OFDM symbol. That is, about 12.5% of totalsub-carriers are assigned for pilot tones. Comparatively,it can be seen that the iterative channel estimation ap-proach outperforms the FDM channel estimator [21] byusing a small pilot power of Ep = 0.05Eb. It reveals thatthe iterative channel estimation achieves a significantimprovement on channel MSE level (as formulated in(48)-(49)) thanks to the effective interference cancellationprocedure. Moreover, it should be noted that since thesuperimposed pilots are spread over the entire band, theproposed ST-based channel estimator is also feasible toestimate the channel with a very long delay spread, i. e.

0 5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Mea

n S

quar

e E

rror

(M

SE

)

Signal−noise Ratio (dB)

Weighted Ave. Ep = 0.05EbWeighted Ave. Ep = 0.5EbIterative Estimation Ep = 0.05EbIterative Estimation Ep = 0.5Eb

Fig. 4. Mean square errors versus SNR for the LTV channel of fn =300Hz, Ω = 13.62ms and NL = 40.

0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

Mea

n S

quar

e E

rror

(M

SE

)

Signal−noise Ratio (dB)

FDM TrainingIterative Est. Ep = 0.05Eb

Fig. 5. Mean square errors versus SNR for the LTV channel of fn =300Hz, Ω = 13.62ms and NL = 40, where FDM training refers to [21].

cluster-based channel.Fig. 6 illustrates the channel MSE of the iterative

method versus the iteration numbers under SNR = 10dB.The simulation shows that the iterative interference can-cellation method can be used for a wide range of thetraining power allocation, i. e. Ep ≤ 0.02Eb. In addition,we note that the required iteration number to achieve asteady-state solution depend on the pilot power. Highertraining power allocation means smaller iteration num-ber, and thus lowers the corresponding computationalcomplexity, and vice versa. Since the iterative approachmay converge to steady-state estimation by only severaliterations, the overall computational complexity will beacceptable for many wireless communication systems.

The simulations above reveal that the channel esti-mation performance can be improved by the weightedaverage estimator along with the increment of trainingpower, whereas for the power limited case, the iterativeapproach provides a valuable alternative at the cost of aslight increment in computational complexity and, offers

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DAI et al.: LINEARLY TIME-VARYING CHANNEL ESTIMATION FOR MIMO/OFDM SYSTEMS USING SUPERIMPOSED TRAINING 691

1 2 3 4 5 6 7 8 9 1010

−3

10−2

10−1

100

Iterative Number

Mea

n S

quar

e E

rror

(M

SE

)

Ep = 0.25EbEp = 0.10EbEp = 0.02Eb

Fig. 6. Mean square errors versus SNR for the LTV channel of fn =300Hz, Ω = 13.62ms and NL = 40, and SNR = 10dB

0 5 10 15 2010

−5

10−4

10−3

10−2

10−1

100

Signal−noise Ratio (dB)

Sym

bol E

rror

Rat

e (S

ER

)

Conventional ST Symbol DetectionNew Iterative Symbol DetectionFDM Training Detection

Fig. 7. Symbol error rate versus SNR, where ’FDM training’ meansthe symbol detection from the OFDM system of FDM training in [21].

a tradeoff between computational complexity and pilotpower.

Test Case 2. Symbol Detection: As aforementioned,symbol detection is affected by the superimposed train-ing sequence. In fact, the interference caused by the su-perimposed training sequence to information sequencerecovery has been effectively cancelled by the iterativecancellation scheme in Section V. Herein we carry outseveral experiments to assess the effectiveness of theinformation sequence recovery.

Fig. 7 depicts SER performance between the conven-tional ST-based demodulator and the proposed iterativescheme corresponding to the same channel estimationerror ρ(m)

n,l,q = 0.005 and pilot power Ep = 0.05Eb. The con-ventional ST-based symbol detector is assumed as thatof in [17], which is affected by the residual pilot error.As shown in Fig. 7, the iterative procedure achieves asignificant improvement on SER performance comparedwith the conventional ST-based demodulator, especiallyat high SNR. An explanation for this behavior is that

1 2 3 4 5 6 7 8 9 1010

−4

10−3

10−2

10−1

100

Iterative Number

Sym

bol E

rror

Rat

e (S

ER

)

NL ≈ 0.05BNL ≈ 0.1BNL ≈ 0.2B

Fig. 8. Symbol error rate of the iterative symbol detection versus theiteration number under SNR=15dB.

for large SNR values, the residual error of pilot exhibitsas the domain interference on symbol recovery. As acomparison, we also list the SER performance basedon the optimal FDM training scheme [21] where infor-mation sequences and pilot symbols are of frequency-division multiplexed and the symbol detection can bethus performed without additional pilot interference. Weobserve that the performance of two demodulators is ingeneral indistinguishable, which confirms that the effectsof the abovementioned residual training on informationsequence recovery have been effectively cancelled by theproposed iterative approach.

Fig. 8 illustrates SER performance under the differentreconstructed signal-size, or the channel unknowns NL.As stated in Section V-A, the number of iteration uti-lized to achieve a steady-state SER performance dependscrucially on the ratio of the reconstructed signal-sizeover the symbol-size, i. e. τ = NL/B. It is observedthat when NL/B ≤ 10%, a significant SER performanceimprovement is achieved in the very first iterations(the first 2 ∼ 3 iterations). Meanwhile, the minimumiterations required to achieve the steady-state solutionincreases along with the increment of . For the situationthat NL/B > 20%, the iterative cancellation may notconvergent and the SER still keeps at a high level. Thus,τ = NL/B ≈ 0.2 is approximately the upper-bound forthe implementation of the proposed iterative approach.

VII. Conclusions

In this paper, we have developed a new method forestimating the LTV channels of MIMO/OFDM systemsby using superimposed training. The LTV channel coef-ficients were firstly modeled by the truncated DFB, andthen a two-step approach was investigated where weestimate the channel by using the user-specific trainingsequence. Using the estimated channel from the firststep, a weighted average procedure was proposed toenhance the channel estimates over multiple OFDMsymbols. We also present a performance analysis of the

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692 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 2, FEBRUARY 2010

channel estimation approach and derive a closed-formexpression for the channel estimation variances. It isshown that the estimation variances, unlike conventionalsuperimposed training, approach to a fixed lower-boundthat can only be reduced by increasing the pilot power.For systems with a limited pilot power, we developed aniterative scheme to enhance the estimation performancewhere the contribution of the information sequence in-terference to channel estimation as well as the trainingeffects on symbol recovery are effectively cancelled. Theiterative approach offers a tradeoff between computa-tional complexity and pilot power. Compared with theexisting ST and FDM trainings, the new estimator doesnot have a loss of rate and requires a lower trainingpower, and thus enables a higher efficiency.

Appendix

Derivation of the upper-bound of β in (43):By (45), we have

2β

∣∣∣∣∣∣∣N∑

n=1

H(m)n,l (i, k)

∣∣∣∣∣∣∣<

∣∣∣∣∣∣∣N∑

n=1

H(m)n,l (i, k)

∣∣∣∣∣∣∣ −∣∣∣∣∣∣∣

N∑n=1

(H(m)

n,l (i, k) −H(m)n,l (i, k)

)∣∣∣∣∣∣∣<

∣∣∣∣∣∣∣N∑

n=1

H(m)n,l −

N∑n=1

(H(m)

n,l (i, k) −H(m)n,l (i, k)

)∣∣∣∣∣∣∣k = 0, · · · ,B − 1. (51)

Therefore, it follows that

2βE

⎧⎪⎪⎪⎨⎪⎪⎪⎩∣∣∣∣∣∣∣

N∑n=1

H(m)n,l (i)

∣∣∣∣∣∣∣2⎫⎪⎪⎪⎬⎪⎪⎪⎭

< E

⎧⎪⎪⎪⎨⎪⎪⎪⎩∣∣∣∣∣∣∣

N∑n=1

H(m)n,l (i) −

N∑n=1

(H(m)

n,l (i) − H(m)n,l (i)

)∣∣∣∣∣∣∣2⎫⎪⎪⎪⎬⎪⎪⎪⎭ (52)

where H(m)n,l (i) = [H(m)

n,l (0), · · · ,H(m)n,l (B − 1)]T and H

(m)n,l (i) =

[H(m)n,l (0), · · · , H(m)

n,l (B − 1)]T. According to the Parseval’sprincipal, (52) can be approximated by

2βE

⎧⎪⎪⎪⎨⎪⎪⎪⎩∣∣∣∣∣∣∣

N∑n=1

�(m)n,l (i)

∣∣∣∣∣∣∣2⎫⎪⎪⎪⎬⎪⎪⎪⎭

< E

⎧⎪⎪⎪⎨⎪⎪⎪⎩∣∣∣∣∣∣∣

N∑n=1

�(m)n,l (i) −

N∑n=1

(�

(m)n,l (i) − �(m)

n,l (i))∣∣∣∣∣∣∣

2⎫⎪⎪⎪⎬⎪⎪⎪⎭ (53)

Invoking the assumption (H3) and (28), we further ob-tain

β <1

2∣∣∣�(m)(i)

∣∣∣2(∣∣∣�(m)(i)

∣∣∣2 − {var(ε(m)n,l )

})

≈ 12

(1 − nvar(ε(m)

n,l ))

≈ 12

⎛⎜⎜⎜⎜⎜⎝1 −√

NL(Q + 1)Ω

Es

Ep

⎞⎟⎟⎟⎟⎟⎠ (54)

From (54), the upper-bound of the initial SER β in (43)is proved.

Acknowledgments

This work is supported by National Natural Sci-ence Foundation of China (NSFC), Grant 60772132, Keyproject of Natural Science Foundation of GuangdongProvince, Grant 8251027501000011, Science and Tech-nology Project of Guangdong Province, Grant 2007-B010200055, and Industry-Universities-Research Coop-eration Project of Guangdong Province and Ministry ofEducation of the P.R. China, Grant 2007A090302116, andalso supported in part by joint foundation of NSFC andGuangdong Province, Grant U0635003.

References

[1] G. B. Giannakis and C. Tepedelenlioglu, “Basis expansion modelsand diversity techniques for blind identification and equalizationof time-varying channels,” Proc. IEEE, vol. 86, pp. 1969-1986, Oct.1998.

[2] X. L. Ma, G. B. Giannakis, and B. Lu, “Block differential encodingfor rapidly fading channels,” IEEE Trans. Commun., vol. 52, no. 3,pp. 416-425, Mar. 2004.

[3] C. Tepedelenlioglu and G. B. Giannakis, “Transmitter redundancyfor blind estimation and equalization of time- and frequency-selectivity channels,” IEEE Trans. Signal Process., vol. 48, no. 7,pp. 2029-2043, 2000.

[4] X. Dai, “Adaptive blind source separation of multiple-inputmultiple-output linearly time-varying FIR system,” IEE Proc.-Vis.Image Signal Process., vol. 151, no. 4, pp. 279-286, Aug. 2004.

[5] T. Zemen and C. F. Mecklenbrauker, “Time-variant channel esti-mation using discrete prolate spheroidal sequences,” IEEE Trans.Signal Process., vol. 53, no. 9, pp. 3597-3607, Sep. 2005.

[6] W. G. Jeon, K. H. Chang, and Y. S. Cho, “An equalization tech-nique for orthogonal frequency-division multiplexing systems intime-variant multipath channel,” IEEE Trans. Commun., vol. 47,no. 1, pp. 27-32, 1999.

[7] W. Hou and B. Chen, “ICI cancellation for OFDM communicationsystems in time-varying multipath fading channels,” IEEE Trans.Wireless Commun., vol. 4, no. 5, pp. 2100-2110, Sep. 2005.

[8] G. T. Zhou, M. Viberg, and T. McKelvey, “A first-order statisticalmethod for channel estimation,” IEEE Signal Process. Lett., vol. 10,no. 3, pp. 57-60, Mar. 2003.

[9] J. K. Tugnait and W. Luo, “On channel estimation using super-imposed training and first-order statistics,” IEEE Commun. Lett.,vol. 7, no. 9, pp. 413-416, Sep. 2003.

[10] M. Ghogho, “Channel and DC-offset estimation using data-dependent superimposed training,” Electron. Lett., vol. 41, no. 22,pp. 1250-1252, Oct. 2005.

[11] F. Mazzenga, “Channel estimation and equalization for M-QAMtransmission with a hidden pilot sequence,” IEEE Trans. Broad-cast., vol. 46, no. 2, pp. 170-176, June 2000.

[12] J. K. Tugnait and X. H. Meng, “On superimposed training forchannel estimation: performance analysis, training power alloca-tion, and frame synchronization,” IEEE Trans. Signal Process., vol.54, no. 2, pp. 752-765, Feb. 2006.

[13] Q. Y. Zhu and Z. Q. Liu, “Optimal pilot superimposition for zero-padded block transmissions,” IEEE Trans. Wireless Commun., vol.5, no. 8, pp. 2194-2201, Aug. 2006.

[14] Q. Yang and K. S. Kwak, “Superimposed-pilot-aided channelestimation for mobile OFDM,” Electron. Lett., vol. 42 no. 12, pp.722-724, June 2006.

[15] D. C. McLernon, E. A. Hernandez, A. G. Orozco-Lugo, and M.M. Lara, “Performance of data-dependent superimposed trainingwithout cyclic prefix,” Electron. Lett., vol. 42, no. 10, pp. 604-606,May 2006.

[16] M. Ghogho, D. C. McLernon, E. A. Hernandez, and A. Swami,“Channel estimation and symbol detection for block transmissionusing data-dependent superimposed training,” IEEE Signal Pro-cess. Lett., vol. 12, no. 3, pp. 226-229, Mar. 2005.

Authorized licensed use limited to: SUN YAT-SEN UNIVERSITY. Downloaded on February 13, 2010 at 03:04 from IEEE Xplore. Restrictions apply.

DAI et al.: LINEARLY TIME-VARYING CHANNEL ESTIMATION FOR MIMO/OFDM SYSTEMS USING SUPERIMPOSED TRAINING 693

[17] N. Chen and G. T. Zhou, “Superimposed training for OFDM: apeak-to-average power ratio analysis,” IEEE Trans. Signal Process.,vol. 54, no. 6, pp. 2277-2287, June 2006.

[18] T. Cui and C. Tellambura, “Pilot symbols for channel estimationin OFDM systems,” in Proc. IEEE Global Telecomm. Conf., vol. 4,Nov. 2005.

[19] T. Keller and L. Hanzo, “Adaptive multicarrier modulation: aconvenient framework for time-frequency processing in wirelesscommunications,” Proc. IEEE, vol. 88, no. 5, pp. 611-640, May2000.

[20] M. Morelli and U. Mengali, “A comparison of pilot-aided chan-nel estimation methods for OFDM systems,” IEEE Trans. SignalProcess., vol. 40, no. 12, pp. 3065-3073, Dec. 2001.

[21] I. Barhumi, G. Leus, and M. Moonen, “Optimal training design forMIMO OFDM systems in mobile wireless channels,” IEEE Trans.Signal Process., vol. 51, no. 6, pp. 1615-1624, June 2003.

[22] L. Wei and C. Schlegel, “Synchronization requirement for multi-user OFDM on satellite mobile and two-path Rayleigh-fadingchannels,” IEEE Trans. Commun., vol. 43, pp. 887-895, Feb. 1995.

[23] K. Choi, K. Kang, and S. Kim, “Peak power reduction schemebased on subcarrier scrambling for MC-CDMA systems,” IEEProc.-Commun., vol. 151, no. 1, pp. 39-43, Feb. 2004.

Xianhua Dai received the M.Sc. degree andPh.D. degree from Southeast University, Nan-jing, China, in 1988 and 1992, respectively, allin electrical engineering. From 1993 to 1994,he was a postdoctoral researcher with SouthChina University of Technology. From 1995 to2003, he was a faculty member with ShantouUniversity. From 2001 to 2002, he was a seniorvisiting scholar with New Jersey Institute ofTechnology. Since 2003, he has been with theDepartment of Electronics Engineering, School

of Information Science & Technology, Sun Yat-Sen University, where heis now a professor. His research interests include 3G and 4G systems,and bioinformatics.

Han Zhang received the M. Sc. degree fromUniversity of Liverpool, UK, in 2005, and thePh. D degree in the department of Electron-ics Engineering, School of Information Science& Technology, Sun Yat-Sen University, in July2009. He is now with the school of Physics andTelecommunication Engineering, South ChinaNormal University, P. R. China, where he isnow a lecture. His research interests includebroadband wireless communications, MIMO/OFDM technology and cognitive radio.

Dong Li received the B.E. degree from YunnanUniversity, Kunming, China, and the M.Sc. de-gree from Sun Yat-Sen University, Guangzhou,China, in 2004 and 2006, respectively, all inelectrical engineering. He is currently workingtoward his Ph.D. degree in the Departmentof Electronics Engineering, Sun Yat-Sen Uni-versity. His research interests include MIMO,OFDM systems, and cognitive radio.

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