Techniques for Suppression of IntercarrierInterference in OFDM Systems
Tiejun (Ronald) Wang, John G. Proakis, and James R. Zeidler∗Center for Wireless CommunicationsUniversity of California, San Diego
La Jolla, CA 92093-04047
Abstract— This paper considers an orthogonal frequency divi-sion multiplexing (OFDM) system over frequency selective time-varying fading channels. The time variations of the channelduring one OFDM frame destroy the orthogonality of differentsubcarriers and results in power leakage among the subcarriers,known as Intercarrier Interference (ICI), which results in adegradation of system performance.
In this paper, channel state information is used to minimize theperformance degradation caused by ICI. A simple and efficientpolynomial surface channel estimation technique is proposed toobtain the necessary channel state information. Based on theestimated channel information, we describe a minimal meansquare error (MMSE) based OFDM detection technique thatreduces the performance degradation caused by ICI distortion.Performance comparisons between conventional OFDM and theproposed MMSE-based OFDM receiver structures under thesame channel conditions are provided in this article. Simulationresults of the system performance further confirm the effective-ness of the new technique over the conventional OFDM receiverin suppressing ICI in OFDM systems.
I. INTRODUCTION
Orthogonal Frequency Division Multiplexing (OFDM) isa widely known modulation scheme in which a serial datastream is split into parallel streams that modulate a groupof orthogonal subcarriers [1]. OFDM is widely used andconsidered a promising technique for high speed data trans-mission in digital broadcasting, wireless LANs, HDTV, andnext generation mobile communications.
OFDM symbols are designed to have a relatively long timeduration, but a narrow bandwidth. Hence OFDM is robustto channel multipath dispersion and results in a decrease inthe complexity of equalizers for high delay spread channelsor high data rates. However, the increased symbol durationmakes an OFDM system more sensitive to the time varia-tions of mobile radio channels. In particular, the effect ofDoppler spreading destroys the orthogonality of the subcar-riers, resulting in intercarrier interference (ICI) due to powerleakage among OFDM subcarriers. In paper [2], the carrier tointerference (C/I) ratio has been introduced to demonstrate theeffect of the ICI under different maximum Doppler spreads anddifferent Doppler spectra. Performance degradation of OFDMsystems due to Dopper spreading is also analyzed in [3].
This work was supported by the Center for Wireless Communications underthe CoRe research grant core 00-10071.
In this paper, the channel state information is assumed to beunavailable at the receiver and has to be estimated in the firstplace. In [4], a time-frequency polynomial model for channelestimation in OFDM systems is proposed, which does not haveto estimate channel statistics such as the channel correlationmatrix and average SNR per bit. However, in practice suchknowledge is usually not available and the channel statisticsmay vary by time. But a large polynomial order is requiredin order to represent the 2-D frequency channel response withsufficient accuracy, since the frequency selectivity makes thechannel changes relatively fast over the frequency domain.Therefore, we need to design a channel estimation methodunder the frequency selective and time-varying fading channelwith low complexity. In this paper, we propose a new modifiedpolynomial channel estimator with better performance. Incontrast to the estimator which estimates the channel responsein the frequency domain, the modified estimation algorithmdirectly estimates the time domain response (has relativelyslower variation and requires lower order to polynomial func-tion representations), and hence achieves better estimationquality. Another point is that in [4], the fading channel ismodeled as constant within one OFDM frame, but changesfrom frame to frame, which is inaccurate for most cases,especially when ICI is to be shown and analyzed. The modifiedpolynomial channel estimator in our paper not only estimatesthe variations frame by frame, but also within one OFDMframe.
We also propose in this paper a minimum mean square error(MMSE) criterion-based OFDM receiver structure that takesinto account both additive noise and the ICI disturbance. Thenumerical simulation results of the system performance thatare provided under various channel conditions confirm thesuperior performance of the MMSE-OFDM receiver over theconventional OFDM receiver.
The rest of the paper is organized as follows: In Section IIwe describe the OFDM system model as well as the frequencyselective time varying fading channel model considered inthis paper. In Section III, the polynomial model is describedfor the channel and an estimation algorithm is provided andapplied to perform the OFDM channel estimation. In SectionIV, two different OFDM receiver structures, the conventionalOFDM receiver and the MMSE-based receiver are descibed. InSection V, the numerical system performance of these differentdetection techniques are presented and compared. Finally, ourconclusions are contained in Section VI.
An OFDM system with N subcarriers is considered in thispaper. A block of N · log2 M bits of data is first mappedinto a sequence {d[k]} of M -ary complex symbols of lengthN , each modulating an orthonormal exponential functionexp(j 2πkn
N ), k = 0, · · · , N − 1. Each data symbol d[k] isnormalized to have unit average signal power E[|d[k]|2] =1. As demonstrated in Fig.1, information bearing sequences{d[k]} is first serial to parallel converted and processed by anIFFT operation, given by
st[n] =1√N
N−1∑k=0
dt[k] · exp(j2πkn
N), 0 ≤ n, k ≤ N − 1 ,
(1)where the subscript t represents the tth OFDM frame. Acyclic prefix is inserted into the transmitted signal to preventpossible intersymbol interference (ISI) between successiveOFDM frames. After parallel to serial conversion, the signalsare transmitted through a frequency selective time varyingfading channel. At the receiver end, after removing the cyclicguard interval, the sampled received signal is characterized inthe following format, by applying the tapped-delay-line model[5]
rt[n] =L−1∑l=0
hlt[n] ·st
[|n− l|N]+n′
t[n], 0 ≤ n ≤ N −1 , (2)
where hlt[n] represents the channel response of the lth path
during the tth OFDM frame, L represents the total numberof paths of the frequency-selective fading channel, n′
t[n]represents the additive Gaussian noise with zero mean andvariance E[|n′
t[n]|2] = σ2 = N0/Es (Es/N0 represents thesystem signal to noise ratio), and | · |N represents the modularN operation.
The fading channel coefficients hlt[n] are modeled as zero
mean complex Gaussian random variables. Based on the WideSense Stationary Uncorrelated Scattering (WSSUS) assump-tion, the fading channel coefficients in different delay taps are
statistically independent. We also assume that they have anexponential power delay profile, which is given by
E[∣∣hl
t[n]∣∣2] = α · exp(−β · l), α =
1 − exp(−β)1 − exp(−L · β)
. (3)
The number of fading taps L is given by τmax/Ts, where τmax
is the maximum multipath delay, and Ts = 1/W , where Wis the channel (OFDM signal) bandwidth. In the time domain,the fading coefficients hl
t[n] are correlated and have a Dopplerpower spectrum density modeled as in Jakes [6], given by
D(f) =
1
πFd·√
1−(
fFd
)2|f | ≤ Fd
0 otherwise, (4)
where Fd is the Doppler bandwidth. Hence hlt[n] has an
autocorrelation function given by
E[hlt[n]·hl
t[m]�] = α·exp(−β ·l)·J0
(2π(n−m)FdTs
). (5)
where J0(·) is the first kind Bessel function of zero order.Written in a concise matrix form, we can represent (2) as,
rt = Ht · st + n′t , (6)
where rt and n′t are vectors of size N × 1, and the channel
matrix Ht is given by
Ht =[h[0]H ,h[1]H , · · · ,h[N − 1]H
]H
, (7)
where h[n]H is the right cyclic shift by n + 1 positions of azero padded vector given by
h[n] =[0, 0, · · · , 0︸ ︷︷ ︸
N−L
, hL−1t [n], hL−2
t [n], · · · , h0t [n]
]∣∣∣∣shift
n+1
.
(8)Hence the received signal rt is related to the data vector dt
as,rt = HtW · dt + n′
t, (9)
where W is the inverse Fourier transformation matrix givenby
W =[wn,m
]N×N
, wn,m = exp(j2πnm
N)/√
N . (10)
From another point of view, the received signal can beexpressed in terms of the equivalent frequency domain channelmodel as
rt[n] =1√N
N−1∑k=0
dt[k] · htk[n] exp
(j2πkn
N
)+ n′
t[n], (11)
where htk[n] is the frequency domain channel response for the
kth subcarrier during the tth OFDM frame. We incorporatethe result given by (2) into (11), then we have the frequencydomain channel response ht
(14)where Rt, dt and nt are column vectors of size N × 1,vector nt is i.i.d complex AWGN noise due to the orthonormaltransformation WH
t of the original noise n′t, and the matrix
Gt is related to the channel frequency response as follow:
Gt = WHHtW =[gt
i,j
]N×N
, (15)
where
gti,j =
1N
N−1∑n=0
hti[n] · exp
(j2π(j − i)n
N
). (16)
The decision symbol Rt[k] at the receiver end is unfortunatelydistorted not only by additive Gaussian noise, but also by ICIas well. We expand (14), and represent Rt[k] in the followingform:
Rt[k] =( 1
N
N−1∑n=0
htk[n]
)· dt[k] +
N−1∑i=0i�=k
( 1N
N−1∑n=0
htk[n] ·
× exp(j2π(i − k)n
N
)) · dt[i] + nt[k], (17)
The first term in (17) is the desired signal, the second termrepresents the ICI from the other subcarriers, and finally thethird term is the additive noise. Our objective is to reduce theeffect of the ICI term by using knowledge of the channel stateinformation.
III. CHANNEL ESTIMATION
In this paper, we assume that perfect channel state infor-mation is not available at the receiver. The channel frequencyresponse ht
k[n] and time domain impulse response hlt[n] are
estimated at the receiver end by inserting pilots at some of thesubcarriers and, thus, estimating the frequency response of thechannel at selected frequencies.
A. A 2-D Polynomial Surface Channel Estimator
We assume that the frequency-selective time-varying chan-nel response is a mathematically smooth function with respectto time and frequency. Hence, we may model the continuouschannel frequency response ht
k[n] as a 2-D polynomial surfacefunction within a certain time-frequency region as in [4]. Thatis,
htk[n] =
∑i+j≤p
ai,j · ki · (t(N + I) + n)j
, (18)
where {ai,j} are the polynomial coefficients up to order p, kand
(t(N + I)+n
)are the frequency and time indexes of the
channel frequency response, and I is the length of the guardinterval inserted into each OFDM frame to avoid intersymbolinterference between consecutive frames. The objective is toestimate the polynomial coefficients {ai,j}.
Estimation may be performed every T OFDM frames,which means that the assumed model region is a N×T (N+I)(frequency × Time) 2-D plane. Pilots are inserted every pf
subcarriers, and every pt OFDM frames. Thus, the overheadratio attributed to the pilots is 1
pf pt.
In order to estimate the coefficients {ai,j} of the channelfrequency response, let us first rewrite (18) in the matrix formas
htk[n] = aH · qt
k,n , (19)
where vector a given by
a =[ap,0, ap−1,1, · · · , ai,j · · · , a0,0
], (20)
is the polynomial channel coefficients to be estimated, andqt
k,n is a vector given by
qtk,n =
[kp, · · · , ki · (t(N + I) + n
)j, · · · , 1
]H. (21)
Since the additive noise in equation (14) is spherically sym-metric and zero mean Gaussian, the ML estimate of thecoefficients vector a is chosen such that∑
(t,k)∈P
∣∣∣Rt[k] −∑i∈Pf
gtk,i · dt[i]
∣∣∣2 , (22)
is minimized, where the noisy estimated channel response gtk,i
is provided in the following form by applying the polynomialchannel model in (18),
gtk,i =
1N
N−1∑n=0
ht
k[n] · exp(j2π(i − k)n
N
)
= aH
(1N
N−1∑n=0
qtk,n · exp
(j2π(i − k)n
N
)). (23)
and the set P in (22) contains all the pilot locations in thedetection region, while Pf contains the frequency locations ofthe pilots,
P ={
(m,n)∣∣∣m = 0, pf , 2pf , · · · , (N/pf − 1)pf ;
n = 0, pt, 2pt, · · · , (T/pt − 1)pt
},
Pf ={
m∣∣∣m = 0, pf , 2pf , · · · , (N/pf − 1)pf
}. (24)
Hence, we can restate the problem of channel estimation,which is equivalent to finding the ML solution of (22), assolving
By taking the complex derivative of (25) with respect to aH
and, after straightforward manipulations, we obtain
a = U−1 · b ,
U =∑
(t,k)∈Put,k · uH
t,k ,
b =∑
(t,k)∈PRH
t [k] · ut,k . (27)
Once the coefficients vector aH is obtained, the estimatedchannel response within the considered region can be obtainedby the following equation,
ht
k[n] = aH · qtk,n , (28)
where a and qtk,n are given by (27) and (21).
B. A Modified Polynomial Channel Estimator
In the estimator described above, a large polynomial orderp may be required in order to represent the 2-D frequencychannel response ht
k[n] with sufficient accuracy. If so, it isquite possible that some numerical problems may result fromthe large condition numbers of the matrix U in (27).
However, if we carefully study equation (12), we will ob-serve that the variation of ht
k[n] with respect to the frequencyindex k is actually caused by the Fourier transformation ofthe time domain channel impulse response hl
t[n]. Therefore,it is much easier to estimate hl
t[n] directly using the samepolynomial channel model. Thus, the time domain channelresponse hl
t[n] requires a small order polynomial function todescribe when the channel fading is slow.
Hence we consider the modified polynomial channel esti-mator based on the model
hlt[n] =
∑i≤p
a′li ·
(t(N + I) + n
)i, 0 ≤ l ≤ L− 1 , (29)
where hlt[n] represent the time-varying fading channel re-
sponse for the lth path, and {a′li} is the corresponding
polynomial coefficient up to order p. Written in a matrix form,we have the following:
hlt[n] = a′lH · qt
n , (30)
where a′l and qtn are column vectors of length (p + 1) given
by,
a′l =[a′
pl,a′
p−1l, · · · ,a′
0l]H
,
qtn =
[(t(N + I) + n
)p,(t(N + I) + n
)p−1, · · · , 1
]H. (31)
By stacking the coefficients a′l as a column vector,
a′ =[a′0H
,a′1H, · · · ,a′L−1H]H
, (32)
and applying the same criterion as in the 2-D channel esti-mator, the optimal polynomial channel coefficient vector a′ ischosen such that equation (22) is minimized, where now the
estimated channel response gtk,i is given by the following form
according to (14),
gtk,i =
L−1∑l=0
a′lH · exp(j−2πil
N
) ·×
( 1N
N−1∑n=0
exp(j2π(i − k)n
N
) · qtn
). (33)
Hence the problem of channel estimation is reduced to findingthe optimal a′ which minimizes the following:
mina′
∑(t,k)∈P
∣∣∣Rt[k] − a′H · vt,k
∣∣∣2 , (34)
where vt,k is given by
vt,k =[v0
t,kH
,v1t,k
H, · · · ,vL−1
t,k
H]H
,
vlt,k =
1N
∑i∈Pf
dt[i] · exp(j−2πil
N
) ·×
( N−1∑n=0
exp(j2π(i − k)n
N
) · qtn
). (35)
Finally, the modified polynomial channel estimator results inthe solution
a′ = V′−1 · b′ ,
V′ =∑
(t,k)∈Pvt,k · vH
t,k ,
b′ =∑
(t,k)∈PRH
t [k] · vt,k . (36)
IV. OFDM DATA DETECTION TECHNIQUES
In this section, we describe a MMSE-based data detectiontechnique and the conventional OFDM signal detection tech-nique. Both detection techniques rely on the channel estimatesthat are obtained as described in section III.
A. Conventional OFDM Signal Detection
According to equation (17), we observe that the simplestand most commonly used OFDM detection technique performsdetection based on the following decision symbol,
dt[k] =N
( ∑N−1n=0
ht
k[n])H∣∣∣ ∑N−1
n=0ht
k[n]∣∣∣2 · Rt[k] , (37)
where ht
k[n] is the estimated frequency-domain channel re-sponse obtained by the estimation algorithm described in theprevious section.
By substituting for Rt[k] from (17) into (37), we obtain
As expected, the decision symbol is distorted not only by theadditive Gaussian noise, but also the intercarrier interference(ICI) from other subcarriers. Hence the system performanceis affected by the time variation of the channel through theintroduction of the ICI. As shown in Section V, there existsan error floor resulting from the ICI, even when the signal toadditive noise ratio is sufficient high.
B. MMSE Based Detection
Motivated by the similarities between ICI distortion inOFDM systems and ISI distortion in single carrier systems,we consider an MMSE based detection technique to suppressICI. By taking both ICI and additive noise into account, theMMSE-based OFDM detection technique is superior to theconventional OFDM detection scheme described above.
Starting from equation (14), suppose the best linear receiverstructure of the data dt[k], which minimizes the mean squareerror, is given by
dt[k] = ctk
H · Rt , (39)
where the optimal coefficient vector ctk is a column vector of
size N × 1, and minimizes the following mean square error,
minct
k
E[∣∣dt[k] − dt[k]
∣∣2] . (40)
By applying the orthogonality principle, we have,
E[Rt · (dt[k] − dt[k])H
]= 0 . (41)
which is equivalent to solving the linear equation,
E[Rt · RH
t
]· ct
k = E[Rt · dt[k]H
]. (42)
It is straightforward to show that,
E[Rt · RH
t
]= GtGH
t + σ2IN , (43)
E[Rt · dt[k]H
]= gt
k , (44)
where gtk is the kth column of matrix Gt, and σ2 is the noise-
to-signal ratio N0/Es. Hence the optimal linear weightingvector ct
k is given by
ctk =
(GtGH
t + σ2IN
)−1 · gtk . (45)
If written in a concise matrix format, the MMSE-basedOFDM estimated symbol vector is given by,
dt = GHt
(GtGH
t + σ2IN
)−1 · Rt . (46)
When we replace Gt in equation (46) by Gt, we obtain
dt = GHt
(GtGH
t + σ2IN
)−1 · Rt . (47)
as the final decision symbol for the new MMSE-based detec-tion technique.
By substituting (45) into (40), we obtain the minimum meansquare error, which is given as
E[∣∣dt[k]− dt[k]
∣∣2] = 1−gtk
H(GtGH
t +σ2IN
)−1gt
k . (48)
Thus, the average mean square error for the tth OFDM frameis
1N
N−1∑k=0
E[∣∣dt[k] − dt[k]
∣∣2]
= 1 − 1N
(tr
[GH
t
(GtGH
t + σ2IN
)−1Gt
]). (49)
V. NUMERICAL RESULTS
In this section, we provide numerical and simulation resultsfor the channel estimation algorithm as well as the perfor-mance of the different OFDM detection techniques describedin the previous sections.
A. Channel Estimation
In general, a multivariate polynomial function of infiniteorder p → ∞ is required to describe a 2-D continuousfrequency selective and time-varying channel ht
k[n]. However,since the channels in which practical OFDM systems operateare slowly time varying and have limited multipath delays,they can be represented by a small order polynomial.
In this paper, we consider an uncoded 16-QAM OFDMsystem that has 16 subcarriers with carrier frequency of 5GHz. This system is transmitting over a 4-path frequencyselective fading channel having an exponential power delayprofile, with a bandwidth W = 1/T = 1MHz and a Dopplerspread Fd = 312.5Hz, corresponding to a mobile speedof 67.5 Km/hr. In the system, one entire OFDM channelestimation block is composed of T = 16 OFDM frames,where each frame has N = 16 orthonormal subcarriers. Pilotsare inserted every pf = 4 subcarriers, and every pt = 4OFDM frames, with pilots overhead ratio 1/(pt ·pf ) = 6.25%.The actually channel as well as the estimated channel areshown in Fig. 2 and Fig. 3 with system signal to noise ratioEs/N0 = 4dB, where the polynomial channel model is oforder up to p = 2. As we can expect, the channel estimationerror will become even smaller with higher order p.
Cha
nnel
Fre
quen
cy R
espo
nse
|H(f
,t)|
Fig. 3. Estimated Channel
As we can see from the above plot when the normalizedDoppler spread is small, the channel experiences slow varia-tions over the time index and hence can be represented by alow order polynomial function. However, frequency-selectivitymakes the channel changes relatively fast over the frequency
Fig. 6. Error rate performance for a known channel
As we observe from Fig. 6, the conventional OFDM receiverstructure has an error floor and its performance is limitedby the ICI. The proposed MMSE-based receiver structureperforms not only slightly better in low signal to noise ratiorange, but is also able to remove the error floor completely inhigh signal to noise range.
C. Performance Comparison under Channel Estimation
When the channel state information is not perfectly knownat the receiver, the modified polynomial channel estimator
is applied to obtain the frequency channel response. Fig. 7demonstrates the system performance curve in terms of av-erage symbol errors versus the signal to noise ratio of con-ventional OFDM and MMSE-based OFDM receiver structureusing the estimated channel state information. The pilotspattern described in Section V-B is used under the sameOFDM system as well as the fading channel parameters.
−10 0 10 20 30 40 50 6010
−6
10−5
10−4
10−3
10−2
10−1
100
Signal−to−Noise Ratio, Eb/N0
Sym
bol−
Err
or P
roba
bilit
y, P
s
Performance Comparison among different Receiver Structures, Estimated Channel State Information
Fig. 7. Error rate performance using modified polynomial channel estimation
As expected, we observe from Fig. 7 that MMSE-basedOFDM receiver structure outperforms the conventional OFDMreceiver over the entire signal to noise range.
VI. CONCLUSIONS
In this paper, in order to combat the ICI distortion causedby Doppler-spread of the time-varying fading channels, theMMSE-based OFDM receiver structure is proposed as adetection technique. System performance is compared withthe conventional OFDM receivers under the same channelconditions.
By applying the a polynomial surface channel estimator,we provide in this paper the simulation results of the overallsystem performance, which further confirms that the newMMSE-based OFDM receiver can reduce the symbol errorrate more than conventional OFDM receivers under the sameDoppler-spread channel environments.
REFERENCES
[1] S. Weinstein and P. Ebert, “Data transmission by frequency-divisionmultiplexing using the discrete Fourier transform,” IEEE Trans. Commun.,vol. 19, pp. 628-634, Oct. 1971.
[2] P. Robertson, and S. Kaiser,“Analysis of the loss of orthogonality throughDoppler spread in OFDM system”, in Proc. Globecom’99, pp. 701-706,Dec., 1999.
[3] T. Wang, J. Proakis, J. Zeidler, “Performance Analysis of High QAMOFDM System Over Frequency Selective Time-Varying Fading Channel,”in Proc. 14th IEEE PIMRC, vol. 1, pp. 793-798, Sept., 2003.
[4] X. Wang and K. J. R. Liu, “OFDM channel estimation based on time-frequency polynomial model of fading multipath channel”, VehicularTechnology Conference 2001, Vol. 1, pp. 460-464, 2001.
[5] J. G. Proakis, Digital Communications, 2nd ed., New York: McGraw-Hill,1989.
[6] W. C. Jakes, Microwave Mobile Communications, IEEE Press, Reprinted,1994.
Fig. 4. Actual Channel
Cha
nnel
Fre
quen
cy R
espo
nse
|H(f
,t)|
Fig. 5. Estimated Channel
B. Performance Comparison under Perfectly Known Channel
Before applying the channel estimation technique proposedin Section III, a comparison among different detection schemeswith perfect channel state information is insightful. Fig. 6demonstrates the system performance curves in terms ofthe average number of symbol errors versus the signal tonoise ratio of conventional OFDM and MMSE-based OFDMreceiver structures. The simulation is performed on the sameOFDM system with 16 and 256 subcarriers over the frequencyselective time varying fading channel with the same parame-ters as is described in Section V-A.
−10 0 10 20 30 40 50 6010
−6
10−5
10−4
10−3
10−2
10−1
100
Signal−to−Noise Ratio, Eb/N0
Sym
bol−
Err
or P
roba
bilit
y, P
s
Performance Comparison among different Receiver Structures, Ideal Known Channel
domain (index). Therefore, the modified polynomial channelestimator can provide better estimation results with a loworder p because it only estimates the time domain polynomialcoefficients. The frequency response of the actual channel aswell as the estimated channel (generated from the estimatedcoefficients a′ of the time domain channel impulse response)are shown in Fig. 4 and Fig. 5, where the polynomial modelis of order up to p = 2.
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