Effect of Carrier Frequency Offset on OFDM Systems for Multipath Fading Channels

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1 Effect of Carrier Frequency Offset on OFDM Systems for Multipath Fading Channels Jungwon Lee, Hui-Ling Lou, Dimitris Toumpakaris and John M. Cioffi Marvell Semiconductor, Inc., 7 First Avenue, Sunnyvale, CA 9489, USA {junglee, hlou, dimitris}@marvell.com Department of Electrical Engineering, Stanford University, Stanford, CA cioffi@stanford.edu Abstract This paper presents an exact analysis of the effect of the carrier frequency offset on orthogonal frequency division multiplexing OFDM systems for a general multipath fading channel. As is well known, the carrier frequency offset attenuates the desired signal and causes inter-carrier interference, thus reducing the signal-to-noise ratio SR. The SR degradation due to the carrier frequency offset is evaluated by deriving the exact SR expression in the presence of the carrier frequency offset. The SR analysis in this paper can be used for the design of a practical OFDM system in determining how small the frequency offset should be in order to maintain the SR degradation to negligible levels. I. ITRODUCTIO Orthogonal frequency division multiplexing OFDM is an important modulation technique for high-speed communications through frequency selective channels [], []. It can easily remove the inter-symbol interference ISI and it can be implemented using computationally efficient fast Fourier transform FFT. Because of these main advantages, OFDM has been chosen as the modulation scheme of various wireless and wireline communication systems such as wireless LA IEEE 8.a and HIPERLA/, digital audio broadcasting DAB, terrestrial digital video broadcasting DVB-T, asymmetric digital subscriber line ADSL, and very high-speed digital subscriber line VDSL systems. Although OFDM has many advantages [3], it is well known that it is susceptible to the carrier frequency offset [4], which results from the Doppler shift or the mismatch between the transmitter and the receiver oscillator frequency. The carrier frequency offset attenuates the desired signal and introduces the inter-carrier interference ICI. Thus, the signal-to-noise ratio SR decreases and the performance of a OFDM system is degraded. To overcome the adverse effect of the carrier frequency offset, various methods of correcting it have been developed [5], [6]. Since these synchronization methods do not remove the frequency offset completely, the synchronization requirements should be determined. To set the synchronization requirements, the effect of the carrier frequency offset on the SR should be quantified for the design of a practical OFDM system. The SR degradation due to the carrier frequency offset has been extensively analyzed in [4], [7], [8]. In [4], an approximate expression for the SR degradation was derived for the additive white Gaussian noise AWG channel. This SR analysis was extended to a time-invariant multipath channel in [7] and a shadowed multipath channel in [8]. However, the SR expression in [7] is complex, and only the approximate expression is available in [8]. This paper analyzes the effect of the carrier frequency offset on the SR for a general multipath fading channel. A concise, exact SR formula is derived for a general multipath fading channel. Then the SR formula for the AWG channel is obtained as a special case. The SR formula is exact for any value of the carrier frequency offset and nominal SR and is simple enough to understand the relationship between the SR degradation and various system parameters in the presence of the carrier frequency offset. The paper is organized as follows: Section II describes the model of OFDM systems. In Section III, the various effects of the carrier frequency offset are explained. In Section IV, the effect of the carrier frequency offset on the SR is analyzed in detail, and the SR expression in the presence of the carrier frequency offset is derived. Section V presents some numerical results, and Section VI concludes the paper. II. SYSTEM MODEL An OFDM system transmits information as a series of OFDM symbols []. Fig. shows the baseband equivalent model of an OFDM system. As is shown in the figure, the inverse discrete Fourier transform IDFT is performed on the information symbols X m [k] for k,,, to produce the time-domain samples x m [n] of the mth OFDM symbol: k X m[k]e jπkn g/, x m [n] if n + g, otherwise, where and g are the numbers of data samples and cyclic prefix samples, respectively. The OFDM symbol x m [n] is transmitted through a channel h m [n] and is affected by a Gaussian noise z m [n]. The channel h m [n] is assumed to be block-stationary, i.e., time-invariant over each OFDM symbol. With this assumption, the output ỹ m [n] of the channel can be represented as a simple convolution operation as follows: ỹ m [n] h m [n] x m [n]+ z m [n], Globecom /4/$. 4 IEEE

2 xm[] transmit signal and the channel. To simplify the notation, c m [n] is defined as Xm[] MUX xm[n] c m [n] ejπɛn/ e jπɛm+α e jθ. 7 Xm[] IDFT hm[n] The received sample y m [n] is then Xm[ ] xm[ + g ] y m [n] c m [n]h m [n] x m [n] + z m [n]. 8 Ym[] Ym[] ym[] ym[g ] DEMUX ym[n] ~ym[n] ~zm[n] j[ß f[n+m+g]t + ] e III. EFFECT OF CARRIER FREQUECY OFFSET In this section, the effect of the carrier frequency offset is analyzed based on the system model in the previous section. The relative frequency offset ɛ can be divided into an integer part l and a non-integer part ɛ such that / ɛ </: DFT ɛ l + ɛ. 9 Ym[ ] Fig.. ym[ + g ] Baseband equivalent model of an OFDM system where h m [n] x m [n] r h m[r]x m [n r], and z m [n] is additive white Gaussian noise with variance σ Z. Since the channel h m [n] in is assumed to be blockstationary, the channel response does not change in each OFDM symbol. However, the channel response h m [n] can change over different OFDM symbols, so it is a function of the OFDM symbol index m. In this paper, it is further assumed that the channel response h m [n] at time n in uncorrelated with the response h m [p] at time p n: E[h m [n]h m[p]] δ[n p]e[ h m [n] ]. 3 This channel model is valid in many practical cases [9] and includes the shadowed multipath channel in [8] as a special case. When the receiver oscillator is not perfectly matched to the transmitter oscillator, there can be a carrier frequency offset f f t f r between the transmitter carrier frequency f t and the receiver carrier frequency f r. In addition, there may be a phase offset θ between the transmitter carrier and the receiver carrier. The received symbol y m [n] is then y m [n] e j[π fn+m+gt +θ] h m [n] x m [n]+ z m [n], 4 where T is the sampling period. The frequency offset f can be represented with respect to the subcarrier bandwidth / T by defining the relative frequency offset ɛ as ɛ f ft 5 / T Using the relative frequency offset ɛ, the received sample y m [n] is expressed as y m [n] e j πɛn e jπɛm+α e jθ h m [n] x m [n]+z m [n], 6 where α g and z m[n] e j πɛn e jπɛm+α e jθ z m [n]. The noise z m [n] is a zero-mean complex-gaussian randomvariable with variance σ Z and is independent of the It can be easily seen from 5 that the absolute frequency offset that corresponds to the above relative frequency offset is l+ ɛ times the subcarrier bandwidth / T. Assuming that H m [k] and X m [k] are periodic with period, the discrete Fourier transform DFT of y m [n] is equal to Y m [k] C m [k] H m [k]x m [k] + Z m [k] C m [l]h m [k l]x m [k l]+z m [k] + r l C m [r]h m [k r]x m [k r], where C m [k], H m [k], and Z m [k] are the DFTs of c m [n], h m [n], and z m [n], respectively. Specifically, the DFT of c m [n], C m [k], can be expressed as C m [k] n c m [n]e jπnk/ e jπɛ kn/ n e jπɛ k e j[πɛm+α+θ] e jπɛ k/ sinπɛ k sinπɛ k/ ejπɛ k / e j[πɛm+α+θ] e j[πɛm+α+θ]. The magnitude of C m [k] is always less than or equal to since C m [k] c m [n]e jπnk/ a n c m [n]e jπnk/ n n where the equality a follows from 7. Globecom /4/$. 4 IEEE

3 From, it can be seen that the desired signal H m [k l]x m [k l] is affected by the frequency offset in the following ways: The signal H m [k l]x m [k l] is received by the kth subcarrier instead of the k lth subcarrier. The magnitude of H m [k l]x m [k l] is attenuated by C m [l]. The phase of H m [k l]x m [k l] is rotated by e jπ[ ɛ / +ɛm+α+θ]. The signal H m [k l]x m [k l] suffers from the ICI I m [k] r l C m[r]h m [k r]x m [k r] in addition to the noise Z m [k]. Since the carrier frequency offset attenuates the desired signal and introduces the ICI, the SR decreases. In the next section, the SR degradation is analyzed in detail. In the SR analysis, it is assumed that l because the integer frequency offset l does not affect the SR. IV. SR AALYSIS This section derives the SR expression for coherent demodulation when the carrier frequency of the receiver is mismatched with that of the transmitter. For coherent demodulation, the receiver should be able to estimate the phase of C m []H m [k] in order to decode the received symbol Y m [k] correctly. One of the ways of estimating the phase is to use training preambles or pilot subcarriers, where known symbols are transmitted. Let C m []H m [k] C m []H m [k] e jθm[k], and assume that the estimate of the phase θ m [k] is perfect. Then, the decision metric M m [k] is obtained by multiplying the received symbol Y m [k] by e jθm[k] : M m [k] e jθm[k] Y m [k] where the ICI I m[k] is C m []H m [k] X m [k]+i m[k]+z m[k],3 I m[k] e jθm[k] and the noise Z m[k] is r C m [r]h m [k r]x m [k r] 4 Z m[k] e jθm[k] Z m [k]. 5 ote that the multiplication factor C m []H m [k] of X m [k] is a random quantity. In some cases such as in [8], C m []H m [k] is factored into a constant component, E[ C m []H m [k]], and a time-varying component, C m []H m [k] E[ C m []H m [k] ]. Then E[ C m []H m [k] ]X m [k] is viewed as a desired signal and C m []H m [k] E[ C m []H m [k] ]X m [k] as an additional noise component. However, in this paper, C m []H m [k] X m [k] is regarded as a desired signal for the following reasons. For the case of constant-magnitude modulation schemes, C m []H m [k] scales the magnitude of X m [k] without altering the phase of X m [k]. Thus, C m []H m [k] X m [k] should be viewed as the desired signal. When a nonconstant-magnitude modulation scheme is employed, the magnitude of C m []H m [k] should be estimated along with the phase of C m []H m [k] for the correct demodulation of the transmitted signals. Since knowledge of C m []H m [k] is used in demodulation, C m []H m [k] X m [k] should be viewed as the desired signal. With the above interpretation on the desired signal, the SR of the k-th subcarrier is given by SR[k] E[ C m[]h m [k]x m [k] ] E[ I m[k]+z m[k]. 6 ] As a first step of deriving a simple SR expression, the power of the desired signal is calculated: E[ C m []H m [k]x m [k] ] C m [] E[ H m [k] ]E[ X m [k] ] C m [] E[ H m [k] ]σ X, 7 because the channel H m [k] is independent of the transmit symbol X m [k]. Since the channel response h m [n] at time n is uncorrelated with the response h m [p] at time p n, E[ H m [k] ] is g g E[ H m [k] ] E h m [n]h m[p]e jπn pk/ g n p E[ h m [n] ] n P h, 8 where P h is the total average power of channel response h m [n]. By combining 7 and 8, the power of the desired signal can be written as E[ C m []H m [k]x m [k] ] C m [] P h σ X. 9 ext, the power of the ICI-plus-noise signal is calculated. The ICI I m[k] and the noise Z m[k] are independent since the noise Z m[k] is independent of the transmitted symbol X m [r] for r,, and the ICI is a linear combination of X m [r] s as can be seen from 4. Thus, the power of the ICI-plus-noise signal is equal to the sum of the power of the ICI and the power of the background noise: E [ I m[k]+z m[k] ] E [ I m[k] ] + E [ Z m[k] ]. Assuming that data symbol in each subcarrier is independent of one another, i.e., E[X m [k r]xm[k s]] σx δ[s r], the ICI power E[ I m[k] ] can be expressed as [ E[ I m[k] ] E C m [r]cm[s]h m [k r] r s ] Hm[k s]x m [k r]xm[k s] r C m [r] E[ H m [k r] ]σ X C m [r] P h σx. r Globecom /4/$. 4 IEEE

4 The above equation can be simplified by noticing that r C m [r] n c m [n], where a basic property of the DFT is used along with the fact that c m [n] / from 7. Then the ICI power E[ I m[k] ] can be expressed as E[ I m[k] ] C m [] P h σ X. 3 The above expression for the ICI power is not an approximation, unlike [4], [8]. The power of the noise Z m[k] is the same as the power of Z m [k]: E[ Z m[k] ]σ Z. 4 Thus, the power of the ICI-plus-noise signal is E[ I m[k]+z m[k] ] C m [] P h σ X + σ Z. 5 From 6, 9, and 5, the SR of the k-th subcarrier can be expressed as where C m [] P h σx SR[k] C m [] P h σx +, 6 σ Z [ ] C m [] sinπɛ 7 sinπɛ/ from. To show the dependence of the SR on the frequency offset explicitly, f ɛ is defined as f ɛ and the SR is expressed as sinπɛ sinπɛ/, 8 f SRɛ ɛp hσx f ɛp hσx +, 9 σ Z where the subcarrier index k is dropped since the SR is the same for all subcarriers. From this SR expression, it is clearly seen that the effect of the frequency offset is to decrease the signal power by f ɛ and to convert the decreased power to interference power. The SR depends not only on the frequency offset ɛ but also on the number of subcarriers,, because f ɛ depends on. However, as increases, f ɛ converges to sinπɛ πɛ sincɛ because sinπɛ/ converges to πɛ. Then the SR in 9 converges to sinc ɛp h σx SRɛ sinc ɛp h σx +, 3 σ Z as tends to infinity. Using the SR expression, the SR degradation Dɛ due to the carrier frequency offset can be found: Dɛ SR SRɛ + f ɛ P hσ X σ Z f ɛ 3 For small ɛ such that ɛ π, the SR degradation can be approximated using the Taylor series expansion Dɛ D + D ɛ + D ɛ as follows: Dɛ + π 3 + SR ɛ, 3 where SR P hσ X is the SR in the absence of the carrier frequency offset. When ɛ 3 π +SR, the SR degradation in db is D db ɛ π ln 3 + SR ɛ, 33 x because log +x ln for x. This expression shows that the SR degradation in db is proportional to the SR in the absence of the frequency offset. In case of the AWG channel, the channel response h m [n] δ[n] for all m. Then the SR expression becomes f SRɛ ɛσ X f ɛσ X σ Z In [4], the approximate expression for the SR degradation in db is derived as D db ɛ π σx ln 3 ɛ. 35 This agrees well with 33 for the AWG channel when and σx /σ Z. However, this approximate expression is 3 valid only for ɛ, whereas 34 is valid for any π σx frequency offset ɛ. V. UMERICAL RESULTS In this section, the SR degradation is numerically evaluated for various values of relative frequency offset ɛ and nominal SR P hσ X. The number of subcarriers is chosen to be equal to 56. Fig. shows the SR degradation due to the carrier frequency offset for the AWG channel. The SR degradation is calculated by Monte Carlos simulation and by the numerical evaluation of the exact expression 3 and the approximate expression of [4]. As can be seen from the figure, the theoretical results using the exact expression agrees well with the simulation results, whereas the approximate expression is not always accurate. For example, when the nominal SR, SR, is db, the approximate expression 33 has an error of.9 db for ɛ.5. This clearly shows that the exact expression should be used in order to assess the effect of the carrier frequency offset accurately. In Fig. 3, the exact SR degradation is plotted as a function of the relative frequency offset ɛ for ɛ.5. The nominal SR P hσ X is chosen as 5 db, db, 5 db and db. As can be seen from the figure, the SR degradation increases as the frequency offset ɛ increases. Furthermore, carrier frequency offset causes more degradation to a system operating at high SR than a system operating at low SR. This can be expected from 9 since the noise power is small Globecom /4/$. 4 IEEE

5 SR Degradation db db, Theory db, Approximation db, Simulation db, Theory db, Approximation db, Simulation db, Theory db, Approximation db, Simulation Frequency offset ε* causing a given degradation in SR db db db. db.5. Relative frequency offset ε Fig.. SR degradation due to carrier frequency offset for the AWG channel when the SR in the absence of the frequency offset is db, db, and db. The SR degradation is calculated by performing the Monte Carlo simulation and by using the exact expression 3 and the approximate expression of [4]. SR Degradation db db 5 db db 5 db Relative frequency offset ε Fig. 3. SR degradation vs. carrier frequency offset when the SR in the absence of the carrier frequency offset is chosen to be 5 db, db, 5 db, and db. for high SR and the effect of the ICI is more visible for small noise. In fact, for high SR and large carrier frequency offset, the SR expression 9 becomes SRɛ f ɛ f 36 ɛ, regardless of the SR in the absence of the carrier frequency offset. Thus, the SR degradation increases as the operating SR in the absence of the frequency offset increases. Fig. 4 shows the frequency offset values ɛ that incur 3 db, db, db, and. db degradation in SR. The frequency offset ɛ indicates the maximum allowable frequency offset that SR in the absence of frequency offset db Fig. 4. The frequency offset values that cause 3 db, db, db, and. db degradation in SR. guarantees less than a given degradation in SR. For example, if the nominal SR is db and only db degradation in SR is allowed, the relative frequency offset should be limited to be less than.8. This figure can be used to determine the target frequency offset for a given SR and SR degradation. VI. COCLUSIO This paper analyzed the effect of the carrier frequency offset on OFDM systems for a general multipath fading channel. The carrier frequency offset attenuates the desired signal and causes inter-carrier interference ICI, thus reducing the SR. A simple yet exact SR expression was derived in the presence of the carrier frequency offset. The SR expression can be used in the design of practical OFDM systems since it is accurate for any frequency offset values. REFERECES [] J. A. C. Bingham, Multicarrier modulation for data transmission: An idea whose time has come, IEEE Commun. Mag., vol. 8, pp. 5-4, May 99. [] R. ee and R. Prasad, OFDM for Wireless Multimedia Communications. orwell, MA: Artech House,. [3] W. Zou and Y. Wu, COFDM: an overview, IEEE Trans. Broadcast., vol. 4, pp. -8, Mar [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. 9-93, Feb./Mar./Apr [5] J. -J. Beek, M. Sandell, and P. O. Börjesson, ML estimation of time and frequency offset in OFDM systems, IEEE Trans. Signal Processing, vol. 48, pp. 8-85, Jul [6] T. Schmidl and D. C. Cox, Robust frequency and timing synchronization for OFDM, IEEE Trans. Commun., vol. 45, pp. 63-6, Dec [7] H. ikookar and R. Prasad, On the sensitivity of multicarrier transmission over multipath channels to phase noise and frequency offsets, in Proc. IEEE GLOBECOM 96, ov. 996, pp [8] W. Hwang, H. Kang, and K. Kim, Approximation of SR degradation due to carrier frequency offset for OFDM in shadowed multipath channels, IEEE Commun. Letters, vol. 7, pp , Dec. 3. [9] T. Rappaport, Wireless Communications. Upper Saddle River, J: Prentice-Hall, 996. Globecom /4/$. 4 IEEE