IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 2, MARCH 2005
`
`765
`
`ICI Mitigation for Pilot-Aided
`OFDM Mobile Systems
`
`Yasamin Mostofi, Member, IEEE and Donald C. Cox, Fellow, IEEE
`
`Abstract—Orthogonal frequency-division multiplexing (OFDM)
`is robust against frequency selective fading due to the increase of
`the symbol duration. However, for mobile applications channel
`time-variations in one OFDM symbol introduce intercarrier-inter-
`ference (ICI) which degrades the performance. This becomes more
`severe as mobile speed, carrier frequency or OFDM symbol dura-
`tion increases. As delay spread increases, symbol duration should
`also increase in order to maintain a near-constant channel in every
`frequency subband. Also, due to the high demand for bandwidth,
`there is a trend toward higher carrier frequencies. Therefore,
`to have an acceptable reception quality for the applications that
`experience high delay and Doppler spread, there is a need for
`ICI mitigation within one OFDM symbol. We introduce two
`new methods to mitigate ICI in an OFDM system with coherent
`channel estimation. Both methods use a piece-wise linear model to
`approximate channel time-variations. The first method extracts
`channel time-variations information from the cyclic prefix. The
`second method estimates these variations using the next symbol.
`We find a closed-form expression for the improvement in average
`signal-to-interference ratio (SIR) when our mitigation methods
`are applied for a narrowband time-variant channel. Finally, our
`simulation results show how these methods would improve the
`performance in a highly time-variant environment with high delay
`spread.
`
`intercarrier-interference
`Index Terms—Channel estimation,
`(ICI) mitigation, mobility, orthogonal frequency-division multi-
`plexing (OFDM).
`
`O RTHOGONAL
`
`I. INTRODUCTION
`multiplexing
`frequency-division
`(OFDM) handles frequency selective fading resulting
`from delay spread by expanding the symbol duration [1]–[4].
`By adding a guard interval to the beginning of each OFDM
`symbol, the effect of delay spread (provided that there is per-
`fect synchronization) would appear as a multiplication in the
`frequency domain for a time-invariant channel.1 This feature
`allows for higher data rates and has resulted in the selection of
`OFDM as a standard for digital audio broadcasting (DAB [5]),
`digital video broadcasting (DVB [6]), and wireless local area
`networks (802.11a).
`
`Manuscript received November 7, 2002; revised July 7, 2003; accepted
`November 7, 2003. The editor coordinating the review of this paper and
`approving it for publication is A. Scaglione. Part of this work was presented in
`at the International Communications Conference 2003.
`Y. Mostofi is with the Department of Electrical Engineering, California Insti-
`tute of Technology, Pasadena, CA 91125 USA (e-mail: yasi@cds.caltech.edu).
`D. C. Cox is with the Department of Electrical Engineering, Stanford Uni-
`versity, Stanford, CA 94305 USA (e-mail: dcox@spark.stanford.edu).
`Digital Object Identifier 10.1109/TWC.2004.840235
`
`Transmission in a mobile communication environment is
`impaired by both delay and Doppler spread. As delay spread
`increases, symbol duration should also increase for two rea-
`sons. First, most receivers require a near-constant channel in
`each frequency subband. As delay spread increases, this can
`be achieved by an increase of the symbol length. Second, to
`prevent inter-OFDM symbol-interference, the length of the
`guard interval should increase as well. Therefore, to reduce
`redundancy, the symbol length should increase [7]. OFDM
`systems become more susceptible to time-variations as symbol
`length increases. Time-variations introduce ICI, which must
`be mitigated to improve the performance in high delay and
`Doppler spread environments.
`In [8] and [9], authors analyzed the effect of ICI by mod-
`eling it as Gaussian noise. A simplified bound on ICI power
`has also been derived [10]. To mitigate the introduced ICI, tech-
`niques using receiver antenna diversity have been proposed [8],
`[11]. However, sensitivity analysis has shown that as normal-
`ized Doppler spread (defined as the maximum Doppler spread
`divided by the sub-carrier spacing) increases, antenna diversity
`becomes less effective in mitigating ICI in OFDM mobile sys-
`tems [12].
`Jeon and Chang have proposed another method for ICI mit-
`igation which assumes a linear model for channel variations
`[13]. However, they assumed that some of the coefficients of the
`channel matrix are negligible, which is only the case under low
`Doppler and delay spread conditions. For instance, their results
`showed performance improvement under normalized Doppler
`of up to 2.72% and delay spread of 2 s for a two-tap channel.
`In high-mobility applications that require ICI mitigation, how-
`ever, delay spread can be much longer. For instance, the delay
`spread can be as high as 40 s for single frequency network
`(SFN) channels2 and 20 s for cellular applications. Further-
`more, normalized Doppler can get as high as 10% depending on
`the carrier frequency. Their method also relies on the informa-
`tion of adjacent OFDM symbols for channel estimation, which
`increases processing delay.
`To improve the performance in high delay and Doppler spread
`environments, we present two new ICI mitigation methods in
`this paper. Unlike the method of Jeon et al., our methods can
`mitigate ICI in considerably high delay and Doppler spread ap-
`plications such as SFN and cellular networks. Furthermore, in
`Method I, we mitigate ICI without relying on the adjacent sym-
`bols. Both of our methods are based on a piece-wise linear ap-
`proximation for channel time-variations.
`
`1Adding the guard interval will also prevent inter-OFDM symbol-interfer-
`ence.
`
`2SFN refers to DAB and DVB type environments in which adjacent base sta-
`tions transmit in the same frequencies to save the bandwidth.
`
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`766
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`IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 2, MARCH 2005
`
`In the absence of time-variations, frequency domain pilot
`tones or differential modulation should be used to remove the
`effect of channel frequency-variations. As the delay spread
`increases, differential modulation across adjacent subbands
`degrades the performance. As mobility and/or the length of
`the OFDM symbol increase, differential modulation across
`adjacent symbols leads to performance loss as well. Therefore,
`we use frequency domain pilot tones in this paper since we
`are dealing with high delay and Doppler spread environments.
`The minimum number of pilot tones required in each symbol
`exceeds normalized channel delay spread3 by one [14]. These
`pilot tones should be equally spaced in the frequency domain
`to minimize noise enhancement [14].
`In the presence of Doppler spread, however, these pilot tones
`can not estimate channel time-variations. In this work, we show
`how to estimate these variations utilizing either the cyclic prefix
`or the next symbol. Finally, our analysis and simulation results
`show performance improvement in high delay and Doppler
`spread environments.
`
`II. SYSTEM MODEL
`
`Fig. 1 shows the discrete baseband equivalent system model.
`We assume perfect timing synchronization in this paper. More
`information on timing synchronization for a pilot-aided OFDM
`system can be found in [15]. The available bandwidth is divided
`into
`subchannels and the guard interval spans
`sampling
`periods. We assume that the normalized length of the channel
`is always less than or equal to
`in this paper.
`represents
`the transmitted data point in the th frequency subband and is
`related to the time domain sequence,
`, as follows:
`
`Fig. 1. Discrete baseband equivalent system model.
`
`, the fast Fourier transform (FFT) of sequence , will be as
`follows:
`
`(4)
`
`where
`and the second term on the right
`denotes the FFT of
`hand side of (4) represents ICI. Define
`as the FFT of the
`channel tap with respect to time-variations:
`
`Then
`
`can be defined as
`
`(5)
`
`(6)
`
`(1)
`
`is the cyclic prefix vector with length
`as follows:
`
`and is related to
`
`(2)
`
`be the time duration of one OFDM symbol after adding
`Let
`represents the
`channel tap at
`the guard interval. Then,
`time instant
`where
`is the sampling
`period. A constant channel is assumed over the time interval
`with
`indicating the start
`for
`and
`of the data part of the symbol.
`represents the th channel tap in the guard and
`data interval respectively.
`The channel output
`can then be expressed as follows:
`
`Furthermore,
`
`where
`channel tap over
`is the average of the
`the time duration of
`. Therefore,
`represents
`the FFT of this average (note that
`solely refers to a time
`averaging over symbol data part and is different from channel
`ensemble average).
`As was noted by previous work, the ICI term on the right-
`hand side of (4) can not be neglected as the maximum Doppler
`shift,
`, increases (e.g., [8]).
`
`III. PILOT EXTRACTION
`
`be the maximum predicted normalized length of
`Let
`the channel. In this paper, we assume that the normalized length
`of the channel is always smaller than . We insert
`for
`equally spaced pilots,
`, at subchannels
`. An estimate of
`can then be acquired at
`pilot tones as follows:
`
`(3)
`
`(7)
`
`and
`represents a cyclic shift in the base of
`In (3),
`represents a sample of additive white Gaussian noise. Then,
`
`3Normalized channel delay spread refers to the channel delay spread divided
`by the sampling period.
`
`In (7),
`denotes ICI [marked in (4)] at
`Through an IFFT of length , the estimate of
`
`th subcarrier.
`would be
`
`(8)
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`MOSTOFI AND COX: ICI MITIGATION FOR PILOT-AIDED OFDM MOBILE SYSTEMS
`
`767
`
`is a diagonal matrix with diagonal elements of
`for
`. It is shown in Appendix C
`that taking an FFT of will result in the following frequency
`domain relationship:
`
`(14)
`
`(15)
`
`(16)
`
`Fig. 2. Piece-wise linear model in one received OFDM symbol. Solid curve:
`real or imaginary part of a channel path. Dashed line: piece-wise linear model.
`
`where
`
`pilots would have been enough
`In the absence of mobility,
`to estimate the channel. However in the presence of Doppler,
`due to the ICI term of (4), using the estimate of
`for data
`detection results in poor performance. This motivates the need
`to mitigate the resultant ICI.
`
`IV. PIECE-WISE LINEAR APPROXIMATION
`
`In this paper, we approximate channel time-variations with a
`piece-wise linear model with a constant slope over the time du-
`ration
`(Fig. 2). For normalized Doppler of up to 20%, linear
`approximation is a good estimate of channel time-variations and
`the effect on correlation characteristics is negligible. To see this,
`Appendix A shows how this approximation affects the correla-
`tion function as normalized Doppler increases.
`In this section, we will derive the frequency domain rela-
`tionship, similar to (4), when the linear approximation is ap-
`plied. Let
`denote the slope of the
`channel tap in the cur-
`rent OFDM symbol. To perform the linearization, knowledge of
`the channel at one time instant in the symbol is necessary. Let
`represent the average of
`. Then for the th channel tap,
`is minimized for
`as is shown in
`
`Appendix B. Therefore, we approximate
`mate of
`. We will have
`
`with the esti-
`
`(9)
`
`Here,
`represents the FFT of the vector
`containing the FFT of noise samples and
`is the FFT of
`and is defined as follows:
`
`.
`
`is a vector
`where
`
`(17)
`
`should be estimated.
`and
`, both
`To solve (14) for
`Matrix
`is a fixed matrix that is precalculated and stored in the
`receiver. An estimate of
`is readily available from (7)–(9)
`and (15). In the following subsections, we show how to estimate
`with our two methods. In Method I, this is done by uti-
`lizing the redundancy of the cyclic prefix while in Method II the
`information of the next symbol is used.
`
`A. Method I: ICI Mitigation Using Cyclic Prefix
`The output prefix vector,
`of Fig. 1, can be written as fol-
`lows:
`
`(18)
`
`(19)
`
`Consider linearization around
`proximated as follows:
`
`. Then,
`
`can be ap-
`
`In (18),
`
`,
`
`contains AWGN samples and
`
`Inserting (10) into (3), we will have
`
`(10)
`
`(11)
`
`, and
`where ,
`for
`, and
`we will have
`
`are
`
`vectors containing samples of
`. Furthermore, for
`
`,
`,
`
`Since
`
`is a
`.
`in matrix
`for
`,
`is similarly defined for the
`vector defined in (2).
`transmitted cyclic prefix of the previous OFDM symbol and is
`already known to the receiver. Define
`as a vector containing
`slopes of all the taps
`
`(20)
`
`Inserting
`from (10) in
`can be written as follows:
`
`(12)
`
`, it can be easily shown that (18)
`
`(21)
`
`(13)
`
`Here,
`
`for
`and can be estimated from (7)–(9).
`
`and
`is a predetermined
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`768
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`IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 2, MARCH 2005
`
`TABLE I
`PROCEDURE FOR METHOD I
`
`diagonal matrix with
`and is stored in the receiver.
`
`for
`is defined as follows:
`
`...
`
`(22)
`
`represents the vector produced by reversing
`where
`through
`of vector
`and
`denotes
`the order of elements
`transpose of
`. Equations (14) and (21) provide enough infor-
`mation to solve for
`. There are two sets of unknowns,
`and
`(
`is formed from FFT of
`). It is possible to combine
`.
`both equations to form a new one that is only a function of
`However, the complexity of solving such an equation would be
`high. Therefore, we use a simpler iterative approach to solve for
`. We start with an initial estimate for
`and . In each itera-
`tion, we improve the estimate of
`using (14) and then improve
`the estimate of
`using (21). This procedure is summarized in
`Table I.
`
`B. Method II: ICI Mitigation Utilizing Adjacent Symbols
`It is possible to acquire channel slopes without using the re-
`dundancy of the cyclic prefix. This can be done by utilizing ei-
`ther the previous symbol or both adjacent symbols. A constant
`slope is assumed over the time duration of
`for the former and
`for the latter. Therefore, the former can
`handle lower Doppler values while adding no processing delay.
`On the other hand, the latter would have a better performance at
`the price of delay of reception of the next symbol. Since we are
`interested in ICI mitigation in high mobility environments, we
`utilize both adjacent symbols to acquire channel slopes. This is
`shown in Fig. 3. Pilots of the current symbol provide an estimate
`of the channel at the mid-point of the current symbol,
`.
`This estimate is stored in the system. Upon processing of the
`next symbol, an estimate of the channel at midpoint of the next
`symbol,
`, becomes available. Estimate of the slopes
`in region 2 (see Fig. 3) can then be obtained as follows:
`
`(23)
`
`Fig. 3. Piece-wise linear model for method II. Solid curve: real or imag. part
`of a channel path. Dashed line: piece-wise linear model.
`
`represents the slope of the th channel tap in region
`where
`2. Similarly,
`, the slope in region 1, is estimated while pro-
`cessing the previous OFDM symbol and is stored in the receiver.
`Utilizing two slopes introduces a minor change in (11). It can be
`shown that in this case, we will have
`
`(24)
`
`In (24),
`region
`
`represents channel slope matrix of (13) in the
`with
`and
`defined as follows:
`
`Following the same procedure of Appendix C, it can be easily
`shown that the frequency domain relationship will be
`
`(25)
`
`(26)
`In (26),
`is the diagonal matrix defined in (16) for the
`region and can be formed from
`.
`and
`slopes of the
`are fixed matrices. It can be easily shown that
`
`(27)
`
`An estimate of
`
`can then be obtained from (26).
`
`C. Complexity Analysis
`In general, solving (14) and (21) in case of Method I and (26)
`or (24) in case of Method II requires matrix inversion which
`could increase receiver complexity. For Method I, since the size
`of (21) is smaller, the main complexity is in solving (14). This
`requires an
`matrix inversion. In general, any matrix in-
`version algorithm can be used. Also, (14) and (26) show a spe-
`cial structure. For instance, in (14) we need to invert a sum of
`. The special structure can be
`used to reduce the complexity in iterative methods. Comparing
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`MOSTOFI AND COX: ICI MITIGATION FOR PILOT-AIDED OFDM MOBILE SYSTEMS
`
`769
`
`with the method proposed in [13], Methods I and II can handle
`considerably higher delay and Doppler spread (see Section I)
`at the price of higher computational complexity (by neglecting
`some of the channel coefficients, the complexity of the method
`proposed in [13] is reduced to
`inversions of a matrix of
`size
`, where
`is smaller than
`). However, depending
`on the computational power of the receiver, other less complex
`methods like conjugate gradient can be used for matrix inver-
`sion. A good survey of such methods and their complexity anal-
`ysis can be found in [16], [17]. Furthermore, Section VI shows
`how adding a noise/interference reduction mechanism can fur-
`ther reduce the complexity.
`Another important
`issue is the convergence property of
`Method I. In general, for the range of Doppler values that the
`piece-wise linear approximation can be applied, channel slopes
`are small enough that the initial estimate of zero in the first step
`of Method I results in the convergence of the method after a few
`iterations. A more detailed analysis of convergence properties
`of such iterative methods is beyond the scope of this paper but
`can be found in [17], [18].
`
`V. MATHEMATICAL ANALYSIS OF THE EFFECT OF
`LINEARIZATION
`
`In this section, we provide a mathematical analysis of the ef-
`fect of piece-wise linear approximation in mitigating ICI. We
`assume a narrowband time-variant channel to make the analysis
`tractable and leave the case of wideband channels to our simu-
`lations in Section VII. We define
`as the ratio of average
`signal power to the average interference power. Our goal is to
`calculate
`when ICI is mitigated and compare it to that
`of the “no mitigation” case. Consider a narrowband time-variant
`channel,
`. Note that we drop the index of
`in this section
`under narrowband channel assumption. Then, in the absence of
`noise, (3) can be simplified as follows:
`
`The estimate of
`
`will be
`
`(28)
`
`(29)
`
`In (29),
`is the sum of a consid-
`. Since
`erable number of uncorrelated random variables as estimated
`from pilot tones, we approximate its distribution with a com-
`plex Gaussian. In practice if
`is having near to zero values,
`the received signal will not be divided by it. From theoretical
`standpoint, if the cases of near to zero
`are not excluded, the
`variance of
`, the estimate of
`, will be infinite (this can be
`seen from the results of this section). Therefore, we need to ex-
`clude the probability of a near to zero
`to make the analysis
`meaningful. This can be done by introducing a slight modifica-
`tion in the pdf of
`. Let
`. Then, for an
`near zero, we take the pdf of
`to be zero for
`.
`Taking an FFT of (29), we will have
`
`(30)
`
`is not purely
`is the FFT of it.
`and
`In (30),
`interference and contains a term that depends on
`as well.
`However, it can be shown that the power of that term is consid-
`erably small. Therefore, to reduce the complexity of the analysis
`we take
`as the interference term which makes the analysis a
`tight approximation.
`can then be defined as follows:
`
`(31)
`
`where
`is the average power of
`as follows:
`
`and
`
`can be calculated
`
`Since
`
`, then
`
`and we will have
`
`(32)
`
`(33)
`
`(34)
`
`Both
`have complex Gaussian distributions. Fur-
`and
`thermore, they are jointly Gaussian (since a linear combination
`of them is the sum of a considerable number of uncorrelated
`random variables) and correlated. Then,
`will be
`as follows (derived in Appendix D):
`
`In (35),
`
`(35)
`and
`,
`. These parameters are functions
`of channel correlation characteristics (or Doppler spectrum)
`and are derived in Appendix D. Also, it can be easily shown
`that
`where
`and
`stand for the exponential integral and logarithm in the base of
`respectively. Inserting (35) in (33) will result in the following
`:
`
`(36)
`
`Fig. 4 shows
`at
`of (36) as a function of
`is defined as
`(maximum Doppler) di-
`.
`vided by the sub-carrier spacing. Channel power spectrum is
`Jakes spectrum [19] for this result. This means that function
`of Appendix D is
`with
`representing
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`
`Fig. 4. Average SIR versus % of f
`
`for a narrowband channel.
`
`zero-order Bessel function. This analytic result matches the
`corresponding simulation as can be seen from the graph. For
`comparison,
`for the case of “no mitigation” is plotted
`as well. Appendix D shows how
`,
`and
`can be
`found for the case of “no mitigation”. The graph indicates how
`ICI mitigation through linearization improves
`.
`
`VI. NOISE/INTERFERENCE REDUCTION
`
`was acquired using equally-
`In Section III, an estimate of
`spaced pilots. In most cases, the number of active channel taps
`will be less than
`, especially in an SFN environment (see
`Fig. 5 as an example). In such cases, some of the estimated
`channel taps would be noise/interference samples after the IFFT
`in (8). Therefore, if these taps can be removed, the effect of
`noise/interference will be reduced. To do so, estimated channel
`taps are compared with a
`. If the value of a tap is
`below the
`, it will be zeroed:
`
`(37)
`The optimum way to define the
`is to relate it to the
`received signal-to-noise plus interference ratio (SNIR) such that
`the taps comparable to or below noise/interference level are ze-
`roed. However, this requires estimation of the power of noise/in-
`terference which may not be feasible in a high mobility envi-
`ronment. Instead, we define a simple but effective
`.
`Upon estimation of
`from (8), the tap with maximum abso-
`lute value is detected. Let
`represent this maximum. Then,
`all the estimated taps with absolute values smaller than
`for some
`will be zeroed. Choosing small
`increases
`the chance of losing channel taps with significant values and
`only improves the performance if the noise/interference level
`is high. On the other hand, choosing a high will reduce the
`risk of losing taps with considerable values at the price of less
`efficiency in high noise/interference cases. In general, in the ab-
`sence of knowledge of SNIR, it is better to choose
`such that
`losing the taps below the
`does not introduce consid-
`erable performance loss. Following the same criteria, we choose
`
`Fig. 5. Power-delay profile of channel#2.
`
`TABLE II
`PARAMETERS OF THE SIMULATED SYSTEM
`
`in our simulations in the next section. In the case of
`losing a tap, the power of such a tap is less than 1% of the power
`of the strongest channel tap. This will lead to a slight perfor-
`mance loss at very high SNIR which should not be a problem
`since these cases already have a very low error rate.
`Furthermore, the number of nonzero channel taps can be es-
`timated from
`after (37) is applied. Let
`represent
`this estimate in the current OFDM symbol. Therefore, we only
`need to estimate
`slopes. This will reduce the complexity of
`both algorithms. For instance it will reduce the number of un-
`knowns from to
`in step 4 of Method I. This reduction can
`be considerable for SFN channels.
`
`VII. SIMULATION RESULTS
`
`We simulate an OFDM system in a time-variant environment
`with high delay spread. System parameters4 are summarized
`in Table II. We simulate two power-delay profiles. The power-
`delay profile of channel#1 has two main taps that are separated
`by 20 s. Power-delay profile of channel#2 is shown in Fig. 5
`
`4Parameters are based on Sirius Radio second-generation system specifica-
`tion proposal for an SFN environment.
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`MOSTOFI AND COX: ICI MITIGATION FOR PILOT-AIDED OFDM MOBILE SYSTEMS
`
`771
`
`Fig. 6. Error floor versus maximum normalized Doppler.
`
`Fig. 7. Average bit-error rate versus average received SN R for f
`6:5%.
`
`=
`
`and has two main clusters with the total delay of 36.5 s to rep-
`resent a case of reception from two adjacent base stations in an
`SFN environment. Each channel tap is generated as a random
`process with Rayleigh distributed amplitude and uniformly dis-
`tributed phase using Jakes model [19]. Therefore, the auto-cor-
`relation of each tap is a zero-order Bessel function. For both
`channels, the power of channel taps is normalized to result in
`a total power of one. To see how ICI mitigation methods re-
`duce the error floor, Fig. 6 shows the average bit-error rate,
`(before decoding), in the absence of noise for both channels. In
`the “no mitigation” case, pilots are used to estimate
`which
`is then used to detect transmitted data without any estimation
`of time-variations. As can be seen from Fig. 6, average
`in-
`creases considerably for the “no mitigation” case. Both of the
`proposed methods reduce the error floor considerably. Method
`II shows a slightly better performance than Method I. This is
`due to the iterative way of solving for unknowns in Method I.
`Also, channel#1 results in a lower error floor due to its shorter
`delay and smaller number of taps, as expected. To see the ef-
`fect of noise, Fig. 7 shows average
`(before decoding) as a
`function of average received
`for
`. Av-
`erage received
`is defined as the ratio of the average total
`signal power received through all the channel paths to the av-
`erage received noise power. Average error rate for the ideal case
`of no Doppler is also plotted for comparison. It can be seen from
`Fig. 7 that ICI mitigation reduces the error rate considerably for
`both channels. In particular for channel#1, the error rate is al-
`most reduced to that of the case with no Doppler.
`To see how ICI mitigation methods reduce the required re-
`ceived
`for achieving a specific pre-decoding bit error rate,
`Fig. 8 shows the required received
`for reaching an av-
`erage
`before decoding. The graph shows how ICI
`mitigation saves power. For comparison, the required received
`for the case of no Doppler is 17.6 dB for both channels.
`It can be seen that both methods reduce the amount of required
`power to a level close to that of the no Doppler case. For in-
`stance, at
`, the amount of power saving is around 4
`dB. Compared to the “no mitigation” case, the amount of power
`saving increases considerably as
`increases.
`
`Fig. 8. Required average received SN R to achieve average P = :02.
`
`VIII. CONCLUSION
`
`In this paper, we proposed two new methods for ICI mit-
`igation in pilot-aided OFDM mobile systems. Both methods
`used a piece-wise linear approximation to estimate channel
`time-variations in each OFDM symbol. Performance improve-
`ment was shown analytically by deriving
`formulas in
`a narrowband mobile environment. In high delay and Doppler
`spread environments, our simulation results showed consider-
`able performance improvement. They illustrated that applying
`these methods would reduce average
`or the required received
`to a value close to that of the case with no Doppler. The
`power savings become considerable as
`increases.
`
`APPENDIX A
`
`and its piece-wise
`Consider a wide-sense stationary process
`linear approximation
`can represent any
`shown in Fig. 9.
`represents
`at time instant
`.
`of the channel taps.
`It is of interest to compare the auto-correlation function of
`,
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`

`772
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`IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 2, MARCH 2005
`
`Fig. 9. Piece-wise linear approximation of a random process h.
`
`Fig. 9,
`
`and
`
`, to that of
`can be expressed as follows:
`
`. From
`
`For simplicity, we assume that
`and
`. Then,
`will be
`
`for5
`
`,
`
`,
`
`We are interested in
`
`which can be defined as follows:
`
`(38)
`
`Fig. 10. P
`
`versus % of f  T .
`
`, we characterize
`For
`Fig. 10 shows
`as a function of
`, so that
`large enough, i.e.
`comes negligible. The graph suggests that for
`20%,
`is negligible. For instance, for
`is 1%.
`
`(39)
`
`(42)
`
`of (42).
`. We pick
`be-
`of up to
`,
`
`Let
`represent the auto-correlation function of process
`will have
`
`. Then, we
`
`Call
`over
`
`APPENDIX B
`
`. The goal is to minimize
`can be written as follows:
`
`.
`
`(40)
`
`Since
`
`power of the difference of
`symbols
`
`is a wide-sense stationary process, we will have
`. Define
`as the average
`over OFDM
`
`and
`
`(43)
`
`(44)
`
`5Due to the presence of noise/interference, h
`for
`may differ from h
`x = m , m , n and n . Extending the analysis to include this difference
`should be a straightforward extension of the work in this appendix.
`
`(41)
`
`Here,
`
`stands for the real part of argument and
`. From (44), minimization of
`is equivalent to
`over
`. Without
`maximization of
`loss of generality, assume
`where
`represents zero-order Bessel function. Since
`(which means that the length of the symbol data part is less
`
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`MOSTOFI AND COX: ICI MITIGATION FOR PILOT-AIDED OFDM MOBILE SYSTEMS
`
`773
`
`is
`than channel coherence time), it can be easily seen that
`maximized for
`and
`(
`is assumed
`even). Therefore, we approximate
`with the estimate of
`.
`
`of
`
`, the correlation coefficient
`and
`(defined in Section V).
`, being equal to
`and
`can be similarly calculated. Inserting
`and
`in (46) will result in (35).
`
`APPENDIX C
`
`Let
`is a circular toeplitz matrix, taking an
`. Since
`FFT of will result in a multiplication by a diagonal matrix in
`the frequency domain. Therefore, we will have
`with
`representing FFT of
`and
`as defined in (16).
`Let
`. Taking an FFT of it, we will have
`where
`is the FFT of
`and
`represents circular
`convolution in the base of
`. Taking an FFT of
`, it can be easily
`calculated that
`is as defined in (17) [20]. Therefore, we will
`have
`
`(45)
`
`APPENDIX D
`
`A. Proof of (35)
`,
`,
`where
`and
`Let
`are zero mean Gaussian variables with independent
`and
`inphase and quadrature parts. We will have
`
`and
`,
`B. Finding
`1) The Case of ICI Mitigation: From (7), (8), in the ab-
`sence of noise and for a narrowband channel, we will have
`, where
`is the estimation noise
`. It can be easily shown, using (7) and
`
`with variance of
`
`is
`.
`(8), that
`the auto-correlation function of the narrowband channel as
`defined in Appendix A. Furthermore, it can be shown that
`due to the independency of the trans-
`mitted data points and channel. Similarly, in the next OFDM
`symbol,
`, where
`is the estimation noise with
`and
`
`. Define
`. Using Method
`
`II, we will have
`
`(47)
`
`A similar formula can be written for estimation in Region 1.
`Using these formulas, after some lengthy but straightforward
`computations, the following formulas can be derived: (see (48)
`located at bottom of page).
`2) The Case of No Mitigation: In this case, we will have
`for
`. Then, the parameters can be
`easily derived as follows:
`
`It can be easily shown [21] that
`with
`
`(46)
`
`,
`
`(49)
`
`(48)
`
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`774
`
`IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 2, MARCH 2005
`
`Donald C. Cox (M’61–SM’72–F’79) received the
`B.S. and M.S. degrees in electrical engineering from
`the University of Nebraska, Lincoln, in 1959 and
`1960, respectively, and the Ph.D. degree in electrical
`engineering from Stanford University, Stanford, CA,
`in 1968. He received an Honorary Dr.Sci. degree
`from the University of Nebraska in 1983.
`From 1960 to 1963, he was with Wright-Patterson
`AFB, OH, where he reasearched microwave commu-
`nications system design. From 1963 to 1968, he was
`with Stanford University researching tunnel diode
`amplifier design and research on microwave propagation in the troposphere.
`From 1968 to 1973, his research a