multiuser/mimo doubly selective fading channel …...ieee transactions on vehicular technology, vol....

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 3, MARCH 2010 1341

Multiuser/MIMO Doubly Selective FadingChannel Estimation Using Superimposed

Training and Slepian SequencesJitendra K. Tugnait, Fellow, IEEE, and Shuangchi He

Abstract—We consider doubly selective multiuser/multiple-input–multiple-output (MIMO) channel estimation and datadetection using superimposed training. The time- and frequency-selective fading channel is assumed to be well described by adiscrete prolate spheroidal basis expansion model (DPS-BEM)using Slepian sequences as basis functions. A user-specific periodic(nonrandom) training sequence is arithmetically added (superim-posed) at low power to each user’s information sequence at thetransmitter before modulation and transmission. A two-step ap-proach is adopted, where, in the first step, we estimate the channelusing only the first-order statistics of the observations. In this step,however, the unknown information sequence acts as interference,resulting in a poor signal-to-noise ratio (SNR). We then iterativelyreduce the interference in the second step by employing an it-erative channel-estimation and data-detection approach, where,by utilizing the detected symbols from the previous iteration, wesequentially improve the multiuser/MIMO channel estimation andsymbol detection. Simulation examples demonstrate that, withoutincurring any transmission data rate loss, the proposed approachis superior to the conventional time-multiplexed (TM) training foruncoordinated users, where the multiuser interference in channelestimation cannot be eliminated and is competitive with the TMtraining for coordinated users, where the TM training designallows for multiuser-interference-free channel estimation.

Index Terms—Basis expansion models (BEMs), discrete prolatespheroidal (DPS) sequences, doubly selective fading channels,multiple-input–multiple-output (MIMO) systems, multiuser chan-nel estimation, superimposed training.

I. INTRODUCTION

THIS paper is concerned with channel estimation and datadetection for multiple-input–multiple-output (MIMO)

doubly selective (time- and frequency-selective) fading chan-nels using superimposed training for both single and multipleusers. The increasing demand for high-speed reliable wirelesscommunications over the limited radio-frequency spectrumhas spurred increasing interest in MIMO systems to achieve

Manuscript received October 7, 2008; revised December 8, 2009. Firstpublished December 18, 2009; current version published March 19, 2010.This work was supported by the National Science Foundation under GrantECS-0424145 and Grant ECCS-0823987. This paper was presented in part atthe IEEE International Conference on Acoustics, Speech, and Signal Process-ing, Honolulu, HI, April 2007. The review of this paper was coordinated byDr. K. Hooli.

J. K. Tugnait is with the Department of Electrical and Computer Engi-neering, Auburn University, Auburn, AL 36849 USA (e-mail: [email protected]).

S. He is with the School of Industrial and Systems Engineering, GeorgiaInstitute of Technology, Atlanta, GA 30332 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TVT.2009.2038786

higher transmission rates [32]. To exploit the enormous capac-ity potential of MIMO communications, accurate knowledgeof channel state information is often a prerequisite for manyphysical-layer approaches. In (conventional) time-multiplexed(TM) training-based approaches, training sequences (known tothe receiver), with one per transmit (Tx) antenna, are TM withthe information sequence and transmitted. This incurs a lossin spectral efficiency, decreasing the effective data transmissionrate. At the receiver, one estimates the channel via least-squaresand related approaches. For time-varying channels, one has tofrequently and periodically send training signals to keep up withthe changing channel. This wastes resources. An alternative isto estimate the channel based solely on noisy data exploitingstatistical and other properties of the information sequences;this is the blind channel-estimation approach [9]. However,blind estimation typically requires longer data records, entailshigher computational complexity, and has more stringent iden-tifiability requirements [47].

More recently, superimposed training-based approaches havebeen explored, where the training sequence is added (super-imposed on) at low power to the information sequence beforemodulation and transmission [6], [14]. In contrast with TMtraining, there is no loss in data transmission rate; on the otherhand, some useful power is wasted in superimposed trainingsequences, which could have otherwise been allocated to infor-mation sequences. In this paper, we consider doubly selectiveMIMO channel estimation using superimposed training with KTx antennas (inputs) and N receive (Rx) antennas (outputs).Our approach applies to both the case of K independent userswith one antenna each and the case of one user with coordinatedtransmissions through K antennas (spatial multiplexing and/orspace-time coding) [32]; a combination of the two cases alsofalls within the scope of this paper. In the simulation resultspresented in this paper, we compare and contrast the results ofsuperimposed training with those of TM training. In the caseof TM training, we will distinguish between two cases: 1) thecase of “coordinated” users, where the users are allowed to co-ordinate the placement of the training symbols in the respectivetransmitted block of symbols, resulting in the elimination ofthe multiuser interference in channel estimation at the receiver(as in [50]), and 2) the case of the “uncoordinated” users,where the users are not allowed to (or cannot) coordinate theplacement of the training symbols, resulting in the presenceof multiuser interference in channel estimation at the receiver.The latter case is likely to arise, for example, when there are

0018-9545/$26.00 © 2010 IEEE

1342 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 3, MARCH 2010

multiple independent users with one transmit antenna each. Aswill become clearer later, such a distinction does not arise inour superimposed training schemes.

In communications, channel variations with time arise dueto relative motion between the transmitter and the receiver, anddue to oscillator drifts and phase noise coupled with multipatheffects [33]. These variations can be captured by statisticalmodels where the time-varying finite impulse response (FIR)channels have their channel taps modeled as uncorrelated sta-tionary random processes (Rayleigh or Ricean fading) [33].Recently, basis expansion models (BEMs) have widely beeninvestigated to represent doubly selective channels in wirelessapplications [3], [10], [22], [35], [51], where the time-varyingtaps are expressed as superpositions of time-varying basisfunctions in modeling Doppler effects, which are weighted bytime-invariant coefficients. Candidate basis functions includecomplex exponential (Fourier) functions [10], [17], [22], poly-nomials [3], and discrete prolate spheroidal (DPS) sequences[51], etc. This paper is concerned with time-varying chan-nels described by BEMs, particularly BEMs using Slepian se-quences (DPS sequences) as basis functions; further details maybe found in Section II. An alternative Gauss–Markov model(which is typically a low-order autoregressive (AR) model)has widely been used (see [18], [20], [21], and the referencestherein), which works well as long as the channel does not fadetoo fast. When TM periodically transmitted training is used,channel tracking may not perform well during informationsymbol transmissions (data sessions) since the informationdata are unknown. During data sessions, channel estimatescan only be obtained based on the results from the previoustraining session [20]. This strategy is not appropriate for a fast-fading channel. Potential solutions lie in exploiting the detectedsymbols for channel tracking; for instance, in [18] and [21],joint channel estimation and data detection is implemented viaextended Kalman filtering and/or turbo techniques. Althoughchannel tracking can be improved by such means during datasessions, error propagation due to incorrect detections can bepronounced for fast-fading channels.

Past work based on AR modeling includes [5], where theperformance bounds, including minimum mean square error(MMSE) and bit error rate (BER), of TM or superimposedtraining-based estimation for time-varying flat-fading channelsare considered. Under the same overall power allocation, itwas shown in [5] that superimposed training performs betterfor fast-fading channels, which confirms the intuition that theconstant presence of training offers considerable benefit. Sim-ilar conclusions have been drawn in [4] via a mutual informa-tion and capacity analysis for flat-fading time-selective MIMOchannels. It is shown in [4] that, when superimposed trainingis used, if one reestimates the channel by using the detectedsymbols, one can achieve a channel capacity that is greater thanthat possible in the case of conventional TM training. Channelreestimation using detected symbols allows one to alleviatethe deleterious effects of the interfering data power during theoriginal estimation of the channel using only superimposedtraining. A channel reestimation approach is also followed inthis paper, except that we also consider frequency-selectivechannels (not just flat fading) and specific approaches rather

than the theoretical upper bound on performance in the form ofchannel capacity.

Periodic superimposed training has been discussed in [7],[31], and [45] for time-invariant channels, in [44] and [46]for time-varying single-input–multiple-output (SIMO) chan-nels based on the complex exponential BEM (CE-BEM),and in [13] for time-varying SIMO channels based on DPS-BEM, all for single-carrier systems. Superimposed training-based channel estimation has been considered in [8], [14],and [25] for time-invariant MIMO systems and in [26] and[34] for CE-BEM-based time-variant MIMO single-carrier sys-tems. Reference [26] is an earlier conference version of thispaper focused on channels modeled by CE-BEM. The first-order statistics-based approach of [26] has differently beenderived (it is based on some large sample considerations),compared with this paper, and more significantly, the lemma in[26, p. 409] is incorrect. In this paper, we have followeda different (least-squares) approach to derive the first-orderstatistics-based channel estimator, and we use the more accurateDPS-BEM. In [34], an iterative turbo algorithm is proposed(following [1]) for coded MIMO systems operating in double-selective environments and employing superimposed training.Reference [34] uses the less accurate CE-BEM, whereas we ex-ploit the more accurate DPS-BEM. Reference [34] performs it-erative channel estimation, equalization, and decoding, whereasin this paper, we only consider iterative channel estimationand equalization without considering channel error-correctioncoding. No comparisons with TM training-based approacheshave been provided in [34], whereas we do so in this paper. Theproblem of superimposed training design has been addressed in[14] and [30] for time-invariant MIMO systems.

Superimposed training for orthogonal frequency-divisionmultiplexing (OFDM) systems has been investigated in [15],[16], [29], and references therein for time-invariant systems.Iterative approaches using detected symbols to improve per-formance have also been employed in [15], [16], and [29].An OFDM modulator usually employs a cyclic prefix or zeropadding to mitigate interblock interference. In [28] (and earlierconference papers by the same authors), it was suggested thatthe zero sequence (or the cyclic prefix) be replaced with aknown pseudorandom postfix (PRP) sequence, leading to aPRP-OFDM system. This allows the receiver to exploit an ad-ditional piece of information for channel estimation, i.e., priorknowledge of part of the transmitted block. Advantages of usingthe PRP technique include improved bandwidth efficiency,because the pilot overhead is avoided, and possible use oflow-complexity first-order statistics-based channel-estimationapproaches. Various aspects of PRP-OFDM channel estimationand equalization, including iterative enhancements and semi-blind implementations, have been investigated in [23], [27],[28], [49], and references therein. In [23], it is concluded thatthe first-order statistics-based channel estimator outperformsthe second-order statistics-based channel estimator. In [49],the previously proposed multisymbol encapsulated OFDM ap-proach of [48] has been modified by replacing the traditionalcyclic prefix with a pseudorandom sequence to enhance band-width efficiency. All these PRP-OFDM schemes can be inter-preted as using superimposed training in the frequency domain

TUGNAIT AND HE: MULTIUSER/MIMO DOUBLY SELECTIVE FADING CHANNEL ESTIMATION USING SUPERIMPOSED TRAINING 1343

since the PRP can be seen as being superimposed on the zerosequence of zero padding. Time-selective channels have notbeen considered in these papers; in contrast, our contributionconsiders both frequency- and time-selective channels, albeitfor single-carrier systems in the time domain.

Objectives and Contributions

As shown in [51], DPS-BEM outperforms other commonlyused BEMs (such as CE-BEM [10], [22], oversampled CE-BEM [19], and polynomial BEM [3]) in approximating a Jakes’channel over a wide range of Doppler spreads for the samenumber of parameters; hence, we focus on DPS-BEMs in thispaper. We first extend the first-order statistics-based approachof [13] pertaining to SIMO doubly selective channels to MIMOsystems, where Slepian sequences are used to model the doublyselective channel. To this end, we exploit the superimposedtraining sequence design of [14] so that the problem of channelestimation is approximately decoupled across various users.This is true, irrespective of any timing synchronization amongthe training sequences of various users at the transmitting end,in contrast to the uncoordinated users’ case for TM training.However, in the first-order statistics-based approach based onsuperimposed training, the information sequences from allusers are viewed as interference in channel estimation. Wetherefore consider an iterative joint channel estimation anddata detection approach where information sequences are ex-ploited to enhance channel estimation and BER performances,instead of being viewed as interference. Two variations areinvestigated: In deterministic maximum likelihood (DML), weuse a Viterbi detector, whereas in another variation, a Kalmandetector is used to reduce computational complexity. Suchtechniques have been considered for the CE-BEM SIMO sys-tems in [24] and the DPS-BEM SIMO systems in [13] usingthe Viterbi detector; Kalman-filtering-based detectors have notbeen considered in these papers. (As noted earlier, iterativeapproaches have been used by others in various contexts.) Notealso that one could view the objective function in (63) as ourprimary optimization cost function, which we seek to “reliably”initialize via the first-order statistics-based channel estimator ofSection III-B.

The rest of this paper is organized as follows: Section IIintroduces the channel model. The first-order statistics-basedchannel estimator using superimposed training and DPS-BEMis the subject of Section III. Section IV focuses on perfor-mance analysis of this estimator. Iterative joint channel esti-mation and data detection is discussed in Section V to reducethe information-induced interference. The DML approach viaViterbi detector is outlined in Section V-A, a Kalman detector-based simplification is presented in Section V-B, and com-putational complexity issues are addressed in Section V-C.Computer simulation examples are presented in Section VI, andSection VII concludes this paper.

Notation

Superscripts H , ∗, T , and † denote the complex conjugatetranspose, complex conjugation, transpose, and Moore–Penrose

pseudoinverse operations, respectively. IN is the N × N iden-tity matrix, the (m1,m2)th entry of a matrix C is denotedby [C]m1,m2 , tr(A) is the trace of a square matrix A, 0m×n

denotes an m × n null matrix, and 0m denotes an m-columnnull vector. The symbols �·�, �·�, and δ(·) stand for integerceiling, integer floor, and the Kronecker delta function, re-spectively. The symbol E{·} denotes expectation, and ⊗ de-notes the Kronecker product. cov(x,y) denotes the covariancematrix of random vectors x and y: cov(x,y) = E{xyH} −E{x}E{yH}. Var(x) is the variance of the random variable x.The notation y = O(x) means that there exists some finite realnumber b > 0, such that |y/x| ≤ b.

II. CHANNEL MODEL

Consider a doubly selective multiuser/MIMO FIR linearchannel with K inputs (users) and N outputs, with the kthuser’s transmitted symbol sequence denoted by {sk(n)} andthe kth user’s discrete-time baseband impulse response denotedby {hk(n; l)}. Then, the symbol-rate channel noise-free outputx(n) and noisy output y(n) are given by

x(n) :=K∑

k=1

L∑l=0

hk(n; l)sk(n − l) (1)

y(n) =x(n) + v(n), n = 0, 1, . . . , T − 1. (2)

A parsimonious representation of time-varying channels isprovided by BEMs, where one assumes that

hk(n; l) =Q∑

q=1

hqk(l)uq(n), n = 0, 1, . . . , T − 1 (3)

where uq(·) is the scalar qth basis function (q = 1, . . . , Q),N -column vectors hqk(l) remain invariant during this datablock, and the basis functions {uq(n)}T−1

n=0(q = 1, 2, . . . , Q)are common to all users for each block. In the CE-BEM[10], [22], for an observation record length of T symbolswith symbol interval Ts s, one chooses uq(n) = ejωqn, ωq :=2π[q − (Q + 1)/2]/T , L := �τd/Ts�, and Q ≥ 2�fdTTs� + 1when the underlying continuous-time channel has a delayspread of τd s and a Doppler spread of fd Hz. In the DPS-BEM, the ith DPS vector ui := [ui(0) ui(1) · · · ui(T −1)]T (which is called Slepian sequence in [51]) is the itheigenvector of a matrix C [37]: Cui = λiui, where the(m1,m2)th entry of the T × T matrix C is [C]m1,m2 =sin[2π(m1 − m2)fdTs]/π(m1 − m2), and λ1 ≥ λ2 ≥ · · · ≥λT are the T eigenvalues of C. The DPS sequences are or-thonormal over the finite time interval n = 0, 1, . . . , T − 1. Inthis paper, we will use the Slepian sequences in (3), where onetakes [51]

Q ≥ �2fdTsT � + 1. (4)

The Slepian sequences (time-limited DPS sequences) are win-dowed (using rectangular windows) versions of infinite DPSsequences that are exactly band limited to the frequency range[−fdTs, fdTs] [37], [51]. As shown in [51], DPS-BEM outper-forms other commonly used BEMs (such as CE-BEM [10], [22]


and polynomial BEM [3]) in approximating a Jakes’ channelover a wide range of Doppler spreads for the same numberof parameters; hence, we focus on DPS-BEM’s in this paper.Note also that whereas, for DPS-BEM (also true for CE-BEM),one can infer the number of basis functions needed from somephysical channel parameters such as the Doppler spread fd [see(4)], no such relationship exists for polynomial BEMs; this isanother point in favor of the use of DPS-BEM.

In superimposed training-based approaches, one takes for thekth user

sk(n) = bk(n) + ck(n) (5)

where {ck(n)} is a training (pilot) sequence added (superim-posed) at low power to the information sequence {bk(n)} atthe transmitter before modulation and transmission. There isno loss in data transmission rate, unlike the conventional TMtraining. Periodic superimposed training has been discussed in[7], [31], and [45] for time-invariant channels and in [44] and[46] for time-varying SIMO channels based on the CE-BEM.Superimposed training-based channel estimation has been con-sidered in [8], [14], and [25] for time-invariant MIMO systemsand in [26] for CE-BEM-based time-variant MIMO systems.

III. FIRST-ORDER STATISTICS-BASED ESTIMATOR

In this section, we extend the first-order statistics-basedapproach of [13] using DPS-BEM for single-user systems tomultiuser/MIMO doubly selective channels. The main idea isto pick user-specific training sequences so that the problem ofchannel estimation is approximately decoupled across varioususers—this allows us to use the SIMO superimposed training-based approach discussed in [13]. The choice of superimposedtraining sequences follows the time-invariant MIMO resultsof [14].

A. Superimposed Training Sequences [14]

Following [14], our approach is to assign distinct cyclefrequencies of the periodic training sequences to distinct users,so that the problem of channel estimation is decoupled acrossvarious users (see Remark 1 later in Section III-B). Supposethat for every user k, {ck(n)} is periodic with period P = PK,where P is a positive integer. Then, in general

ck(n) =P−1∑m′=0

cm′kej(2πm′/P )n ∀n (6)

where cm′k := P−1∑P−1

n=0 ck(n)e−j(2πm′/P )n. Pick {ck(n)}so that only P coefficients (out of the total P ) cm′k thatare associated with P distinct frequencies are nonzeros. Forinstance, we may choose

ck(n)=P−1∑m=0

c′mkejαmkn, αmk := 2π(Km+k−1)/P (7)

for suitably chosen c′mk �= 0 for 1 ≤ k ≤ K and 0 ≤ m ≤P − 1. (Under this choice, we have αm1k1 �= αm2k2 for any

m1,m2 ∈ {0, 1, . . . , P − 1}.) One way to accomplish this goalis to first pick a periodic “base” sequence {co(n)} (= {co(n +k′P )} for any integers k′ and n) with period P such that

cm0 =1P

P−1∑n=0

co(n)e−j(2πm/P )n (8)

and P−1∑P−1

n=0 |co(n)|2 = 1. Define {c1(n)}P−1n=0 as K repeti-

tions of {co(n)}. Pick the superimposed training sequence foruser k as

ck(n) := σck c1(n)ej(2π/P )(k−1)n (9)

for k = 1, 2, . . . ,K so that P−1∑P−1

n=0 |ck(n)|2 = σ2ck. Then,

by [14], we have

cm′k =1P

P−1∑n=0

ck(n)e−j(2πm′/P )n

=

{σck cm0, if m′ = k − 1 + Km,m =

⌊m′−k+1

K

⌋0, otherwise.

(10)

Thus, the preceding choice satisfies (7) for k = 1, 2, . . . ,K.Candidate training sequences include m-sequences (maximal-length pseudorandom sequences) [33] and discrete chirp se-quences [31].

B. First-Order Statistics-Based MIMO Channel EstimatorUsing Superimposed Training

We now extend the first-order statistics-based approach of[13] for single-user systems using DPS-BEM to multiuser/MIMO doubly selective channels. Four model assumptionsare made.

H1) The time-varying channel {hk(n; l)} satisfies (3), with{uq(n)}Q

q=1 being the first Q DPS sequences. In addi-tion, N ≥ 1.

H2) The information sequences {bk(n)} are of zero meanand finite alphabet, i.i.d. with E{|bk(n)|2} = σ2

bk, andmutually independent for k = 1, 2, . . . ,K.

H3) The measurement noise {v(n)} in (1) is of zero meanand white complex Gaussian and is independent of{bk(n)}, with E{[v(n + τ)][v(n)]H} = σ2

vINδ(τ).H4) The superimposed training sequences ck(n) = ck(n +

mP ) ∀m,n are nonrandom periodic sequences withperiod P and average power σ2

ck :=∑P−1

n=0 |ck(n)|2/P ,satisfying (7) such that c′mk �= 0 for 1 ≤ k ≤ K and0 ≤ m ≤ P − 1, and P is an integer with P = PK. Thetraining sequences at the receiver are synchronized withtheir respective counterparts at the transmitter.

The choice of training sequences discussed in Section III-Asatisfies H4. Assumptions H2 and H3 are standard for digitalcommunications signals.


Using (1), (3), (5), and (7) and taking expectation over thebk’s and v, we obtain

E {y(n)} =K∑

k=1

P−1∑m=0

Q∑q=1

[L∑

l=0

hqk(l)c′mke−jαmkl

]︸︷︷︸

=:dmqk

× uq(n)ejαmkn ∀n. (11)

For 0 ≤ m ≤ P − 1, 1 ≤ k ≤ K, and 0 ≤ l ≤ L, we define

Dmk := [dTm1k · · · dT

mQk ]T (12)

Hkl := [hH1k(l) · · · hH

Qk(l) ]H (13)

Vk :=

⎡⎢⎢⎣

1 e−jα0k · · · e−jα0kL

1 e−jα1k · · · e−jα1kL

......

. . ....

1 e−jα(P−1)k · · · e−jα(P−1)kL

⎤⎥⎥⎦

P×(L+1)

(14)

Ck := diag{

c′0k, c′1k, . . . , c′(P−1)k

}Vk (15)

Ck := Ck ⊗ INQ (16)

Hk = [HHk0 · · · HH

kL ]H (17)

Dk =[DH

0k · · · DH(P−1)k

]H(18)

where Dmk is [NQ] × 1, Hkl is [NQ] × 1, Ck is P × (L + 1),Ck is [PNQ] × [(L + 1)NQ], Hk is [(L + 1)NQ] × 1, and Dk

is [PNQ] × 1. By the definition of dmqk in (11), we have

CkHk = Dk. (19)

Since αmk’s are distinct and c′mk �= 0 for 0 ≤ m ≤ P − 1 and1 ≤ k ≤ K, rank(Ck) = NQ(L + 1) if P ≥ L + 1; hence, wecan uniquely determine the hqk(l)’s from (19). This requiresknowledge of Dk, whose estimation we discuss next.

It follows from (1), (3), (5), and (11) that

y(n) =E {y(n)} + e(n)

=K∑

k=1

Q∑q=1

P−1∑m=0

dmqkuq(n)ejαmkn + e(n) (20)

where {e(n)} is a zero-mean random sequence. Equation (20)forms the basis for estimating the BEM coefficients hqk(l) fromthe received signal y(n) by first estimating the dmqk’s via aleast-squares approach and then using (19) with known super-imposed training to estimate the hqk(l)’s. Given the receivedsignal over n = 0, 1, . . . , T − 1, define the cost function

J :=T−1∑n=0

‖e(n)‖2 . (21)

Choose the dmqk’s to minimize J . We must have (∂J/∂d∗

mqk)|dmqk=dmqk= 0, which leads to

K∑k′=1

Q∑q′=1

P−1∑m′=0

dm′q′k′

[T−1∑n=0

uq′(n)uq(n)ej(αm′k′−αmk)n

]

=T−1∑n=0

y(n)uq(n)e−jαmkn

︸︷︷︸=:gmqk

. (22)

Define Dmk as in (13), with dmqk replaced with dmqk,and define Dk as in (18), with Dmk replaced with Dmk;similarly, define Gmk as in (13), with dmqk replaced withgmqk, and define Gk as in (18), with Dmk replaced with Gmk,i.e., Gmk := [gT

m1k, . . . ,gTmQk]T , etc. Furthermore, define

D :=

⎡⎢⎢⎣D1

D2...

DK

⎤⎥⎥⎦

[KPNQ]×1

G :=

⎡⎢⎢⎣G1

G2...

GK

⎤⎥⎥⎦

[KPNQ]×1

(23)

Ψ :=

⎡⎢⎢⎣

Ψ11 Ψ12 · · · Ψ1K

Ψ21 Ψ22 · · · Ψ2K...

.... . .

...ΨK1 ΨK2 · · · ΨKK

⎤⎥⎥⎦

[KPQ]×[KPQ]

(24)

where the PQ × PQ matrix Ψk′k has entries (m,m′ =0, 1, . . . , P − 1; q, q′ = 1, 2, . . . , Q)

[Ψk′k]mQ+q,m′Q+q′ =T−1∑n=0


k, k′ = 1, 2, . . . ,K. (25)

Then, (22) leads to

(Ψ ⊗ IN )D = G ⇒ D = (Ψ−1 ⊗ IN )G. (26)

Then, the estimate of Hk is given by

Hk =(CH

k Ck

)−1 CHk Dk. (27)

Denote the corresponding estimate of hqk(l) as hqk(l). Follow-ing the DPS-BEM representation (3), the estimate of the time-varying channel is given by

hk(n; l) =Q∑

q=1

hqk(l)uq(n). (28)

Remark 1: In the succeeding remarks, m ∈ {0, 1, . . . , P −1}, and k = 1, 2, . . . ,K. Following the arguments of [13,Remark 1], {uq(n)e−jαmkn} is approximately band limited to[

−fdTs −k

T+

Km + k − 1P

, fdTs +k

T+

Km + k − 1P

]


with the integer k approximately as 1 ≤ k ≤ 3. Therefore, forfdTs � 1/P , {uq(n)e−jαmkn} has (approximately) vanishingzero-frequency content when Km + k − 1 �= 0, leading to

T−1∑n=0

uq(n)e−jαmkn ≈ 0 ∀αmk �= 0. (29)

By similar arguments, the discrete-time Fourier transformsof {uq(n)e−jαmkn} and {uq(n)e−jαm′k′n} are nonzero overnonoverlapping frequency bands, leading to

T−1∑n=0


≈ δ(k′ − k)δ(m′ − m)δ(q′ − q). (30)

In this case, we have Ψk′k ≈ IPQδ(k′ − k) and Ψ ≈ IKPQ.

Then, the estimate dmqk of dmqk is

dmqk ≈T−1∑n=0

y(n)uq(n)e−jαmkn (31)

which, via (27), leads to channel estimates decoupled acrossdifferent users. It is not too hard to show that timing synchro-nization among the superimposed training sequences of varioususers is not required for (30) to hold true, i.e., (30) holds, even ifthere is a relative time shift between ck(n) and ck′(n), becausethe cycle frequencies of the various users are unchanged, andby choice, they are distinct.

IV. PERFORMANCE ANALYSIS

In this section, we present a performance analysis of theapproach discussed in Section III-B. To obtain tractable results,we assume that the MIMO channel is complex Gaussian, satis-fying the assumption given here.

H5) The time-varying channels {hk(n; l)} are of zero mean,complex Gaussian with correlation E{hk(n; l)hH

k (n;l)} = σ2

hklIN , and mutually independent for distinct l’sand different users.

This assumption applies to this section only; the algorithmsproposed in this paper are not based on this assumption.The widely used wide-sense stationary uncorrelated scatteringchannel model [33], combined with the independently fading(sub)channel between any transmit–receive antenna pair, satis-fies H5. The assumption of independently fading links betweenany transmit–receive antenna pair has widely been used inMIMO literature (see [32], [43], and references therein).

By (22), gmqk has contributions from the information se-quences {bk(n)} unknown at the receiver, the superimposedtraining {ck(n)} known at the receiver, and the measurementnoise v(n). It follows from (1)–(5), H2, and (22) that

gmqk =T−1∑n=0

[E {y(n)} + v(n) +

K∑k′=1

L∑l=0

hk′(n; l)bk′(n − l)

]

× uq(n)e−jαmkn. (32)

Then, by (H2), (H3), and (11)

E{gmqk} =T−1∑n=0

E {y(n)}uq(n)e−jαmkn =: dmqk. (33)

It follows that

gmqk = dmqk + smqk + wmqk (34)

where

wmqk :=T−1∑n=0

v(n)uq(n)e−jαmkn (35)

smqk :=T−1∑n=0

K∑k′=1

L∑l=0

hk′(n; l)bk′(n − l)

× uq(n)e−jαmkn. (36)

We then have

E{wm′q′k′wH

mqk

}= σ2

v

[T−1∑n=0

uq′(n)uq(n)ej(αmk−αm′k′ )n

]IN (37)

E{sm′q′k′sH

mqk

}= (L + 1)

(K∑

p=1

(L∑

l=0

σ2hpl

)σ2

bp

)

×[

T−1∑n=0

uq′(n)uq(n)ej(αmk−αm′k′ )n

]IN . (38)

Therefore, it follows that

cov(gm′q′k′ ,gmqk) = E{sm′q′k′sH

mqk

}+ E

{wm′q′k′wH

mqk

}.

(39)

Hence

cov(D, D) = (Ψ−1 ⊗ IN )cov(G,G)((Ψ−1)H ⊗ IN

)(40)

where various entries in cov(G,G) follow fromcov(gmqk,gmqk).

A. Simplification Under Remark 1

To obtain a simpler more “interpretable” covariance expres-sion, we will invoke the conditions of Remark 1. Under (29)and (30), it easily follows that dmqk = gmqk, dmqk = dmqk,Ψ ≈ IKPQ, and

E{wm′q′k′wH

mqk

}≈ σ2

vINδ(m′ − m)δ(q′ − q)δ(k′ − k) (41)

E{sm′q′k′sH

mqk

}≈ (L + 1)

(K∑

p=1

σ2hpσ

2bp

)δ(m′ − m)δ(q′ − q)δ(k′ − k)IN

(42)


where

σ2hp :=

L∑l=0

E{‖hp(n; l)‖2

}=

L∑l=0

σ2hpl. (43)

Hence

cov(Dk, Dk) ≈[(L + 1)

K∑p=1

σ2hpσ

2bp + σ2

v

]INQP (44)

cov(D, D) ≈diag{

cov(Dk, Dk), k = 1, 2, . . . ,K}

.

(45)

Using (16) and the channel estimator (27), it follows that

cov(Hk, Hk) ≈[(L + 1)

K∑p=1

(L∑

l=0

σ2hpl

)σ2

bp + σ2v

]

×([

CHk Ck

]⊗ INQ

)−1

. (46)

The mean square error (MSE) in channel estimation for user kis defined by

MSEk :=1T

T−1∑n=0

L∑l=0

E

{∥∥∥hk(n; l) − hk(n; l)∥∥∥2

}. (47)

If the true channel follows (3), using the orthonormality of theSlepian sequences, we have

MSEk =1T

E

{L∑

l=0

Q∑q=1

∥∥∥hqk(l) − hqk(l)∥∥∥2

}(48)

=1T

tr{

cov(Hk, Hk)}

(49)

≈ 1T

[K∑

p=1

(L∑

l=0

σ2hpl

)σ2

bp + σ2v

]NQtr

{(CH

k Ck

)−1}

.

(50)

Thus, the interference from all the users’ information se-quences {bk(n)} contribute to a major part of the MSE inthe first-order statistics-based estimator, even when the channelestimation has been decoupled for each user.

Following [13, Sec. IV-D1], tr{(CHk Ck)−1} is minimized if

and only if CHk Ck is diagonal, with all its diagonal elements

equal. There is no unique choice of superimposed training se-quence that minimizes tr{(CH

k Ck)−1}; however, discrete chirpsequences [31] meet the requirement exactly [13], [14], andm-sequences [33] meet the requirement approximately [13].For such sequences, one obtains (CH

k Ck)−1 = σ−2ck IL+1 [13,

eq. (50)]. Then, the optimized MSE for user k is given by

MSEok ≈ 1σ2

ckT

[K∑

p=1

(L∑

l=0

σ2hpl

)σ2

bp + σ2v

]NQ(L + 1).

(51)

For a given channel, users’ power, and noise power σ2hpl, σ2

bp,and σ2

v , respectively, using (4), we may rewrite (51) as

MSEok = O(

Q

σ2ckT

)= O

(�2fdTsT � + 1

σ2ckT

). (52)

If fd = 0 (time-invariant channels, as in [14]), one haslimT→∞ MSEok = 0. For time-varying channels with fd > 0,it follows from (52) that

limT→∞

MSEok ≈ O(

2fdTs

σ2ck

). (53)

Thus, whereas the MSE for user k can be decreased by in-creasing the training power σ2

ck, it does not offer a “practical”solution since, for a fixed transmitted power, increasing σ2

ck

would imply decreasing the signal power σ2bk, which, in turn,

would lead to a deteriorated bit error performance. In addition,in accordance with one’s intuition, a smaller Doppler spread fd

(slower time variations) leads to a smaller MSE, and vice versa.

V. ITERATIVE JOINT CHANNEL ESTIMATION

AND DATA DETECTION

The first-order statistics-based approach of Section III-Bviews the information sequences from all users as interferences.Since the training and information sequences pass through anidentical channel, this fact can be exploited to enhance channelestimation performance (and, hence, BER performance). Wenow consider joint channel and information sequence esti-mation via an iterative approach. Such techniques have beenconsidered for the CE-BEM SIMO systems in [24] and theDPS-BEM SIMO systems in [13], using the Viterbi detector;the Kalman-filtering-based detectors have not been consideredin these papers. One could view the objective function in (63)as our primary optimization cost function, which we seek to“reliably” initialize via the first-order statistics-based channelestimator of Section III-B.

A. Iterative Enhancement via Viterbi Algorithm:DML Approach

In this section, we consider joint channel and informationsequence estimation via an iterative DML (since the infor-mation sequence is modeled as unknown but deterministic)approach. The cost in (63) is proportional to the negative log-likelihood function for the noisy data jointly conditioned on thechannel and the information sequence; hence, one may also callthis approach as an iterative conditional maximum-likelihood(CML) approach. However, our assumption H2 still holds truealthough it is not exploited. The DML formulation has beenexploited by various authors in a variety of contexts (see [2],[42], and references therein); it is known to be statisticallyefficient at high SNRs [42]. If we consider a CML approachconditioned only on the channel, then one gets an intractableoptimization problem; this formulation is called statisticalML [42].


Given the received signal y(n) for n = 0, 1, . . . , T − 1, wedefine (54)–(59), shown at the bottom of the page, where Yis [(T − L)N ] × 1, s is [KT ] × 1, Σn is N × [NQ], T (s) is[(T − L)N ] × [K(L + 1)NQ], H is [K(L + 1)NQ] × 1, andV is [(T − L)N ] × 1.

By (1) and (3), we have the linear model

Y = T (s)H + V. (60)

If we further define (61), shown at the bottom of the page, whereF(H) is [(T − L)N ] × [(T − L)N ], we obtain another linearmodel as

Y = F(H)s + V. (62)

We consider joint (ML) estimation

{H, s} = arg minH,s∈S

‖Y − T (s)H‖2 (63)

initialized by the first-order statistics-based channel estimator(27), with Dk calculated via (31), where S is the (discrete)domain of s. Under a white Gaussian noise assumption, theDML estimators are obtained by the nonlinear least-squaresoptimization (63). Using (60) and (62), we have a separablenonlinear least-squares problem that can sequentially be solvedas follows: At iteration j, with an initial guess of the channelH(j), the algorithm estimates the input sequence s(j) and thechannel H(j+1) for the next iteration by

s(j) = arg mins∈S

∥∥∥Y −F(H(j)

)s∥∥∥2

(64)

H(j+1) = arg minH

∥∥∥Y − T(s(j)

)H∥∥∥2

. (65)

The optimization in (65) is a linear least-squares problemhaving the solution

H(j+1) = T †(s(j)

)Y (66)

whereas the optimization in (64) can be achieved by using thevector Viterbi algorithm [41, Sec. 7.8.4]. Since the precedingiterative procedure involving (64) and (65) decreases the cost atevery iteration, one achieves a local minimum of the nonlinearleast-squares cost (local maximum of DML function). If weinitialize (64) with our superimposed training-based solution,one expects to reach the global extremum (minimum errorprobability sequence estimator) if the superimposed training-based solution is “good.”

B. Iterative Enhancement via Linear MMSE Equalization

The computational complexity of the Viterbi algorithm ex-ponentially grows with the length of channel, the number ofusers, and the constellation of transmitted symbols [33, p. 681].We may use other symbol detectors whose computational com-plexity linearly grows, e.g., the Kalman filter. It can be viewedas a suboptimal approximation to the DML approach, withmuch lower computational burden. Define the state vector forthe kth user consisting of the transmitted symbols as

Sk(n) := [sk(n) sk(n − 1) · · · sk(n − d)]T (67)

where d ≥ L is also the equalization delay, and Sk(n) is (d +1) × 1. The “overall” K(d + 1)-state vector is defined as

S(n) :=[ST

1 (n) ST2 (n) · · · ST

K(n)]T

. (68)

Define the K × 1 “input” w(n) as

w(n) := [s1(n + 1) s2(n + 1) · · · sK(n + 1)]T . (69)

Then, the state and the measurement equations of the state-space model of interest for the received signal are given by

S(n + 1) =ΦS(n) + Γw(n) (70)

y(n) =H(n)S(n) + v(n) (71)

Y := [yT (T − 1) · · · yT (L) ]T (54)

s := [ s1(T − 1) · · · sK(T − 1) s1(T − 2) · · · sK(0) ]T (55)

Σn := [u1(n)IN · · · uQ(n)IN ] (56)

T (s) :=

⎡⎢⎢⎣

s1(T − 1)ΣT−1 · · · s1(T − L − 1)ΣT−1 · · · sK(T − L − 1)ΣT−1

s1(T − 2)ΣT−2 · · · s1(T − L − 2)ΣT−2 · · · sK(T − L − 2)ΣT−2

.... . .

.... . .

...s1(L)ΣL · · · s1(0)ΣL · · · sK(0)ΣL

⎤⎥⎥⎦ (57)

H := [HT1 · · · HT

K ]T (58)

V := [vT (T − 1) · · · vT (L) ]T (59)

F(H) :=

⎡⎢⎣h1(T − 1; 0) · · · hK(T − 1; 0) · · · hK(T − 1;L)

. . .. . .

. . .h1(L; 0) · · · hK(L; 0) · · · hK(L;L)

⎤⎥⎦ (61)


respectively, where

Φ := IK⊗[01×d 0Id 0d×1

]Γ := IK⊗[ 1 01×d ]T (72)

H(n) := [H1(n) H2(n) · · · HK(n) ] (73)

Hk(n) := [hk(n; 0) hk(n; 1) · · · hk(n;L) 0N×(d−L) ]

(74)

with Φ being [K(d + 1)] × [K(d + 1)], Γ being [(K(d +1)] × 1, Hk(n) being N × [d + 1], and H(n) being N × [(d +1)]. The Kalman-filtering algorithm for estimating sk(n − d),given y(m) (m ≤ n), for a chosen value of equalization delayd based on (71) can be found in [39]. We use the estimatedchannel in (73) and (74), using (28), and estimated parametervector H.

In (69), for superimposed training, we take E{sk(n)} =ck(n) and var(sk(n)) = var(bk(n)). For TM training, we takeE{sk(n)} = 0 and var(sk(n)) = var(bk(n)) if sk(n) is aninformation symbol, whereas we take E{sk(n)} = ck(n) andvar(sk(n)) = 0 if sk(n) is a training symbol.

Compared with the approach of Section V-A, here, iteration(64) is replaced with the hard-quantized Kalman equalizer out-put with equalization delay d, whereas optimization specifiedby (65) remains unchanged.

C. Computational Complexity

By [33, p. 681] and [41, Sec. 7.8.4], the computationalcomplexity of the Viterbi algorithm, as applied to our problem,is O([KM ]LNT ), where M is the signal constellation size, andas defined earlier, K is the number of users (Tx antennas), Nis the numbers of Rx antennas, L + 1 is the channel length, andT is the number of measurement samples at the receiver. By[11, Tab. 6.5], the computational complexity of the Kalman fil-ter, as applied to our problem, is O((K(d + 1))2NT + K(d +1)N2T ), where we have assumed that the equalization delayd ≥ L. The computational complexity of the Kalman detectorwould be much less than that of the Viterbi detector, partic-ularly for higher alphabet size, channel length, and numberof Tx antennas. Channel reestimation via (66) has compu-tational complexity O([K(L + 1)NQ]3 + T ). Therefore, theoverall complexity of the iterative enhancement would be thenumber of iterations times the sum of the complexities ofthe data-detection and channel-estimation steps; comparatively,the first-order statistics-based channel-estimation approach has“negligible” computational requirement.

Comparing TM training-based approaches (see alsoSection VI-A) with superimposed training-based approaches,we note that, in the TM case, one estimates the channelbased solely on training and then detects the data; this iswhat we consider in Section VI. Thus, the computationalcomplexity of TM training-based approaches is approximatelyequal to that of the data-detection step, which, in turn,is either O([KM ]LNT ) if a Viterbi detector is used orO((K(d + 1))2NT + K(d + 1)N2T ) if a Kalman detector isused with equalization delay d (≥ L).

Let I denote the number of iterations executed. Then,approximately, the computational complexity of the proposed

superimposed training-based iterative approaches is I + 1times that of the TM case, where we have counted the first-order statistics-based step in addition to the I iterations. In thesimulations section (see Section VI), we have taken I = 3.

VI. SIMULATION EXAMPLES

We now illustrate our approaches, with two simulation ex-amples dealing with a two-user (K = 2) and multiple-receiver-antennas scenario, where the received signals from N ≥ 1receive antennas are jointly processed. We assume thatboth users have σ2

b1 = σ2b2 = σ2

b and σ2c1 = σ2

c2 = σ2c with a

training-to-information-power ratio (TIR) σ2b/σ2

c of 0.3. Con-sidering a random doubly selective Rayleigh fading channel,we take L = 2 in (1) with hk(n; l) as in H5 and having a uni-form power delay profile with σ2

h1 = σ2h2 = 1, satisfying Jakes’

model (recall that σ2hk =

∑Ll=0 E{‖hk(n; l)‖2}). To this end,

we simulate each single tap of a doubly selective channel fol-lowing [52] (with a correction in [51, App.]). We consider bothDPS-BEM and the widely used CE-BEM representations of thedoubly selective channels. The methods of this paper also applyto the CE-BEMs, provided we set uq(n) = (1/

√T )ejωqn.

We consider a system with a carrier frequency of 2 GHz,a data rate of 40 kBd (therefore, Ts = 25 μs), and a Dopplerspread of fd = 50 or 100 Hz. At the receiver, we take the recordlength (block size) of T = 420 symbols. We emphasize thatthe DPS-BEM is used only for processing at the receiver; therandom channels are generated by Jakes’ model and not theDPS-BEM in (3). The estimated channel is used in a symboldetector, which could be a Viterbi detector or a Kalman filter inour examples, to calculate the BERs. We assume that the addi-tive noise is zero-mean complex white Gaussian. The (receiver)SNR refers to the energy per bit per user (Tx) per Rx over one-sided noise spectral density with both information and super-imposed training sequence counting toward the bit energy. Anm-sequence is used to generate the superimposed training se-quence for each user. We hence take P = 7 and P = 14 in H4.The training sequence for the first user is

{c1(n)}13n=0 = {1,−1,−1, 1, 1, 1,−1, 1,−1,−1, 1, 1, 1,−1}

(75)

which is two repetitions of an m-sequence of period P = 7, andc2(n) satisfies (7). The sequences ck(n) are scaled to achieveTIR = 0.3.

A. TM Training

1) Coordinated Users: For comparison, we consider a CE-or DPS-BEM-based periodically placed TM training with zeropadding, following the CE-BEM-based design of [50]; we usedit for DPS-BEMs as well. In [50], each transmitted blockof symbols {sk(n)}T−1

n=0 from the kth user is segmented intoJ subblocks of training and information symbols ck(n) andbk(n). Each subblock is of equal length with Nb informationsymbols and Nc training symbols. If sk denotes a column vectorcomposed of {sk(n)}T−1

n=0, then sk is arranged as

sk :=[bT

k1, cTk1,b

Tk2, c

Tk2, . . . ,b

TkJ , cT

kJ

]T(76)


where bkj is a column of Nb information symbols, and ckj is acolumn of Nc training symbols. We clearly have T = J(Nb +Nc). For CE-BEM channels, [50] has shown that (76) is an op-timum structure with Nc = (K + 1)L + K, J = Q, and ckj =[0T

k(L+1)−1, γk,0TL+(K−k)(L+1)]

T (γk > 0), where 0L denotesa null column of size L. (A justification of (76) for DPS-BEMfor the single-user case is given in [38].) Thus, given a trans-mission block of size T , ((K + 1)L + K)J symbols have to bedevoted to training, and the remaining T − ((K + 1)L + K)Jare available for information symbols. Thus, we have a training-to-information rate overhead of (((K + 1)L + K)J)[T −(((K + 1)L + K)J ]−1 for the scheme of [50]. Zero padding inthe design of [50] allows for multiuser-interference-free chan-nel estimation. The results of [50] extend the single-user resultsof [22] to the multiuser/MIMO case. The design of [50] is pos-sible only in the case of the coordinated users, as discussed inSection I. To carry out a fair comparison between superimposedand TM training-based approaches, for a given transmissionblock size T , we keep the training-to-information rate over-head of (((K + 1)L + K)J)[T − (((K + 1)L + K)J ]−1 =Nc/Nb, as well as the TIR of γ2

kJσ−2bk [T − ((K + 1)L +

K)J ]−1 = γ2k[σ2

bkNb]−1 (with J = Q whenever possible) per-taining to the TM training scheme of [50], to be equal to theTIR of σ2

ck/σ2bk for the superimposed training-based schemes.

(We take γk = γ ∀k when the channels for the individualusers have the same statistics, as in the example consideredhere.) For the two-user case (K = 2), we take a trainingsession of length eight symbols with the first user’s trainingsequence {0, 0,

√(K + 1)L + K, 0, 0, 0, 0, 0} and the second

user’s {0, 0, 0, 0, 0,√

(K + 1)L + K, 0, 0} so that the trainingsessions have the same average power as the information datasessions. An information data session of 27 symbols with unitpower is inserted between two such training sessions to forma frame of length 35 symbols. Such a frame is repeated overa record length of 420 symbols (12 frames). Thus, we have atraining-to-information bit ratio of about 0.3. Using (3) and thetraining sequence, we can uniquely determine the hq(l)’s via aleast-squares approach. The Viterbi detector or the Kalman de-tector was used for data detection using the estimated channel.

2) Uncoordinated Users: In the case of the uncoordinatedusers, there is no coordination among the various users regard-ing the placement of the training symbols in the transmittedblock. Hence, multiuser interference in channel estimation can-not be avoided; therefore, zero padding in the training designdoes not make sense. For this reason, while the alternatingarrangement of the training and information symbols for userk is like that in (76), the placement and the choice of the ckj’swill be different: ckj consists of a random binary sequence thatis scaled by γk, of length Nc for each k and independent forvarious k’s and j’s, and for different k’s, their placement mayor may not overlap from run to run. As in Section VI-A1, thisleads to a training-to-information rate overhead of Nc/Nb, andunlike Section VI-A1, the TIR turns out be [γ2

kNc][σ2bkNb]−1.

If σ2bk = 1 and γk = 1 ∀k, then Nc/Nb equal to the TIR of

the superimposed training leads to a fair comparison. For thetwo-user case considered in the simulations, we take γk = 1,Nc = 8, and Nb = 25 as in Section VI-A1, leading to atraining-to-information bit ratio of about 0.3. As before, using

Fig. 1. Example 1. NCMSE: (77) versus SNR, averaged over 500 MonteCarlo runs for CE-BEM- and DPS-BEM-based DML channel estimators, withtwo users (K = 2), two receive antennas (N = 2), fd = 50 Hz, Viterbidetector, TIR = 0.3, binary signals, and T = 420 bits. Step 1 refers to theapproach of Remark 1 in Section III-B. “3rd iter.” refers to the third iter-ation of the DML approach. SI-CE, SI-DPS, “TM-CE: coord,” “TM-DPS:coord,” and “TM-DPS: uncoord” refer to superimposed training with CE-BEM representation, superimposed training with DPS-BEM representation,TM training for coordinated users with CE-BEM representation [50], TMtraining for coordinated users with DPS-BEM representation, and TM trainingfor uncoordinated users with DPS-BEM representation, respectively.

(3) and the training sequence, we can uniquely determinehq(l)’s via a least-squares approach, and then, the Viterbidetector or the Kalman detector can be used for data detectionusing the estimated channel.

B. Example 1

In this example, a two-user two-receiver (K = N = 2) sce-nario is considered. For each user, the information sequencesare drawn from the binary alphabet {±1}. Both cases, i.e., coor-dinated users and uncoordinated users, are considered. This af-fects only the TM training design, as discussed in Section VI-A.A Viterbi detector is used for symbol detection. The BER andnormalized channel MSE (NCMSE) results are shown in Figs. 1and 2 for a Doppler spread of fd = 50 Hz and in Figs. 3 and 4for a Doppler spread of fd = 100 Hz, where the NCMSE isdefined as

NCMSE =(TMr)−1

Mr∑i=1

K∑k=1

T−1∑n=0

L∑l=0

∥∥∥h(i)k (n; l) − h(i)

k (n; l)∥∥∥2

(TMr)−1Mr∑i=1

K∑k=1

T−1∑n=0

L∑l=0

∥∥∥h(i)k (n; l)

∥∥∥2

(77)

where h(i)k (n; l) is the true channel, and h(i)

k (n; l) is theestimated channel at the ith run, among the total Mr runs.The corresponding detection results are based on the Viterbialgorithm utilizing the estimated channel. The iterations followour DML approach in Section V-A. For fd = 50 and 100 Hz,we choose the number of basis functions Q = 3 and 5 byQ ≥ 2�fdTTs� + 1 for CE-BEM and Q = 3 and 4 by (4) forDPS-BEM, respectively.

The first-order statistics-based channel estimator (27)[see also (28)] was implemented with Dk calculated via (31).


Fig. 2. As in Fig. 1, except that BER versus SNR is shown.

Fig. 3. As in Fig. 1, except that fd = 100 Hz.


We show the results from the first-order statistics-based es-timator using superimposed training (denoted by “step 1” inthe figures), the third iteration of the DML approach (denotedby “3rd iter.” in the figures), and the TM training. (In addi-tion, in the figures, “SI-CE” denotes the estimators based on

Fig. 5. BER comparison between the MIMO (K = 2, N = 2) and SISO(K = 1, N = 1) systems. The results are based on 500 runs and show theoutcome of the third iteration of the DML algorithm.

superimposed training and CE-BEM, and “SI-DPS” denotesthose based on superimposed training and DPS-BEM; “TM-CE” and “TM-DPS” follow the similar terminology.) As notedin [51], the DPS-BEM efficiently reduces the spectral leakageinduced by CE-BEM, leading to a much smaller modelingerror; the BER and NCMSE curves both exhibit this advantage.It is also seen that the DML algorithm, whether it is CE orDPS-BEM based, significantly reduces the interference fromthe information sequence, which is induced by its first step,which is the first-order statistics-based approach. The BERperformance after three DML iterations is superior to that dueto the TM training for uncoordinated users and is competitivewith the TM training for coordinated users, without incurringthe 30% training overhead penalty.

1) MIMO versus SISO Comparison: In Fig. 5, we comparethe BER performance of the MIMO case (two Tx antennas,two Rx antennas, and spatial multiplexing with “odd” bits sentthrough Tx 1 and “even” bits through Tx 2) with the SISOcase of one Tx antenna and one Rx antenna. All (sub)channelscorresponding to each Tx–Rx pair are mutually independent,identically distributed, and as described earlier for Example 1.Note that the total transmit power in both cases is the same,i.e., in the MIMO case, each of the two Tx antennas transmitshalf the power, compared with the SISO case, for a fair com-parison between the MIMO and SISO cases [32]. The SNRshown in Fig. 5 is the total SNR at an Rx antenna resultingfrom the received signal from both Tx antennas; therefore, werefer to it in Fig. 5 as the total SNR to contrast it from theearlier results where the SNR includes the signal power fromjust one Tx antenna, as would be appropriate in a multiuserscenario. Only DPS-BEM representation is considered. It isseen that the MIMO case outperforms the SISO scenario whilehaving an overall transmission rate that is double that for SISOtransmission.

2) Performance Analysis Verification: For the first-orderstatistics-based DPS-BEM estimator, we also plotted the the-oretical channel MSE of (50), after summing over the two usersand normalizing as in (77), in Fig. 6. The theoretical expressionand the simulation-based MSE results agree quite well.


Fig. 6. Performance analysis comparison. Example 1. “SI-DPS: analytical”shows the theoretical channel MSE of (50) after summing over the two usersand normalizing as in (77).

Fig. 7. Example 2. Normalized channel mean-square error [NCMSE: (77)]versus SNR, which is averaged over 1000 Monte Carlo runs, for DPS-BEM-based DML channel estimators, with two users (K = 2), multiple receiveantennas (N = 2, 3, or 4), fd = 100 Hz, Kalman detector with equalizationdelay d = 5 symbols, TIR = 0.3, four-level complex-valued signals, and T =420 symbols. Step 1 refers to the approach of Remark 1 in Section III-B.“3rd iter.” refers to the third iteration of the DML approach. SI is the super-imposed training with DPS-BEM representation, and TM is the TM trainingwith DPS-BEM representation.

C. Example 2

In this example, we compare the improvement of perfor-mance as the number of receive antennas N increases. OnlyDPS-BEM is considered here. For each user, the informationsequences are drawn from the alphabet {(±1 ± j)/

√2} of size

four. To reduce the computational complexity, we employ aKalman filter, together with a quantizer, as the symbol detectorexploiting the estimated channel, following the model (71),with an equalization delay d = 5 symbols. In Figs. 7 and 8, wecompare the DPS-BEM-based estimators using superimposedand TM training (coordinated users only) for fd = 100 Hz. ForN = 2, 3, and 4, the BER performance benefits from employ-ing more receivers. Although the Kalman filter can only offer


us an “approximation” of the DML approach, the enhancementafter iterations is still significant. As N increases, the gapbetween the BERs for the TM (coordinated users) and the DPS-BEM-based approaches rapidly narrows, with the DPS-BEMapproach being competitive with the TM approach for N = 3without the 30% data rate loss.

VII. CONCLUSION

Channel estimation and symbol detection for multiuser/MIMO doubly selective fading channels using superimposedtraining and DPS-BEM have been considered. A user-specificperiodic training sequence has been superimposed on eachuser’s information sequence. We have first employed a first-order statistics-based estimator to estimate the channel, wherethe information sequences from all users act as interference.We have then presented an iterative joint channel estimationand data-detection approach exploiting the detected symbolsfrom the previous iteration to enhance channel estimation.Simulation results have shown that, without incurring any lossin data transmission rate, the proposed approach is superior tothe TM training for uncoordinated users, where the multiuserinterference in channel estimation cannot be eliminated (as ina general multiuser scenario), and is competitive with the TMtraining for coordinated users, where the TM training designallows for multiuser-interference-free channel estimation (as ina MIMO scenario).

REFERENCES

[1] T. Abe and T. Matsumoto, “Space time turbo equalization in frequency-selective MIMO channels,” IEEE Trans. Veh. Technol., vol. 52, no. 3,pp. 469–475, May 2003.

[2] F. Alberge, M. Nikolova, and P. Duhamel, “Blind identification/equalization using deterministic maximum likelihood and a partial prioron the input,” IEEE Trans. Signal Process., vol. 54, no. 2, pp. 724–737,Feb. 2006.

[3] D. K. Borah and B. D. Hart, “Frequency-selective fading channel es-timation with a polynomial time-varying channel model,” IEEE Trans.Commun., vol. 47, no. 6, pp. 862–873, Jun. 1999.


[4] M. Coldrey and P. Bohlin, “Training-based MIMO systems—Part II:Improvements using detected symbol information,” IEEE Trans. SignalProcess., vol. 56, no. 1, pp. 296–303, Jan. 2008.

[5] M. Dong, L. Tong, and B. M. Sadler, “Optimal insertion of pilot symbolsfor transmissions over time-varying flat fading channels,” IEEE Trans.Signal Process., vol. 52, no. 5, pp. 1403–1418, May 2004.

[6] B. Farhang-Boroujeny, “Pilot-based channel identification: Proposal forsemi-blind identification of communications channels,” Electron. Lett.,vol. 31, no. 15, pp. 1044–1046, Jun. 1995.

[7] M. Ghogho, D. McLernon, E. Alamdea-Hernandez, and A. Swami,“Channel estimation and symbol detection for block transmission us-ing data-dependent superimposed training,” IEEE Signal Process. Lett.,vol. 12, no. 3, pp. 226–229, Mar. 2005.

[8] M. Ghogho, D. McLernon, E. Alamdea-Hernandez, and A. Swami,“SISO and MIMO channel estimation and symbol detection using data-dependent superimposed training,” in Proc. IEEE ICASSP, Philadelphia,PA, Mar. 2005, pp. 461–464.

[9] G. B. Giannakis, Y. Hua, P. Stoica, and L. Tong, Eds., Signal ProcessingAdvances in Wireless & Mobile Communications, vol. 1, Trends in Chan-nel Estimation and Equalization. Upper Saddle River, NJ: Prentice-Hall,2001.

[10] G. B. Giannakis and C. Tepedelenlioðlu, “Basis expansion models anddiversity techniques for blind identification and equalization of time-varying channels,” Proc. IEEE, vol. 86, no. 10, pp. 1969–1986, Oct. 1998.

[11] M. S. Grewal and A. P. Andrews, Kalman Filtering Theory and Practice.Englewood Cliffs, NJ: Prentice-Hall, 1993.

[12] S. He and J. K. Tugnait, “Doubly-selective multiuser channel estimationusing superimposed training and discrete prolate spheroidal basis expan-sion models,” in Proc. IEEE Int. Conf. Acoust., Speech Signal Process.,Honolulu, HI, Apr. 2007, vol. II, pp. 861–864.

[13] S. He and J. K. Tugnait, “On doubly selective channel estimation usingsuperimposed training and discrete prolate spheroidal sequences,” IEEETrans. Signal Process., vol. 56, pt. 2, no. 7, pp. 3214–3228, Jul. 2008.

[14] S. He, J. K. Tugnait, and X. Meng, “On superimposed training for MIMOchannel estimation and symbol detection,” IEEE Trans. Signal Process.,vol. 55, pt. 2, no. 6, pp. 3007–3021, Jun. 2007.

[15] K. Josiam and D. Rajan, “Bandwidth efficient channel estimation usingsuperimposed pilots in OFDM systems,” IEEE Trans. Wireless Commun.,vol. 6, no. 6, pp. 2234–2245, Jun. 2007.

[16] B. W. Kim, S. Y. Jung, J. Kim, and D. J. Park, “Hidden pilot based pre-coder design for MIMO-OFDM systems,” IEEE Commun. Lett., vol. 12,no. 9, pp. 657–659, Sep. 2008.

[17] H. Kim and J. K. Tugnait, “Doubly-selective MIMO channel estimationusing exponential basis models and subblock tracking,” in Proc. 42ndAnnu. Conf. Inf. Sci. Syst., Mar. 19–21, 2008, pp. 1258–1261.

[18] C. Komninakis, C. Fragouli, A. H. Sayed, and R. D. Wesel, “Multi-input multi-output fading channel tracking and equalization using Kalmanestimation,” IEEE Trans. Signal Process., vol. 50, no. 5, pp. 1065–1076,May 2002.

[19] G. Leus, “On the estimation of rapidly time-varying channels,” inProc. Eur. Signal Process. Conf., Vienna, Austria, Sep. 6–10, 2004,pp. 2227–2230.

[20] Z. Liu, X. Ma, and G. B. Giannakis, “Space-time coding and Kalman fil-tering for time-selective fading channels,” IEEE Trans. Commun., vol. 50,no. 2, pp. 183–186, Feb. 2002.

[21] X. Li and T. F. Wong, “Turbo equalization with nonlinear Kalman filter-ing for time-varying frequency-selective fading channels,” IEEE Trans.Wireless Commun., vol. 6, no. 2, pp. 691–700, Feb. 2007.

[22] X. Ma, G. B. Giannakis, and S. Ohno, “Optimal training for block trans-missions over doubly selective wireless fading channels,” IEEE Trans.Signal Process., vol. 51, no. 5, pp. 1351–1366, May 2003.

[23] Y. Ma, N. Yi, and R. Tafazolli, “Channel estimation for PRP-OFDM inslowly time-varying channel: First-order or second-order statistics?” IEEESignal Process. Lett., vol. 13, no. 3, pp. 129–132, Mar. 2006.

[24] X. Meng and J. K. Tugnait, “Semi-blind time-varying channel estimationusing superimposed training,” in Proc. IEEE Int. Conf. Acoust., Speech,Signal Process., Montreal, QC, Canada, May 2004, vol. 3, pp. 797–800.

[25] X. Meng and J. K. Tugnait, “MIMO channel estimation using super-imposed training,” in Proc. IEEE Int. Conf. Commun., Paris, France,Jun. 2004, pp. 2663–2667.

[26] X. Meng and J. K. Tugnait, “Doubly-selective MIMO channel estimationusing superimposed training,” in Proc. IEEE Sensor Array MultichannelSignal Process. Workshop, Barcelona, Spain, Jul. 2004, pp. 407–411.

[27] M. Muck, M. de Courville, X. Miet, and P. Duhamel, “Iterative inter-ference suppression for pseudo random postfix OFDM based channelestimation,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process.,Philadelphia, PA, Mar. 2004, vol. 3, pp. 765–768.

[28] M. Muck, M. de Courville, X. Miet, and P. Duhamel, “A pseudo ran-dom postfix OFDM modulator-semi-blind channel estimation and equal-ization,” IEEE Trans. Signal Process., vol. 54, no. 3, pp. 1005–1017,Mar. 2006.

[29] J. P. Nair and R. V. Raja Kumar, “An iterative channel estimationmethod using superimposed training in OFDM systems,” in Proc. IEEEVTC—Fall, Calgary, AB, Canada, Dec. 21–24, 2008, pp. 1–5.

[30] V. Nguyen, H. D. Tuan, H. H. Nguyen, and N. N. Tran, “Optimal su-perimposed training design for spatially correlated fading MIMO chan-nels,” IEEE Trans. Wireless Commun., vol. 7, no. 8, pp. 3206–3217,Aug. 2008.

[31] A. G. Orozco-Lugo, M. M. Lara, and D. C. McLernon, “Channel estima-tion using implicit training,” IEEE Trans. Signal Process., vol. 52, no. 1,pp. 240–254, Jan. 2004.

[32] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-TimeWireless Communications. Cambridge, U.K.: Cambridge Univ. Press,2003.

[33] J. G. Proakis, Digital Communications, 4th ed. New York: McGraw-Hill, 2001.

[34] M. Qaisrani and S. Lambotharan, “Estimation of doubly-selective MIMOchannels using superimposed training and turbo equalization,” in Proc.IEEE VTC, Singapore, May 2008, pp. 1316–1319.

[35] A. M. Sayeed and B. Aazhang, “Joint multipath-Doppler diversity inmobile wireless communications,” IEEE Trans. Commun., vol. 47, no. 1,pp. 123–132, Jan. 1999.

[36] N. Seshadri, “Joint data and channel estimation using blind trellis searchtechniques,” IEEE Trans. Commun., vol. 42, no. 234, pp. 1000–1011,Mar. 1994.

[37] D. Slepian, “Prolate spheroidal wave functions, Fourier analysis, anduncertainty—V: The discrete case,” Bell Syst. Tech. J., vol. 57, pp. 1371–1430, May/Jun. 1978.

[38] L. Song and J. K. Tugnait, “On designing time-multiplexed pilots fordoubly-selective channel estimation using discrete prolate spheroidal ba-sis expansion models,” in Proc. IEEE ICASSP Conf., Honolulu, HI,Apr. 2007, pp. III-433–III-436.

[39] M. D. Srinath, P. K. Rajasekaran, and R. Viswanathan, Introduction toStatistical Signal Processing With Applications. Upper Saddle River,NJ: Prentice-Hall, 1996.

[40] P. Stoica and R. L. Moses, Introduction to Spectral Analysis. UpperSaddle River, NJ: Prentice-Hall, 1997.

[41] G. L. Stuber, Principles of Mobile Communication, 2nd ed. Boston, MA:Kluwer, 2001.

[42] L. Tong and S. Perreau, “Multichannel blind identification: From subspaceto maximum likelihood methods,” Proc. IEEE, vol. 86, no. 10, pp. 1951–1968, Oct. 1998.

[43] D. Tse and P. Viswanath, Fundamentals of Wireless Communication.Cambridge, U.K.: Cambridge Univ. Press, 2005.

[44] J. K. Tugnait and W. Luo, “On channel estimation using superim-posed training and first-order statistics,” in Proc. ICASSP, Hong Kong,Apr. 2003, vol. 4, pp. 624–627.

[45] J. K. Tugnait and X. Meng, “On superimposed training for channelestimation: Performance analysis, training power allocation and framesynchronization,” IEEE Trans. Signal Process., vol. 54, no. 2, pp. 752–765, Feb. 2006.

[46] J. K. Tugnait, X. Meng, and S. He, “Doubly-selective channel estimationusing superimposed training and exponential bases models,” EURASIPJ. Appl. Signal Process.—Special Issue Reliable Commun. Over RapidlyTime-Varying Channels, vol. 2006, p. 252, Jul. 2006.

[47] J. K. Tugnait, L. Tong, and Z. Ding, “Single-user channel estimationand equalization,” IEEE Signal Process. Mag., vol. 17, no. 3, pp. 16–28,May 2000.

[48] X. Wang, Y. Wu, J.-Y. Chouinard, and H-C. Wu, “On the design and per-formance analysis of multisymbol encapsulated OFDM systems,” IEEETrans. Veh. Technol., vol. 55, no. 3, pp. 990–1002, May 2006.

[49] X. Wang, Y. Wu, H.-C. Wu, and G. Gagnon, “An MSE-OFDM systemwith reduced implementation complexity using pseudo random prefix,”in Proc. IEEE GLOBECOM Conf., Washington, DC, Nov. 26–30, 2007,pp. 2836–2840.

[50] L. Yang, X. Ma, and G. B. Giannakis, “Optimal training for MIMO fadingchannels with time- and frequency-selectivity,” in Proc. Int. Conf. Acoust,Speech, Signal Process., May 17–21, 2004, vol. 3, pp. 821–824.

[51] T. Zemen and C. F. Mecklenbräuker, “Time-variant channel estimation us-ing discrete prolate spheroidal sequences,” IEEE Trans. Signal Process.,vol. 53, no. 9, pp. 3597–3607, Sep. 2005.

[52] Y. R. Zheng and C. Xiao, “Simulation models with correct statisticalproperties for Rayleigh fading channels,” IEEE Trans. Commun., vol. 51,no. 6, pp. 920–928, Jun. 2003.


Jitendra K. Tugnait (M’79–SM’93–F’94) was bornin Jabalpur, India, on December 3, 1950. He re-ceived the B.Sc. degree (with honors) in electronicsand electrical communication engineering from thePunjab Engineering College, Chandigarh, India, in1971, the M.S. and E.E. degrees in electrical engi-neering from Syracuse University, Syracuse, NY, in1973 and 1974, respectively, and the Ph.D. degree inelectrical engineering from the University of Illinois,Urbana, in 1978.

From 1978 to 1982, he was an Assistant Professorof electrical and computer engineering with the University of Iowa, IowaCity. From June 1982 to September 1989, he was with the Long RangeResearch Division, Exxon Production Research Company, Houston, TX. InSeptember 1989, he joined the Department of Electrical and Computer Engi-neering, Auburn University, Auburn, AL, as a Professor, where he is currentlythe James B. Davis Professor. His current research interests are statisticalsignal processing, wireless and wireline digital communications, multiple-sensor–multiple-target tracking, and stochastic systems analysis.

Dr. Tugnait is a past Associate Editor for the IEEE TRANSACTIONS ON

AUTOMATIC CONTROL, the IEEE TRANSACTIONS ON SIGNAL PROCESS-ING, and the IEEE SIGNAL PROCESSING LETTERS. He is currently an Editorfor the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS.

Shuangchi He received the B.E. and M.S. degreesin electronic engineering from Tsinghua Univer-sity, Beijing, China, in 2000 and 2003, respectively,and the Ph.D. degree in electrical engineering fromAuburn University, Auburn, AL, in 2007.

From August 2003 to August 2007, he was a Grad-uate Research Assistant and then a Vodafone Fellowwith the Department of Electrical and ComputerEngineering, Auburn University. He is currently withthe School of Industrial and Systems Engineering,Georgia Institute of Technology, Atlanta. His re-

search interests include channel estimation and equalization, multiuser detec-tion, and statistical and adaptive signal processing and analysis.

multiuser/mimo doubly selective fading channel …...ieee transactions on vehicular technology, vol....

Documents