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Pilot-aided angle-domain channel estimation techniques forMIMO-OFDM systemsCitation for published version (APA):Huang, L., Ho, C. K., Bergmans, J. W. M., & Willems, F. M. J. (2008). Pilot-aided angle-domain channelestimation techniques for MIMO-OFDM systems. IEEE Transactions on Vehicular Technology, 57(2), 906-920.https://doi.org/10.1109/TVT.2007.905621

DOI:10.1109/TVT.2007.905621

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https://doi.org/10.1109/TVT.2007.905621

https://doi.org/10.1109/TVT.2007.905621

https://research.tue.nl/en/publications/pilotaided-angledomain-channel-estimation-techniques-for-mimoofdm-systems(b2e7f5c3-5e67-4426-97a3-d0497aec49d9).html

906 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 2, MARCH 2008

Pilot-Aided Angle-Domain Channel EstimationTechniques for MIMO-OFDM Systems

Li Huang, Member, IEEE, Chin Keong Ho, Member, IEEE,J. W. M. Bergmans, Senior Member, IEEE, and Frans M. J. Willems, Fellow, IEEE

Abstract—Early multiple-input multiple-output with orthog-onal frequency division multiplexing (MIMO-OFDM) channelestimation techniques treat channels as spatially uncorrelated.However, in many situations, MIMO-OFDM channels tend to bespatially correlated, for example, due to limited scattering. Forsuch channels, estimation performance can be improved throughexploitation of prior knowledge of the channel spatial correla-tion, for example, by means of the linear multiple mean squareerror (MMSE) technique. This knowledge is, however, not alwaysavailable. As an alternative, we investigate techniques in the angledomain, where the MIMO-OFDM channel model lends itself toa physical interpretation. Our theoretical analysis and simulationresults indicate that the proposed angle-domain approximatedMMSE (AMMSE) channel estimation technique performs well interms of the mean square error (mse) for various channel mod-els representing different indoor environments. When a suitablethreshold is chosen, we can use the angle-domain most-significant-taps selection technique instead of the angle-domain AMMSEtechnique to simplify the channel estimation procedure with littleperformance loss.

Index Terms—Channel estimation, multiple-input multiple-output (MIMO), orthogonal frequency division multiplexing(OFDM), spatial correlation.

I. INTRODUCTION

R ECENT research trends have shown that combining themultiple-input multiple-output (MIMO) approach with

orthogonal frequency division multiplexing (OFDM) can helpto achieve spatial diversity and/or space-division multiplexinggain [1]–[4]. As coherent demodulation, which requires and uti-lizes the knowledge of channel coefficients, can achieve a 3-dBperformance gain compared with differential demodulation[5], it is quite commonly adopted in MIMO-OFDM systems.Therefore, accurate and robust channel estimation1 that permits

Manuscript received September 20, 2006; revised May 9, 2007 andJune 17, 2007. The review of this paper was coordinated by Prof. X.-G. Xia.

L. Huang is with the Interuniversity Microelectronics Centre Nederland(IMEC-NL), Holst Centre, 5605 KN Eindhoven, The Netherlands, the De-partment of Electrical and Computer Engineering, National University ofSingapore, Singapore 117576, and the Department of Electrical Engineering,Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands(e-mail: [email protected]).

C. K. Ho is with the Institute for Infocomm Research, Singapore 119613.J. W. M. Bergmans and F. M. J. Willems are with the Department of Elec-

trical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven,The Netherlands.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2007.905621

1In this paper, we refer to the channel as the physical fading channelexclusive of the addictive electronics noise. This notation is conventionally usedin the field of channel estimation techniques.

the realization of coherent demodulation is very important toensure reliable data recovery.

Generally speaking, pilot-aided channel estimation is basedon either the least squares (LS) [6] or the linear MMSE(LMMSE) technique [7], [8]. The essential difference betweenthese two types of techniques is that the channel coefficientsare treated as deterministic but unknown constants in the for-mer, and as random variables of a stochastic process in thelatter. Compared with LS-based techniques, LMMSE-basedtechniques yield better performance because they additionallyexploit and require prior knowledge of the channel correlation.However, the channel correlation is sometimes not a prioriknown, which makes LMMSE-based techniques infeasible. Toconsider wider applications, we will focus on techniques thatdo not require prior knowledge of the channel correlation.

A typical MIMO-OFDM channel is conceived of as theunique link between the transmitted and noiseless receivedsignals, and is referred to as the array-domain channel. Thisarray-domain channel is treated as spatially uncorrelated inmost previous pilot-aided channel estimation techniques forMIMO-OFDM systems (e.g., [4], [9], and [10]), possibly dueto the fact that early MIMO studies assume the array-domainchannel to be spatially uncorrelated (e.g., [11] and [12]). Wecall these techniques LS-based techniques in the array domain.However, in many realistic scenarios, the MIMO-OFDM chan-nel tends to be spatially correlated, for example, due to antennaspacing constraints and limited scattering [13]–[15]. For thesespatially correlated MIMO-OFDM systems, the LMMSE-basedtechniques in the array domain, which exploit and requireprior knowledge of the channel spatial correlation, yield betterperformance than LS-based techniques in the array domain[16]–[18]. However, when the channel spatial correlation is nota priori known, which is the assumption made in this paper,these techniques are not applicable. In such cases, to improvethe performance of conventional LS-based techniques in thearray domain, we investigate techniques in the angle domain,where the channel model lends itself to a simple physicalinterpretation.

In the angle domain, beamforming patterns with differentmain lobes are used to characterize the physical propagationenvironment [19], [20]. For a MIMO system with Nt transmitand Nr receive antennas, the beamforming patterns have Nt

transmit lobes and Nr receive lobes. A pair of transmit andreceive lobes forms one angle-domain bin, and, thus, the angledomain is partitioned into (Nt × Nr) angle-domain bins. Forexample, as shown in Fig. 1, the transmit lobe 0 together withreceive lobe 0 corresponds to the angle-domain bin (0, 0).

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HUANG et al.: PILOT-AIDED ANGLE-DOMAIN CHANNEL ESTIMATION TECHNIQUES FOR MIMO-OFDM SYSTEMS 907

Fig. 1. Schematic angle-domain representation of a MIMO channel with four transmit and four receive antennas.

Then, multiple unresolvable physical paths (e.g., paths 1 and 2)that occur in the angle-domain bin (0, 0) can be approxi-mately aggregated into one resolvable path, and the paths fromother directions (e.g., paths 3 and 4) will have little effecton this resolvable path because they originate or end at otherlobes. Consequently, different physical paths approximatelycontribute to different angle-domain bins, and the channelcoefficients in different angle-domain bins can be assumed tobe approximately spatially uncorrelated. Furthermore, whensome angle-domain bins contain few physical paths due to lim-ited scattering, the corresponding channel coefficients shouldapproach zero. Based on these two special properties for theangle-domain channel coefficients, we introduce three novelchannel estimation techniques for MIMO-OFDM systems inthis paper.

Note that in MIMO-OFDM systems, we classify the signalsand channels in two domain types. The first type is representedby either array or angle domain, and the latter one is representedby either time or frequency domain. The channel correlationin the two domain types is referred to as the channel spatialcorrelation and the channel frequency correlation, respectively.Additionally, we use the term channel correlation for boththe channel spatial and frequency correlation in this paper.Hereinafter, we will explicitly state which representation isused for each domain type. For example, the angle–time domainmeans that the angle and time representations are used forthe above two domain types, respectively. When we state therepresentation for only one domain type, we mean that bothrepresentations for the other domain type are applicable unlessindicated otherwise. For example, the angle domain refers toeither the angle–time or angle–frequency domain.

We focus on angle-domain channel estimation techniques inthis paper. More specifically, we consider the angle–frequencyand angle–time domain techniques. In the angle–frequencydomain, when some angle–frequency domain bins contain fewphysical paths due to limited scattering, the correspondingchannel coefficients should approach zero. This allows us tochoose a suitable threshold for ignoring the small-valued chan-nel taps and retaining only the most significant taps (MST) toreduce the effect of noise on the estimates, thereby improvingthe performance of the channel estimation technique. We callthis technique of retaining channel coefficients of sufficientlylarge power as the MST selection technique. In the angle–time

domain, we can also use the MST selection technique. Fur-thermore, the channel coefficients in the angle–time domainare approximately uncorrelated. Thus, we may use the channelpower instead of the channel correlation to approximately per-form the angle–time domain LMMSE technique when the SNRis known or reliably estimated. We call the resulting techniquethe approximated MMSE (AMMSE) technique. Note that werefer to the channel power as the channel average power in thispaper. When the channel power is not available, we can use thechannel instantaneous power (i.e., the instantaneous power ofestimated channel coefficients) to estimate the channel power.In such cases, to maintain the estimated channel power positiveand the estimation reliable, a threshold is required to ignorecoefficients with low instantaneous power. Furthermore, wewill not consider the AMMSE technique in the angle–frequencydomain because the channel correlation in the correspondingfrequency domain might be too high to be reasonably replacedby the channel power.

To our best knowledge, this paper is the first to systemat-ically study angle-domain channel estimation techniques forMIMO-OFDM systems. We note that one angle-domain chan-nel estimation technique has been investigated for single-inputmultiple-output OFDM systems [23]. In this technique, theangle–time domain bins that correspond to the identical angularlobes (but with different time indexes) are grouped into oneangle–time domain beam. Examining all the estimated channelcoefficients in one beam, only the beams that contain significantpeak values along the time axis are identified as the signalbeams and, thus, retained. All the remaining beams are con-sidered as noise (or interference) beams and are ignored. Thistechnique was shown to be effective when the channel power isconcentrated in a few beams. However, when the channel poweris distributed over all the beams (e.g., the channel model Ein [24]), this technique may hardly improve the performance.To overcome this problem and investigate the techniques forMIMO-OFDM systems, we do not group the angle–time do-main bins into beams. Instead, we independently filter the noisein each angle–time domain bin based on the fact that the mul-tipath components are approximately disjoint in the angle–timedomain. Then, as introduced, we use either the MST selectiontechnique or the AMMSE technique to estimate channels in theangle–time domain. Furthermore, we also investigate the MSTselection technique in the angle–frequency domain.



The three angle-domain channel estimation techniques pro-posed in this paper have two main advantages. First, theachieved performance gain over the conventional LS techniquedoes not require prior knowledge of the channel correlation andeven the channel power. Therefore, the techniques are applica-ble to various propagation environments. Second, they can useconventional array-domain estimators as the coarse estimatorsand perform postprocessing in the angle domain. Thus, they arewell suited for chip design because integrating these estimatorswill not significantly change the existing system architecture.It should be noted that materialization as a postprocessor isnot a requirement for the angle-domain techniques. They canbe also directly implemented in the angle domain by firsttransforming the transmitted and received signals from the arraydomain into the angle domain. However, this may modify theexisting system architecture and, thus, will not be a focus in thispaper. Instead, we concentrate on describing the angle-domaintechniques in the framework of a postprocessor.

The major contributions and results of this paper are asfollows.

1) We systematically develop the channel estimation tech-niques in the angle–time and angle–frequency domainsfor MIMO-OFDM systems. We find that the proposedtechniques perform particularly well in the angle–timedomain.

2) We develop a unified approach to analyze the perfor-mance of the MST and AMMSE channel estimation tech-niques in terms of the mse. Based on this approach, wealso develop a simple way to compare the performanceof different angle-domain techniques with the help of thefirst derivative test [25].

3) The performance of all the angle-domain techniques isdependent on the respective thresholds. Nevertheless, wefind that setting the threshold to be two times of noisevariance is sufficient for the angle–time domain MSTselection and AMMSE techniques to yield better perfor-mance than the conventional LS technique at all the SNRsfor various IEEE 802.11n channel models [24].

4) Of all the proposed angle-domain techniques, our theoret-ical analysis and simulation results demonstrate that theangle–time domain AMMSE technique results in the bestperformance and achieves up to 8-dB performance gainwhen the mse is 10−2 compared to the conventional LStechnique.

Throughout this paper, we make four assumptions. First, allthe transmit and receive antennas have the same polarizationand radiation patterns, and the geometry of the antenna arrayis the uniform linear array. Second, the spacing between theantennas at the transmitter or the receiver is much smallerthan the distances between the scatterers and antenna arraysso that the paths from a given scatterer to all the transmitor receive antennas are approximated to be parallel. This is atypical assumption for the analysis of MIMO-OFDM systems[20]–[22]. Third, as our main concern is the indoor propagationenvironment, we assume the channel to be time invariant over agiven training period. Fourth, the channel correlation and eventhe channel power are not available to the receiver.

This paper is organized as follows. Section II describesconventional MIMO-OFDM systems. In Section III, the angle-domain representation of MIMO-OFDM systems is presented.In Section IV, three angle-domain techniques are proposed toestimate the MIMO-OFDM channels, and their performance isanalyzed in Section V. We evaluate the performance of differenttechniques in Section VI by simulating typical IEEE 802.11nchannel models. Last, we conclude this paper in Section VII.

II. MIMO-OFDM SYSTEM MODEL

In this section, we first introduce the array-domain repre-sentation of MIMO-OFDM systems. Therefore, for the sake ofconvenience, we refer to the time and frequency domains as thearray–time and array–frequency domains, respectively, in thissection. Then, in the next section, we describe the angle-domainrepresentation of MIMO-OFDM systems.

In a typical MIMO-OFDM system with Nt transmit andNr receive antennas, the high rate symbols to be trans-mitted are first grouped into blocks of Nd data symbolsat the transmitter. These groups are called the frequency-domain OFDM symbols, and the nth group at the (it)thtransmitter is represented by the vector xit

(n) = [xit(0, n),

xit(1, n), . . . , xit

(Nd − 1, n)]T , where it and n denote theindexes of the transmitter and the OFDM symbol, respectively.Next, an inverse discrete Fourier transform (IDFT) block isapplied to each OFDM symbol at each transmitter. The IDFTblock at the transmitter and the discrete Fourier transform(DFT) block at the receiver serve to modulate and demodu-late the data on the orthogonal subcarriers, respectively. Atthe IDFT output (i.e., in the time domain), a cyclic prefix(CP) of length Ng , as a copy of the last part of the currentOFDM symbol, is inserted at the beginning of each symbolto avoid ISI, and its length Ng is assumed to be not shorterthan the channel length. The resulting nth time-domain OFDMsymbol at the (it)th transmitter is represented by sit

(n) =[sit

(0, n), sit(1, n), . . . , sit

(Nd + Ng − 1, n)]T , where

sit(m,n) =

1Nd

Nd−1∑l=0

xit(l, n)ej2πl(m−Ng)/Nd (1)

for m = 0, 1, . . . , Nd + Ng − 1.The samples sit

(m,n) are sent through a frequency-selective fading channel, which can be represented by anequivalent discrete-time linear finite duration channel impulseresponse (CIR) given by a sequence of channel matrices C(l)for l = 0, 1, . . . , Nh − 1, where Nh is the time span of theMIMO channel, and C(l) is an Nr × Nt matrix whose (ir, it)thelement cir,it

(l) represents the channel coefficient from the(it)th transmit antenna to the (ir)th receive antenna at delay l.2

Assuming that the transmitter and the receiver are perfectlysynchronized, the received OFDM symbols become free ofISI when the CP is removed from each symbol at all the

2For the sake of simple description, C(l) represents the channel matrix ateach integer time index. This representation is obvious for the sample-spacedchannels. For the nonsample-spaced channels, C(l) can be obtained via theinterpolation, as shown in [6].



Nr receivers. By performing an Nd-point DFT on the result-ing symbols at each receiver, we obtain the correspondingfrequency-domain symbols. The resulting frequency-domaindata sample at the kth subcarrier and (ir)th receiver in the nthOFDM symbol is given by

yir(k, n) =

Nt−1∑it=0

hir,it(k)xit

(k, n) + ϑir(k, n) (2)

where hir,it(k) is the channel transfer function (CTF) at the kth

subcarrier from the (it)th transmitter to the (ir)th receiver, andthe noise ϑir

(k, n) is assumed to be additive white Gaussianwith variance σ2

f . In a compact notation, the received block ofthe nth OFDM symbols at the kth subcarrier can be written as

y(k, n) = H(k)x(k, n) + ϑ(k, n) (3)

where y(k, n) = [y0(k, n) , y1(k, n) , . . . , yNr−1(k, n)]T ,x(k, n) = [x0(k, n), x1(k, n), . . . , xNt−1(k, n)]T , ϑ(k, n) =[ϑ0(k, n), ϑ1(k, n), . . . , ϑNr−1(k, n)]T , and

H(k) =

h0,0(k) h0,1(k) · · · h0,Nt−1(k)h1,0(k) h1,1(k) · · · h1,Nt−1(k)

......

. . ....

hNr−1,0(k) hNr−1,1(k) · · · hNr−1,Nt−1(k)

is the CTF matrix at the kth subcarrier. Furthermore, thereceived nth OFDM symbol in the time domain is given by

z(m,n) =Nh−1∑l=0

C(l)s(m − l, n) + u(m,n) (4)

where s(m, n) = [s0(m, n) , s1(m,n), . . . , sNt−1(m,n)]T ,z(m,n) = [z0(m, n) , z1(m, n) , . . . , zNr−1(m,n)]T , andu(m,n) are the transmitted, received, and noise vectors,respectively, at the mth sample of the nth OFDM symbol. Ifthe columns of C(l) and H(k) are stacked into the vectorsc(l) and h(k), respectively, the time- and frequency-domainvariables are related by

y(n) = (F ⊗ INr)z(n) (5)

x(n) = (F ⊗ INt)s(n) (6)

h = (F ⊗ INr×Nt)c (7)

where y(n)=[yT(0, n),yT(1, n), . . . ,yT(Nd−1, n)]T, z(n)=[zT (0, n), zT (1, n), . . . , zT (Nd − 1, n)]T , x(n) = [xT (0, n),xT (1, n), . . . ,xT (Nd − 1, n)]T , s(n) = [sT (0, n), sT (1, n),. . . , sT(Nd−1, n)]T , h=[hT(0),hT(1), . . . ,hT(Nd−1)]T ,c = [cT (0), cT (1), . . . , cT (Nh − 1),01×[(Nd −Nh )Nt Nr]]T ,⊗denotes the Kronecker product, and F, IN , and 0N1×N2 are theNd × Nd unitary Fourier matrix, N × N identity matrix, andN1 × N2 zero matrix, respectively.

III. ANGLE-DOMAIN MIMO-OFDM SYSTEMS

In this section, we represent MIMO-OFDM systems in theangle domain. This is an extension of the work for MIMO flat-fading systems, as shown in [20].

As the channel models for MIMO-OFDM systems are com-monly introduced in the array–time domain [24], [26], westart to represent the corresponding angle–time domain MIMO-OFDM systems from (6). Suppose that there is an arbitrarynumber of physical paths (four paths are illustrated in Fig. 1)between the transmit and receive antennas at time l; the ithpath has an attenuation of ai with an angle φt

i (Ωti := sin φt

i)and φr

i (Ωri := sin φr

i ) for the transmit and receive antennas,respectively. Then, C(l) is given by

C(l) =∑

i

abier (Ωr

i ) eHt

(Ωt

i

)(8)

where3

abi := ai

√NtNr exp

(j2πdi

λc

)(9)

er (Ωri ) :=

1√Nr

1exp [j (2π∆rΩr

i )]...

exp [j(Nr − 1) (2π∆rΩri )]

(10)

et

(Ωt

i

):=

1√Nt

1exp [j (2π∆tΩt

i)]...

exp [j(Nt − 1) (2π∆tΩti)]

(11)

where the superscript H denotes the Hermitian transpose;Nt,∆t, et(Ωt

i) and Nr,∆r, er(Ωri ) are the number of

antennas, the separation between adjacent antennas normalizedby λc, and the array response vectors,4 respectively, for thetransmit and receive antennas, respectively; λc is the carrierwavelength; and di is the distance between the last transmit andreceive antennas along path i.

As from [20], the orthonormal bases for the angle–timetransmitted and received signals are given by

ξt :=et(0), et

(1Lt

), . . . , et

(Nt − 1

Lt

)(12)

ξr :=er(0), er

(1Lr

), . . . , er

(Nr − 1

Lr

)(13)

respectively, where Lt = Nt∆t and Lr = Nr∆r are the nor-malized antenna array lengths of the transmitter and the re-ceiver, respectively. Let Ut and Ur be the unitary matriceswhose columns are the basis vectors in (12) and (13), respec-tively. Then, we can transform the mth samples of the nth

3For notational convenience, we ignore the time index l in these threevariables. We also assume that the fractional bandwidth is small such thatλc, er(Ωr

i ), and et(Ωti) are approximated to be unchanged over the whole

signal bandwidth.4In the literature, the array response vector has also been alternatively called

the array vector, the array steering vector, the array propagation vector, and thearray manifold vector.



transmitted and received OFDM symbol from the array–timedomain into the angle–time domain by

sa(m,n) :=UHt s(m,n) (14)

za(m,n) :=UHr z(m,n) (15)

respectively, where the superscript “a” denotes the angle-domain variables. From (4), we obtain the angle–time domainMIMO-OFDM system equation as

za(m,n) =Nh−1∑l=0

Ca(l)sa(m − l, n) + ua(m,n) (16)

where

Ca(l) :=

ca0,0(l) ca

0,1(l) · · · ca0,Nt−1(l)

ca1,0(l) ca

1,1(l) · · · ca1,Nt−1(l)

......

. . ....

caNr−1,0(l) ca

Nr−1,1(l) · · · caNr−1,Nt−1(l)

=UHr C(l)Ut (17)

is the angle–time domain channel matrix, and

ua(m,n) = UHr u(m,n) (18)

is the angle–time domain noise vector that still satisfies an in-dependent identically distributed multivariate complex normaldistribution. As explained in [20], due to the finite number ofantennas, multiple unresolvable physical paths can be appropri-ately aggregated into one resolvable path with gain ca

kr,kt(l).

This gain uniquely corresponds to the channel coefficient forthe (kr, kt) angle-domain bin at delay l. Hence, different phys-ical paths (approximately) contribute to different elements ofCa(l). This means that the angle–time domain channel matrixCa(l) lends itself to a physical interpretation. For example, inFig. 1, paths 1 and 2 contribute to the element ca

0,0(l). Paths 3and 4 contribute to the elements ca

3,3(l) and ca1,2(l), respectively.

All the other elements of Ca(l) approach zero because no pathscontribute to these elements.

By analogy to the angle–time domain case, from (3), we ob-tain the angle–frequency domain MIMO-OFDM system equa-tion at the kth subcarrier as

ya(k, n) = Ha(k)xa(k, n) + ϑa(k, n) (19)

where

xa(k, n) :=UHt x(k, n) (20)

ya(k, n) :=UHr y(k, n) (21)

ϑa(k, n) :=UHr ϑ(k, n) (22)

Ha(k) :=UHr H(k)Ut. (23)

IV. ANGLE-DOMAIN CHANNEL ESTIMATION

As the physical path within one angle–time domain beamhas most of its energy within this beam, the elements within

one angle-domain channel matrix Ca(l) or Ha(k) maintainlow spatial correlation. Furthermore, various elements in theangle-domain channel matrices tend to approach zero due tolimited scattering. Therefore, we may independently performnoise filtering in each angle–time or angle–frequency domainbin to improve the estimation performance. The angle-domainchannel can be directly estimated in the angle domain [27] byfirst transforming the transmitted and received signal from thearray–frequency domain into the angle–frequency domain asshown in (20) and (21). However, this may modify the existingsystem architecture and, thus, will not be a focus in this pa-per. Instead, without affecting the estimation performance, weconcentrate on describing the angle-domain techniques in theframework of a postprocessor. This follows three steps: 1) per-forming the coarse channel estimation in the array domain; 2) inthe postprocessor, transforming the estimated channel from thearray domain into the angle–time or angle–frequency domainwhere the noise filtering process in each angle-domain bin isindependently performed; and 3) transforming back the filteredestimated channel into the array domain.

In the literature, there exist two types of frequency-domainpilot arrangements in OFDM systems for the implementationof pilot-aided channel estimation techniques—the block-typeand comb-type pilot arrangements. The first type is realized byinserting pilots into all of the subcarriers within a periodicaltime interval, and the estimated channel coefficients that areobtained from one period are used for further signal processing(e.g., equalization, detection, etc.) until the next period of pilotsis received. This type of pilot arrangement is particularly suit-able for slowly fading channels. The second type is performedby inserting pilots into a certain number of subcarriers of eachOFDM symbol and using the estimated channel coefficients atthese pilot subcarriers to interpolate the channel coefficientsat other subcarriers. Compared with the first type, this typeis more suitable for rapidly time-varying channels in that itcan track time variation within each OFDM symbol. However,it may suffer an irreducible estimation error floor due to theinterpolation.

In typical MIMO-OFDM systems, only one transmit antennasends pilot symbols in a given time or frequency position [3],[10], [28]. This technique can lead to very simple MIMO-OFDM channel estimation because the received signals at agiven time or frequency position correspond to one uniquechannel coefficient from a given transmit antenna to a givenreceive antenna. In such cases, the channel estimation fora MIMO-OFDM system with Nt transmit and Nr receivedantennas becomes the channel estimation for the total Nt × Nr

single-input single-output (SISO) OFDM systems. Thus, thewell-developed SISO-OFDM channel estimation techniques(e.g., [6], [7], [29], and [31]) are directly applicable to theestimation of MIMO-OFDM channels. In this paper, we usethe conventional array–frequency domain LS technique [6] tocoarsely estimate the array–frequency domain hir,it

(k) be-cause the knowledge of channel correlation is assumed to benot available. For the sake of a simple description, as shown inFig. 2, we assume that pilots from different transmit antennasare time orthogonal to each other. Thus, only one transmitantenna is used to transmit pilots in each OFDM training



Fig. 2. Time-orthogonal pilot pattern for a MIMO-OFDM system with Nt

transmit antennas.

symbol period as introduced in [3]. The proposed techniquescan be easily extended to other pilot transmission schemes(e.g., [28]) but will not be covered in this paper.

A. Angle–Frequency Domain Technique

From (3) and (7) and using the assumption that the pilotsfrom different transmit antennas are time orthogonal to eachother, we obtain

Y = Xh + ϑ (24)

where Y=[YT(n), Y

T(n+Nt), . . . , Y

T(n + (Nc − 1)Nt)]T

with Y(n) = [yT (0, n),yT (0, n+1), . . . ,yT (0, n+Nt−1),yT(1, n), . . . ,yT(1, n+Nt−1), . . . ,yT(Nd−1, n+Nt − 1)]T ,X = [X (n), X (n + Nt), . . . , X (n + (Nc − 1)Nt)]T withX(n) = diag[xT (0, n),xT (0, n + 1), . . . ,xT (0, n + Nt − 1),xT (1, n), . . . ,xT (1, n+Nt−1), . . . ,xT (Nd−1, n + Nt−1)],and ϑ=[ϑ

T(n), ϑ

T(n+Nt), . . . , ϑ

T(n + (Nc−1)Nt)]T with

ϑT(n) = [ϑT (0, n), ϑT (0, n + 1), . . . , ϑT (0, n + Nt − 1),

ϑT (1, n), . . . , ϑT (1, n+Nt−1), . . . , ϑT (Nd−1, n+Nt−1)]T

are the received signal vector, the transmitted signal matrix,and the noise vector, respectively, and Nc is the number ofpilots used for each channel coefficient in the LS channelestimation. Note that X is a block diagonal matrix. Then, thearray–frequency domain LS estimator is given by

hLS =(XHX

)−1XHY. (25)

By rearranging the vector form hLS into its matrix form, inthis first step, we obtain the coarsely estimated array–frequencydomain channel matrix H(k) for k = 0, 1, . . . , Nd − 1.

In the second step, we transform the estimated array–frequency domain channel matrix from the array–frequencydomain H(k) into the angle–frequency domain Ha(k) by theuse of (23) for all the subcarriers under consideration. Letha

ir,it(k) denote the (ir, it)th element of Ha(k). Then, the

filtered angle–frequency domain channel coefficient is given bycomparing the power of ha

ir,it(k) with a threshold η as follows:

hair,it,MST(k) =

ha

irit(k), if |ha

ir,it(k)|2 ≥ η

0, otherwise.(26)

Last, the filtered angle–frequency domain channel matri-ces are transformed back to the array–frequency domain. We

call this technique the angle–frequency domain MST selectiontechnique.

To make a fair complexity comparison among all the tech-niques, we assume that LS-estimated array–frequency do-main channel coefficients are available beforehand. Then, from(23), obtaining the angle–frequency domain Ha(k) from thearray–frequency domain H(k) requires Nt + Nr complex mul-tiplication operations for each channel coefficient. Furthermore,transforming the estimated angle–frequency domain into thearray–frequency domain also requires Nt + Nr complex multi-plication operations for each channel coefficient. Therefore, thetotal required number of complex multiplication operations foreach channel coefficient is 2(Nt + Nr).

B. Angle–Time Domain Techniques

In many cases such as the nonline-of-sight (NLOS) sce-nario, the mean angle of departure (AoD) and angle of arrival(AoA) of the clusters of multipath components tend to beuniformly distributed over all angles [26], [33]. Thus, whenthe number of clusters is relatively large (e.g., the channelmodel E in [24]), the above angle–frequency domain channelestimation technique may hardly improve over the conventionalarray–frequency domain LS technique because nearly all theelements of angle–frequency domain channel matrices may notapproach zero. As the clusters of multipath components aredisjoint in the angle–time domain, we may perform the noisefiltering in this domain instead of the angle–frequency domain.

For the implementation of angle–time domain techniques,we first transform the estimated channel coefficients from thearray–frequency domain into the array–time domain by the useof DFT [31]. Then, we transform the channel matrices into theangle–time domain by using (17). In the angle–time domain, wecan select the MST in channel matrices to reduce the effect ofnoise on the estimates. Now, the estimated angle–time domainchannel coefficient becomes

cair,it,MST(l) =

cair,it

(l), if |cair,it

(l)|2 ≥ η0, otherwise

(27)

where cair,it

(l) is the coarsely estimated angle–time domainchannel coefficient.

Note that we presume that the channel spatial correlation isnot available to the receiver in this paper. Therefore, the con-ventional LMMSE technique that utilizes the channel spatialcorrelation is not applicable here. However, as discussed, thechannel coefficients in the angle domain at a given time areapproximately spatially uncorrelated. Therefore, we may usethe channel instantaneous power to approximate the channelcorrelation (here, the channel power is also assumed to be notavailable). As the approximated channel correlation matrix isa diagonal matrix, the LMMSE technique that jointly filtersall the channel coefficients becomes the independent spatialfiltering for each channel coefficient. Furthermore, the channelcoefficient is uncorrelated to the noise. Therefore, we mayestimate the channel instantaneous power for each coefficient as(|ca(l)|2 − σ2

f ). The noise variance σ2f is assumed to be known

here. In practice, it can be estimated during periods when no



transmitted signal is detected or at virtual carriers where nodata are transmitted. Then, the angle–time domain AMMSEtechnique is realized as

cair,it,AMMSE(l)

=

|ca

ir,it(l)|2−σ2

f

|cair,it

(l)|2 cair,it

(l), if |cair,it

(l)|2 ≥ η

0, otherwise.(28)

Here, the threshold η is usually chosen to be smaller than σ2f .

Otherwise, the approximated channel power (|cair,it

(l)|2 − σ2f )

becomes negative. In comparison with the MST selection tech-nique, the difference is the use of a dynamic multiplicationfactor instead of a constant one. The factor turns out to becrucial in improving the performance of channel estimation, asshown in the following two sections. However, the complexityis only increased by only an additional complex multiplicationfor each channel coefficient.

After the noise filtering in the angle–time domain either by(27) or (28), we transform back the estimated channel into thearray–time domain and then into the array–frequency domain.

Similar to the angle–frequency domain MST technique, weassume that LS-estimated array–frequency domain channel co-efficients are available beforehand to make a fair complexitycomparison. Then, from (17), we know that the transformationsbetween the array–time domain and the angle–time domainrequire a total of 2(Nt + Nr) complex multiplication opera-tions for each channel coefficient. In addition, the angle–timedomain techniques require the transformations between thearray–frequency domain and the array–time domain. This re-quires a total of 2Nd complex multiplication operations foreach channel coefficient. Typically, Nd is a power of 2. Then,using the FFT and the IFFT [35] for transformations betweenthe array–time and array–frequency domains, the total complexmultiplication operations required for each channel coefficientis reduced to log2 Nd complex multiplication operations foreach channel coefficient. From the above, the total requiredcomplex multiplication operations for each channel coeffi-cient in the angle–time domain MST selection technique is2(Nt + Nr) + log2 Nd. For the angle–time domain AMMSEtechnique, an additional complex multiplication is needed.Therefore, the total required complex multiplication operationsfor each channel coefficient in the angle–time domain AMMSEtechnique is 2(Nt + Nr) + 1 + log2 Nd. Compared with theangle–frequency domain MST technique, these angle–time do-main techniques always have higher complexity. Nevertheless,our results show that these angle–time domain techniques al-ways outperform the angle–frequency domain MST techniquefor all the channel models under consideration.

Note that the angle–time domain MST selection andAMMSE techniques are well suited for the sample-spacedchannels. The performance will be degraded for the nonsample-spaced channels due to the power leakage [6]. Nevertheless,our results show that the angle–time domain techniques stilloutperform the array–frequency domain LS technique for allthe channel models under consideration.

V. PERFORMANCE ANALYSIS OF ANGLE-DOMAIN

CHANNEL ESTIMATION TECHNIQUES

For channel estimation techniques, one of the most importantperformance measures is the mse, which measures the averagemean squared deviation of the estimator from the true value[30]. In this section, we present a unified approach to com-puting the mse of the angle-domain (i.e., either angle–time orangle–frequency domain) MST selection and AMMSE tech-niques. Note that the array–time and array–frequency domainsare related by a unitary transformation. Thus, of a given estima-tion technique, the mse represented in either the array–time orarray–frequency domain yields the same result and is given by

mse =1

NdNtNrE

[‖h − h‖2

](29)

where h is either h or c, and h is the estimated h.Here, we use the array–frequency domain LS technique to

perform the coarse channel estimation. Then, the resulting mseof the coarse estimation is given by

mseLS =1

NdNtNrE

[‖hLS − h‖2

]

=σ2

f

NdNtNrtrace

E

[(XHX)−1

]. (30)

In many cases such as in the IEEE 802.11a standard [32], thepowers of all pilots are unity. Therefore, we also assume thatE[(XHX)−1] is the identity matrix. Then, since X is diagonal,(30) becomes

mseLS = σ2f . (31)

Let ha represent the stacked angle-domain channel vector.Then, from (23), we obtain

ha = Bh (32)

where

B = INd⊗

[(INt

⊗ UHr

) (UT

t ⊗ INr

)](33)

is an (NdNtNr × NdNtNr) matrix, and the superscript Tdenotes the transpose. It is easily verified that the matrix B isunitary, i.e.,

BHB = INr×Nt. (34)

Then, the angle-domain MST selection or AMMSE techniqueis given by

ha

= MhaLS (35)

where M is a diagonal (NdNtNr × NdNtNr) matrix thatrepresents either the MST selection or AMMSE process, andha

LS is the stacked LS-estimated channel vector in the angledomain. Note that the ith diagonal element of M (denotedas mi) is dependent on the channel coefficient and the noise.



Denote ri as the instantaneous power of the ith element of haLS.

In the MST selection techniques, we have

mi =

1, if ri ≥ η0, otherwise.

(36)

In the AMMSE technique, from (28), we have

mi =

ri−σ2f

ri, if ri ≥ η

0, otherwise(37)

where the threshold η is chosen not to be smaller than σ2f .

From (29), the mse of the angle-domain techniques isgiven by

mse =1

NdNtNrE

[‖BHMBhLS − h‖2

](38)

where hLS is the LS-estimated h. Since M is diagonal and real,using (34), we obtain

mse =1

NdNtNrtrace

E[Mha(ha)HMH + ha(ha)H

− 2Mha(ha)H + Mva(va)HMH]

(39)

where va is the angle-domain noise vector. Denoting hi and vi

as the instantaneous power of the ith element of ha and va,respectively, we rewrite (39) as

mse =1

NdNtNr

NdNtNr∑i=1

E[(mi − 1)2hi + m2

i vi

]. (40)

More specifically

E[(mi−1)2hi

]=

∞∫0

hiPh(hi)

∞∫η

(mi−1)2Pr|h(ri|hi)dridhi

+

∞∫0

hiPh(hi)

η∫0

Pr|h(ri|hi)dridhi (41)

is the mse part due to the filtered instantaneous power of the ithchannel coefficient, and

E[m2

i vi

]=

∞∫0

viPv(vi)

∞∫η

m2i Pr|v(ri|vi)dridvi (42)

is the mse part due to the unfiltered noise components inthe ith estimated channel coefficient, where Ph(hi), Pv(vi),Pr|h(ri|hi), and Pr|v(ri|vi) are the probability density func-

tions of hi and vi, and the conditional probability density func-tions of ri conditioned on hi and vi, respectively. The channel

and the noise have complex normal distributions. Thus, Ph(hi)and Pv(vi) have exponential distributions, and Pr|h(ri|hi) andPr|v(ri|vi) have noncentral chi-square distributions [25].

Let σ2i denote the channel power of the ith element of ha.

When σ2i is not equal to zero, (41) and (42) become

E[(mi − 1)2hi

]=

∞∫0

hi

σ2i

e− hi

σ2i

∞∫η

(mi − 1)2e− ri+hi

σ2f

σ2f

× J0

(2√

rihi

σ2f

)dridhi +

∞∫0

hi

σ2i

e− hi

σ2i

×η∫

0

e− ri+hi

σ2f

σ2f

J0

(2√

rihi

σ2f

)dridhi

(43)

E[m2

i vi

]=

∞∫0

vi

σ2f

e− vi

σ2f

∞∫η

m2i

e− ri+vi

σ2i

σ2i

× J0

(2√

rivi

σ2i

)dridvi (44)

respectively, where J0(z) is the modified Bessel function of thefirst kind.

When σ2i is equal to zero, hi = 0, and ri = vi, then E[(mi −

1)2hi] = 0, and the mse of the ith estimated channel coefficientis simplified as

msei = E[m2

i vi

]=

∞∫η

m2i

vi

σ2f

e− vi

σ2f dvi. (45)

A. Performance of the MST Selection Techniques

In this subsection, we analyze the performance of the MSTselection techniques and show that the optimal strategy tominimize the mse for the MST selection techniques is toignore all the array–frequency domain LS-estimated channelcoefficients, whose corresponding channel powers are smallerthan the noise variance, and retain all the remaining estimatedchannel coefficients.1) When σ2

i Is Not Equal to Zero: From (43) and (44), themse of the ith estimated channel coefficient is given by

msei = E[(mi − 1)2hi

]+ E

[m2

i vi

]

= σ2f +

∞∫0

hi

σ2i

e− hi

σ2i

η∫0

e− ri+hi

σ2f

σ2f

× J0

(2√

rihi

σ2f

)dridhi −

∞∫0

vi

σ2f

e− vi

σ2f

×η∫

0

e− ri+vi

σ2i

σ2i

J0

(2√

rivi

σ2i

)dridvi. (46)



Differentiating msei with respect to η, we obtain

∂msei

∂η=

(σ2

i − σ2f

)e− η

σ2i+σ2

f η(σ2

i + σ2f

)2 . (47)

From (47), we find that although msei is difficult to analyticallycalculate, its gradient is surprisingly simple.

• When σ2i = σ2

f , the last two terms in (46) become identi-cal. This is reasonable since setting the estimated channelcoefficient as zero or retaining the original coarse estimateyields the same mse. Therefore, msei will always be σ2

f nomatter what threshold is chosen.

• When σ2i > σ2

f , from (47) and the first derivative test[25], we conclude that msei reaches its extremum whenη = 0 or η = ∞.5 Since (47) is always positive, msei

is monotonically increasing and has its minimum σ2f at

η = 0. This result indicates that the LS-estimated channelcoefficient should be retained when the correspondingchannel power is larger than the noise variance. It isreasonable since when we ignore the estimated channelcoefficient, the resulting performance loss due to the ig-norance of instantaneous power will be larger than theperformance gain due to the removal of the noise.

• When σ2i < σ2

f , (47) is always negative. Thus, msei

reaches its minimum σ2i at η = ∞. This result indicates

that the LS-estimated channel coefficient is required to beignored when the corresponding channel power is smallerthan the noise variance. It is reasonable as the performancegain surpasses the performance loss when the correspond-ing estimated coefficient is ignored.

2) When σ2i Is Equal to Zero: In this case, (45) becomes

msei =

∞∫η

vi

σ2f

e− vi

σ2f dvi = e

− η

σ2f

(η + σ2

f

). (48)

Differentiating msei with respect to η, we obtain

∂msei

∂η= −ηe

− η

σ2f

σ2f

≤ 0. (49)

Similar to the discussion above, we find that msei reachesits minimum at η = ∞. This result indicates that theLS-estimated channel coefficient is required to be ignored whenthe corresponding channel power is smaller than the noisevariance. It is reasonable since the corresponding estimatedchannel coefficient only contains the noise, and ignoring thisestimated coefficient will result in no performance loss.

From the discussion above, we conclude that the choiceof threshold η is dependent on σ2

i and σ2f , as shown in

Table I, because we need to balance the performance gain andloss when we ignore the estimated channel coefficients. Thus,the optimum threshold should be obtained for each channel

5For σ2i = σ2

f , ∂msei/∂η is always zero, hence, indicating that msei isconstant, as claimed earlier.

TABLE IMAXIMUM AND MINIMUM MSEi AND THE CORRESPONDING

THRESHOLDS FOR THE MST SELECTION TECHNIQUES

coefficient. Since the channel is independent of noise, theaverage power of the LS-estimated channel coefficient is thesum of the channel power and the noise variance (i.e., σ2

i + σ2f ).

For this reason, we may set the threshold η to be 2σ2f . When

the average power of the ith LS-estimated channel coefficientexceeds this threshold 2σ2

f , it means that σ2i > σ2

f . Therefore,we set mi to be 1 [see (36)] to retain the ith LS-estimatedchannel coefficient. This is optimum to minimize the msei, aswe discussed above. Similarly, this threshold is optimum to theminimization of msei for the case when σ2

i < σ2f . Therefore,

when the average power of the LS-estimated channel coefficientis available, we can still obtain the corresponding optimalthreshold.

As σ2i is assumed to be not available in this paper, the

average power of the corresponding LS-estimated channel co-efficient may also not be available. Then, we may use thechannel instantaneous power (i.e., the instantaneous power ofthe estimated channel coefficient) to approximate the averagepower of the LS-estimated channel coefficient. Due to themonotone property of msei with the increase in η, Table Iimplies that for the given threshold η = 2σ2

f , msei is alwayssmaller than σ2

f when σ2i < σ2

f , and larger than σ2f when σ2

i >

σ2f . Therefore, the overall performance of the MST selection

techniques is dependent on the portion6 of the number ofchannel coefficients, whose average powers are below σ2

f , to thetotal number of channel coefficients. When a majority of chan-nel coefficients have average powers that are below σ2

f (suchas in the angle–time domain), the MST selection techniquecan improve over the array–frequency domain LS technique.This is also verified in our simulation results presented inSection VI.

B. Performance of the AMMSE Technique

As mi is dependent on the channel instantaneous power,directly analyzing the performance of the AMMSE techniqueresults in high computational complexity. Therefore, in thissubsection, we only compare the AMMSE technique with theMST selection technique to provide some general understand-ing on the performance trends.1) When σ2

i Is Not Equal to Zero: As shown in theAppendix, the AMMSE technique performs better than theMST selection technique when σ2

i is not equal to zero.

6η = 2σ2f may not be the optimum choice but is reasonable when no prior

information of σ2i is available to the receiver. When additional information

(e.g., the portion) is available, we may moderately adjust the threshold toimprove the estimation performance, as shown in Section VI.



Fig. 3. Relation of the clustered model and the angle-domain representation.

2) When σ2i Is Equal to Zero: In this case, (45) becomes

msei =

∞∫η

m2i

vi

σ2f

e− vi

σ2f dvi <

∞∫η

vi

σ2f

e− vi

σ2f dvi (50)

because mi is always smaller than one. This implies that theAMMSE technique performs better than the MST selectiontechnique when σ2

i is equal to zero.Therefore, we conclude that the AMMSE technique always

performs better than the MST selection technique. Since theangle–time domain MST selection technique yields better per-formance than the array–frequency domain LS technique whenthe threshold is 2σ2

f , as discussed in the previous subsection, theangle–time domain AMMSE technique should achieve furtherperformance gain.

VI. SIMULATION RESULTS

Computer simulations are carried out for the IEEE 802.11nchannel models [24], [34]. As similar conclusions can be drawnfrom both the line-of-sight and NLOS scenarios based on oursimulations, we only present the results for the NLOS scenario.We evaluated different channel estimation techniques for thefive IEEE 802.11n channel models that represent various in-door environments such as residential homes and small offices.Model A corresponds to the MIMO flat-fading channel witha single cluster. This simple model serves as the basis toinvestigate the characteristics of cluster-based channel modelsin the angle domain. Here, AoDm, ASt and AoAm, ASrrefer to the mean angle of clusters and angular spread ofclusters, respectively, for the transmit and receive antennas,respectively, as illustrated in Fig. 3. All these four parametersare physically determined for a given propagation scenarioand will affect the relative average power for each angle–timedomain beam. In the simulation, we assign different values tothese parameters for the model A to represent various propa-gation environments. This model is usually used for stressingthe detection performance. It occurs only a small percentageof time in reality for the systems under consideration [24].Models B–E correspond to 15-, 30-, 50-, and 100-ns root-mean-square delay spread, respectively. All these four modelsrepresent nonsample-spaced channels. As we found that theperformance trends in different channel estimation techniques

are similar in all these four models, we only present the resultsfor models B and E because of their minimum and maximumdegrees of freedom, respectively. In the simulations, we assignthe values indicated in [24, App. C] to AoDm, ASt andAoAm,ASr to represent typical small environments. Wealso evaluated different channel estimation techniques for theideally assumed channel model that is spatially uncorrelatedand sample spaced [4], [9], [10]. This channel is set to havethe exponential power delay profile with the channel length Nh

being 12. Throughout the simulations, we assume that Nd = 64and Nt = Nr = 4, and that the normalized separation betweenadjacent antennas ∆t = ∆r = 0.5. Furthermore, we assumethat the channel power for each link between one transmitand one receive antenna is normalized to one throughout thesimulations.

A. Channel Model A

As discussed, we assign various values to AoDm, ASt andAoAm, ASr in channel model A to investigate the perfor-mance of the angle-domain channel estimation techniques inthe presence of a single cluster. As the angular spread is usuallynot smaller than 40 in indoor environments [24], we consider40 for both ASt and ASr as the worst case in performing angle-domain channel estimations. We also consider the angularspread to be 2, which is valid for outdoor environments, forboth ASt and ASr as the best case. Furthermore, the energyof multipath components from 45 leaks into more than oneangle–time domain beam, which is undesirable in the angle-domain channel estimations. Thus, we consider 45 to be theAoDm and the AoAm of the cluster for the worse-case consid-eration. We also consider 0, from which most of the energy ofmultipath components concentrates on one angle–time domainbeam, to be the AoDm and the AoAm. In the following figures,which represent the channel power for each angle–time domainbeam (see Figs. 4 and 6), the areas of square and circle areproportional to the angle- and array-domain normalized averagepower, respectively, with respect to the corresponding maxi-mum average power in the angle domain.

Fig. 4 corresponds to the best case where the angular spreadis 2. As expected, we see that the angle-domain channelpower is concentrated in the (1, 1)th angle–time domain beam,whereas the array-domain channel power is uniformly distrib-uted over all the array–time domain beams. These observations



Fig. 4. Normalized channel power for each angle-time domain beam ofmodel A with AoDm = 0, ASt = 2, AoAm = 0, and ASr = 2.

Fig. 5. Performance of different channel estimation techniques for model Awith AoDm = 0, ASt = 2, AoAm = 0, and ASr = 2.

indicate the effectiveness of the MST selection and AMMSEtechniques. From Fig. 5, it is clear that all the angle-domaintechniques improve over the array–frequency domain LS tech-nique at all the SNRs under consideration. We also find thatthe angle–time domain AMMSE technique improves over theangle–time domain MST selection technique by around 3.5 dBbecause the former is more like an LMMSE technique. Al-though the MST selection is a nonlinear process, it is interestingto find that the performance of the angle-domain techniquesis proportional to that of the array–frequency domain LStechnique in Fig. 5 because the ignored channel instantaneouspower is not significant.

We consider the worst case where the angular spread is 40

in Figs. 6 and 7. Compared to Fig. 4, the angle-domain channelpower tends to leak into other angle–time domain beams.Therefore, the channel powers in all angle–time domain beamsare relatively high. From Fig. 7, we find that all the angle–timedomain techniques still outperform the array–frequency domain

Fig. 6. Normalized channel power of model A for each angle-time domainbeam with AoDm = 45, ASt = 40, AoAm = 45, and ASr = 40.

Fig. 7. Performance of different channel estimation techniques for model Awith AoDm = 45, ASt = 40, AoAm = 45, and ASr = 40.

LS technique for all the SNRs. However, the angle–frequencydomain MST selection technique does not perform well be-cause the number of channel coefficients whose channel powersare above the noise variance is relatively large. We also observethat the angle–time domain AMMSE technique achieves thebest performance, as expected.

B. Typical Channel Models

We consider typical nonsample-spaced indoor channels inFigs. 8 and 9. In such cases, AoDm and AoAm of clustersof multipath components tend to be uniformly distributedover all the angles. Then, the technique proposed in [23]will be the same as the array–frequency LS technique be-cause all the angle–time domain beams are identified as signalbeams. For this reason, its simulation results are not presentedhere. Compared with Figs. 5 and 7, we find that unlike inmodel A, the achieved performance gain of the angle-domain



Fig. 8. Performance of different channel estimation techniques for model B.

Fig. 9. Performance of different channel estimation techniques for model E.

techniques over the array–frequency domain LS technique isnot significant at high SNRs in models B and E. This is becausethe ignored channel instantaneous powers are always larger inthe nonsample-spaced channels. Nevertheless, the angle–timedomain AMMSE technique still achieves the best performancefor models B and E. Therefore, we can choose the angle–timedomain AMMSE technique to perform channel estimation inthe IEEE-802.11-based MIMO-OFDM systems because of itssuperior performance as well as its robustness.

As the channel power tends to be concentrated over a shorttime interval in the angle–time domain,7 the number of channelcoefficients whose corresponding channel powers are belowthe noise variance is relatively large. For such channel coeffi-cients, the optimum threshold should be infinite, as discussed inSection V. Intuitively, increasing the threshold will improve theestimation performance for these channel coefficients. How-ever, this will result in an adverse effect on the estimation

7The time intervals for different angle-time domain beams are not likely tobe the same because the first paths falling in each beam may asynchronouslyarrive.

Fig. 10. Performance of the angle-domain channel estimation techniques withdifferent thresholds for model B. The number α shown in the bracket indicatesthat the threshold is set to ασ2

f . Otherwise, the threshold is set to 2σ2f .

performance for the channel coefficients whose channel pow-ers exceed σ2

f . As the number of channel coefficients whosechannel powers are below σ2

f is larger compared to that ofthe remaining channel coefficients, it should be interesting toinvestigate the overall performance of the angle–time domaintechniques when the threshold is increased. Fig. 10 showsthat the angle–time MST selection and AMMSE techniquesare improved in terms of performance when the thresholdis increased to 3σ2

f at nearly all the SNRs. Note that theselection of threshold is a tradeoff between the performanceloss due to the ignorance of channel instantaneous power andthe performance gain due to the removal of noise. Therefore,we find that further increasing the threshold to 6σ2

f degradesthe performance at high SNRs because the performance losswill dominate. Note that when the threshold is relatively high,we only reserve the channel coefficients whose corresponding[|ca

ir,it(l)|2 − σ2

f ]/[|cair,it

(l)|2] shown in (28) approaches one.This implies that the performance of the angle–time domainMST selection technique should gradually approach that of theangle–time domain AMMSE technique with the increase inthresholds. This implication is verified in Fig. 10. Therefore,we may use the angle–time domain MST selection techniquewhen the threshold is relatively high for typical MIMO-OFDMsystems.

Note that the noise variance σ2f is required to be priorly

known to decide the threshold in the above simulations. How-ever, the exact knowledge of σ2

f may not always be available.Therefore, for a robust estimator design, we should fix thethreshold for a target range of SNRs. As illustrated in Fig. 10,increasing the threshold will even improve the performanceof angle–time domain techniques particularly at low SNRs.Therefore, for a given target SNR range, we may use 2σ2

f

at the lowest SNR (because of the largest σ2f ) as the fixed

threshold. In Fig. 11, we divide the whole SNR range into fivedisjoint groups. Each corresponds to one target SNR range,within which the threshold is fixed. Fig. 10 indicates thatthe threshold should be smaller (or at most slightly larger)



Fig. 11. Performance of the angle-domain channel estimation techniques formodel B. “Fixed” in the brackets indicates that the threshold is fixed for a givenSNR range.

than the corresponding 6σ2f at low SNRs and smaller than the

corresponding 3σ2f at high SNRs. These results imply that we

could choose a larger factor of σ2f (and, thus, more elements for

each group) at low SNRs than at high SNRs. From Fig. 11, weobserve that the performance of angle–time domain estimationsis always better than that of the array–frequency domain LStechnique. Furthermore, we find that the performance of theangle–time domain MST selection technique is close to that ofthe angle–time domain AMMSE technique at the highest SNRin each target range (see the points when SNR = 10, 15, 19,22, and 25 dB in Fig. 11). These results are consistent with theobservations in Fig. 10, which shows that increasing the thresh-old makes the performance of the angle–time domain MSTselection technique approach that of the angle–time domainAMMSE technique. However, the angle–frequency domainMST selection technique may not perform well at the highestSNR for each target range because of the relatively large perfor-mance loss due to the channel instantaneous power ignorance.As the angle–time domain AMMSE technique performs bestat all the SNRs under consideration, we may conclude thatthe angle–time domain AMMSE technique is suitable for thetypical IEEE 802.11n MIMO-OFDM systems when the targetrange of SNRs is available.

Note that at higher SNRs, the threshold η approacheszero, and thus, almost all the channel coefficients will notbe filtered. Therefore, the angle–time domain MST selectionand angle–frequency domain MST selection techniques willperform more similarly to the array–frequency domain LStechnique as the SNR increases. In addition, the angle–timeAMMSE technique will also perform more similarly to theangle–time domain MST selection technique as the SNRincreases because the multiplication factor shown in (28) ap-proaches to one for each angle–time domain channel coeffi-cient. In summary, all the channel estimation techniques willconverge at a relatively high SNR, as implied in all the fig-ures shown in this subsection. Thus, the angle–time domain

Fig. 12. Performances of angle-domain channel estimation techniques for thespatially uncorrelated channel.

AMMSE technique is more preferable at lower SNRs than athigher SNRs because of the larger achievable performance gain.

C. Spatially Uncorrelated Channel

In Sections VI-A and B, we consider IEEE 802.11n chan-nel models that are spatially correlated and show that theangle-domain channel estimation techniques can achieve goodperformance. In this subsection, we consider the sample-spaced spatially uncorrelated channel. In such a channel, theangle–time domain and array–time domain channel coefficientshave the same statistics. Still, as the channel coefficients inthe angle–time domain are uncorrelated with each other, fromFig. 12, we find that the AMMSE technique in this domaincan achieve the best performance. Therefore, compared withthe array–frequency domain LS technique, we can always usethe angle-domain AMMSE technique for different channels toachieve higher performance at the price of higher complexity.

VII. CONCLUSION

In this paper, the angle-domain representation is used todescribe MIMO-OFDM systems. Based on this representa-tion, we propose the angle–frequency domain MST selectiontechnique, the angle–time domain MST selection technique,and the angle–time domain AMMSE technique. These threetechniques do not require prior knowledge of the channelcorrelation and are shown to be effective when the angularspread of clusters of multipath components is small. More im-portantly, the angle–time domain techniques can improve overthe array–frequency domain LS technique in all the cases underconsideration even when the angular spread is large, at the priceof higher complexity. Furthermore, our theoretical analysis andsimulation results indicate that the angle–time domain AMMSEtechnique achieves the best performance and is robust to thechoice of threshold and mismatch of operating an SNR. Thus,when only the target SNR range is available to the receiver,



the angle–time domain AMMSE technique is a potential can-didate for typical IEEE 802.11n MIMO-OFDM systems. Inaddition, we also find that with a suitable threshold and aknown operating SNR, the angle–time domain MST selectiontechnique results in little performance degradation compared tothe angle–time domain AMMSE technique. Therefore, in suchcases, the angle–time domain MST selection technique may beused for IEEE 802.11n MIMO-OFDM systems because of itslower computational complexity compared to the angle–timedomain AMMSE technique.

APPENDIX

PERFORMANCE COMPARISON OF AMMSE AND MSTSELECTION TECHNIQUES

In this appendix, we analytically compare the performanceof the AMMSE and MST selection techniques. From (43) and(44), the mse due to the ith estimated channel coefficient isgiven by

msei =

∞∫0

vi

σ2f

e− vi

σ2f

∞∫η

(ri − σ2f )2

r2i

e− ri+vi

σ2i

σ2i

× J0

(2√

rivi

σ2i

)dridvi +

∞∫0

hi

σ2i

e− hi

σ2i

×η∫

0

σ4f

r2i

e− ri+hi

σ2f

σ2f

J0

(2√

rihi

σ2f

)dridhi

+

∞∫0

hi

σ2i

e− hi

σ2i

η∫0

e− ri+hi

σ2f

σ2f

J0

(2√

rihi

σ2f

)dridhi.

(51)

Compared to (46), the difference of msei, which is contributedby the ith channel coefficient, between the AMMSE techniqueand the MST selection technique is given by

DIFi =

∞∫0

vi

σ2f

e− vi

σ2f

∞∫η

σ4f

r2i

e− ri+vi

σ2i

σ2i

× J0

(2√

rivi

σ2i

)dridvi +

∞∫0

hi

σ2i

e− hi

σ2i

×∞∫

η

σ4f

r2i

e− ri+hi

σ2f

σ2f

J0

(2√

rihi

σ2f

)dridhi

− 2

∞∫0

vi

σ2f

e− vi

σ2f

∞∫η

σ2f

ri

e− ri+vi

σ2i

σ2i

× J0

(2√

rivi

σ2i

)dridvi. (52)

Differentiating DIFi with respect to η, we obtain

∂DIFi

∂η=

σ4fe

− η

σ2i+σ2

f f(σ2i , σ2

f , η)(σ2

i + σ2f )3η2

(53)

where

f(σ2

i , σ2f , η

)= 2σ2

fη2 +(σ4

i + 2σ2i σ2

f − σ4f

)η − 2σ2

i σ2f

(σ2

i + σ2f

)

= 2σ2f

(η −

σ4f − 2σ2

fσ2i − σ4

i

4σ2f

)2

− 2σ2i σ2

f

(σ2

i + σ2f

)

−

(σ4

f − 2σ2fσ2

i − σ4i

)2

8σ2f

. (54)

As the threshold η is not smaller than σ2f , which is larger than

(σ4f − 2σ2

fσ2i − σ4

i )/(4σ2f ), (54) reaches its minimum 0 when

the threshold η is equal to σ2f . Therefore, when the threshold

η is larger than σ2f , (54) is larger than 0 (i.e., positive), and

so is (53). Therefore, DIFi is monotonically increasing withthe increase in η and reaches its maximum 0 when η = ∞.Consequently, DIFi is always negative, which implies that theAMMSE technique performs better than the MST selectiontechnique when σ2

i is not equal to zero.

ACKNOWLEDGMENT

The authors would like to thank Cellular Systems Divi-sion, Aalborg University (AAU-CSys), Aalborg, Denmark,and Computer Science Institute, The University of Namur(FUNDP-INFO), Namur, Belgium, for their parenthood on theMATLAB packages for the simulation of MIMO-OFDM chan-nel models. The authors would also like to thank Project IST-2000-30148 I-METRA, which permitted the use of MATLABpackages of channel models.

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Li Huang (S’04–M’08) received the B.Eng. de-gree in electronics and information engineering fromHuazhong University of Science and Technology,Wuhan, China, in 2002. After 2002, he was initiallyenrolled in an M.Eng. program at the National Uni-versity of Singapore (NUS), Singapore, under theData Storage Institute (DSI) Scholarship and waslater selected to be part of the joint Ph.D. programbetween NUS and Eindhoven University of Technol-ogy, Eindhoven, The Netherlands, in 2004 under theDesign Technology Institute (DSI) Scholarship.

Since 2006, he has been with the Interuniversity Microelectronics CentreNederland (IMEC-NL), Holst Centre, Eindhoven, working on physical layerdesign for ultralow power wireless systems. His main research interests includestatistical signal processing, wireless communications, magnetics, and opticalrecording.

Chin Keong Ho (M’08) received the B.Eng. (first-class honors) and M.Eng. degrees from the NationalUniversity of Singapore, Singapore, in 1999 and2001, respectively. He is currently working towardthe Ph.D. degree at the Eindhoven University ofTechnology, Eindhoven, The Netherlands.

He is also currently with the Institute for In-focomm Research, Singapore. His research in-terests include signal processing and wirelesscommunications for multicarrier and space-timecommunications.

J. W. M. Bergmans (M’95–SM’91) received the In-genieur and Ph.D. degrees from the Eindhoven Uni-versity of Technology, Eindhoven, The Netherlands,in 1981 and 1987, respectively.

From 1982 to 1999, he was with Philips ResearchLabs, Eindhoven, working on signal processing fordigital transmission and recording systems. In 1988and 1989, he was an Exchange Researcher withHitachi Central Research Labs, Tokyo, Japan. He iscurrently a Professor of signal processing systemswith the Eindhoven University of Technology.

Frans M. J. Willems (F’05) was born in Stein,The Netherlands, on June 26, 1954. He receivedthe M.Sc. degree in electrical engineering from theEindhoven University of Technology, Eindhoven,The Netherlands, and the Ph.D. degree from theCatholic University of Louvain, Louvain, Belgium,in 1979 and 1982, respectively.

From 1979 to 1982 he was a Research Assistantwith the Catholic University of Louvain. Since 1982,he has been a Staff Member with the Departmentof Electrical Engineering, Eindhoven University of

Technology. Since 1999, he has been connected with Philips Research Labo-ratories as an Advisor. His research contributions are in the areas of multiuserinformation theory and noiseless source coding.

Dr. Willems received the Marconi Young Scientist Award in 1982 and wasa corecipient of the IEEE Information Theory Society Paper Award in 1996.From 1988 to 1990, he served as an Associate Editor for Shannon Theory forthe IEEE TRANSACTIONS ON INFORMATION THEORY. From 1998 to 2000,he was a member of the Board of Governors of the IEEE Information TheorySociety. From 2001 to 2004, he served as an Associate Editor for informationtheory for the European Transactions on Telecommunications.


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