Enhanced Channel-Estimation Technique for MIMO-OFDM Systems
Correspondences______________________________________________________________________
Enhanced Channel-Estimation Technique
for MIMO-OFDM Systems
Myeongchoel Shin,Hakju Lee,andChungyong Lee Abstract—In multi-input–multi-output orthogonal frequency-division multiplexing systems,conventional channel-estimation techniques using comb-type training symbols give relatively large mean squared errors (MSEs)at the edge subcarriers.To reduce the MSEs at these subcarriers, a cyclic comb-type training structure is proposed.In the proposed cyclic training structure,all types of training symbols are transmitted cyclically at each antenna.At the receiver,the channel frequency responses that are estimated using each training symbol are averaged with weights obtained from the corresponding https://www.sodocs.net/doc/c417132541.html,puter simulations showed that the proposed cyclic training structure gives more signal-to-noise ratio gain than the conventional training structure.
Index Terms—Channel estimation,cyclic training structure,multi-input–multi-output orthogonal frequency-division multiplexing systems (MIMO-OFDM),training symbols.
I.I NTRODUCTION
The orthogonal frequency-division multiplexing(OFDM)technique has recently attractedconsid erable interest as an effective method for high-rate communication systems[1].Since the OFDM system can efficiently combat intersymbol interference(ISI),it has been adopted as a standard for digital audio broadcasting,digital video broadcasting,and broad-band indoor wireless systems.On the other hand,information theory indicates that a multi-input–multi-output (MIMO)system is able to support enormous capacities[2],[3],pro-vided the multipath scattering of a wireless channel is exploited with appropriate space–time signal-processing techniques.However,in a frequency-selective broad-band channel,the MIMO system requires a complicatedchannel-equalization technique in ord er to eliminate the ISI.To alleviate this problem,the use of an OFDM technique for MIMO systems would be desirable.Recent studies have shown that combining the OFDM technique with a MIMO system can provide high-performance transmission[4],[5].
In a MIMO-OFDM system,the receiver shouldknow the frequency responses of the spectral andspatial channels between the transmit andreceive antennas to achieve coherent signal d etection.For reliable channel estimation,the channel estimators exploit training symbols from each transmit antenna,which are known by the receivers.These training symbols must be orthogonal with respect to each other[5],[6]. The commonly usedtraining symbol for a MIMO-OFDM system is the “comb-type training symbol”in which each transmitter uses a different subset of subcarriers for transmitting its training symbols[6].In this method,the training symbols of the transmit antennas are orthogonal with respect to each other in the frequency domain,thus permitting the channel-frequency responses to be effectively estimated.
Manuscript receivedFebruary17,2003;revisedJuly15,2003,andSeptember 9,2003.This work was supportedin part by Samsung Electronics und er the project on4G wireless communication systems.
The authors are with the Department of Electrical andElectronic Engi-neering,Yonsei University,andalso with Center for Information Technology of Yonsei(CITY)University,Seoul120-749,Korea(e-mail:somyoung@ mcsp.yonsei.ac.kr).
Digital Object Identifier10.1109/TVT.2003.822002
However,the conventional comb-type training symbol produces high mean squarederrors(MSEs)at the ed ge subcarriers[7],so that the overall bit error rate(BER)is increased.To reduce the MSEs at such subcarriers,we propose a cyclic comb-type training structure for the training symbols.In the proposedcyclic training structure,all types of training symbols are transmittedcyclically at each transmitter. At the receiver,the channel-frequency responses are estimatedusing each training symbol andare then averagedwith weights obtained from the corresponding MSEs.
II.MIMO-OFDM S YSTEM M ODEL
A simplifiedMIMO-OFDM system is shown in Fig.1.If the time-varying and frequency-selective fading channel is considered,the re-ceivedsignal vector y q(n)at the q th antenna can be expressedas y q(n)=
N
p=1
X P(n)F[1:L]h p;q t(n)+w q(n)(1)
[5]where N t indicates the number of transmit antennas,h p;q t(n)=
[h p;q t(0)h p;q t(1)111h p;q t(L01)]is the channel impulse-response vector between the p th and q th antennas,and w q(n)is a noise vector at the q th receive antenna with variance 2per component.X p(n)is diagonalized OFDM symbol matrix at the p th transmit antenna and can be expressedas
X p(n)=diag01110x p1111x p
N
0x p
N+1
111x p2
N
01110(2)
where N denotes the number of one-sided available subcarriers so that the total number of subcarriers for data transmission is2N ,the subscript indicates the available subcarrier index,and diag(1)denotes diagonalize operator.In addition,F describes the K2K Fourier trans-form matrix and F[1:L]indicates the first L columns of F.Assume that the virtual subcarriers at the edges of the spectrum are not used to avoid aliasing problems at the receiver andthe center subcarrier is not used to avoid intermodulation effects and difficulties in D/A and A/D con-version[8].
III.C ONVENTIONAL C HANNEL-E STIMATION T ECHNIQUE
In the MIMO-OFDM system,the comb-type training symbol is a well-known training structure for channel estimation.In the comb-type training structure,it is assumedthat each transmit antenna has the same number of training subcarriers,so that the total number of available subcarriers is
2N =N c N t(3)
where N c denotes the number of pilot subcarriers reserved for the training symbols of each transmit antenna andis assumedto be not smaller than the channel-dispersion length L.Then,the matrix of comb-type training symbols at the p th transmit antenna,X p comb has diagonal elements x p i,which satisfies
x p i
=c i;i=(m01)N t+p
0;otherwise(4)
0018-9545/04$20.00?2004IEEE
Fig.1.SimplifiedMIMO-OFDM
system.
Fig.2.Conventional training symbol structure.
where c i denotes an arbitrary complex number with magnitude p
N t and m indicates an arbitrary positive integer that is not greater than N c .In the comb-type training structure,X p comb is repeatedly transmitted N t times,as depicted in Fig.2,for a four-transmit antenna system,for example.
At the receiver,each channel-frequency response is estimatedind e-pendently using multiple training symbols with an identical structure andthe estimatedchannel-frequency responses are averagedto red uce noise variance.
In the comb-type structure,the training symbol transmittedat the p th antenna can be separatedfrom the receivedsignal vector y q (n ),be-cause the comb-type training symbols are orthogonal in the frequency domain.The received training symbol vector of the q th receive antenna at the n th OFDM block can then be modified from (1)as
y p;q (n )=X p comb (n )F [1:L ]h p;q t (n )+w q
(n )
(5)
where X p
comb
(n )does not have full rank because of virtual subcarriers andDC subcarrier.The d imension of X p
comb (n )can be reduced to an
N c 2N c matrix,_X p comb (n ),by removing the null diagonal elements.
In the same manner,y p;q (n )and F [1:L ]can be reduced according to
the dimension of _X p comb
(n ).Consequently,the reduced receive vector _y
p;q (n )can be expressedas _y p;q (n )=_X p comb (n )_F p [1:L ]
h p;q t (n )+w q (n )(6)
where _F p [1:L ]denotes the reduced Fourier transform matrix.From (6),the channel impulse response between the p th transmit antenna andthe q th receive antenna can be then estimatedby the least-square technique
^h p;q t (n )
=_X p comb (n )_F p [1:L
]
y
_y
p;q (n )(7)
[9]where (1)y indicates a pseudoinverse operation.However,the channel-frequency response h p:q f (n )is actually requiredfor MIMO-OFDM channel equalization andcan be estimatedby
^h p;q f
(n )=_F [1:L ]^h p;q t (n )=h p;q f (n )+_F [1:L
]_X p comb (n )_F p [1:L
]
y
w q (n )
=h p;q f (n )+e
p;q
(n )(8)
where _F
[1:L ]is the reduced Fourier transform matrix of 2N 2L di-mension with rows corresponding to data subcarriers and e p;q (n )de-notes the estimation error vector.The mean squarederror (MSE)vector can then be obtainedas
2p;q (n )=
9
E e
p;q
(n )e p;q (n )
H =9 2_F [1:L
]_F p [1:L ]
H _X p comb H _X p comb _F p [1:L
]01
2_F
[1:L ]
H = 2
9U p
D
p 01
U
p
= 2
N t
9U p U
p
(9)
where 9(1)denotes the dediagonalization operator that creates a
vector from a matrix by taking the diagonal term of the matrix,
U p =_F [1:L ](_F p [1:L ]
)y and D p =_X p comb H _X p comb =N t I .In the comb-type training symbol,the channel-frequency responses between the transmit andreceive antennas are sampledby sparse training subcarriers andthe sampledchannel-frequency responses are interpolatedto estimate the channel coefficient of all subcarriers.In this case,there are no training symbols to obtain channel information at the edge subcarriers,so that the channel-estimation errors are in-evitably increasedin such subcarriers.Fig.3shows a typical example of the MSE of each subcarrier when four transmit antennas are used in a MIMO-OFDM system with 64subcarriers.It clearly shows that the MSEs of the edge subcarriers are relatively large.
In the comb-type training structure,the replica of _X p comb is trans-mitted N t times andthe estimatedchannel-frequency responses are
averaged under the assumption that the channel is invariant during the training phase.The averagedchannel-frequency response can be ex-pressedas
h p;q f
=1N t
N
n =1
^h p;q f (n )=h p;q f
+1N t
N n =1
e p;q (n ):
(10)
In this case,since e p;q (n )is identical with respect to n ,the
channel-frequency response can be obtainedby averaging with equal
Fig.3.Example of the MSE of each data subcarrier using the conventional training symbol at SNR10dB.(a)First,(b)second,(c)third,and(d)fourth Tx antennas.
weight.Accordingly,the averaged MSE vector 2p;q can be expressed as
2p;q=
9
E1
N t
N
n=1
e p;q(n
)1
N t
N
n=1
e p;q(n
)
H
= 2
N2t
9U p U
p
=1
N t
2p;q(n):(11)
Note that 2p;q is dependent,not on the receive antenna,but on the transmit antenna,i.e.,the averagedMSE of the channel estimation be-tween the p th transmit antenna andany receive antenna is id entical ( 2p;1= 2p;2=111= 2p;N).Therefore,the overall normalized MSE(NMSE),which is defined by the sum of all MSEs normalized by the number of antennas andd ata subcarriers,can be expressedas
NMSE conv=
1
2N N t N r
N
p=1
N
q=1
trace diag 2p;q
= 2
2N N3t
N
p=1
trace U p U
p(12)
where trace(1)indicates the trace
operator.
Fig.4.Proposedcyclic training symbol structure.
IV.P ROPOSED C HANNEL-E STIMATION T ECHNIQUE
As discussed above,in the conventional training structure,the esti-
matedchannel-frequency responses have estimation-error variances of
equal patterns,so that the averagedMSEs are relatively large at the ed ge
subcarriers.Therefore,an approach in which the same training symbols
are sent repeatedly to each transmitter and the estimated channel-fre-
quency responses are averagedwith equal weight is not d esirable.
To alleviate this problem,we propose“the cyclic comb-type training
structure.”In the proposedstructure,a comb-type training symbol,
transmittedthrough the first antenna,is transmittedthrough the second
antenna andwill be transmittedthrough the thirdantenna the next time,
andso https://www.sodocs.net/doc/c417132541.html,ing this cyclical scheme,all types of training symbols are
transmittedat each antenna.Fig.4d escribes the proposedstructure in
the case of a four-transmit antenna system.
In the proposedstructure,the estimatedchannel-frequency re-
sponses^h p;q f(n)shouldnot be averagedusing(10),because 2p;q(n)
has different patterns according to n.Therefore,it wouldbe better
to weight^h p;q f(n)according to the estimation error in the averaging
process,so as to ensure that the averagedchannel-frequency response
h p;q
f
has a minimum error variance
h p;q
f
=
N
n=1
C p;q(n
)
01
N
n=1
C p;q(n)^h p;q f(n)
=h p;q f
+
N
n=1
C p;q(n
)
01
N
n=1
C p;q(n)e p;q(n)
=h p;q f+ p;q(13)
where C p;q(n)is a diagonal weight matrix to minimize the MSEs and
p;q denotes the channel-estimation error.C p;q(n)can be obtainedby
solving the following minimization problem[9]:
min
C
E p;q p;q:(14)
The weight matrix satisfying(14)is then given by
C p;q(n)=diag 2p;q(n)01:(15)
The corresponding minimum MSE of each subcarrier is expressed as
2p;q=
9
N
n=1
diag 2p;q(n)0
1
1
:(16)
Consequently,the normalizedMSE using the proposedcyclic training
structure can be expressedas
NMSE cyclic=
2
2N N t
2
trace
N
p=1
diag
9U p U
p0
1
1
:(17)
Fig.5gives an example of typical MSEs of channel estimation for
the conventional andproposedstructures at10-d B signal-to-noise ratio
Fig.5.Example of averagedMSEs of channel estimation using the conven-tional training symbol andthe proposedcyclic training symbol at SNR 10d
B.Fig.6.NMSER as a function of the number of transmit antennas.
(SNR).This clearly shows that the proposedstructure is capable of reducing MSEs at the edge subcarriers.
In addition,the performance of the proposed structure can be evalu-atedby the NMSE ratio (NMSER),d efinedas
NMSER
=
NMSE conv cyclic
=
N t 2
N p =1
trace U p U
p
trace
N p =1
diag 9U p U
p
1
:(18)
Fig.6shows the NMSER with respect to the number of transmit antennas.This indicates that the NMSE of the proposed structure is al-ways less than that of the conventional structure,when the number of transmit antennas is greater than one.It also indicates that the NMSER tends to be increased,with increasing the number of transmit antennas.From this result,we can infer that the proposedstructure becomes in-creasingly useful as the number of antennas increases.
V .S IMULATION R ESULTS
It was assumedthat each transmit antenna of the target MIMO-OFDM system has an individual OFDM block,which is based on the IEEE 802.11a system [8].Each OFDM system has 64subcarriers in-cluding 52data subcarriers,11virtual subcarriers at the edge band,
and
Fig.7.NormalizedMSE of the conventional training symbol andthe proposed
training symbol for 424antenna
system.
Fig.8.BER of the conventional training symbol andthe proposedtraining symbol for 424antenna system.
a dc subcarrier.Four of the data carriers are dedicated as pilot signals for the robust coherent detection against the frequency offset and phase noise.Although the IEEE 802.11a system supports variable data rates,only the 12Mb/s rate case,which uses a quaternary phase-shift keying (QPSK)symbol set and 1/2coding rate,was considered.As a MIMO configuration,a 424system is considered.For channel estimation,the training symbols were transmittedfour times in a preamble to 16 s.At the receive end,the V-BLAST algorithm [3]was used to detect spa-tially multiplexedsignals.
In order to evaluate the performance of the channel estimation in the MIMO-OFDM system,the channel-dispersion length was assumed to be 12samples.It was assumedthat the power d elay profile of the channel decays exponentially [7]
E (h p;q
t (n ))
2=e
;n =0;1;...;L 01:(19)
It was also assumedthat the channel state is static d uring the transmis-sion of the OFDM symbol block.
Computer simulations were performedto verify the effects of the proposedcyclic training structure.Fig.7shows the NMSE of channel estimation for a 424system.It can be seen that the proposedstructure has a 2.5-dB SNR gain over the conventional structure.
Fig.8compares BER performance andind icates that the cyclic struc-ture has about a 1.3-dB SNR gain at BER =1003over the conventional structure.
VI.C ONCLUSION
A cyclic comb-type training symbol structure for training sym-bols is proposedto enhance the channel-estimation performance of MIMO-OFDM https://www.sodocs.net/doc/c417132541.html,puter simulations indicate that the proposedstructure gives a higher SNR gain than the conventional structure.Therefore,the proposedmethodeffectively enhances the channel estimation performance of MIMO-OFDM systems.
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“V-BLAST:An architecture for realizing very high data rates over the rich-scattering wireless channel,”in Proc.ISSSE,Pisa,Italy,Sept.1998, pp.295–300.
[4]Y.Li,N.Seshadri,and S.Ariyavisitakul,“Channel estimation for
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[6]W.G.Jeon,K.H.Paik,andY.S.Cho,“An efficient channel estima-
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[7]M.Michele Morelli andU.Umberto Mengali,“A comparison of pilot-
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(PHY)specifications:High speedphysical layer in the5GHz band,in IEEE Standard,1999.
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Theory.EnglewoodCliffs,NJ:Prentice-Hall,1993.
Sigmoid-Basis Nonlinear Power-Control
Algorithm for Mobile Radio Systems
Zekeriya Uykan andHeikki N.K oivo
Abstract—Convergence speed and distributiveness are important prop-erties of a power-control algorithm in order to evaluate its potential for use in cellular radio systems.Most of the power-control algorithms in lit-erature are derived from numerical linear algebra or linear control theory and,consequently,are in linear form.This paper,on the other hand,pro-poses a(sigmoid-basis)nonlinear power-control algorithm that is fully dis-tributed and first order.The algorithm is obtained by discretization of the differential equation forms of the algorithm shown to be stable in the case of a feasible system.It is shown to be quadratically convergent in the neigh-borhood of its fixed point.We carried out computational experiments on a code-division multiple-access system.The results indicate that our algo-rithm significantly enhances the convergence speed of power control in an estimation error-free scenario and is more robust against estimation errors as compared with the linear distributed power-control algorithm of Fos-chini and Miljanic as a reference algorithm.The proposed algorithm was also verified with an advanced dynamic system simulator.
Index Terms—Code-division multiple-access(CDMA)radio systems,dis-tributed power-control algorithms,sigmoid function.
I.I NTRODUCTION
Effective transmitter power control is essential for high-capacity cel-lular radio systems.The power-control(PC)problem has drawn much attention since Zander’s works on centralized[19]and distributed[20] carrier-to-interference+noise ratio(CIR)balancing.CIR balancing was further investigatedby Grand hi et al.[2],[3].In[1],Foschini andMiljanic consid ereda more general andrealistic mod el,in which a positive receiver noise anda respective target SIR were taken into account.Foschini andMiljanic’s d istributedalgorithm(FMA)was shown to converge either synchronously[1]or asynchronously[10]to
a fixedpoint of a feasible system.Basedon the FMA,Grand hi et al.
[4]suggested distributed constrained power control(DCPC),in which
a transmission upper limit was considered.DCPC has become one of the most widely accepted algorithms by the academic community. Meanwhile,a framework on convergence of the generalized uplink power control was provided by Yates[18]and has been extended by Huang andYates[9].The results in[9]and[18]have presenteda framework for designing and analyzing new algorithms.Moreover, recently some second-order power-control algorithms requiring power levels of current andprevious iterations were proposedin[7]and[13] to enhance the convergence speedof the PC.
So far,most of the PC algorithms suggestedin literature,includ ing the ones above,are derived from numerical linear algebra or linear control theory and,consequently,are in linear form.The advantage of linear algorithms is that we have a direct link between the parameters of the algorithm andthe feasibility of the system,namely the spectral Manuscript receivedMarch6,2001;revisedDecember15,2001,September 10,2002,andNovember9,2003.This work was supportedin part by GETA, Finland.Z.Uykan was supported by SONERA(Finland Telecom)Foundation andELISA Communications Found ation.Parts of this paper were presentedat IEEE Vehicular Technology Conference,September2000,Boston,MA.
Z.Uykan is with the Radio Communications Laboratory,Nokia Research Center,Helsinki00180,Finland(e-mail:zekeriya.uykan@https://www.sodocs.net/doc/c417132541.html,).
H.N.Koivo is with the Control Engineering Laboratory,Helsinki University of Technology,Helsinki02015,Finland(e-mail:heikki.koivo@hut.fi). Digital Object Identifier10.1109/TVT.2003.822327
0018-9545/04$20.00?2004IEEE