simplified linear mmse algorithm for ofdm sytems

7/27/2019 Simplified Linear MMSE algorithm for OFDM sytems

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A Simplified LMMSE Channel Estimation Algorithm

for OFDM Systems

Hu Feng , Li JianpingSchool of Information Engineering

Communication University of China

Beijing, China

Email: [email protected]

AbstractOrthogonal frequency division multiplexing (OFDM)

is a key technique of multi-carrier transmission to meet high data

rate requirement of wireless communication. Channel state

information (CSI) derived from channel estimation is necessary

for receivers to accurately recover the transmitted signals. It is

well known that the LMMSE (Linear Minimum Mean SquareError) algorithm is one of the best linear receivers for OFDM

channel estimation, which provides better mean squared error

(MSE) performance but requires more computations than others.

In this paper, a simplified LMMSE channel estimation algorithm

using Fourier Transform technique and an appropriate

training-sequences-aided is proposed without loss of MSE

performance. Simulation results show that the proposed LMMSE

algorithm can effectively lower computational complexity. When

adopting 256 pilot symbols in OFDM systems with Rayleigh

fading channels, the time spending in channel estimation reduces

about a quarter.

Keywords-channel estimation; LMMSE; OFDM; FFT;

computational complexity

I. INTRODUCTIONAs a hotspot and a promising technique for future high data

rate systems, OFDM has been adopted by many wirelesscommunications. In wireless systems, transmitted informationreaches receivers after passing through Rayleigh fadingchannels. For conventional coherent receivers, the effect of thechannel on the transmitted signal must be estimated to recoverthe transmitted information. As long as the receiver accuratelyestimates how the channel modifies the transmitted signal, itcan recover the transmitted information [1]. The capability toestimate the varying channels accurately and effectivelyremains a challenging topic.

LMMSE is widely used in the OFDM channel estimationsince it is optimum in minimizing the MSE of the channelestimates in the presence of Rayleigh fading. LMMSE usesadditional information like the operating SNR and the otherchannel statistics. LMMSE is a smoother/interpolater, andhence is very attractive for the channel estimation of OFDMsystems with pilot subcarriers. However, the computationalcomplexity of LMMSE is very high due to extra informationincorporated in the estimation technique [1] [2]. Thus, amodified LMMSE estimator is proposed in this paper in order

to lower computational complexity without loss of MSEperformance.

The remainder of this paper is organized as follows. InSection II, the pilot-based OFDM system model is described.

Section III discusses LS channel estimators, and ahigh-complexity LMMSE estimator is studied. Interpolationmethods and a way to mitigate the problem of complexity aredeliberated in Section IV. Section V presents the simulationresults, which indicate the BER and MSE improvements [3].Section VI concludes the paper.

II. CHANNEL AND SYSTEM MODELA. system model

Fig. 1 shows a typical block diagram of OFDM system withpilot signal assisted. The binary information data are groupedand mapped into multi-amplitude-multi-phase signals,depending on the modulation type. The serial data symbols are

then converted to parallel blocks. In order to eliminateinterference between parallel data streams, each low-rate datastream modulates orthogonal subcarriers by means of the IFFT.

A cyclic prefix is then added to eliminate the effect of theISI. For proper digital-to-analog (D/A) conversion and lowpassfiltering (LPF), the unused subcarriers (virtual subcarriers)which are contiguous with the occupied subcarriers should beincluded in the cyclic prefix [2]. For an OFDM system with Nsubcarriers, the equivalent low-pass signal in time domain can

be represented as:

Figure 1. A typical pilot-based OFDM system

The project sponsored by the Scientific Research Foundation for theReturned Overseas Chinese Scholars, State Education Ministry (2007[24])

978-1-4244-4639-1/09/$25.00 2009 IEEE


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12 /

0

( ) , 1P

j nk Nsk

k

EX n d e L n N

N

=

= (1)

where Es is the symbol energy per subcarrier, N is the FFTsize, P is the number of subcarriers, L is the length of the guardinterval inserted to prevent possible inter-symbol interferencein OFDM systems.

After the addition of CP, which is larger than the expectedmaximum excess delay of the channel, and D/A conversion,the transmitted signal is then sent to Rayleigh multi-path fadingchannels.

To the contrary, the proper analog-to-digital (A/D)conversion and lowpass filtering (LPF) are intercalated in thereceiver. After removing the guard interval from yg(n) thereceived samples y(n) are sent to a DFT block to demultiplexthe multicarrier signals.

B. channel modelThe aim of this section is to explain how to obtain the SNR

for OFDM systems using a conventional channel estimationmethod when n2 is estimated. In many radio channels, theremay be more than one path from transmitter to receiver.

We will consider the system shown in Fig. 1, where X(k)are the transmitted symbols, h(k) is the channel impulseresponse, w(k) is the white complex Gaussian channel noiseand y(k) are the received symbols. And then the received signaly(k) can be expressed as

1

0

( ) ( , ) ( ) ( )L

l

y k h k l X k l w k

=

= +(2)

The received signal is not corrupted by previous

multicarrier symbols, due to the presence of the guard interval.Thus, in this interval the received signal becomes

Figure 2. Comparisons of MSEs in Rayleigh fading channel

Figure 3. Comparisons of MSEs in different multi-path channels

12 / 2 /

0

( ) ( ) ( , ) e ( )eL

j lk N j k N

l

y n X n h n l w n

=

= +

( ) ( ) ( )X n H n W n= + (3)

which is the Fourier transform of the channel impulse responseat time n. Thus, the received signal at the output of the FFT

processor for the OFDM symbol can be written as

( ) ( ) ( ) ( )Y k X k H k W k = + ( ) ( ) ( ) ( ) ( )X k F k h k F k w k= + (4)

III. CHANNEL ESTIMATIONBefore going into the details of the estimation techniques, it

is necessary to give the LS estimation technique as it is neededby many estimation techniques as an initial estimation. Startingfrom system model of SISO-OFDM given in Eq. 17 as [4]

Y XH W = + (5)

The LS estimator for the cyclic impulse response h

minimizes ( ) ( )HY XFh Y XFh and generates

1 H H( ) F X YH HLS

H F F X XF

=(6)

Note that HLS

also corresponds to the estimator structure inthe appendix. Since (6) reduces to

H

LSH X Y= (7)

The LS estimate of HLS is susceptible to Gaussian noise andinter-carrier interference (ICI). Because the channel responsesof data subcarriers are obtained by interpolating the channelresponses of pilot subcarriers, the performance of OFDMsystems based on comb-type pilot arrangement is highlydependent on the rigorousness of estimate of pilot signals.


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Thus an estimate better than the LS estimate is required. In

(8), a low-rank approximation is applied to a linear minimummean-squared error (LMMSE) estimator that uses thefrequency correlation of the channel. The key idea to reducethe complexity is using some special algorithms to derive anoptimal low-rank estimator, where performance is essentially

preserved.

1LS HHLS

lmmse HH LS H R R H=(8)

IV. LMMSE CHANNEL ESTIMATIONA. LMMSE channel estimation method

In this subsection, the unification of LMMSE with theFourier Transform technique will be presented. The unificationof LMMSE with the other transform domain techniques is also

possible if the LMMSE estimation is performed in thecorresponding transform domain technique. As most of thetransform domain techniques are exploited for the pilotsubcarriers aided OFDM channel estimation, the LMMSEformulation for the pilot subcarriers with equal spacing will beused, and the case of non-equal pilot spacing is very similar[1]. Starting with the matrix equation,

H Fh= (9)

The auto-covariance matrix of H when all the subcarriersare used as the pilots can be expressed as

{ } { ( ) }H H

H

HH hh

E HH E Fh Fh

R FR F

=

=(10)

Based on the LMMSE criterion, the estimated channel canbe written as

2 1 1( ( ) )Hlmmse HH HH n LS H R R XX H

= +(11)

For positive definite M M matrix A with its mth diagonalelement given by am, the following inequality holds:

M-1

m=1 m

1tr(A )

a

(12)

Where equality holds if and only if A is diagonal [5]. Based on

this lemma, the minimum of MSE is achieved if and only ifXHX is diagonal [6]. Therefore, the optimal training scheme is

H

NX X I=

(13)

B. Low-complexity LMMSE channel estimatorNote that the LMMSE estimators have been derived under

the assumption of known channel correlation and noisevariance. In practice these quantities, Rhh and n

2, are either

taken fixed or estimated, possibly in an adaptive way, it will

increase the estimator complexity and reduce the performanceslightly [7] [8].

H

NFF NI= (14)

The LMMSE channel estimator (11) is of considerablecomplexity since a matrix inversion is involved. To simplifythis estimator, we exploit the optimal training scheme (13) toget an optimal low-complexity LMMSE channel estimator.Then, the estimated channel (11) can be written as

21 1( )nlmmse hh hh N LS H FR R I F H

N

= +

(15)

An important difference to the equivalent system modeldevised by (15) is denoting the diagonal entry of Rhh and n

2IN

as and .

Exploiting the noncorrelation property of the Rayleighfading channels of different links [9], we can obtain theautocorrelation matrix of the channel vector h:

( 1)

( 1)

2 2

0

2 2

0 1,

{ } ([ ... ])

([ ... 0 ])

h L

h L

H

hh h

h N L

R E hh diag

diag

= =

=(16)

Using (16) and (14), LMMSE channel estimator (15) canbe rewritten as

1

lmmse LS

H

LS

H F F H

N

F F HN

=

+

=+

(17)

Figure 4. Comparisons ofSERs in Rayleigh fading channels


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V. SIMULATION RESULTS AND DISCUSSIONThe performance of the proposed estimator is evaluated by

computer simulations using mean and mean-square error(MSE). In the simulations we consider a system operating witha bandwidth of 500 kHz, divided into 64 tones with a totalsymbol period of 138 ps, of which 10 ps is a cyclic prefix.Sampling is performed with a 500 kHz rate [10].

From MSE curves in Fig. 2, it is clearly observed that theimproved estimator has much smaller MSE than that of LSestimator and has almost the same MSE as the conventionalLMMSE estimator. When SNRs becomes more than 25 dB, the

proposed estimator is very approximate to LS methodestimators.

MSE curves based on different multi-path channel modelsare shown in Fig. 3. The more the number of multi-pathincreases, the better performance of the proposed estimator will

be achieved.

The symbol-error rate (SER) curves presented in Fig. 4 arebased on the mean-square errors of the channel estimations.Depending upon admissible complexity, up to 5dB gain can beobtained at particular SNRs by using a modified LMMSEestimator instead of the LS estimator. The gain in SNRdeclines larger compared with the LS estimator.

VI. CONCLUSIONSThe improved LMMSE channel estimation proposed in this

paper can be used to efficiently estimate the channels in OFDMsystems. The advantage of LMMSE estimate over Rayleighchannels can be shown in the simulation. However, itscomplexity is higher compared with that of the LS estimator.By exploiting the inherent orthogonal characteristic of theoptimal training scheme, a low complexity LMMSE estimate

based on Fourier Transform technique without loss of MSE

performance is introduced. Due to less computation, themodified LMMSE estimator costs 531 ms to deal with 256 bitsymbols, resulting in a deduction of 141 ms.

APPENDIX

PROOFOF(15)

Substituting (14) and (13) into (15) yields

2 1 1

2 1

2 1

21

21 1

( ( ) )

( )

( )

( )

( )

H

lmmse HH HH n LS

HH HH n N LS

H H

hh hh n N LS

H H Hnhh hh LS

nhh hh N LS

H R R X X H

R R I H

FR F FR F I H

FR F FR F FF HN

FR R I F HN

= +

= +

= +

= +

= +

This completes the proof.

REFERENCES

[1] Mehmet Kemal OZDEMIR and Huseyin Arslan, CHANNELESTIMATION FOR WIRELESS OFDM SYSTEMS, IEEE

TRANSACTIONS ON COMMUNICATIONS 2ND QUARTER,VOLUME 9, NO. 2, 2007.

[2] Yang-Seok Choi and Peter J. Voltz, On Channel Estimation andDetection for Multicarrier Signals in Fast and Selective Rayleigh FadingChannels, IEEE TRANSACTIONS ON COMMUNICATIONS, VOL.49, NO. 8, AUGUST 2001.

[3] Tuncer Can Aysal, and Kenneth E. Barner, Constrained DecentralizedEstimation Over NoisyChannels for Sensor Networks, IEEETRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 4,

APRIL 2008.[4] W. G. Jeon, K. H. Paik, and Y. S. Cho, An Efficient Channel

Estimation Technique for OFDM Systems with Transmitter Diversity,Proc. IEEE Intl. Symp. Personal, Indoor and Mobile Radio Commun.,vol. 2, London, UK, Sept. 2000, pp. 124650, Sept. 2000.

[5] S. M. Kay, Fundamentals of Statistical Signal Processing:EstimationTheory, Prentice-Hall, Englewood Cliffs, NJ, USA,1993.

[6] Hua Zhang, Ye (Geoffrey) Li, and Yi Yuan-Wu, PracticalConsiderations on Channel Estimation for Up-Link MC-CDMASystems, IEEE TRANSACTIONS ON WIRELESSCOMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008.

[7] Kai Yan, Sheng Ding and Yunzhou Qiu, A Low-Complexity LMMSEChannel EstimationMethod for OFDM-Based Cooperative DiversitySystems with Multiple Amplify-and-Forward Relays, EURASIPJournal on Wireless Communications and NetworkingVolume 2008.

[8] F. Gao, T. Cui, and A. Nallanathan, On channel estimation and optimaltraining design for amplify and forward relay networks, IEEETransactions onWireless Communications, vol.7, no. 5, part 2, pp.19071916, 2008.

[9] K. Kim, H. Kim, and H. Park, OFDM channel estimation for theamply-and-forward cooperative channel, in Proceedings of the 65thIEEE Vehicular Technology Conference (VTC 07), pp.16421646,Dublin, Ireland, April 2007.

[10] Jan-Jaap van de Beek, Ove Edfors and Magnus Sandell, In Proceedingsof Vehicular Technology Conference (VTC095), vol. 2, pp. 815-819,Chicago, USA, September 1995.

simplified linear mmse algorithm for ofdm sytems

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