IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 3, MARCH 1999
`
`365
`
`Analysis of New and Existing Methods of Reducing Intercarrier
`Interference Due to Carrier Frequency Offset in OFDM
`Jean Armstrong
`
`Abstract—Orthogonal frequency division multiplexing (OFDM)
`is very sensitive to frequency errors caused by frequency differ-
`ences between transmitter and receiver local oscillators. In this
`paper, this sensitivity is analyzed in terms of the complex weight-
`ing coefficients which give the contribution of each transmitter
`subcarrier to each demodulated subcarrier. Previously described
`windowing and self-intercarrier interference (ICI) cancellation
`methods are analyzed in terms of these weighting coefficients.
`New ICI cancellation schemes with very much improved perfor-
`mance are described. A condition for orthogonality of windowing
`schemes is derived in terms of the discrete Fourier transform
`(DFT) of the windowing function.
`
`Index Terms— Discrete Fourier transform, intercarrier inter-
`ference, interference suppression, orthogonal frequency division
`multiplexing, synchronization.
`
`ORTHOGONAL
`
`I. INTRODUCTION
`frequency
`division multiplexing
`is being considered for data transmission
`(OFDM)
`in a number of environments [1], [2]. One limitation of
`OFDM in many applications is that it is very sensitive to
`frequency errors caused by frequency differences between the
`local oscillators in the transmitter and the receiver [3]–[5].
`Carrier frequency offset causes a number of impairments
`including attenuation and rotation of each of the subcarriers
`and intercarrier interference (ICI) between subcarriers [4].
`A number of methods have been developed to reduce this
`sensitivity to frequency offset, including windowing of the
`transmitted signal [6], [7] and use of self ICI cancellation
`schemes [8].
`This paper analyzes in detail, for a perfect Nyquist channel,
`the ICI resulting from carrier frequency offset. Expressions
`are derived for each demodulated subcarrier at the receiver in
`terms of each transmitted subcarrier and
`complex weighting
`factors. Windowing and ICI cancellation schemes can be
`related and described in terms of these complex weighting
`factors. New ICI cancellation schemes which give greater
`ICI cancellation are developed. A new condition for the
`orthogonality of windowing schemes is derived in terms of the
`discrete Fourier transform (DFT) of the windowing function.
`
`II. ANALYSIS OF ICI
`
`A. Structure of OFDM System
`Fig. 1 shows the structure of the OFDM communication
`system being considered. In this OFDM system there are
`
`Paper approved by S. B. Gelfand, the Editor for Transmission Systems
`of the IEEE Communications Society. Manuscript received March 5, 1997;
`revised February 16, 1998.
`The author is with the School of Electronic Engineering, La Trobe Univer-
`sity, Bundoora, VIC 3083, Australia (e-mail: j.armstrong@ee.latrobe.edu.au).
`Publisher Item Identifier S 0090-6778(99)01925-X.
`
`Fig. 1. Structure of an OFDM communication system.
`
`. In the th symbol
`modulate the
`
`subcarriers and the symbol period is
`period, the
`complex values
`subcarriers.
`Systems using a number of different types of modulation
`of subcarriers within OFDM, such as phase shift keying
`(PSK) and quadrature amplitude modulation (QAM) have been
`described in the literature. This analysis does not depend on
`the mapping of the data to be transmitted to the complex
`values
`, and is therefore applicable to all forms
`of modulation which can be used within OFDM.
`This analysis considers only the impairments due to carrier
`frequency offset. Other authors [9] have analyzed more general
`models, but these do not clearly reveal the structure on which
`ICI cancellation depends. Frequency offset alone does not
`cause intersymbol interference (ISI). Often a cyclic prefix is
`used in OFDM to eliminate the ISI and ICI caused by errors
`in sampling time or distortion in the channel. This use of a
`cyclic prefix is not considered in this analysis.
`
`B. Derivation of Expressions for the
`Complex Weighting Coefficients
`The signal at the output of the OFDM transmitter resulting
`from the th transmitted symbol is given by
`
`(1)
`
`0090–6778/99$10.00 ª
`
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`366
`
`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 3, MARCH 1999
`
`is the impulse
`is the carrier frequency and
`where
`response of the low-pass filter in the transmitter. At
`the
`receiver, the signal is mixed with a local oscillator signal which
`is
`above the correct frequency
`. Ignoring the effects of
`noise, the demodulated signal is then given by
`
`(2)
`
`is the combined impulse response of the channel
`where
`and of the transmitter and receiver filters.
`is the phase
`offset between the phase of the receiver local oscillator and
`the carrier phase at the start of the received symbol.
`Assuming that
`satisfies the Nyquist criterion for sam-
`ples taken at intervals
`and that
`is sampled at the
`optimum instants, then the samples input to the receiver DFT
`are given by
`
`Fig. 2. Real and imaginary components of the complex weighting factors
`c0 cN 1 for the case of f T = 0:2 and N = 16.
`
`where
`
`The result of the DFT of these samples is given by
`
`(3)
`
`(4)
`
`The decoded complex value
`therefore consists of a wanted
`component which is due to
`but which is subject to a
`change in amplitude and phase given by
`, where
`
`(8)
`
`Substituting the value of
`ulation, it can be shown that
`
`from (3) and after some manip-
`
`(9)
`
`Using the properties of geometric series, this can alternatively
`be expressed as
`
`(5)
`
`(6)
`
`, and each decoded
`, then
`If
`complex value is simply the phase rotated version of the
`transmitted value. The amount of rotation depends on the phase
`offset between the transmitter and receiver local oscillators.
`If
`, then ICI will occur and each output data symbol
`will depend on all of the input values. The analysis of ICI can
`be simplified by defining
`complex weighting coefficients,
`, which give the contribution of each of the
`input
`to the output value
`
`values
`
`(7)
`
`but is
`depends on the normalized frequency offset
`independent of
`. In other words, all subcarriers experience
`the same degree of attenuation and rotation of the wanted
`component.
`In addition, the decoded complex value is subject to ICI.
`This is the sum of components dependent on each of values
`. The contribution of each
`depends on the
`normalized frequency offset
`and on
`. It
`does not depend directly on
`.
`Fig. 2 shows the complex weighing factors
`. Note that
`for the case of
`and
`as the coefficients depend on the distance mod
`between the
`subcarriers, there are only
`.
`distinct coefficients,
`The graphs are smooth, there are no sudden changes in the
`weighting coefficients as the distance moves from 15 to 15,
`except between
`1 and 0, and between 0 and 1.
`Fig. 3 shows how, for
`depends
`, the value of
`on the number of subcarriers in the OFDM system. For
`there is little change in the power of the wanted subcarrier.
`Because carrier frequency offset does not change the total
`power in the received signal, this also means that the total
`ICI power changes little with
`.
`
`III. SELF ICI CANCELLATION SCHEMES
`Zhao and H¨aggman [8] have described a method of reduc-
`ing sensitivity to frequency errors which they call self ICI
`cancellation. The method maps the data to be transmitted onto
`
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`ARMSTRONG: ANALYSIS METHODS OF REDUCING INTERCARRIER INTERFERENCE IN OFDM
`
`367
`
`scheme gives no reduction in overall SNR compared with
`normal OFDM. A disadvantage of the method is that it is
`less bandwidth efficient than normal OFDM as only half as
`many complex values can be transmitted per symbol.
`The ICI cancellation scheme also results in a form of
`windowing and the overall transmitted signal has a sinusoidal
`envelope of period
`. This is because pairs of sinusoids of
`frequency difference
`are being subtracted.
`
`IV. WINDOWING
`
`A. Windowing to Reduce Sensitivity to Linear Distortions
`A number of authors [6], [7], [10], [11] have described
`the use of windowing in OFDM. These applications can be
`divided into two groups. In the first group, windowing is used
`to reduce the sensitivity to linear distortions [10], [11]. In
`the second, windowing is used to reduce the sensitivity to
`frequency errors [6], [7]. In the first group, the signal at the
`output of the IDFT in the transmitter is cyclically extended.
`The windowing function shapes the cyclic extension, but the
`original part of the signal remains unchanged. Where the only
`impairments in the system are due to frequency differences
`between the local oscillators, this form of windowing has no
`effect on the system performance. This form of windowing
`will not be considered further.
`
`B. Windowing to Reduce Sensitivity to Frequency Offset
`The second form of windowing involves cyclically extend-
`ing by
`samples the time domain signal associated with each
`symbol. The whole of the resulting signal is then shaped
`with the window function. Fig. 4(a) shows a block diagram
`of a typical system. Note that the transform in the receiver is
`point whereas that in the transmitter is
`point. The
`inputs to the transmitter transform have been labeled
`so that windowing can be more readily
`analyzed and related to ICI cancellation. If
`, then
`points of the received signal are used as input to the DFT, if
`, then the signal corresponding to each symbol is zero
`padded at the receiver to give length
`. The outputs of the
`DFT with even-numbered subscripts are used as estimates of
`the transmitted data and the odd-numbered ones are discarded.
`Because not all of the received signal power is being used in
`generating data estimates, the method has a reduced overall
`SNR compared with OFDM without windowing. The value of
`the SNR loss depends on the form of windowing.
`A number of different windows, including the Hanning
`window [6], windows satisfying the Nyquist criterion [7], and
`the Kaiser window [7] have been described in the literature.
`All of these windows give some reduction in the sensitivity
`to frequency offset. But only Nyquist windows (of which the
`Hanning window is one particular example) have no ICI for
`the case of no frequency offset [7].
`
`C. Windows Which Preserve Orthogonality
`The conditions under which a window preserves orthog-
`onality can be derived by considering the block diagram
`of Fig. 4(b). In the absence of noise and distortion, this is
`
`Fig. 3.
`
`c0 as function of number of subcarriers for f T = 0:1.
`
`adjacent pairs of subcarriers rather than onto single subcarriers,
`.
`so that
`This results in cancellation of most of the ICI in the values
`.
`For example, the decoded value for the zeroth carrier is
`given by
`
`(10)
`The ICI now depends on the difference between the ad-
`jacent weighting coefficients rather than on the coefficients
`themselves. As the difference between adjacent coefficients is
`small, this results in substantial reduction in ICI. If adjacent
`coefficients were equal, then the ICI would be completely
`cancelled. Thus this process can be considered as cancelling
`out the component of ICI which is constant between adjacent
`pairs of coefficients. ICI cancellation depends only on the
`coefficients being slowly varying functions of offset. It does
`depend not on the absolute values of the coefficients and so
`improves the performance for any frequency offset.
`To maximize the overall SNR, the values
`should be subtracted in pairs, because this results in the
`addition of the wanted signal components. This also further
`reduces the ICI
`
`(11)
`
`The remaining ICI depends on factors of the form
`. If the three weighting coefficients in each factor
`were linear functions of offset, for example, if
`and
`, where
`is any constant as well as the gradient
`of the linear function, these factors would all be zero. This
`cancellation could be considered as canceling out the compo-
`nent of ICI which is due to the linear variation in weighting
`coefficient over groups of three adjacent coefficients.
`When there is no carrier frequency offset
`
`(12)
`Thus, in the absence of other impairments, all of the received
`power is decoded into wanted signal, and the ICI cancellation
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`368
`
`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 3, MARCH 1999
`
`For the Hanning window, which is a cosine roll-off window
`with roll-off factor of o,
`
`(14)
`
`This window results in linear cancellation of the ICI in
`, but each of
`the even-numbered outputs,
`the odd numbered outputs depend on two of the inputs
`. Therefore, these odd-numbered outputs
`are discarded and are not used to contribute to the useful signal.
`This windowing has the same performance with respect to ICI
`as the self ICI cancellation scheme with linear cancellation
`but has worse performance with respect to noise added in the
`channel.
`
`V. NEW HIGHER ORDER ICI CANCELLATION SCHEMES
`The concept of self ICI cancellation can be extended. In the
`method of Zhao and H¨aggman [8], data is mapped onto pairs
`of subcarriers. This results in cancellation of the component of
`ICI due to the linear variation in weighting coefficients over
`groups of three adjacent coefficients. By mapping data onto
`larger groups of subcarriers, higher order ICI cancellation can
`be achieved. For the general case of mapping onto groups
`of
`subcarriers, the relative weightings of the subcarriers
`in the group are given by the coefficients of the polynomial
`expansion of
`. Using a similar analysis to that for the
`linear case, it can be shown that when groups of
`subcarriers
`are weighted in this way in both the transmitter and receiver,
`the component of ICI which is due to the variation in weighting
`coefficients described by a polynomial of order
`over
`groups of
`adjacent coefficients is cancelled.
`For example, by using groups of three subcarriers, cancel-
`lation of the component of ICI which is due to the cubic
`variation in weighting coefficients over groups of five adjacent
`coefficients can be achieved. In this case,
`At the receiver, the data
`is estimated from weighted sums of the form
`
`If the low-pass filter in the transmitter and the receiver are
`designed so that the receiver filter is matched to the transmitted
`filter, this form of weighting also results in overall matched
`filtering where the filtering is matched to the data which is
`being mapped onto the
`.
`
`VI. COMPARISON OF PERFORMANCE OF DIFFERENT
`METHODS OF REDUCING ICI DUE TO FREQUENCY OFFSET
`Fig. 5 shows how the ratio of mean wanted power to mean
`uncancelled ICI power varies as a function of normalized
`frequency offset for four different systems: standard OFDM
`and OFDM with cancellation of the constant, linear, and cubic
`components of ICI. The graphs are for
`, but graphs
`for any
`would have almost identical form.
`When ICI cancellation schemes are used, weighted groups
`of subcarriers are modulated rather than individual subcarriers.
`
`(a)
`
`(b)
`
`(a) Windowing to reduce sensitivity to frequency offset. (b) Simpli-
`Fig. 4.
`fied block diagram of windowing.
`
`equivalent to Fig. 4(a). The combination of cyclic extension,
`windowing, and zero padding is equivalent to a cyclic exten-
`sion of length
`and windowing of length
`, where some
`of the window coefficients may be zero. The
`point IDFT
`followed by the cyclic extension can be further simplified
`to an equivalent
`point IDFT in which all the inputs with
`odd-numbered subscripts are zero.
`For no distortion and orthogonality of the wanted outputs,
`that is those outputs with even subscripts, then
`for
`even. For
`odd,
`can take any value, and may depend
`on any of the inputs
`as these
`values are discarded.
`The windowing is a multiplication of the output of the IDFT
`output values with the windowing values. A well-known
`property of the DFT is that multiplication in the discrete time
`domain is equivalent to circular convolution in the discrete
`frequency domain and vice versa. Let
`be the
`point DFT of
`. Then
`is the circular
`convolution of
`with
`. Using the
`fact that
`for
`odd it can readily be shown that the
`orthogonality condition is met if
`
`any value.
`
`(13)
`
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`ARMSTRONG: ANALYSIS METHODS OF REDUCING INTERCARRIER INTERFERENCE IN OFDM
`
`369
`
`cancel higher order components. These have been shown
`to give very great reductions in the ICI due to frequency
`offset. These ICI cancellation methods map each complex
`value to be transmitted onto weighted groups of subcarriers. A
`disadvantage of these ICI methods is that fewer complex data
`values are transmitted per symbol period. A condition for the
`orthogonality of windowing schemes in terms of the DFT of
`the windowing function has been derived.
`
`ACKNOWLEDGMENT
`The author wishes to thank the anonymous reviewers and
`Dr. J. M. Badcock, Prof. P. M. Grant, Dr. K. A. Seaton, and
`Prof. E. R. Smith for their helpful comments and suggestions.
`
`REFERENCES
`
`[1] B. Hirosaki, S. Hasegawa, and A. Sabato, “Advanced group-band
`modems using orthogonally multiplexed QAM technique,” IEEE Trans.
`Commun., vol. 34, pp. 587–592, June 1986.
`[2] M. Alard and R. Lassalle, “Principles of modulation and channel coding
`for digital broadcasting for mobile receivers,” EBU Tech. Rev., no. 256,
`pp. 168–190, 1987.
`[3] H. Sari, G. Karam, and I. Jeanclaude, “Channel equalization and carrier
`synchronization in OFDM systems,” presented at 1993 Tirrenia Int.
`Workshop on Digital Communications, Tirrenia, Italy, Sept. 1993.
`[4] T. Pollet, M. Van Bladel, and M. Moeneclaey, “BER sensitivity of
`OFDM systems to carrier frequency offset and Wiener phase noise,”
`IEEE Trans. Commun., vol. 43, pp. 191–193, Feb./Mar./Apr. 1995.
`[5] P. H. Moose, “A technique for orthogonal frequency division multi-
`plexing frequency offset correction,” IEEE Trans. Commun., vol. 42,
`pp. 2908–2914, Oct. 1994.
`[6] M. Gudmundson and P.-O. Anderson, “Adjacent channel interference in
`an OFDM system,” in IEEE 46th Vehicular Technology Conf., Atlanta,
`GA, Apr. 1996, pp. 918–922.
`[7] C. Muschallik, “Improving an OFDM reception using an adaptive
`Nyquist windowing,” IEEE Trans. Consumer Electron., vol. 42, Aug.
`1996.
`[8] Y. Zhao and S.-G. H¨aggman, “Sensitivity to Doppler shift and carrier
`frequency errors in OFDM systems—The consequences and solutions,”
`in IEEE 46th Vehicular Technology Conf., Atlanta, GA, Apr. 1996, pp.
`1564–1568.
`[9] M. Russell and G. L. Stuber, “Interchannel interference analysis of
`OFDM in a mobile environment,” in IEEE 45th Vehicular Technology
`Conf., July 1995, 1996, pp. 820–824.
`[10] S. B. Weinstein and P. M. Ebert, “Data transmission by frequency-
`division multiplexing using the discrete Fourier transform,” IEEE Trans.
`Commun. Technol., vol. COM-19, pp. 628–634, Oct. 1971.
`[11] L. J. Cimini, “Analysis and simulation of a digital mobile channel using
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`vol. COM-33, pp. 665–675, July 1985.
`
`Fig. 5. Wanted signal power/uncancelled ICI power as a function of
`=Deltaf T .
`
`The graphs describe the performance for any modulation
`scheme which meets the condition that the data being mapped
`onto the weighted groups are independent
`identically dis-
`tributed random variables and when
`is integral so that
`all subcarriers are used. They do not describe exactly the
`case where different weighted groups of subcarriers are being
`modulated with different average powers, for example where
`spectral shaping is used, or where there is correlation between
`the variables modulating different groups of subcarriers. For
`not integral, the ICI is less for subcarriers close to the
`unused subcarriers. The graphs shows the worst case ICI. All
`of the methods give a higher signal-to-uncancelled-ICI ratio
`than standard OFDM.
`
`VII. CONCLUSION
`An analysis of the effect of frequency errors in OFDM has
`been presented. The ICI due to carrier frequency offset has
`been described in terms of complex weighting coefficients.
`The self ICI cancellation schemes and windowing schemes
`described by other authors have been analyzed in terms of the
`complex weighting factors. It is shown that the ICI cancellation
`scheme and cosine roll-off windowing, with a roll-off factor
`of one, cancel the component of ICI due to the linear variation
`of weighting coefficients over groups of three coefficients.
`New self ICI cancellation schemes have been derived which
`
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