International Conference on Communication Technology ICCT'98 October 22-24, 1998 Beijing, China BER Analysis of OFDM Communication Systems with Intercarrier Interference Yuping Zhao") and Sven-Gustav Haggman(2) ")Nokia Research Centre, P.O. Box 407, FIN-00045 NOKIA GROUP, Finland e-mail: yuping.zhao@research.nokia.com '"IRC / Communications Laboratory, Helsinki University of Technology P.O. Box 3000, FIN-02015, Finland. e-mail: sgh@hut.fi Abstract -- For the Orthogonal Frequency Division Multiplexing communication systems (OFDM), the theoretical Bit Error Rate (BER) is difficult to obtain when Intercarrier Interference (ICI) is taken into account. The IC1 is a random process caused by channel frequency errors, its distribution function is unknown. This paper gives theoretical BER expressions of OFDM systems, by exploring IC1 performance in comparison with conventional single carrier communication systems. First, BER upper bound is obtained by assuming received signals to suffer maximum ICI. Then by using the sequence development BER calculation method, the accurate theoretical BER expression for OFDM systems is derived and validated by simulations. In addition, such a BER calculation method is applied to the "IC1 self cancellation OFDM system", which is a system proposed for reducing system IC1 signal level. It shows that BER of IC1 self cancellation OFDM systems is rather robust to channel frequency errors, while BER of normal OFDM systems increases rapidly with increasing frequency error. I. INTRODUCTIONS The most meaningful criterion to evaluate performance of communication systems is the Bit Error Rate (BER). For the Orthogonal Frequency Division Multiplexing (OFDM) communication systems [ 1,2], the main sources affecting its BER performances are Additive White Gaussian Noise (AWGN) and Intercarrier Interference (ICI) [3]. A theoretical BER derivation needs the knowledge of the distribution function of the IC1 signal. Unfortunately, it is normally unknown. In many papers, the IC1 is assumed to have a normal distribution, however such assumption is not accurate theoretically. With the help of well developed conventional single carrier communication systems [4,5], the BER expression of OFDM systems can be obtained without knowing the IC1 distribution function. In section 11, the frequency domain IC1 performance of OFDM systems is compared with the time domain intersymbol interference (ISI) of single carrier systems. It shows that IC1 coefficients given in the paper play similar role as the pulse shape function of single carrier systems. Section ID gives a BER upper bound of OFDM systems with respect to a given frequency error, by assuming the system to suffer maximum ICI. Section lv gives the accurate BER expression when considering both IC1 and AWGN. The BER is expressed in terms of first n'th order moments of the random IC1 signal, to removing the need to know the IC1 distribution function. The results are validated by simulations. In Section V, this BER calculation method is expanded to a special OFDM system, the IC1 self cancellation OFDM system [6]. Simulations given in [6] show that by using the IC1 self cancellation scheme, IC1 level of OFDM systems can be compressed significantly. The accurate BER values will be calculated in this section to evaluate system S38-02-1
performance with respect to different frequency errors. 11. REPRESENTATION OF IC1 SIGNALS It will be a good way to analyse IC1 of OFDM systems in the similar way as IS1 in single carrier systems. Assume that the channel has an available bandwidth B, and BPSK modulation is chosen. For a single carrier system, each transmitted symbol occupies the entire bandwidth, and the symbol duration is T=lIB. In such a system, the receiver sampling time must be very accurate since timing offset will cause ISI. However if OFDM system with N subcarriers is used, then each symbol occupies equivalently BIN bandwidth, and symbol duration is 7kN. Obviously for OFDM systems, the receiver frequency synchronisation is crucial. The IS1 signal level in the single carrier systems is determined by the sampling time error and the time domain pulse shape. For the ideal band-limit rectangular lowpass filter, the received pulse is a sinc(x) function sin(n(k + dt)) rk = (1) n(k + dt) ' where k denotes the sampling instance and dt the normalised sampling time error. Here k takes integer values, the IS1 has strong impact to the signal train of infinite length. subcarrier frequency separation and denoted by d! At the receiver side, DFT is performed to get the received data on each subcarrier Yk When assuming a0=1 and ak=o, k=1,... N-1 in Eq.(2), then the Yk is known as IC1 coefficient function and it is denoted by sk (3) Equation (3) indicates ICI signal values on all other subcarriers when transmitting a signal on subcarrier 0. When transmitting signals on the other subcarriers, a similar IC1 function is obtained. sk is a periodic function with period of N. Obviously, the role of this function is comparable with the pulse shape function given in Eq.( I). Fig.1 gives amplitudes of rk and sk, under the condition dt=d!0.2. The value N=I6 is used in the OFDM system. When k=o, lrkl and ISk( takes nearly same value. For k#o, Irk1 will simply decrease when k getting larger. For minimum value takes place when k=n/2. To analyse IC1 level with respect to the frequency error, it is necessary to have a corresponding basic IC1 function with respect to the system frequency error. In this paper, such a function is called IC1 coefficient function and it can be derived as follows. At the transmission side of OFDM systems, signals ak, k=o,... N-1 are modulated onto N subcarriers. It can be done by performing Inverse Discrete Fourier Transform (IDFT) to the signal sequence ak. To make a general analysis, the channel frequency error is normalised by the S38-02-2 I1 + r, (Single Carrier) "0 5 10 15 20 Subcarrier or Sampling Time Index Fig. 1 Comparison of rk and &. I
111. BER UPPER BOUND When a frequency error df exists, then without considering AWGN, the received signal on each subcarrier can be recognised as a sum of the expected signal d and the IC1 signal I, which can be calculated by value I, is only 1/2N, therefore this BER upper bound is rather loose. d = Souo, (4) N-I I = CS,a,. The amplitude of the received signals lies between two limit values Id+Im( and Id- Imw;l, where Inlax is the maximum IC1 signal with respect to d! N-1 L a x = CIS, I * If the frequency error is sufficiently large, it is possible that Zmx>d occurs. In such a case, a data decision error can be made even in the absence of AWGN. The df value where the condition Im,=d holds is called critical frequency error. This critical frequency error is a function of N. For example, it takes the value dfio.17 for N=32 and d!o. 10 for N=256. When the condition Z,<d holds, the upper BER bound of OFDM systems can be obtained by assuming that the received signal takes values rk=dk 1". For a BPSK- OFDM system, this BER upper bound is (7) Since I,, is a function of N, therefore this upper bound also varies as a function of N. Fig2 shows the cases when different N value is chosen (d!o.os), and the BER simulation results is plotted by "*'". Obviously, since the probability that I takes 1u 0 2 4 6 8 10 12 Fig. 2 BER upper bound for variant N values (dfi0.05). IV. ACCURATE BER OF OFDM SYSTEMS In the AWGN channel, BER of BPSK modulation OFDM systems is calculated by When df#o, the IC1 signals are added to the received signal. Since the IC1 signal is a random variable, therefore the accurate BER representation is the expectation of BER with respect to IC1 distributions. It can be expressed as (9) However since the distribution function of IC1 signals is unknown, therefore the closed form of integration is difficult to obtain. In [4], a sequence development method has been used for calculating BER of single carrier systems. Applying such a method to our case, the BER of an OFDM system can be obtained. The sequence development method uses the form [4,5] S38-02-3
erfc(x + y ) = erfc(x) + where H,(x) is the Hermite polynomial of order n. Substituting ~=d/fio and y = z/fio into Eq.( lo), the BER expression of the OFDM systems becomes where M2n is the 2n'th order moment of the IC1 signal and it can be calculated by [4] M2n = L L where B, is a Bernoulli number. Fig. 3 gives BER versus &/No with respect to different df values. The solid line shows the BER of an OFDM system in the AWGN channel given by Eq.(8), it is the same as single carrier BPSK system. For different df values, theoretical results are validated simulations given by "+" in the figure. The BER of an OFDM systems increases rapidly when frequency error increases. Normally, the condition df<0.05 is necessary to maintain acceptable system performance. V. BER OF IC1 SELF CANCELLATION OFDM SYSTEMS In order to reduce IC1 influence due to the frequency errors, the self IC1 cancellation scheme has been proposed [6]. The main idea of the IC1 self cancellation scheme is to modulate the same data onto two adjacent subcarriers with opposite phase. Even though this scheme reduces system bandwidth efficiency by factor two, however the reduction of IC1 by 15dB still makes the system attractive. To calculate BER performance of an OFDM system with IC1 cancellation scheme, the key point is to obtain corresponding IC1 coefficients. At the transmitter side of such a system, the condition uk=-uk+] (k=0,2,... N- 2) holds for each pair of (k,k+l) subcarriers. The received signals still can be calculated using Eq.(2), however the exact received information signal is the sum of the each pair of signals (Yk,Yk+l). That is the signals for making decision is Yk' given by Yk'=Yk-YkCl, k=0,2,..., N-2 (13) Therefore the corresponding IC1 coefficient can be derived as loo I I S, '=-Sk-, + 2Sk - Sk+l, k = 0,2,..., N - 2. (14) For IC1 self cancellation OFDM systems, the expected signals then is given as d'= (- SN-] + 2S0 -- S,)U,, (15) and IC1 signals are IU 0 2 4 6 8 1 0 1 2 1 4 Fig. 3. BER of OFDM systems. S38-02-4 N -2 Z'= E(- sk-l + 2s, - S,+,)U,.
The Maximum IC1 signal in this case can be calculated by Eq.(6) when using Sk in stead of Sk, and k only takes even integer values. Therefore the upper bound of IC1 self cancellation OFDM systems can be obtained as well. The accurate BER can also be derived by using Eq.(l 1) and Eq.(12), in the case of using d and Sk as d and Sk respectively. Fig.4 shows theoretical and simulation results for some frequency error values. Evidently, the system BER only shows slightly increases even in the case dfi0.1 to dfi0.25. 0.140 2 4 6 8 10 12 14 Fig. 4 BER curves of IC1 self cancellation OFDM system. A comparison between Fig. 3 and Fig.4 shows that the self IC1 cancellation OFDM system is rather robust to system frequency errors. The BER values in the IC1 self cancellation scheme on deo.2 is of similar order as that the df=0.05 in the normal OFDM system. Therefore it can be used in the case where channel frequency error is large. VI CONCLUSIONS This paper gives theoretical BER analysis of OFDM communication systems when both IC1 and AWGN signals are considered. Even though the BER upper bound gives a rather simple expression, however, it is not tight enough to estimate system BER. The sequence development method gives accurate BER result and it can be used for the case where the number of subcarriers larger than 8. In addition, the proposed BER calculation methods are also applied to a special case, the OFDM system using IC1 self cancellation scheme. The results show that BER of OFDM system with IC1 self cancellation scheme is rather robust to channel frequency errors, while BER of normal OFDM systems is rather sensitive to channel frequency error. 1. 2. 3. 4. 5. 6. REFERENCES R.W.Chang, R.A.Gibby: A Theoretical Study of Performance of an Orthogonal Multiplexing Data Transmission Scheme. IEEE Trans. Comm. Vol. com- 16, NO. 4, pp.529-540. August, 1968. Alard, R. Lassalle: Principles of modulation and channel coding for digital broadcasting for mobile receivers. EBU Review, technical. No. 224, pp.168-190. August, 1987. P. H. Moose, A technique for Orthogonal Frequency Division Multiplexing Frequency Offset Correction. ZEEE Trans. Commun., vol. 42, pp. 2908-2914. Oct. 1994. E. Y. Ho, Y. S. Yeh: A new approach for evaluating the error probability in the presence of intersymbol interference and additive Gaussian noise. BSTJ, Vol. 49, November 1970, pp2249-2265. J. C. Bic, D. Duponteil and J. C. Imbeaux: Elements of Digitul Communication. Y. Zhao, S-G. Haggman: Sensitivity to Doppler Shift and Carrier Frequency Errors in OFDM Systems -- The Consequences and Solutions. EEE 46th Vehicular Technology Conference, (VTC 96), pp.1564-1568. April 28-May 1, 1996, Atlanta, GA, USA. S38-02-5