Channel Estimation for Wireless OFDM Communication Systems · Channel Estimation for Wireless OFDM Communication Systems ... data rate wireless communications because of its advantages

Channel Estimation for Wireless OFDM Communication Systems

EROL ONEN1, AYDIN AKAN1 and LUIS F. CHAPARRO2

1 Department of Electrical and Electronics Engineering, Istanbul UniversityAvcilar, 34320, Istanbul, TURKEY

e-mail: eeerol,[email protected] Department of Electrical and Computer Engineering, University of Pittsburgh

Pittsburgh, PA 15261, USAe-mail: [email protected]

Abstract: - Orthogonal Frequency Division Multiplexing (OFDM) has become a very popular method for highdata rate wireless communications because of its advantages over single carrier modulation schemes on multi-path,frequency selective fading channels. However, inter-carrier interference due to Doppler frequency shifts, and multi-path fading severely degrades the performance of OFDM systems. Estimation of channel parameters is requiredat the receiver. In this paper, we present a channel modeling and estimation method based on time-frequencyrepresentation of the received signal. The Discrete Evolutionary Transform provides a time-frequency procedureto obtain a complete characterization of the multi-path, fading and frequency selective channel. Performance of theproposed method is tested on different levels of channel noise, Doppler frequency shifts, and jamming interferencepowers.

Key-Words: - Wireless communications, OFDM, Channel estimation, Evolutionary spectrum.

1 IntroductionOrthogonal Frequency Division Multiplexing (OFDM)is considered an effective technique for broadband wire-less communications because of its great immunity tofast fading channels and inter-symbol interference (ISI).It has been adopted in several wireless standards suchas digital audio broadcasting (DAB), digital video broad-casting (DVB-T), the wireless local area network (W-LAN) standard; IEEE 802.11a, and the metropolitanarea network (W-MAN) standard; IEEE 802.16a [1,2]. OFDM partitions the entire bandwidth into paral-lel subchannels by dividing the transmit data bitstreaminto parallel, low bit rate data streams to modulate thesubcarriers of those subchannels. As such OFDM hasa relatively longer symbol duration than single car-rier systems (due to the lower bit rate of subchannels)which makes it very immune to fast channel fading andimpulse noise. The independence among the subchan-nels simplifies the design of the equalizer. Because ofall these advantages, OFDM is becoming a standard

1 This work was supported by The Scientific and Technical Re-search Council of Turkey, TUBITAK and by The Research Fund ofThe University of Istanbul, Project numbers: UDP-414/25012005,UDP-491/24052005, and 323/03062005.

in digital audio / video broadcasting and wireless com-munications. However, inter-carrier interference (ICI)due to Doppler shifts, phase offset, local oscillator fre-quency shifts, and multi-path fading severely degradesthe performance of multi-carrier communication sys-tems [1, 3]. For fast-varying channels, especially inmobile systems, large fluctuations of the channel pa-rameters are expected between consecutive transmitsymbols. Estimation of the channel parameters is re-quired to employ coherent receivers. Most of the chan-nel estimation methods assume a linear time–invariantmodel for the channel, which is not valid for fast vary-ing environments [4, 5]. A complete time-varying char-acterization of the channel can be obtained by employ-ing time-frequency representation methods.

We present a time–varying channel modeling andestimation method based on the time–frequency repre-sentation of channel output. The Discrete Evolution-ary Transform (DET) provides a time-frequency rep-resentation of the received signal by means of whichthe spreading function of the multi-path, fading andfrequency-selective channel can be modeled and esti-mated.

Proceedings of the 5th WSEAS International Conference on Signal Processing, Istanbul, Turkey, May 27-29, 2006 (pp222-227)

2 OFDM System ModelIn an OFDM communication system, the available band-width Bd is divided into K subchannels. The inputdata is also divided into K-bit parallel bit streams. Thesebit streams are mapped into some complex constella-tion points: Xn,k, k = 0, 1, · · ·K−1 where n is thetime index and k is the frequency index. Blocks of dataare modulated onto a set of subcarriers of correspond-ing subchannels with bandwidth ∆f = Bd/K. Themodulation is efficiently implemented using a K-pointInverse Discrete Fourier Transform (IDFT). Then thedata are passed through a Parallel/Serial (P/S) converterto form a serial data stream xn,k. Before sending thexn,k’s to the channel, last LCP samples are insertedin front and called the Cyclic Prefix (CP). This is doneto mitigate the effects of intersymbol interference (ISI)caused by the channel time spread [1, 2]. The length ofthe CP is taken at least equal to the length of the chan-nel impulse response.As a result, the effects of the ISIare easily and completely eliminated. Furthermore, thereceiver can implement demodulation of the OFDM byusing fast signal processing algorithms such as FFT.

For a channel of bandwidth Bd and K subchan-nels, symbol duration is T = 1

∆f= 1

Bd/K = KBd

.

However, the actual block duration is Tf = K+LCPBd

.For a system with Bd = 800kHz., K = 512 andLCP = 64, Tf = 512+64

800kHz = 720µs. At the receiver,the CP part is eliminated. Demodulation is performedby a K-point DFT operation on xn,k to get Rn,k. Ifthe CP is long enough, the interference between twoOFDM blocks is eliminated and the subchannels canbe viewed as independent of each other, i.e., Rn,k =Hn,k Xn,k+Nn,k, where Hn,k are the samples of chan-nel frequency response at n∆f of the nth block, andNn,k is the Fourier transform of the additive Gaussianwhite (AGWN) channel noise with zero mean and σ2

variance. A simple equalizer is sufficient for each sub-channel at the receiver, i.e., Xn,k = Rn,k/Hn,k. Thenthe decision is made upon Xn,k. The channel estima-tion problem is to obtain the channel parameters Hn,k.

2.1 Channel ModelIn wireless communications, the multi-path, fading chan-nel with Doppler frequency shifts is generally modeledas a linear time-varying system with the following im-pulse response [6, 7]:

h(t, τ) =L−1∑

i=0

γi(t)δ(τ − τi) (1)

where γi(t) are independent Gaussian processes withzero mean, σ2

i variance, and normalized overall power,τi are delay profiles describing the channel dispersionwith τmax as the maximum delay and L is the totalnumber of paths. The variance σ2

i is a measure ofthe average signal power received at path i, character-ized by the relative attenuation of that path, αi. In thedicrete–time, the channel can be modeled by

h(m, `) =L−1∑

i=0

αi ejψim δ(`−Ni) (2)

where ψi represents the Doppler frequency, αi is therelative attenuation, and Ni is the time delay causedby path i. The Doppler frequency shift ψi, on the car-rier frequency ωc, is caused by an object with radialvelocity υ and can be approximated by

ψi∼= υ

cωc, (3)

where c is the speed of light in the transmission medium[8]. In wireless mobile communication systems, withhigh carrier frequencies, Doppler shifts become sig-nificant and have to be taken into consideration. Thechannel parameters cannot be easily estimated fromthe impulse response, however the estimation problemcan be solved in the time-frequency plane by meansof the so called spreading function which is related tothe generalized transfer function and the bi-frequencyfunction of the channel. The generalized transfer func-tion of this linear time–varying channel is obtained bytaking the discrete Fourier transform (DFT) with re-spect to `, i.e.,

H(m,ωk) =L−1∑

i=0

αi ejψim e−jωkNi (4)

where ωk = 2πK k, k = 0, 1, · · · ,K − 1. Now, the bi-

frequency function is found by computing the discreteFourier transform of H(m,ωk) with respect to timevariable, m:

B(Ωs, ωk) =L−1∑

i=0

αie−jωkNiδ(Ωs − ψi). (5)

Furthermore, the spreading function of the channel isobtained by calculating the DFT of h(m, `) with re-spect to m, or by taking the inverse DFT of B(Ωs, ωk)with respect to ωk;

S(Ωs, `) =L−1∑

i=0

αiδ(Ωs − ψi)δ(`−Ni) (6)


which displays peaks located at the time-frequency po-sitions determined by the delays and the correspondingDoppler frequencies, and with αi as their amplitudes[8]. If we extract this information from the receivedsignal, we should then be able to figure out the trans-mitted data symbol.

3 OFDM Channel EstimationAssume we are given bit stream bn converted into N -bit parallel blocks, and then mapped onto some trans-mit symbols Xn,k drawn from an arbitrary constella-tion points where n ∈ Z is the time index, Z is theset of integers, and k = 0, 1, · · · ,K − 1, denotes thefrequency or subcarrier index. We then insert some pi-lot symbols, pn,k ∈ −1, 1 at some pilot positions(n′, k′), known to the receiver: (n′, k′) ∈P = (n′, k′)|n′ ∈ Z, k′ = iS + (n′mod(S)), i ∈[0, P − 1] where P is the number of pilots, and theinteger S = K/P is the distance between adjacent pi-lots in an OFDM symbol [6].

The nth OFDM symbol sn(m) is obtained by tak-ing the inverse discrete Fourier transform (IDFT) andthen adding a cyclic prefix of length LCP (where LCP

is chosen such that L ≤ LCP + 1, and L is the time-support of the channel impulse response.)

sn(m) =1√K

K−1∑

k=0

Xn,kejωkm (7)

m = −LCP ,−LCP + 1, · · · , 0, · · · ,K − 1 whereagain ωk = 2π

K k, and each OFDM symbol has N =K + LCP length. The overall transmit symbol is thengiven by s(m) =

∑n sn(m− nN). The channel out-

put suffers from multi-path propagation, fading andDoppler frequency shifts introduced by the nature ofthe wireless channel:

yn(m) =L−1∑

`=0

h(m, `) sn(m− `)

=L−1∑

i=0

αi ejψim sn(m−Ni)

=1√K

K−1∑

k=0

Xn,k

L−1∑

i=0

αi ejψim ejωk(m−Ni)

The transmit signal is also corrupted by Additive WhiteGaussian Noise η(m) over the channel. The receivedsignal for the nth frame can then be written as rn(m) =

yn(m) + ηn(m). The receiver discards the Cyclic Pre-fix and demodulates the signal using a K-point DFTas

Rn,k =1√K

K−1∑

m=0

[yn(m) + ηn(m)] e−jωkm

=1K

K−1∑

s=0

Xn,s

L−1∑

i=0

αi e−jωsNi

×K−1∑

m=0

ejψim ej(ωs−ωk)m + Zn,k (8)

If the Doppler effects in all the channel paths are neg-ligible, ψi = 0, ∀i, then the channel is almost time–invariant within one OFDM symbol. In that case, thelast summation in the above equation gives K δ(s−k),and

Rn,k = Xn,k

L−1∑

i=0

αi e−jωkNi + Zn,k

= Xn,k Hn,k + Zn,k (9)

where the channel frequency response Hn,k is the dis-crete Fourier transform of h(nN, `), and Zn,k is theFourier transform of the noise, η(nN + m). By es-timating the channel frequency response coefficientsHn,k, data symbols, Xn,k, can be recovered accordingto equation (9). However, if there are large Dopplerfrequency shifts in the channel, then the time–invarianceassumption above is no longer valid. Here we considertime–varying channel modeling and estimation and ap-proach the problem from a time–frequency point ofview [7, 8]. In the following we briefly explain the Dis-crete Evolutionary Transform as a tool for the time–frequency representation of non–stationary signals.

3.1 The Discrete Evolutionary TransformA non-stationary signal, x(n), 0 ≤ n ≤ N−1, may berepresented in terms of a time-varying kernel X(n, ωk)or its corresponding bi-frequency kernel X(Ωs, ωk).The time–frequency discrete evolutionary representa-tion of x(n) is given by [9],

x(n) =K−1∑

k=0

X(n, ωk)ejωkn, (10)

where ωk = 2πk/K, K is the number of frequencysamples, and X(n, ωk) is the evolutionary kernel.

The discrete evolutionary transformation (DET) isobtained by expressing the kernel X(n, ωk) in terms


of the signal. This is done by using conventional sig-nal representations [9]. Thus, for the representationin (10) the DET that provides the evolutionary kernelX(n, ωk), 0 ≤ k ≤ K − 1, is given by

X(n, ωk) =N−1∑

`=0

x(`)wk(n, `)e−jωk`, (11)

where wk(n, `) is, in general, a time and frequency de-pendent window. The DET can be seen as a general-ization of the short-time Fourier transform, where thewindows are constant. The windows wk(n, `) can beobtained from either the Gabor representation that usesnon-orthogonal bases, or the Malvar wavelet represen-tation that uses orthogonal bases [9]. Details of howthe windows can be obtained for the Gabor and Mal-var representations are given in [9]. However, for therepresentation of multipath wireless channel outputs,we need to consider signal-dependent windows that areadapted to the Doppler frequencies of the channel.

3.2 Channel Estimation using DETWe will now consider the computation of the spread-ing function by means of the evolutionary transforma-tion of the received signal. The output of the channelyn(m) for the nth OFDM symbol can be written as,

yn(m) =1√K

L−1∑

i=0

K−1∑

k=0

αi ejψim ejωk(m−Ni)Xn,k

Now calculating the discrete evolutionary representa-tion of yn(m):

yn(m) =K−1∑

k=0

Yn(m,ωk)ejωkm

=1√K

K−1∑

k=0

Hn(m,ωk)Xn,kejωkm (12)

By comparing the above two representations of yn(m),we get the corresponding evolutionary kernel as

Yn(m,ωk) =1√K

L−1∑

i=0

αi ejψim e−jωkNiXn,k (13)

Finally, the channel frequency response for the nth

OFDM symbol can be obtained by

Hn(m,ωk) =√

K Yn(m, ωk)Xn,k

(14)

The evolutionary kernel Yn(m,ωk) can be calculateddirectly form yn(m) [9] and channel parameters α`, ψ`,and N` can be obtained form the spreading functionS(Ωs, `). However, (14) indicates that to estimate thechannel frequency response, we need the input datasymbols Xn,k at pilot positions. Two possible solu-tions can be implemented:

1. One complete OFDM symbol, after every C sym-bols can be sent as pilot so that Xn,k = pn,k ∈−1, 1, ∀k, n = rC, r ∈ Z . In this case, thespreading function can be calculated by these pi-lot values, and can be used until the next pilotOFDM symbol.

2. Other pilot symbol patterns [2, 5, 6] can be usedand data symbols Xn,k can be detected usingany of the pilot aided channel estimation and fil-tering methods [2, 5] by Xn,k = Rn,k/Hn,k.Then the detected data can be used for the es-timation of the spreading function via DET.

Using either of these approaches, the DFT of H(m,ωk)with respect to m, and the inverse DFT with respectto ωk, gives us the spreading function S(Ωs, `) fromwhich all the parameters of the channel will be ob-tained and the transmitted data symbol will be detected.

The time-frequency evolutionary kernel of the chan-nel output is obtained by replacing yn(m) in equation(11), or

Yn(m,ωk) =K−1∑

`=0

yn(`)wk(m, `)e−jωk`

=1√K

K−1∑

s=0

Xn,s

L−1∑

i=0

αie−jωsNi

×N−1∑

`=0

wk(m, `)ej(ψi+ωs−ωk)` (15)

We consider windows of the form wp(m, `) = ejψp(m−`),for 0 ≤ ψp ≤ π presented in [7] that depends on theDoppler frequency ψp. This window will give us thecorrect representation of Yn(m,ωk) only when ψp =ψi, in fact, using the window wi(m, `) = ejψi(m−`),above representation of Yn(m,ωk) becomes,

Yn(m,ωk) =√

KL−1∑

i=0

αiej(ψim−ωkNi) Xn,k

which is the expected result multiplied by K.


After estimating the spreading and the correspondingfrequency response Hn(m,ωk) of the channel, datasymbols Xn,k can be detected using a time-frequencyreceiver. Consider the DET of the received signal,r(m) for any OFDM frame n,

R(m,ωk) =1√K

H(m,ωk)Xn,k + Z(m,ωk)

where Z(m,ωk) is the DET of the additive noise η(m).Data symbols can be recovered by

Xk =√

KR(m0, ωk)H(m0, ωk)

= Xn,k +√

KZ(m0, ωk)H(m0, ωk)

for any value of m0 ∈ [0,K − 1]. In our previouswork, we picked and used any arbitrary time instantm0 to recover data symbols. Here we present resultsobtained by repeating this procedure for all values ofm0 and averaging to get a better estimate of Xk as:

Xn,k =1√K

K−1∑

i=0

R(mi, ωk)H(mi, ωk)

4 Simulation ResultsIn the experiments, the wireless channel is simulatedrandomly, i.e, the number of paths, 1 ≤ L ≤ 5, thedelays, 0 ≤ Ni ≤ LCP − 1 and the doppler frequencyshift 0 ≤ ψi ≤ ψmax, i = 0, 1, · · · , L−1 of each pathare picked randomly. Input data is BPSK coded andmodulated onto K = 128 sub-carriers, 12 % of whichis assigned to the pilot symbols. The OFDM sym-bol duration is chosen to be T = 200µs, and TCP =50µs. Frequency spacing between the sub-carriers isF = 5kHz. First, the Signal-to-Noise Ratio (SNR)of the channel noise is changed between 0 and 15dB,for fixed values of the maximum doppler ψmax oneach path, and the bit error rate (BER) is calculatedby four different approaches: 1) No Channel Estima-tion, 2) Pilot Symbol Assisted (PSA) Channel Equal-ization 3) Proposed Approach, and 4) Known Channelparameters. The spreading function, hence all the pa-rameters of the channel are estimated by the proposedmethod. Figures 1 and 2 show the BER versus SNRfor maximum Doppler frequency ψmax = 50Hz andψmax = 500Hz respectively. Notice that our pro-posed method outperforms the PSA channel estimationeven with low SNR values. Finally, the SNR is fixedto 15dB while the normalized Doppler frequency ischanged between 50Hz and 500Hz (see Fig. 3) then

between 500Hz and 3500Hz, (see Fig. 4) and BER iscalculated for each of the above methods.

5 ConclusionsIn this work, we present a complete modeling of themulti-path, fading OFDM channels with Doppler fre-quency shifts by means of discrete evolutionary trans-form of the channel output. This approach allows us toobtain a representation of the time-dependent channeltransfer function from the noisy channel output. At thesame time, using the estimated channel parameters, abetter detection of the input data can be achieved. Ex-amples show that, our method has a considerably bet-ter BER performance than PSA channel estimation.

References:

[1] Petropulu, A., Zhang, R., and Lin, R., “BlindOFDM Channel Estimation Through Simple Lin-ear Precoding,” IEEE Trans. on Wireless Com.,Vol. 3, No. 2, pp. 647–655, Mar. 2004.

[2] Stuber, G.L., Barry, J.R., McLoughlin, S.W.,Li, Y.G., Ingram, M.A., Pratt, T.G., “BroadbandMIMO-OFDM wireless communications,” IEEEProceedings, Vol. 92, No. 2, pp. 271–294, Feb.2004.

[3] Cai, X., Giannakis, G.B., “Error probability mini-mizing pilots for OFDM with M-PSK modulationover Rayleigh-fading channels,” IEEE Trans. onVehicular Tech., Vol. 53, No. 1, pp. 146–155, Jan.2004.

[4] Simeone, O., Bar-Ness, Y., and Spagnolini, U.,“Pilot-Based Channel Estimation for OFDM Sys-tems by Tracking the Delay-Subspace,” IEEETrans. on Wireless Com., Vol. 3 , No. 1, pp. 315–325, Jan. 2004.

[5] Kang, S.G., Ha, Y.M., Joo, E.K., “A comparativeinvestigation on channel estimation algortihms forOFDM in mobile communications,” IEEE Trans.on Broadcasting, Vol. 49, No. 2, pp. 142–149,June 2003.

[6] Schafhuber, D., Matz, G., Hlawatsch, F., “Adap-tive Wiener filters for time–varying channel es-timation in wireless OFDM systems,” IEEEICASSP’03, Vol. 4 , pp. IV - 688–691, Hong Kong,Apr. 6-10, 2003.


[7] Chaparro, L.F., and Alshehri, A.A., “ChannelModeling for Spread Spectrum via EvolutionaryTransform,” IEEE ICASSP’04, Vol. II, pp. 609–612, Montreal, Quebec, Canada, May 17–21,2004.

[8] Hahm, M.D., Mitrovski, Z.I., and Titlebaum, E.L.,“Deconvolution in the Presence of Doppler withApplication to Specular Multipath Parameter Es-timation,” IEEE Trans. on Signal Proc., Vol. 45,No. 9, pp. 2203–2219, Sep. 1997.

[9] Suleesathira, R., Akan, A. and Chaparro, L.F., “Discrete Evolutionary Transform for Time-Frequency Signal Analysis,” J. Franklin Institute,pp. 347-364, Vol. 337, No. 4, Jul. 2000.

[10] Garcia, M.J.F.-G.; Zazo, S.; Paez-Borrallo, J.M.;“Pilot patterns for channel estimation in OFDM,”Electronics Letters, Vol. 36, No. 12, pp. 1049–1050, June 2000.

[11] Sayeed, A.M., Aazhang, B., “Joint Multipath-Doppler Diversity in Mobile Wireless Communi-cations,” IEEE Trans. on Com., Vol. 47, No. 1, pp.123 - 132, Jan. 1999.

[12] Bello, P.A., “Characterization of RandomlyTime-Variant Linear Channels,” IEEE Trans. onCom. Systems, Vol. CS.11, pp. 360–393, 1963.

0 2 4 6 8 10 12 1410

−4

10−3

10−2

10−1

SNR

BE

R

BER vs SNR for ψmax

=50 Hz. Doppler frequency

no channel estPSA channel estTF channel estknown channel

Fig. 1. BER versus SNR for ψmax = 50Hz.

0 2 4 6 8 10 12 1410

−4

10−3

10−2

10−1

SNR

BE

R

BER vs SNR for ψmax

=500 Hz. Doppler frequency


Fig. 2. BER versus SNR for ψmax = 500Hz.

50 100 150 200 250 300 350 400 450 50010

−4

10−3

10−2

10−1

Maximum Doppler frequency, ψmax

[Hz]

BE

R

BER vs Maximum Doppler frequency for SNR = 15dB


Fig. 3. BER versus ψmax = [50− 500] Hz.

500 1000 1500 2000 2500 3000 350010

−4

10−3

10−2

10−1

Maximum Doppler frequency, ψmax

[Hz]

BE

R

BER vs Maximum Doppler frequency for SNR = 15dB


Fig. 4. BER versus ψmax = [500− 3500] Hz.


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Channel Estimation for Wireless OFDM Communication Systems · Channel Estimation for Wireless OFDM Communication Systems ... data rate wireless communications because of its advantages