Effect of Carrier Frequency Offset on Frequency-Domain Differential Demodulation in OFDM Systems

Transcription

1 Effect of Carrier Frequency Offset on Frequency-Domain Differential Demodulation in OFDM Systems Jungwon Lee, Hui-Ling Lou, and Dimitris Toumpakaris Marvell Semiconductor, Inc., 7 First Avenue, Sunnyvale, CA 9489, USA {junglee, hlou, dimitris}@marvell.com Abstract This paper analyzes the effect of the carrier frequency offset on frequency-domain differential demodulation in orthogonal frequency division multiplexing OFDM systems. The exact signal-to-noise ratio SR expression is derived in the presence of the frequency offset. Using this exact SR expression, the SR degradation due to the frequency offset is quantified and the symbol error rate SER performance is evaluated. The effect of the frequency offset on frequency-domain differential demodulation is compared with that on coherent demodulation. It is found that frequency-domain differential demodulation is less immune to the carrier frequency offset compared to coherent demodulation. The analysis in this paper not only provides insights into how the carrier frequency offset affects the system performance, but also provides guidelines on the amount of tolerable frequency offset to obtain the desired system performance. I. ITRODUCTIO There is growing interest in using orthogonal frequency division multiplexing OFDM for high-speed communication systems in frequency selective channels []. By dividing a frequency selective channel into many subchannels, OFDM systems can remove inter-symbol interference ISI readily. Moreover, the modulation of an OFDM system can be implemented using the computationally efficient fast Fourier transform FFT. In fact, OFDM systems are already widely used in practical wireless systems including wireless local area networks WLA, fixed wireless access systems, digital audio broadcasting systems DAB and digital video broadcasting systems DVB. Similar to single-carrier systems, differential demodulation in OFDM systems eliminates the need for channel estimation, resulting in a simpler receiver and less training overhead compared to coherent demodulation, but at the expense of approximately 3 db degradation in signal-to-noise ratio SR []. Differential encoding can be done in the time-domain or the frequency-domain. The time-domain approach encodes a symbol differentially over two consecutive OFDM symbols at the same subcarrier, as the frequency-domain approach encodes a symbol differentially over two adjacent subcarriers of the same OFDM symbol. Time-domain differential demodulation performs well when the channel changes slowly over time so that the channel phase of two consecutive OFDM symbols is about the same. On the other hand, frequencydomain differential demodulation is a good choice when the multipath spread is small compared to the length of one OFDM symbol so that the channel phase between two adjacent subcarriers in the same OFDM symbol is about the same. Although much research has been done on the effect of the multipath and Doppler spread on differential demodulation [], [3], little is known about the effect of the carrier frequency offset, i.e., the difference between the transmitter and the receiver carrier frequency. The receiver carrier frequency should be synchronized with that of the transmitter to avoid performance degradation even though a differential demodulation recevier does not need to track the transmitter carrier phase. However, previous research on carrier frequency offset has mostly focused on coherent demodulation [4] [7]. The effect of the carrier frequency offset on the time-domain differential demodulation is studied in [8], while this paper focuses on the frequency-domain differential demodulation. A simple yet exact SR expression for frequency-domain differential modulation OFDM systems is derived in the presence of the carrier frequency offset. Using this SR expression, the SR degradation due to the carrier frequency offset is evaluated, and the symbol error rate SER is calculated. In the analysis of SR, the channel is assumed to be a flat fading channel in order to isolate the effect of the carrier frequency offset from the unequal channel response between adjacent subcarriers. The paper is organized as follows. Section II describes the OFDM system model. In Section III, the SR and SER expressions in the presence of frequency offset are derived for frequency-domain differential demodulation, and the effect of the frequency offset on frequency-domain differential demodulation is compared with that on coherent demodulation. Section IV presents some numerical results on the SR degradation and the SER increase due to carrier frequency offset. Section V concludes the paper. II. SYSTEM MODEL An OFDM system transmits information as a series of OFDM symbols []. Fig. depicts 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 m-th OFDM

2 Xm[] Xm[] Xm[ ] Ym[] Ym[] Ym[ ] Fig.. IDFT DFT xm[] xm[ + g ] ym[] ym[g ] ym[ + g ] MUX DEMUX xm[n] ym[n] ~ym[n] hm[n] ~zm[n] Baseband equivalent model of an OFDM system j[ß f[n+m+g]t + ] e symbol: k X m[k]e jπkn g/, x m [n] if n + g, otherwise, 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 corrupted by Gaussian noise z m [n]. The channel h m [n] is assumed to be blockstationary, i.e., time-invariant during 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], denotes the convolution operation, i.e., 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. When the receiver oscillator is not perfectly matched to the transmitter oscillator, there can be a carrier frequency offset Δf 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 and the receiver carrier. The received symbol y m [n] is then y m [n] e j[πδfn+m+gt +θ] h m [n] x m [n]+ z m [n], 3 T is the sampling period. The frequency offset Δf can be represented with respect to the subcarrier bandwidth / T by defining the relative frequency offset ɛ as ɛ Δf ΔfT 4 / T Using the relative frequency offset ɛ, the received symbol 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], 5 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 σz σ Z and is independent of the transmit signal and the channel. To simplify the notation, c m [n] is defined as c m [n] ejπɛn/ e jπɛm+α e jθ. 6 The received sample y m [n] is then y m [n] c m [n]h m [n] x m [n] + z m [n]. 7 III. FREQUECY-DOMAI DIFFERETIAL DEMODULATIO This section quantifies the effect of the frequency offset on the SR and SER of differential phase-shift-keying DPSK modulation in OFDM systems. The relative frequency offset ɛ can be divided into an integer part l and a non-integer part ɛ such that / ɛ </: ɛ l + ɛ. 8 It can be easily seen from 4 that the absolute frequency offset that corresponds to the above relative frequency offset is l+ ɛ times the subcarrier bandwidth / T. To simplify the notation, H m [k] and X m [k] are assumed to be periodic with period. The discrete Fourier transform DFT of y m [n] in the presence of the carrier frequency offset ɛ l + ɛ is then Y m [k] C m [l]h m [k l]x m [k l]+i m [k]+z m [k], 9 I m [k] r l C m [r]x m [k r], and C m [k], H m [k], and Z m [k] are the DFTs of c m [n], h m [n], and z m [n]. From the definition of the DFT, it can be derived in a straightforward way [7] that C m [k] n c m [n]e πnk/ sinπɛ k sinπɛ k/ ejπɛ k / e j[πɛm+α+θ]. The integer frequency offset l causes the cyclic shift of the subcarriers, as the residual frequency offset ɛ reduces the desired signal and introduces ICI, resulting in SR degradation. It is assumed in the rest of the paper that the integer frequency offset l is zero for the sake of simplicity since l does not affect the SR and SER analysis. The decision metric M m [k] for frequency-domain differential demodulation is then M m [k] Ym[k ]Y m [k] C m [] Hm [k ]H m[k]xm [k ]X m[k] +I m[k]+z m[k],

3 the ICI I m[k] is I m [k] C m []H m [k ]X m [k ]I m[k] +C m []H m [k]x m [k]im[k ] +Im [k ]I m[k], 3 and the noise Z m[k] is Z m[k] Ym[k ]Z m [k]+y m [k]zm[k ] Zm [k ]Z m[k]. 4 Thus, the signal Hm[k ]H m [k]xm[k ]X m [k] is reduced by C m [] and corrupted by the ICI I m [k] and the noise Z m [k], but the frequency offset does not introduce any phase change in the desired signal. It is assumed that the adjacent subcarriers have the same channel response, i.e., H m [k ] H m [k] for all k, in order to focus on the effect of the frequency offset. The effect of the channel difference between adjacent subcarriers in differential demodulation has been investigated in [3]; only the effect of the frequency offset will be analyzed in this paper. With the assumption that H m [k ] H m [k] for all k, the channel becomes a flat-fading channel and can be represented as { hm for n h m [n] 5 otherwise and H m [k] h m for all k. 6 The power of the desired signal is [ Cm E [] Hm [k ]H m[k]xm [k ]X m[k] ] a C m [] 4 E[ h m 4 ]E[ Xm [k ]X m[k] ] C m [] 4 E[ h m 4 ]σx 4 7 a follows from the fact that the channel is independent of the transmit symbol. By defining P h E[ h m 4 ] and sinπɛ f ɛ C m [] sinπɛ/, 8 the power of the desired signal can be expressed as [ E C m [] Hm [k ]H m[k]xm [k ]X m[k] ] fɛp 4 h σx. 4 9 Both the ICI and the noise are zero-mean, and they are independent from each other because the noise is independent from the channel and the transmit symbols. Thus, the power of the ICI-plus-noise signal I m[k]+z m[k] is given as E[ I m[k]+z m[k] ]σi + σ Z, σi and σ Z are the variance of I m [k] and Z m [k], respectively. The power of the ICI can be calculated by subtracting the power of the desired signal and the power of the noise from the power of the received signal, since the ICI is uncorrelated with the desired signal and the noise. Thus, the ICI power can be expressed as σi E[ Y m[k ]Y m [k] Z m[k] ] f 4 ɛphσ X. 4 In the Appendix, the ICI power is shown to be equal to σ I [ a cosπɛ f 4 ɛ ] P hσ 4 X, a 3 k From 4, the noise variance is k kcos πk. 3 σ Z σ Xσ Z + σ 4 Z. 4 Thus, from 9,,, and 4, the SR for frequency-domain differential demodulation is SRɛ f 4 ɛp hσx [ a cosπɛ f 4 ɛ] P hσx +σ Z +. σ4 Z P h σx 5 The SR depends on the number of subcarriers,, since f ɛ and a depends on. As increases to infinity, the SR converges to SRɛ sinc 4 ɛp h σ X cosπɛ π sinc 4 ɛ P h σx +σ Z + σ4 Z P h σ X, 6 because f ɛ converges to sincɛ and a converges to π as is shown below: lim a lim k k k cos x xcosπxdx πk π, 7 the integral is easily calculated by the method of integration by parts. By evaluating a numerically, it is found that it can be approximated as π for 64 with less than.% error. Similarly, it is found that f ɛ can be approximated as sincɛ for 64 with less than.% error. Thus, the SR does not depend on the number of subcarriers much as long as 64, and 6 can be used to determine the SR with sufficient accuracy for 64. The SR degradation Dɛ due to the frequency offset ɛ is Dɛ SR SRɛ +[ a cosπɛ f 4 ɛ] P hσx σz f 4 ɛ 8 Integration by parts: R uv uv R u v

4 for σ X P h. The SR degradation can be approximated σz using the Taylor series expansion Dɛ D + D ɛ + D ɛ for ɛ : Dɛ + 3π π P h σx 3 σz ɛ, 9 3σZ for 6. When ɛ π P h, the SR degradation in σx db is approximated as D db ɛ 3 π π ln 3 P h σx σz ɛ. 3 Table I summarizes the effect of the frequency offset on the SR for coherent demodulation and frequency-domain differential demodulation. The values for coherent demodulation are quoted from [7]. As can be seen from the expression of the SR, frequency-domain differential demodulation has approximately 3 db lower SR than coherent demodulation in the absence of the carrier frequency offset. However, it can be seen from the SR degradation expression that the SR degradation of frequency-domain differential demodulation is less than twice of the SR degradation of coherent demodulation when the frequency offset ɛ is small. This fact can also be seen from the SR expression because f ɛ and f 4 ɛ f ɛ. Ifa cosπɛ did not exist, the SR of differential demodulation for small ɛ would remain to be half of the SR of coherent demodulation in the presence of the small frequency offset. However, a cosπɛ reduces the power of the ICI in differential demodulation, resulting in SR degradation that is less than twice of the SR degradation of coherent demodulation. For the AWG channel, a SER expression can be found by approximating the ICI-plus-noise signal I m[k] +Z m[k] as a zero-mean Gaussian random variable with variance σ I + σ Z. The SER for binary differential phase shift keying BDPSK is P e Q SR, 3 as the SER for M-ary differential phase shift keying MDPSK for M 4 is P e Q sin π SR. 3 M SR is given by 5 [9]. However, for small constellation sizes such as M 4, the SR should be adjusted to obtain the accurate SER. Although differential PSK suffers from 3 db loss in SR compared to coherent PSK for M>4, the SR loss for four-phase DPSK is approximately.3 db, and the SR loss for binary DSPK varies from less than db to db depending on the SR [9]. Thus, the SR in 5 should be increased by.7 db for four-phase DPSK and db to db for BDPSK before putting it in 3 and 3, respectively, since 5 assumes 3 db loss. For a flat-fading channel, the average SER for BDPSK can be calculated as P e Q SRp h 4αdα, 33 SR Degradation db SR 5 db, FDDD SR 5 db, Coherent SR db, FDDD SR db, Coherent SR 5 db, FDDD SR 5 db, Coherent Relative frequency offset ε Fig.. SR degradation of coherent demodulation and frequency-domain differential demodulation FDDD schemes due to the frequency offset. The SR in the absence of the carrier frequency offset, SR is chosen to be 5,, and 5 db. SR is the same as 5 with P h replaced by α, and p h 4α is the probability density function of h. The average SER for MDPSK can be calculated in a similar way. IV. UMERICAL RESULTS Since the SR degradation expression in Table I is approximate and is valid for small frequency offsets, the exact SR degradation values are plotted as a function of the frequency offset ɛ for ɛ.5 in Fig.. For comparison, the SR degradation for coherent demodulation is also presented. For both demodulation schemes, the SR degradation increases as the frequency offset increases. Moreover, frequency offset causes higher SR degradation to the systems operating at high SR than at low SR. Frequency-domain differential demodulation is less immune to the frequency offset than coherent demodulation for a given SR. Since the frequency offset has different effects on SR for different nominal SR, the frequency offset value ɛ that causes a given SR degradation is plotted as a function of nominal SR in Fig. 3. As is expected, ɛ decreases as the nominal SR increases in both demodulation schemes. However, ɛ is smaller for the case of frequency-domain differential demodulation than coherent demodulation. Therefore, it can be concluded that the frequency synchronization requirement of differential demodulation is more stringent than that of coherent demodulation. Fig. 4 shows the SER plot in the presence of the frequency offset for frequency-domain differential demodulation and four-phase DPSK. The number of subcarriers,, is chosen to be 64. The lines were calculated using the theoretical expression 3, while the discrete points were found by simulation. As can be seen in the figure, the simulation results agree well with the results obtained by evaluating the theoretical expression even when the number of subcarriers,, is only

5 TABLE I SR AD SR DEGRADATIO FOR COHERET DEMODULATIO AD FREQUECY-DOMAI DIFFERETIAL DEMODULATIO FDDD. Coherent FDDD SR with no frequency offset SR with frequency offset ɛ SR Degradation db SR SRɛ for small ɛ and large SR E[ h m ]σx f ɛsr σ Z f ɛsr π + ln 3 SR ɛ E[ h m 4 ]σx 4 f 4 ɛsr [ a cosπɛ f 4 ɛ]sr π + ln 3 3a SR ɛ E[ h m 4 σ X σ Z +σ4 Z Frequency offset ε* causing a given degradation in SR.3.. db, Coherent db, FDDD db, Coherent db, FDDD. db, Coherent. db, FDDD ominal SR db Symbol Error Rate ε ε.5 ε. ε.5 ε SR db Fig. 3. The frequency offset that causes db, db, and. db SR degradation. Symbol error rate SER for frequency-domain differential demodu- Fig. 4. lation 64. Since the frequency offset has a more adverse effect at low target SERs, the frequency offset correction should be more accurate for low target SERs than for high target SERs. V. COCLUSIO This paper analyzed the effect of the carrier frequency offset on frequency-domain differential demodulation in OFDM systems. The exact SR expression as a function of the carrier frequency offset was derived, and it was used to calculate the SR degradation due to the frequency offset. The exact SR expression was also used to compute the SER under the assumption that the ICI-plus-noise signal is Gaussian. The effect of the carrier frequency offset on frequency-domain differential demodulation was compared to that on coherent demodulation. The SR degradation was found to increase as the nominal SR increases, similar to the case of coherent demodulation. However, compared to coherent demodulation, the frequency-domain differential demodulation was shown to be less immune to the carrier frequency offset. APPEDIX DERIVATIO OF THE ICI POWER σi In this Appendix, a closed-form expression for σ I is derived. To simplify the notation, the OFDM symbol index m is omitted. As stated in Section III, the variance of the ICI can be expressed as σ I E[ Y [k ]Y [k] Z [k] ] f 4 ɛσ4 X. 34 The first term of the right-hand side can be expressed as follows. P h E[ Y [k ]Y [k] Z [k] ] q r s t C[q]C [r]c [s]c[t] E[X[k q]x [k r ]X [k s]x[k t ]] Ph σ4 X C[q]C[r] + q q s r C[q]C [q ]C [s]c[s ] C[q]C [q ], 35 q the following fact was used: E[X[k q]x [k r ]X [k s]x[k t ]] δ[q s]δ[r t]+δ[q r ]δ[s t ] δ[q s]δ[r t]δ[s t ] σx 4. 36

6 The first term of 35 is equal to, since q r C[q]C[r] q C[q] r C[r], 37 as the second term of 35 is equal to, because q C[q]C [q ] q m m m n m πm e e jπɛ qn/ e πɛ q+m/ n πm e πm e n e j πɛn m q πqn m e e j πɛn m δ[n m]. 38 Finally, as will be shown below, the third term can be expressed as q C[q]C [q ] a cosπɛ, 39 a 3 k Thus, 35 can be simplified to k kcos πk. 4 E[ Y [k ]Y [k] Z [k] ] [ a cosπɛ] Phσ X 4 4 By substituting 4 into 34, the variance of the ICI is obtained: σ I [ a cosπɛ f 4 ɛ] P h σ4 X 4 ow, this Appendix is concluded with the proof of 39: q 4 q C[q]C [q ] n m s t e j πɛn s m+t πm t e e j πqm t+n s 43 Since πqm t+n s q ej δ[t m + n s ]+δ[t m + n s]+δ[t m + n s + ], q 3 C[q]C [q ] n s πn s { e n s + e jπɛ + [ n s + s n + ]+s n + e πɛ} 3 n s πn s { e n s + e jπɛ s n + e πɛ }, 44 x + It can be easily seen that n q C[q]C [q ] { x for x otherwise. 45 πn s s e. Thus, [ πn s 3 e n s + e jπɛ n s ] +s n + e πɛ [ πn s 3 e n s + cosπɛ n s ] sinπɛ + s n + cosπɛ+jsinπɛ 3 n s e πn s n s cosπɛ a cosπɛ, 46 a 3 a 3 n s πn s e n s. 47 By manipulating the above equation, it can be shown that [ 3 k k ke j πk k kcos k REFERECES πk + k πk k ke ]. 48 [] 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] M. Lott, Comparison of frequency and time domain differential modulation in an OFDM system for wireless ATM, in Proc. IEEE 49th Veh. Technol. Conf., 999, pp

7 [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] 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 [6] 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. [7] J. Lee, H. Lou, D. Toumpakaris, and J. M. Cioffi, Effect of carrier frequency offset on OFDM systems for multipath fading channels, in Proc. IEEE GLOBECOM 4, ov. 4. [8] J. Lee, H. Lou, D. Toumpakaris, and J. M. Cioffi, Effect of carrier frequency offset on time-domain differential demodulation in OFDM systems, in Proc. IEEE Veh. Technol. Conf. 4 Fall, Sep. 4. [9] J. G. Proakis, Digital Communications, 4th ed. ew York:McGraw-Hill,.