Volker Aue and Gerhard P. Fettweis Dresden University of Technology, Germany Reinaldo Valenzuela Lucent Technologies, Holmdel, NJ, USA

Transcription

1 A Comparison of the Performance of Linearly qualized Single Carrier And Coded OFDM Over Frequency Selective Fading Channels Using the Random Coding Technique Volker Aue and Gerhard P. Fettweis Dresden University of Technology, Germany Reinaldo Valenzuela Lucent Technologies, Holmdel, NJ, USA Astract: Different attempts have een taken to compare performances of coded orthogonal frequency division multiplexing (COFDM) and a corresponding single carrier system employing a linear equalizer often y comparing average it error rates. In this paper, the random coding technique is used to analyze the performances of these systems for signaling over piecewise constant s. The proailities that the cutoff rates are less than a desired rate are calculated and compared. It is shown that for low SNRs and low code rates, the cutoff rates of oth systems are approximately the same. With an increasing SNR the cutoff rate for the linearly equalized system, however, converges faster to its maximum value than the cutoff rate for the COFDM system. Simulation results for outage proaility are presented for the two-ray Rayleigh fading with QPSK as the fundamental modulation technique. For code rates close to one or for uncoded transmission, the linearly equalized single carrier system sustantially outperforms the COFDM system. For low to medium code rates COFDM and a linearly equalized single carrier system perform approximately identically. I. Introduction Orthogonal frequency division multiplexing has received a lot of attention, as a method for comating intersymol interference (ISI) due to signaling over time-dispersive communications s. The conventional single carrier system employs an equalizer at the receiver which attempts to mitigate the multipath effects y reconstructing the transmitted sequence. Linear equalization, decision feedack equalization (DF) or maximum likelihood sequence estimation (MLS) are most commonly used []. The principal idea ehind OFDM is to transmit locks of data over a piecewise constant, i.e., the is assumed to e stationary during the transmission of a lock. A stream of high rate data is demultiplexed into several streams of lower rate which are then transmitted in parallel over a set of orthogonal carriers. If a sufficient numer of carriers is used, each su can e regarded flat. A guard interval consisting of a cyclic prefix (suffix) larger than the maximum delay span precedes (succeeds) each symol such that all ISI is removed. The optimum carrier assignment makes use of water pouring [], ut, as the response needs to e known to the transmitter, feedack from the receiver is required. Zervos and Kalet [3] give a theoretical evaluation of this technique assuming a uniform power distriution. It is shown that for high signal-to-noise ratios (SNRs) the performance of OFDM is approximately that of a single carrier system using a DF [3,4]. Willing and Wittke [5] show that OFDM can outperform a single carrier system with DF, if the constraint that the it error rate (BR) is uniformly distriuted over carriers is removed and a different optimization method is used. A general treatment on DF including lockwise transmission using cyclic prefixes is given in [6]. A asic evaluation of the proaility of system outage for a typical office-uilding indoor environment of OFDM with different carrier assignments has een carried out y Cimini [7]. If no knowledge aout the response can e otained y the transmitter, e.g., for rapidly fading s or in a roadcast environment, the water pouring concept is infeasile. Henceforth, a way to reduce the proaility of it error is to use extensive coding to provide the redundancy needed to overcome losses which occur on carriers where the frequency response is suject to deep fades. This technique usually referred to as coded OFDM (COFDM) has een proposed for different roadcasting systems [8] and is currently under investigation for moile wireless systems [9,]. Sari et al. have investigated the performance of a COFDM system over two different frequency selective s and report a performance similar to that of an equivalent single carrier system employing a linear equalizer. It is also noted y the same authors that coding is a necessity in COFDM whereas in a single carrier system it is not. The random coding technique [] provides a general means to compare performances of systems which use coding. For environments that vary over time or place, as it is the case for many wireless applications, it is meaningful to consider the proaility that a certain requirement is met (or not met) rather than looking at average error rates. Assuming the is constant during the transmission period of a lock, the proaility that the cutoff rate is less than a desired rate then corresponds to the proaility of detecting erroneous locks. If locations of transmitter and receiver are fixed, e.g. in fixed wireless, this proaility directly corresponds to the proaility of system outage for the anticipated code rate. In the following section we first derive general expressions for the cutoff rate for oth systems. For a simple frequencyselective fading model, the two-ray fading, further results are derived. For different code rates the proaility of outage is then evaluated y means of simulations. Results are shown and discussed.

2 encoder decoder encoder decoder symol mapping interleaver IFFT add guard interval deinterleaver symol mapping deinterleaver (a) COFDM system model. interleaver IFFT equalizer FFT FFT () Single carrier system model. remove guard interval add guard interval remove guard interval are equally likely, the cutoff rate is found to e q q ( q exp ) s l s k, () 4N where s l s k is the uclidean distance etween a symol pair s k,s l. N is the noise power spectral density. For a flat fading AWGN, the derivation can e carried out in an analogous manner [], and it can e shown that q q q [ )] exp ( α s l s k, 4N where α is a random variale according to the distriution of the magnitude of the normalized. denotes expectation. () Fig.. System models. II. Analysis In order to e as general as possile we neglect implementation issues as we assume perfect interleaving, perfect synchronization, and that the receiver has perfect knowledge of the. Coherent detection is assumed. The model for the COFDM system is shown in Fig. (a). The corresponding model for the linearly equalized single carrier system for which equalization is carried out in the frequency domain is shown in Fig. (). Both systems use a lock signaling approach, where each lock precedes a cyclic prefix. Furthermore it is assumed that the is constant over the regarded time interval. Thus, the received signal can e viewed as eing cyclicly convolved with the impulse response. As we further generalize our approach we assume an infinite lock length, i.e., an infinite numer of carriers for the COFDM system, and an infinite numer of symols for the single carrier system. For each transmitted lock the overhead introduced y the guard interval is neglected as it is identical for oth systems. Furthermore, it is assumed that the linear equalizer has infinite length or directly operates on the frequency response. A. Analysis Using the Random Coding Technique The random coding technique is ased on ounding the average proaility of it error over all possile lock codes. The cutoff rate can e defined as a parameter of this ound for which the ensemle average proaility of error over all possile lock codes goes to zero as the lock length reaches infinity. The advantage of comparing cutoff rate rather than capacity is that the former additionally takes the modulation technique into consideration. For q-ary signaling and for the AWGN, the cutoff rate is derived from an exponential ound []. If all symols Derivation of the Cutoff Rates for COFDM and Single Carrier () and () form the asis of our comparison. For the COFDM system, each su can e viewed as a flat AWGN. Assuming perfect interleaving, all coefficients can e assumed independent and identically distriuted. Thus, we can use () to calculate the cutoff rate for the system shown in Fig. (a). The remaining task is to find the proaility density function (pdf) of the power density spectrum H(f). For some models, the pdf can e descried analytically and () can e calculated in closed form. In Section II.B we derive an expression for the two-ray. In a single carrier system each transmitted symol occupies the entire andwidth and thus each symol is suject to the same fading. The two major criteria for adjusting the filter coefficients are zero ISI or minimum mean square error (MMS). The MMS criterion trades off noise enhancement for residual ISI, and is known to yield a etter performance in s which exhiit deep spectral fades. For an infinite tap linear filter output the MMS is well known []. The MMS can then e used to calculate the SNR at the output of the equalizer [3]. For the linear equalizer, the SNR at the equalizer output is γ e =, (3) +γ H(f) where γ = s N is the received SNR. s is the average symol energy. If we assume perfect interleaving and a Gaussian distriution of the output noise (3) can e used in () to compute the cutoff rate for the coded linearly equalized single carrier system. We are aware that, in general, the residual ISI is not Gaussian distriuted. By means of simulating the uncoded single carrier system employing the linear equalizer with MMS criterion, however, we verified that the Gaussian distriution is a valid assumption for all s under consideration. As the exponential ound is well aove the

3 Q-function which gives the exact result for the error integral, we can assume the validity of (). For the zero-forcing solution, the output noise is exactly Gaussian distriuted, and the output SNR γ e is γ e = γ. (4) H(f) B. Special Case: Two Ray Channel For the two-ray fading () and () can e calculated in closed form as shown as follows. The impulse response for the piecewise constant is h(t) =h e jϕ δ(t)+h e jϕ δ(t τ d ), (5) where h,h are the magnitudes of the two rays. τ d is the delay of the second path with respect to the first path. The frequency response of this is given y H(f) =h e jϕ + h e j(ϕ πfτ d) and its power spectral density is (6) H(f) = h + h +h h cos(πτ d f + ϕ ϕ ). (7) If the andwidth occupied y the signal is large with respect to /τ d, the distriution of H(f) is p x (x) = π ( x a ), a x a+, (8) where a = h + h and =h h. We introduce the normalized pairwise distance parameter d l,k = s l s k 4 s and calculate the expectations in () and (3). For COFDM we find exp [ γ H(f) ] d l,k = π a+ a wherewehavesetξ= x a. For the single carrier system +γ H(f) e xγd l,k ( ) dx x a = e (a+ξ)γd l,k π dξ ξ = e aγd l,k I (γd l,k ) = π = π a+ a +γx ( ) dx x a ( +aγ γ = ( ). +aγ γ ) dξ + ξ ξ (9) () Sustituting the result in (3) yields γ e = ( + aγ) γ. () Finally, we otain the general expressions for the cutoff rates for the two-ray. For COFDM q q e aγd l,k I (γ d l,k ). () q and for the linearly equalized single carrier system R = log q Discussion q q [ ( ) ] exp ( + aγ) γ d l,k. (3) For simplicity we set x i = h i γ,i =,, and define ã = x + x and =x x, and consider the terms and B(ã,, d l,k )=exp A(ã,, d l,k )=e ãd l,k I ( d l,k ) (4) [ ] ( + ã) d l,k (5) The cutoff rate increases as these terms decrease. As expected, in case of a single path, either h or h are zero, and thus =, for which () and (3) yield the same result. For dl,k close to zero, I ( d l,k ) and(+ã) ( + ã), and hence A B. For and d l,k large, we can use the asymptotic formula for which I ( d l,k ) A(ã,, d l,k ) π d l,k e d l,k, (6) π d l,k e (ã )d l,k It is easily verified that the negative exponent in B is larger than (ã ) d l,k. Therefore, it is expected that B<Afor large, i.e., we can expect the cutoff rate to converge faster to its maximum value for the linear equalizer, as the SNR increases. In case that oth rays carry the same power, i.e., h = h, =ã,aconverges only with whereas B πãdl,k converges with e ãd l,k for large. Note that the equivalent term for the zero-forcing equalizer is exp (ã ) d l,k which equals one for ã = and does not converge. For d l,k =,A(x,x )andb(x,x ) are analyzed numerically. The ratio B(x,x) A(x,x ) is shown in Fig.. It is seen that In all practical systems, the normalized uclidean distance d l,k is consideraly larger than zero. Thus, it is sufficient to assume large.

4 proaility that cutoff rate is less than desired rate L R=. OFDM R=. L R=/ OFDM R=/ L R=3/4 OFDM R=3/4 L R=.999 OFDM R=.999 x 3.5 x SNR per it [db] Fig. 3. Two-ray Rayleigh fading. Fig.. Ratio B(x,x ) A(x,x ). for x,x large, B(x,x ) converges consideraly faster to zero than A(x,x ) does as the ratio goes to zero as either x or x ecome large. For small values of x,x and for x = x, B(x,x ) A(x,x ). It can e seen that for values around x,x.6, A(x,x ) >B(x,x ). It has een found numerically that the maximum relative deviation occurs at x i =.654, for which A(x,x ) exceeds B(x,x )y 3.4 percent. Thus, if a high likelihood exists for x,x to e in that range, the cutoff rate for COFDM can exceed the cutoff rate of the single carrier system. We can conclude that for low SNRs, the cutoff rate of the COFDM system can e expected to e similar to the cutoff rate of the equivalent linearly equalized single carrier system using the MMS criterion. For higher SNRs, the cutoff rate of the single carrier system can e expected to e higher. C. Results for QPSK As an example, we consider the cutoff rates for COFDM and the equivalent single carrier system with quarternary phase shift keying (QPSK) for signaling over two-ray Rayleigh fading s, and calculate the proaility that the cutoff rate is elow a desired rate, i.e., the outage proailities of these ideal systems. For QPSK, the cutoff rate for COFDM is 4 + [ e γ H(f) ] + 4 [e γ H(f) ], and for the two-ray, 4 + ( ) e γa I γ + 4 e γa I (γ). (7) (8) For the linearly equalized single carrier system, the cutoff rate is given y 4 + e γe + 4 e γe, (9) where γ e is given y (3) and (), respectively. For the tworay Rayleigh fading, the magnitudes h and h have the pdf p h (h) = h σ h e h σ h, h, () where σh =/4 for the normalized, i.e., for (h + h )=. Here, we consider code rates of R c =., /, 3/4 and.999. R c =/and3/4 are chosen as an example for code rates which are often associated with convolutional codes. R c =.999 in practice corresponds to systems that have to operate with little or without coding. R c =. has een chosen for the limiting case where almost all transmitted its are used for coding. The desired transmission rate (in its/dimension) is R d =R c, since two coded its are transmitted on each QPSK symol. The symol energy s =R c,where is the it energy. The results otained y Monte Carlo simulations are shown in Fig. 3. As expected, for the very low code rate (R c =.) the performance of COFDM and the equivalent single carrier system is equal. For the code rate of /, the COFDM system outperforms the single carrier system y an insignificant factor. For an SNR per it of 5 db, the cutoff rate of the COFDM system falls only in percent of all cases elow the desired rate. For a code rate of 3/4, db SNR more is required for the COFDM system for the same outage proaility. At the other extreme where almost no coding is used, the single carrier system can operate at db SNR per it whereas for the COFDM system 33 db are needed. This confirms the well known fact that OFDM without water pouring should not e used without coding [8].

5 We have also computed the outage proailities of COFDM and single carrier with QPSK over three and five ray Rayleigh fading s with same average power, i.e., h(t) dt =. For these s, (7) and (3) have een computed numerically. The results show the same characteristics as the results for the two-ray of Fig. 3 with the only difference that the required SNR per it for a desired outage proaility is less than for the two-ray Rayleigh fading. This, however, is expected, as the proaility, that a signal level falls elow a certain threshold decreases as the numer of rays increases. D. Other Signal Sets mploying signal sets other than QPSK as the fundamental modulation technique, e.g., to enale trellis coded modulation can usually yield different cutoff rates for oth systems. As the asic terms (4) and (5) are independent of the modulation technique, it is expected that the ehavior of COFDM with regard to the single carrier system does not change significantly. Moving more points ã,, d l,k for which A(ã,, d l,k ) B(ã,, d l,k ) into a region for which B(ã,, d l,k ) <A(ã,, d l,k ) can increase the difference in cutoff rate etween the COFDM system and the single carrier system. III. Conclusions We have analyzed the performance of coded OFDM and the corresponding linearly equalized single carrier system for piecewise constant frequency-selective s using the random coding technique for which we calculated the proailities that the cutoff rate is less than a desired threshold. For linear equalizer, the MMS criterion should e used as the preferred criterion for adjusting the filter coefficients. We have shown that despite the differences in signaling and signal processing, the performance of oth systems is approximately identical for low to medium code rates. For code rates close to one, the linearly equalized system using the MMS criterion sustantially outperforms the COFDM system. This, however, is expected, since in a single carrier system, the SNR is evenly distriuted over symols whereas in a COFDM system it is not. Hence, a high error proaility exists for symols on carriers for which the SNR is low. Simulations showed that the outage proaility is approximately the same for oth systems and for code rates up to one half. Further simulations reveal that these results can e generalized for s with more than two rays. Thus, it is shown that for low SNRs preferring COFDM over a linearly equalized single carrier system is an implementation issue. If less transmitted its are to e used for coding or at high SNRs, a single carrier system employing a linear equalizer with the MMS criterion can perform significantly etter, and should e the preferred solution. As the linear equalizer can e implemented in the frequency domain the complexity of the single carrier system is approximately the same as of the COFDM system. The overall system complexity further depends on synchronization issues, estimation, decoder complexity and on linearity requirements for the amplifiers. Acknowledgments The authors gratefully thank G. J. Foschini for his suggestions on carrying out this work. References [] S. U. H. Qureshi, Adaptive equalization, Proceedings of the I, vol. 53, pp , Sept [] J. A. Bingham, Multicarrier modulation for data transmission: An idea whose time has come, I Communications Magazine, pp. 5, May 99. [3] N. A. Zervos and I. Kalet, Optimized decision feedack equalization versus optimized orthogonal frequency division multiplexing for high-speed data transmission over the local cale network, I International Conference on Communications, vol., (Boston), pp. 8 85, June 989. [4] I. Kalet, The multitone, I Transactions on Communications, vol. 37, no. 3, pp. 9 4, Fe [5] T. J. Willink and P. H. 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