Dual-Diagonal LMMSE Channel Estimation for OFDM Systemsliping/Research/Journal/30 Dual-Diagonal LMMSE Channel Estimation...Dual-Diagonal LMMSE Channel Estimation for OFDM Systems Nian

4734 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 9, SEPTEMBER 2012

Dual-Diagonal LMMSE Channel Estimation forOFDM Systems

Nian Geng, Student Member, IEEE, Xiaojun Yuan, Member, IEEE, and Li Ping, Fellow, IEEE

Abstract—We propose a low-complexity dual-diagonal (DD-)linear-minimum-mean-square-error (LMMSE) channel estima-tion algorithm for orthogonal frequency-division multiplexing(OFDM) systems involving iterative channel estimation and signaldetection. Computational complexity and mean-square-error(MSE) analysis are presented to evaluate the efficiency of theproposed algorithm. A closed-form expression is derived for theasymptotic MSE of the DD-LMMSE channel estimator. Bothanalysis and numerical results show that DD-LMMSE performsclose to the well-known LMMSE estimator with much lowercomplexity.

Index Terms—Channel estimation, dual-diagonal LMMSE(DD-LMMSE) estimation, linear-minimum-mean-square-error(LMMSE), orthogonal frequency division multiplexing (OFDM).

I. INTRODUCTION

C HANNEL estimation is a key operation in an orthogonalfrequency division multiplexing (OFDM) system to ac-

quire channel state information (CSI) at the receiver [1]–[3].Pilot signals are commonly employed to facilitate channel es-timation. However, pilot signals constitute extra overheads onboth transmission power and available bandwidth.Intuitively, data signals, if known or partially known at the

receiver, may have the same function as pilot signals. This mo-tivates the research on iterative channel estimation and signaldetection, in which detection feedbacks are used together withpilots in channel estimation [5]–[8]. Furthermore, the accuracyof channel estimation can be improved if certain channel statis-tics, such as the channel power delay profile (PDP) [4], is avail-able in prior.The above mentioned joint channel estimation and signal

detection technique involves combining information fromdifferent sources, namely, pilot signals, PDP, and decoder

Manuscript received September 10, 2011; revised January 19, 2012 and May02, 2012; accepted May 06, 2012. Date of publication June 01, 2012; dateof current version August 07, 2012. The associate editor coordinating the re-view of this manuscript and approving it for publication was Dr. Milica Sto-janovic. This work has been performed in the framework of the ICT projectsICT-217033 WHERE and ICT-248894 WHERE2, which are partly funded bythe European Union. The material in this paper was partially presented at theIEEE GLOBECOM, Miami, Florida, December 6–10, 2010.N. Geng was with the Department of Electronic Engineering, City University

of Hong Kong (e-mail: [email protected]).X. Yuan was with the Department of Electronic Engineering, City Univer-

sity of Hong Kong. He is now with the Institute of Network Coding, the De-partment of Information Engineering, the Chinese University of Hong Kong(e-mail: [email protected]).L. Ping is with the Department of Electronic Engineering, City University of

Hong Kong (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2012.2202112

feedback. The linear-minimum-mean-square-error (LMMSE)estimation has been extensively studied for this purpose. Sig-nificant performance improvement has been reported, e.g., in[6] and [8]. However, in its straightforward form, an LMMSEchannel estimator involves complexity per iteration,where is the number of OFDM subcarriers. This can be verycostly (e.g., for in the 3GPP-LTE system [9]).Low-complexity channel estimation algorithms have been

studied in the literature. For example, the windowed discreteFourier transform (WDFT) methods discussed in [10]–[12]reduce complexity to . However, the suboptimalinformation combining techniques used in the WDFT methodmay lead to a considerable performance loss compared withthe LMMSE method.In this paper, we address the complexity problem for the

LMMSE method and propose a novel dual-diagonal LMMSE(DD-LMMSE) channel estimation algorithm. The new algo-rithm involves two diagonal LMMSE (D-LMMSE) operations,one in the frequency domain and the other in the time domain.Fast Fourier transform (FFT) is used to convert information be-tween the time and frequency domains. The total computationalcomplexity of the proposed algorithm is only .We analyze the aforementioned options for channel estima-

tion, namely, LMMSE, DD-LMMSE, and WDFT. We deriveclosed-form asymptotic mean-square error (MSE) bounds forthese options, based on which their performances are compared.We show that DD-LMMSE outperforms WDFT considerably,although they have comparable complexity. We also show thatDD-LMMSE performs closely to LMMSE, although the formerrequires much lower complexity.DD-LMMSE provides an attractive solution for an iterative

receiver where pilots as well as partially detected data are usedtogether to improve the performance of channel estimation. Inthis scenario, the complexity of LMMSE may become a se-rious issue, while DD-LMMSE provides a more cost-effectivealternative. We provide the simulation results of the differentchannel estimators mentioned above in coded OFDM systemswith iterative channel estimation and signal detection. The sim-ulation results agree well with our analysis.The notations used in this paper are listed as follows. Bold

uppercase letters represent matrices, and bold lowercase lettersrepresent column vectors. Notations “ ,” “ ,” and “ ” repre-sent complex conjugate, transpose, and Hermitian transpose, re-spectively. Let denote the expectation operation, thevariance, the magnitude, the trace, “ ” the entry-wiseproduct, and “ ” the circular convolution. Function re-turns a diagonal matrix with being the main diagonal;returns with the off-diagonal entries of set to zero; and

represents the vector formed by the diagonal of .represents the th element of matrix , and the

1053-587X/$31.00 © 2012 IEEE

GENG et al.: DUAL-DIAGONAL LMMSE CHANNEL ESTIMATION 4735

th element of vector . Notation “ ” represents convergence,

“ ” represents convergence in probability, “ ” represents noless than in probability.

II. PRELIMINARIES

A. System Model

Consider an OFDM system in which cyclic prefixes (CP)are inserted to ensure the orthogonality among the subcarriers.Let be a received signal vector,

be a transmitted signal vector (referred toas an OFDM symbol), be a vectorof independent and identically distributed (i.i.d.) circular com-plex additive white Gaussian noise (AWGN) samples with zeromean and variance , and be a vectorof channel coefficients in the frequency domain. Denote bythe entry-wise product between and . The system can be rep-resented in the frequency domain by a set of parallel subcar-riers as

(1)

or equivalently

(2)

where . We assume that the entries of are inde-pendently and uniformly taken over a signaling constellation .We denote by the average signal power, i.e.

(3)

The channel coefficients in the time domain can be repre-sented as

where is related to via the discrete Fourier transform (DFT),i.e.

or in a vector form as

(4)

where is the normalized DFT matrix with the th ele-ment given by

(5)

with . The channel coefficients have zero meanand covariance

where is the average power of the th delay path. The di-agonal matrix is referred to as the channel PDP. The totalchannel power is denoted as

(6)

We denote by the number of nonzero channel delay taps(i.e., the number of nonzero entries of ). In practice, is usu-ally much less than . For convenience of discussion, we as-sume that is perfectly known at the receiver until Section IV.The extension to a partially known is addressed in Section V.

B. LMMSE Estimation

Let be an estimator of . The efficiency of can be mea-sured by the MSE defined as

(7)

Note that is assumed perfectly known in the above expecta-tion. Denote by the covariance matrix of and , i.e.

(8)

From (4), the auto-covariance of is given by

(9)

The well-known LMMSE estimator is given by [15]

(10)

where

(11)

(12)

The above LMMSE estimator is optimal among all linear esti-mators in minimizing theMSE in (7), and it is generally optimalif the elements of are Gaussian distributed [15].Substituting (11) and (12) into (10), we obtain

(13)The corresponding MSE is given by [15]

(14)

The complexity of LMMSE estimator in (10) is dominatedby the inversion of . For constant modulus signaling (i.e.,

), we obtain

(15)

As is diagonal, the inverse of can be efficiently calculatedusing FFT with complexity .For nonconstant signaling, however, cannot be written in

the form (15). Then, its inversion generally involves complexity. This is quite high for a large (or even moderate) . The


objective of this paper is to study alternative approaches with agood tradeoff between complexity and performance.

C. D-LMMSE Estimation

Define the diagonal estimator as

(16)

where is restricted to be a diagonal matrix. The correspondingMSE is given by

(17)To find the optimal solution for , we take the partial derivativeof (17) with respect to for each and set the result tozero. Solving these equations, we obtain the optimal as

or in a matrix form as

(18)

Substituting (11) and (12) into (18), we obtain

(19)

The diagonal LMMSE (D-LMMSE) estimator is defined aswith given by (19). Substituting (18) into

(17), we obtain

(20)

Using (9), (11), and (12), we further obtain

(21)

Since is diagonal, the computational complexity involved in(19) is very low. However, as seen later, the MSE in (21) ismuch higher than that of LMMSE. This motivates us to seekbetter solutions.

III. DUAL-DIAGONAL LMMSE ESTIMATION

In this section, we propose a novel DD-LMMSE estimatorwith very low complexity. We show that the DD-LMMSE esti-mator can achieve performance close to the LMMSE one.

A. DD-LMMSE Estimation

From (4), we have

(22)

Correspondingly, the time domain representation of the diag-onal estimator in (16) is defined as

(23)

Note that and are related by

(24)

where follows from (16), from (2), and from (4).We now apply the diagonal estimation technique to estimateby treating as an equivalent observation. The resulting

estimator is expressed as

(25)

where is diagonal, and the subscript “ ” means that twodiagonal estimators are involved. We aim at finding a diagonalthat minimizes the MSE. This is analogous to the problem in

Section II-C, and so the result in (18) can be borrowed, yielding

(26)

With the above approach, we construct an estimator of as

(27)

Particularly, with given by (18) and by (26) is referredto as the DD-LMMSE estimator (denoted by ), asthe D-LMMSE estimation is involved twice.In the DD-LMMSE estimation, we first transform the fre-

quency-domain D-LMMSE estimator to the time-domain, and then apply again the D-LMMSE principle to estimate

by treating as the equivalent observation. Conceptually, wemay directly treat as an equivalent observation andapply the D-LMMSE principle to in estimating . However,the latter approach cannot provide any improvement, sincegiven in (18) is already optimal among all diagonal matrixes.This explains the necessity of the DFT operation in (23) (a rota-tion in the linear space), which, as will be shown, leads to goodperformance as well as low complexity.

B. Computational Complexity

Now we consider the complexity involved in DD-LMMSE.Given and , the DD-LMMSE estimator canbe fast computed using the FFT algorithm. As is diagonal,the calculation of in (19) is straightforward.What remains is the calculation of , which involves the two

covariance matrixes in (26). Applying the definition in (8), weobtain from (24) that

(28)

and

(29)As discussed in Appendix I, the diagonals of the above two co-variance matrices can be fast calculated based on the FFT. Theoverall complexity is only .


C. Alternative Dual-Diagonal Estimators

In general, a dual-diagonal (DD) estimator is defined as

where and are two diagonal matrixes. The DD-LMMSEtechnique (with and , respectively, given by (18) and (26))is a special case of the DD estimation.Two alternative choices of and are as follows. The first

is the WDFT technique discussed in [10]–[12] (referred to asWDFT-I) with and , respectively, given by

(30)where the indicator function is defined as for

, and for . The second is a modificationof (30) (referred to as WDFT-II) as

(31a)

(31b)

DD-LMMSE considerably outperforms the twoWDFT options,as will be detailed later.

D. Treatment for Guard Subcarriers

In practical OFDM systems, some subcarriers are turned offfor band protection. These subcarriers are referred to as guardsubcarriers [9], [28]. The DD estimation can be directly appliedto OFDM systems with guard subcarriers. However, as pointedout in [29], DFT-based channel estimators (including the DDestimators) suffer from an aliasing problem. It is easy to see thisproblem using WDFT-I as an example. Assume andsuppose that all the subcarriers are used in transmission. It iseasy to show that

which is error free, as expected. However, if some subcarriersare turned off, theWDFT-I result is no longer error free as(by noting that for each guard subcarrier ). This is the

so-called aliasing problem.A remedy for the above problem was proposed in [21]. The

main idea is to estimate the channel coefficients and then to re-construct the virtual observations on the guard subcarrier (usingthe coarse channel estimates) in a recursive fashion. The discus-sions in [21] are limited to the least-square (LS) estimation. Wecombine this idea with the DD estimators as follows.With guardsubcarriers, the system in (1) should be modified as

We calculate and as follows. For each guard subcarrier ,we generate a virtual pilot symbol (uniformly drawn fromthe constellation ). We form an OFDM symbol using boththe actual and virtual pilot symbols. Then, we calculate andas if is transmitted over the system in (1). We set the initial

estimate . Then, we repeatedly execute the followingtwo steps.

Fig. 1. The MSE performance of various channel estimation methodsin the OFDM system with QPSK modulation. The total number of sub-carriers is . The channel power delay profile is given by

in dB with relative path delay {0, 310,710, 1090, 1730, 2510} in ns.

Step 1) For , if subcarrier is a guard sub-carrier, reconstruct the virtual observation as

.Step 2) Compute the DD estimator: .Here, the DD estimator can be DD-LMMSE, WDFT-I orWDFT-II. This algorithm can efficiently compensate thealiasing error, as seen in Figs. 3 and 4.

E. Numerical Examples

In simulation, we mainly follow the settings used in [6].Specifically, the carrier frequency is 5GHz, and the bandwidthof the simulated OFDM system is 5MHz. We use the IMT-2000vehicular-A channel [30] with the exponentially decayedchannel power profile given byin dB and with relative path delay {0, 310, 710, 1090, 1730,2510} in ns. The total channel power is normalized to1. Jakes model [31] is used in channel generation with the

, corresponding to a Dopplerfrequency shift of 462.5 Hz. The total number of OFDMsubcarriers is . The subcarrier spacing is approxi-mately 5.12 kHz. The cyclic-prefix duration is 16 samples.The channel SNR is defined as . All the simulationresults provided are averaged over no less than 10000 channelrealizations.First, we consider quadrature phase-shift keying (QPSK)

modulation in which the entries of have constant modulus.Fig. 1 shows the MSE of various channel estimators. We seethat the curves of DD-LMMSE and LMMSE coincide (asanalyzed in Appendix VI). Both of them considerably out-perform D-LMMSE and WDFT-II. We also see that WDFT-Iprovides quite good performance in this case, except in thelow-SNR region. In addition, the conventional LS estimation[21] is also included for comparison. From Fig. 1, we see thatLMMSE/DD-LMMSE outperforms LS by over 3 dB in theentire SNR range of interest.


Fig. 2. The MSE performance of various channel estimators in the OFDMsystem with standard 16-QAM signaling. The system settings follow Fig. 1,except that QPSK is changed to 16-QAM.

Second, we consider standard 16 quadrature amplitude mod-ulation (16-QAM) signaling (cf., SCM-1 in [17]). Other set-tings remain the same as in Fig. 1. The numerical results areshown in Fig. 2.We see that, for nonconstant modulus signaling,DD-LMMSE performs slightly worse than LMMSE in the highSNR region. This phenomenon will be analyzed in detail in thenext section. The other performance trends remain roughly thesame as those in Fig. 1.Now we study the impact of guard subcarriers. We still use

the settings in Fig. 2, except that 28 lower-frequency and 27upper-frequency guard subcarriers, as well as the subcarrier atthe carrier frequency, are turned off (following IEEE 802.16-2004 [28]). The recursive DD-LMMSE estimator is referred toas R-DD-LMMSE. The .From Fig. 3, we see that the performances of DD-LMMSE

and the WDFT methods are severely degraded due to thealiasing effect. We also see that R-DD-LMMSE can effi-ciently mitigate this effect. The performance curves of theLMMSE estimator, the LS estimator, and the modified LSestimator proposed in [21] are also included for comparison.From Fig. 3, R-DD-LMMSE performs close to LMMSE inthe low-to-median SNR region, but slightly worse in the highSNR region. (Note that similar performance trends have beenobserved in Fig. 2.) Moreover, it is demonstrated in Fig. 3that R-DD-LMMSE considerably outperforms the modified LSestimator at a relatively low SNR.The performance of R-DD-LMMSE against the number of

recursions is illustrated in Fig. 4. We see that R-DD-LMMSEconverges quite fast in the low-to-median SNR region. For ex-ample, at , only 4 recursions are required to ap-proach convergence.

IV. ASYMPTOTIC MSE ANALYSIS

In this section, we analyze and compare the performance ofDD-LMMSE with other alternatives. It turns out the problemis very complicated for a finite . We circumvent the problemby studying the asymptotic limit as the subcarrier number. We will show that the analysis agrees well with numerical

results even for practically used values.

Fig. 3. The MSE performance of various channel estimators in the OFDMsystem with guard subcarriers. The simulation settings are the same as in Fig. 2,except that 55 subcarriers are turned off as guard subcarriers.

Fig. 4. The MSE performance of R-DD-LMMSE against the number of recur-sions. The simulation settings are exactly the same as in Fig. 3.

A. Preliminaries

Recall that is the number of nonzero entries in the channelPDP . When remains finite as (implying), an infinite number of observations are used to estimate a fi-nite number of unknown variables. In this case, it can be shownthat all the channel estimators listed in Section III-C have a van-ishing MSE (normalized by the channel power ). Therefore,it is more interesting to study the nontrivial situation whengoes to infinity together with . This is specified in the fol-lowing definition.Definition 1 (Sufficiently Dispersive Channel (SDC)): A

channel is said to be sufficiently dispersive if is uniformlyupper bounded by a finite positive constant , i.e.

(32)

where is the supremum of the set in the brackets.


In practice, a communication environment with rich scat-tering and reflections (such as the case in typical suburban andurban areas [4]) is a good approximate of SDC.Let be the diagonal of in (27), and be the th diagonal

entry of . The following lemma holds for an SDC, with theproof given in Appendix II.Lemma 1: For an SDC, if are i.i.d., then, as ,

(33a)

where1

(33b)

(33c)

(33d)

and the convergence in (33a) is in an entry-by-entry manner.Lemma 1 states that the diagonal entries of

are approximately deterministic for a sufficiently large .The condition of Lemma 1 (i.e., are i.i.d.) is valid forDD-LMMSE, WDFT-I, and WDFT-II described in Section III.Therefore, (33a) holds for all the three DD estimators.The discussions below are mostly for SDCs. Empirically, we

have observed that Lemma 1 approximately holds even if thechannel is not sufficiently dispersive.We briefly explain this ob-servation in Appendix II. Lemma 1 is useful in establishing theasymptotic MSE bound for the DD estimators, as seen below.

B. Asymptotic MSE of the DD Estimators

We now compare the asymptotic performance of the DD tech-niques described in Section III-C. Define a parameter

(34)

Note that can be treated as a measure of the variability of ,with when are equal, and otherwise (cf.,Proposition 1). We are now ready to present the main theorem.Theorem 1: For an SDC, the three DD estimators discussed

in Section III satisfy

(35)In particular, the equality holds for DD-LMMSE.The proof of Theorem 1 can be found in Appendix III. From

Theorem 1, DD-LMMSE asymptotically outperforms the twoWDFT alternatives. This is reasonable since, for DD-LMMSE,in (18) and in (26) are both optimal diagonal estimators.It is also worth mentioning that, from the discussions in

Appendix II-D, Theorem 1 approximately holds even withoutimposing the SDC assumption.

1The subscript “ ” is dropped from , , and , since arei.i.d.

C. Comparison With LMMSE

DD-LMMSE is in general a suboptimal option. Naturally, weare interested in its performance loss compared with the optimalLMMSE estimator. This is the focus of the discussions below.We cite from [22, Theorem 1] the following result: for the

system in (1) with , the MSE of a linearchannel estimator is minimized when the entries of have aconstant modulus (i.e., ), i.e.

(36)

where the equality holds when , and the right-handside (RHS) of (36) is obtained by substitutinginto (14).Comparing (35) and (36), we see that DD-LMMSE for (1) has

the same asymptotic performance as LMMSE for the system

(37)

where represents the signal for subcarrier with a constantmodulus (i.e., , for any index ), and is an AWGNsample with variance . Comparing (37) and (1), we see thatdetermines the maximum performance loss of DD-LMMSE

compared to LMMSE. The following properties of are useful,with the proof given in Appendix IV.Proposition 1 (Properties of ): defined in (34) is mono-

tonically decreasing in with

From (35), (36), and Proposition 1, the maximum asymp-totic MSE gap between DD-LMMSE and LMMSE is at most

dB in SNR (as ); and this gap monotoni-cally reduces to zero as . This indicates that the twoestimators perform nearly the same in the relatively low SNRregion (cf., Figs. 2, 5, and 6).

D. MSE Bounds in the High SNR Region

Now we consider the MSE performance of the aforemen-tioned channel estimators in the high SNR region. Recall thatchannel taps are to be estimated based on the observations on

the subcarriers. For an SDC, is bounded away from zerosince

(38)

where step follows from (6), and from (32). We assumethat the limit of exists as , denoted by

(39)

Then, we have the following result.


Fig. 5. The MSE bounds of various channel estimation methods. The simu-lation settings follow Fig. 2. The MSE bounds of LMMSE and DD-LMMSEare, respectively, given by (36) and (35). The MSE bounds of D-LMMSE andWDFT-I are given in Theorem 2.

Fig. 6. The MSE bounds of various channel estimation methods. The simula-tion settings are the same as in Fig. 5, except that here the uniform PDP with

is employed.

Theorem 2: For an SDC, as and (in thatorder) while is kept constant, the following limits hold:

and

The proof of Theorem 2 can be found in Appendix V. FromTheorem 2, in the high SNR region, DD-LMMSE performs

dB worse than LMMSE, but dB betterthan D-LMMSE. Also, it is interesting to see that WDFT-I hasthe same asymptotic MSE bound as DD-LMMSE.It is worth mentioning that DD-LMMSE is actually equiva-

lent to LMMSE if the entries of are of constant modulus, i.e.,(which holds, e.g., for phase-shift-keying mod-

ulated signaling). This equivalence is verified in Appendix VI.Later in Section V, we will see that this equivalence does nothold for the case of partially known , no matter constant ornonconstant modulus signaling is employed.

E. Numerical Results

We present numerical examples to verify the above analysis.In simulation, the system settings remain the same as in Fig. 2.The MSE bound of DD-LMMSE is given by Theorem 1. Thatof LMMSE is given by (36). The MSE bounds of D-LMMSEand WDFT-I are given by Theorem 2.From Fig. 5, we see that the MSE bounds agree well with the

simulated performance curves. Since nonconstant modulus sig-naling is employed here, DD-LMMSE performs slightly worsethan LMMSE in the high SNR region. As stated in Theorem2, the asymptotic gap between DD-LMMSE and LMMSE for

is . (Note: forstandard 16-QAM.)In Fig. 6, we replace the exponential channel PDP in Fig. 5

by a uniform one with (i.e., , for , andotherwise). Again, we see that the MSE bounds agree

well with the simulated performances. Comparing Figs. 5 and6, we see that the performances of the concerned estimators arenot sensitive to the shape of the channel PDP.

V. APPLICATIONS IN ITERATIVE RECEIVERS

From the discussions below (15), although the LMMSE com-plexity is in general, it can be reduced to bychoosing a constant-modulus . If this is the case, DD-LMMSEhas no advantage over LMMSE.2

DD-LMMSE becomes an attractive option in an iterative re-ceiver, where partially estimated data are used to improve theaccuracy of channel estimation. In this case, contains bothpilot and partially known data, and so LMMSE involves com-plexity . Then, DD-LMMSE stands out for its low com-plexity.In this section, we briefly outline iterative channel estimation

and detection principles. Similar techniques can be found in,e.g., [13], [14], [16]–[20], and the references therein. We willshow that DD-LMMSE can provide efficient tradeoff betweenperformance and complexity in iterative systems.

A. Transceiver Structure

The transceiver structure is illustrated in Fig. 7. At the trans-mitter side, the binary data stream is encoded by a generic en-coder (ENC). The ENC includes the forward error correctioncoding, random interleaving and multiary modulation, as wellas pilot insertion. The entries of the transmitted signal are as-sumed to be independently and uniformly taken over a signalingconstellation (that may or may not be of constantmodulus).

2Actually, the two estimators are equivalent to each other in this case, as statedin Appendix VI.


Fig. 7. The flow chart of an OFDM system involving iterative channel estima-tion and symbol detection.

The iterative receiver consists of two main modules, namely,the channel estimator (CE) and the signal detector (SD).• The CE estimates based on the feedback from the SD.We initialize the distribution of as follows. For pilotpositions, and . For datapositions, and datapower (assuming a common modulation scheme with zeromean).

• The SD performs a posteriori probability (APP) demod-ulation and decoding of based on , assuming that thechannel is given by the CE output . Its outputs the aposteriori mean and variance of each . Forconvenience, we write

(40a)

(40b)

The CE and SD are executed iteratively until convergence. Initeration, and are fed back to the CE and used as the apriori means and variances. We always assume that are apriori uncorrelated. This is approximately ensured by randominterleaving [25], [26].

B. DD-LMMSE for the CE Module

Recall the channel model in (2): . The CE esti-mates based on . This problem is the same as that discussedpreviously, except that each is now characterized by a dis-tribution instead of perfectly known. The results in Sections IIand III can be applied with some minor modifications, takinginto account the distribution of .For simplicity, we only discuss DD-LMMSE. The DD

estimator is still given in (27) as . ForDD-LMMSE, and are, respectively, given by

(41)

where the covariance matrixes are given by

(42a)

(42b)

(42c)

and

(42d)

with . Similar to the discussions in Appendix I,we can show that the complexity involved above is still

. However, the LMMSE complexity is ingeneral, as cannot be written in the form of (15). This isbecause now (consisting of the mean of ) is generally notof a constant modulus, even if itself is.Moreover, the MSE analysis in Section IV can be readily ex-

tended as follows. Assume that are i.i.d. random vari-ables, and so are (as approximately ensured by randominterleaving [25], [26]). Similarly to (34), define

(43)

Then, the results of Theorem 1 still hold, except that is re-placed by in (43). The proof is similar to that of Theorem 1.The details are omitted for simplicity.

C. Numerical Results

Numerical results are provided to demonstrate the perfor-mance of the transceiver scheme in Fig. 7. The channel remainsthe same as in Fig. 1. Other settings are described as follows. AnOFDM system with 256 subcarriers is considered. 16 equallyspaced subcarriers are selected as the pilot tones. Each trans-mission frame contains 32 OFDM symbols. Hence the framelength is 8192. The ENC employs a rate-1/2 (3, 6) low-den-sity-parity-check (LDPC) code. The output of the LDPC en-coder is randomly interleaved and QPSK-Gray modulated toform a transmission frame. At the receiver side, the CE moduleemploys the various channel estimators involved in Fig. 1.3 Asa benchmark, we also include the performance of the noniter-ative system in which the channel is estimated solely based onthe pilots (following the LMMSE principle).The bit error rate (BER) of the above scheme is illustrated in

Fig. 8. Every simulated point in Fig. 8 is obtained by averagingover at least 1000 blocks. We see that DD-LMMSE performsclose to LMMSE, and achieves about 1.2 dB gain at

compared with the noniterative system that employs onlythe pilot for channel estimation. For DD-LMMSE, the perfor-mance loss incurred by the channel uncertainty is about 0.5 dBat . We also see that, compared with the nonit-erative system, WDFT-II can achieve marginal gain. These ob-servations demonstrate the advantage of DD-LMMSE. It is alsoworth mentioning that the BER performance of WDFT-I is con-siderably worse than the other estimators in the low SNR region.The reason is that, as illustrated in Figs. 1 and 2, the MSE ofWDFT-I is unbounded in the low SNR region.In Fig. 9, we follow the same settings as in Fig. 8, except that

now the coded bits are Gray mapped onto the standard 16-QAMconstellation and the frame length is increased to 65536. Thecurves in Fig. 9 are obtained by averaging over 500 blocks. Wesee from Fig. 9 that, at , the DD-LMMSE curve isabout 0.6 dB away from the known-channel case, 0.2 dB awayfrom the LMMSE curve, and 0.8 dB better than the noniterativesystem. The gap between DD-LMMSE and LMMSE in Fig. 9is slightly larger than that in Fig. 8. The reason is that, as illus-trated in Fig. 1 and Fig. 2, the performance loss of DD-LMMSE

3In the simulation, the LMMSE estimation solely based on the pilots is alwaysused in the CE to obtain the initial channel estimates.


Fig. 8. The performance of the OFDM system involving iterative channel es-timation and signal detection. The channel settings are the same as in Fig. 1. 16equally spaced subcarriers are used as the pilot tones. The information is en-coded by a rate-1/2 (3, 6) LDPC code, and then Gray-QPSK modulated. Theframe length is 8192. The iteration number is 8.

Fig. 9. The performance of the OFDM systems involving iterative channel es-timation and signal detection. The same settings as in Fig. 8 are used, except thatstandard 16-QAM modulation with Gray mapping is employed and the framelength is 65536.

slightly increases when does not have a constant modulus. Itis also interesting to see that WDFT-I and WDFT-II performeven worse than the noniterative system. This implies that theLMMSE estimation based solely on pilots can provide more ac-curate channel estimates than WDFT-I and WDFT-II based onboth pilots and signals.

VI. CONCLUSION

In this paper, we propose a novel DD-LMMSE channel esti-mation method for OFDM systems. The complexity of the pro-posed method is significantly lower than the optimal LMMSEestimation. We derive an asymptotic MSE bound for the pro-posed method as the subcarrier number tends to infinity. Wealso consider the application of DD-LMMSE to OFDM sys-

tems involving iterative channel estimation and signal detec-tion. Numerical results demonstrate that DD-LMMSE performsclose to LMMSE at a much lower computational cost. More-over, DD-LMMSE considerably outperforms the other subop-timal alternatives with comparable complexity.

APPENDIX IFAST ALGORITHM OF DD-LMMSE ESTIMATION

A. Brief Summary

Here is a summary of the discussions in Section III-A. TheDD-LMMSE estimator is defined as

(A1)

with and are given (cf., (18) and (26))

(A2)

(A3)

The covariance matrixes in (A2) and (A3) are computed as fol-lows [cf., (11), (12), (28), and (29)]:

(A4)

(A5)

(A6)

(A7)

B. Fast Algorithm

With defined in (5), we have (for any diagonal ):

(A8)

Based on (A8), the diagonals of (A4)–(A6) can be efficientlycomputed as follows (as , , and are all diagonal).

(A9)

(A10)

(A11)

We now focus on (A7). Denote

(A12)

where represents the vector formed by the diagonalof the matrix in the brackets. Using (A8) and (A12), we canwrite the diagonal of (A7) as

(A13)

where is an -by-1 all-one column vector. Let, and . With

some straightforward manipulations, it can be shown that

(A14)

where “ ” is for complex conjugate, “ ” for entry-wise product,and “ ” for circulate convolution.


Note that can be calculated efficiently using theFFT with complexity . The circular convolutionof two vectors can also be realized efficiently using the FFT.Therefore, the overall cost involved in (A14) is ,which dominates the complexity for computing (A2) and (A3).

APPENDIX IIPROOF OF LEMMA 1

Noting , we equivalently express (33a) as

(A15)To prove (A15), we start with defined in (A12).

A. Mean of

We here derive the mean of . Define

(A16)

Then

(A17)

The circular auto-correlation of the vector is defined as

(A18)

Note that is the energy spectrum of . From the circularconvolution theorem of the DFT, we have

(A19)

Now denote the impulse sequence as

Let be th entry of , and be that of . By definition

Since are i.i.d., the mean of is given by

(A20)

or equivalently

(A21)

where and are defined in (33). From (A19), we obtain

(A22)

and hence

(A23)

B. Variances of the Entries of

We now show that the variance of the entries of tends tozero as . We describe some properties of defined in(A16). Let be the th entry of . From the definition in (A16),we have

(A24)

The following properties can be verified based on the standardspectral analysis [27].(i) For , , , andare asymptotically uncorrelated as .

(ii) For , , , we have.4

(iii) The variance of , denoted by , is invariant withrespect to the index .

(iv) remains finite as .Let be the th entry of . From the above properties, thevariance of , denoted by , is given by

(A25)

For an SDC defined in (32)

Thus, we obtain that, for any index ,

(A26)

Combining (A25) and (A26), we obtain

(A27)

Note that may remain finite if the SDC condition in (32) isnot met. More discussions on the violation of the SDC conditioncan be found in Subsection D.

C. Proof of (A15)

Denote

(A28)

4Two underlying assumptions are imposed here. First, it is assumed thatare real-valued. Without this assumption (i.e., for complex-valued), (A27) still holds, and so does Lemma 1. We leave the details to

interested readers. Second, it is assumed here that is even. The extension ofthe proof to the case of an odd is straightforward.


Fig. 10. The ratio against the channel SNR with given by (19). ForGaussian signaling, are independently drawn from aGaussian distribution,and for 16-QAM signaling, are independently and uniformly drawn fromthe standard 16-QAM constellation. and .

As are i.i.d., we have

(A29a)

(A29b)

Combining mai(A23), (A27)–(A29), we obtain that, for a SDC

(A30a)

(A30b)

With (A30), we can obtain (A15) from the Chebyshev’s in-equality which completes the proof of Lemma 1.

D. Further Discussions

The above discussions are limited to the case of SDC.We nowshow that Lemma 1 approximately holds for more general cases.From the above discussions, it suffices to show that [cf.,(A27)].Note that

(A31)

Then, from (A25), we obtain

(A32)

Denote

(A33)

From (A23), is the average deterministic part of . Then, thevariation of can be measured by (which is the power ratio

of the stochastic part over the deterministic part). Consider thefollowing upper-bound of as

(A34)

Note that is a function of , while is not. Thus, we use the

upper bound to measure the variation of .If is draw from a constellation with constant modulus (e.g.,

QPSK), then

Therefore, in this case.For other common distributions of , is also very

small compared with . For example, the curves against thechannel SNR for Gaussian and 16-QAM signaling are shownin Fig. 10. We see that , implying that is dominantby its mean in (A23). Consequently, Lemma 1 approximatelyholds even for non-SDC channels.

APPENDIX IIIPROOF OF THEOREM 1

From the definition in (7), the MSE of the DD estimator isgiven by

(A35)

where step follows from and, and by substituting together with

the definition of in (8).Clearly, the MSE in (A35) is a function of both and . For

an arbitrarily given , the minimum of (A35) over all possiblechoices of is achieved with given in (26). Similar to (20),the corresponding minimum MSE can be computed as

(A36)

Recall from (6), (28), and (A30) that

and


Thus, as

(A37)

where step follows from Lemma 1, and from (33c).Consider the Cauchy-Schwarz inequality:

(A38)where the equality holds for the given in (19), i.e.

(A39)

Substituting (A39) into (A37), we obtain

(A40)

Recall that, to arrive at the minimum MSE in (A40), we haveused given in (26) and given in (19). This choice of andyields DD-LMMSE, hence the proof of Theorem 1.

APPENDIX IVPROOF OF PROPOSITION 1

The derivative of with respect to is given by

Thus, is monotonically decreasing in . Also

and

APPENDIX VPROOF OF THEOREM 2

The MSE bounds for the D-LMMSE, DD-LMMSE, andLMMSE estimators are obtained immediately by letting

and then in (21), (35), and (36), respectively.Now consider the WDFT-I estimator with and given by(30). From the definition in (7), we obtain

(A41)

where step utilizes the fact that , andstep follows from that, for in (36b)

(A42)

Letting and in (A41), we obtain

APPENDIX VIEQUIVALENCE OF DD-LMMSE AND LMMSE

In this appendix, we show that DD-LMMSE is equivalent toLMMSE if . We rewrite LMMSE in (10) as


where step utilizes the definition of , and steputilizes (22) and (23). Since the entries of have the same

modulus, both and are scaled identity matrices. Then,with (28) and (29), it can be readily verified that

Together with (26) and (27), we have .

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers fortheir invaluable comments and suggestions that have consider-ably improved the presentation of this paper.

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Nian Geng (S’10) received the B.E. and M.E.degrees in communication engineering from WuhanUniversity in 2005 and 2007, respectively. Shereceived the Ph.D. degree from the Department ofElectronic Engineering, City University of HongKong, 2011.Her research interests include channel estimation,

signal processing, and iterative detection.

Xiaojun Yuan (S’04–M’09) received the Ph.D. de-gree in electrical engineering from the Departmentof Electronic Engineering, City University of HongKong, 2008.During 2009–2010, he was working as a Post-

doctoral Fellow with the Department of ElectricEngineering, University of Hawaii at Manoa. He isnow a Research Assistant Professor with the Instituteof Network Coding, the Chinese University of HongKong. His research interests are in the generalarea of wireless communications and information

theory, including coding and coded modulation, iterative detection, cooperativecommunications, and physical-layer network coding.

Li Ping (S’87–M’91–SM’06–F’10) received thePh.D. degree from Glasgow University, Scotland, in1990.He lectured at the Department of Electronic Engi-

neering, Melbourne University, Australia, from 1990to 1992, and worked at Telecom Australia ResearchLaboratories from 1993 to 1995. He has been with theDepartment of Electronic Engineering, City Univer-sity of Hong Kong, since January 1996, where he isnow a Chair Professor. His research interests includeiterative signal processing, mobile communications,

coding and modulation, information theory, and numerical methods.Dr. Ping is now serving as a member of the Board of Governors of the IEEE

Information Theory Society from 2010 to 2012.

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