Optimum Pre-DFT Combining with Cyclic Delay Diversity for OFDM Based WLAN Systems

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1 Optimum Pre-DFT Combining with Cyclic Delay Diversity for OFD Based WLAN Systems uhammad Imadur Rahman, Klaus Witrisal, Suvra Sekhar Das, Frank H.P. Fitzek, Ole Olsen, Ramjee Prasad Center for TeleInFrastructure (CTiF), Aalborg University Neils Jernes Vej, 90 Aalborg Øst, Denmark; ph: , Institute for Telecommunications and Wave Propagation, Graz University of Technology Inffeldgasse, A-800 Graz, Austria; ph: , Abstract The cyclic delay diversity is applied in OFD receiver to increase the frequency-selectivity of the channel seen at the receiver for flat (or less frequency-selective) channels. The diversity combining is performed prior to the Discrete Fourier Transform (DFT) operation. A new method has been studied to optimize the Pre-DFT diversity combining by selecting the cyclic shifts and weight factors based on known channel state information. The optimum signal to noise ratio for maximum cyclic delay and optimum weight factors for multiple antenna receiver diversity is analyzed. Performance results in terms of bit error probability is presented. Finally, it was found that the scheme advocated in this paper drastically reduces the computational complexity comparing to traditional receiver diversity schemes. Fig.. S/P DFT S/P S/P DFT DFT r, r,k r, r,k r, r,k g, g, g, R Combiner r k Data Detection Output ultiple Antenna Receiver Diversity with RC at subcarrier level I. INTRODUCTION Orthogonal Frequency Division ultiplexing (OFD) is very effective in mitigating adverse multipath effects of a broadband wireless channel []. OFD has been successfully used in Wireless Local Area Networks (WLANs), such as IEEE 80.a, European HiperLAN/ or Japanese AC standards as high-data rate physical layer transmission scheme for local area coverage. The IEEE 80.a WLAN standard specifies channel coding and frequency interleaving to exploit the frequency diversity of the wideband channel. Efficiency can only be achieved if the channel is sufficiently frequency-selective, corresponding to long channel delay spreads. In a flat fading situation (or in relatively lesser frequency-selective fading situation which we often encounter in indoor wireless scenario), all or most subcarriers are attenuated simultaneously leading to long error bursts. In this case, frequency interleaving does not provide enough diversity to significantly improve the decoding performance as reported in [], [3]. Traditionally space domain is exploited at the receiver to obtain multipath diversity, so schemes like aximum Ratio Combining (RC), Equal Gain Combining (EGC), or Selection Combining (SC) are used to obtain a better link quality. For an RC-OFD system as shown in Figure, the combining operations are performed at subcarrier level after the DFT operation, thus we denote the process as Post-DFT RC or subcarrier combining receiver [4]. The received OFD signals at different antenna branches are first transformed via separate DFTs. Their outputs are assigned to N diversity combiners where N refers to number of OFD subcarriers. Note that, similar to RC, all of the above spatial diversity schemes in an OFD system requires multiple DFT blocks in the receiver. Cyclic Delay Diversity (CDD) is proposed in [3], [5] as a transmit diversity solution. CDD concept can also be implemented in the receiver [], to obtain receiver diversity like RC. In CDD, the signal is not truely delayed, but cyclically shifted between respective antennas. All the signal processing needed is performed in time domain, so the duplication of the DFT operation for each receiving antenna branch is not a requirement any more, thus the receiver has lower computational cost compared to Post-DFT RC (Figure ). In this work, we have studied a scheme named Pre-DFT aximum Average Ratio Combing (Pre-DFT ARC), which is basically application of CDD in an OFD receiver. A detailed discussion on the scheme is presented, where the optimum weighting factors and cyclic shifts are derived for the multiple antenna case, based on the estimated CSI. In this paper, we analyze the optimum Signal-to-Noise Ratio (SNR) for maximum cyclic shift and optimum gain factors in Section II. Performance results and discussions are placed in Section III. This section also describes the comparison of computational complexity between Post-DFT RC and Pre-DFT ARC techniques and the effect of time-variance in the combining scheme. II. PRE-DFT AXIU AVERAGE RATIO COBINING Introducing CDD in an OFD system (either in the transmitter or in the receiver) amounts to increasing the frequency-selectivity of a relatively flat fading channel seen from the receiver side [], [3]. When we shift the OFD signal cyclically and add them up in the receiver linearly, we actually insert some virtual echoes on the channel response. This effect increases the channel frequencyselectivity, thus higher order frequency diversity can be achieved, which is effectively exploited by a Coded OFD (COFD) system. When CDD is introduced at the receiver, the diversity combining is performed prior to the DFT operation [3, Section 8.3], as shown in Figure. At the receiver, the antenna branch signals can be used for estimating the channel responses for each individual receiver antenna in order to optimize the diversity combining based on the instantaneous channel behavior. This allows for an optimized diversity combining using cyclic delays, τ n (m) in received data samples and complex gain factors, g m a me jφm, where a m g m, φ m g m and P m gm, for m,,..., and is the number of diversity branches. We denote this combining technique in the OFD receiver as Pre-DFT aximum Average (signal-to-noise) Ratio Combing (Pre- DFT ARC). If we select equal values for the gain magnitudes (i.e.

2 Channel estimation g Calculated combined channel transfer function after diversity combining as SNR w[0], where w[l] is Cyclic delay g g ARC/EGC DFT Data detection FEC Output data stream Fig.. OFD receiver with Pre-DFT Combining CDD. The instantaneous channel is estimated from the received signals to determine the optimum cyclic shifts (and gain factors, if ARC combining is performed). q g m ), the combining technique is named as Pre-DFT Equal Gain Combining (Pre-DFT EGC). A. Optimum SNR for the Combined Signal We denote the discrete time and discrete frequency index as l and k respectively; and define, r m(l), r m,cdd(l) and r comb (l) as received signal in time-domain at m th receive antenna, signal after applying CDD at m th diversity branch and combined signal after the combining respectively. All of these vectors are defined for one OFD symbol, so they have a dimension of [N, ]. Denoting complex valued time-invariant channel impulse response (CIR) of m th diversity branch as c m(l), we can write that [6] r m(l) d(l) N c m(l) () where N denotes N-point circular convolution, and by definition, 0 l N. The length of c m(l) is actually the length of CIR for m th diversity branch which is obviously less than N, so the length is extended to N by padding zeros at the end of the vector. The signals at each diversity branch after applying CDD can be written as r m,cdd(l) d(l) N c m((l τ n (m) )) N () where c m((l)) N c m((l)modn). The CDD signal, r m,cdd(l) is multiplied with the gain factor g m to obtain the combined signal, r comb (l) X m g m d(l) N c m((l τ n (m) )) N The data part of the convolution in (3) is same for all diversity branches, so we can write the effective channel impulse response of the combined channel as c comb m (3) g mc m((l τ (m) n )) N (4) We assume, independent additive white gaussian noise with equal powers are present at the branches, σ σ... σ m σ. After diversity combing, the noise powers are scaled by the squared magnitude of the gain factors and summed up. Since we choose P m g m, the resulting noise level after diversity combining is constant and equal to the noise power of each antenna branches []. Therefore we can derive a measure that is proportional to the SNR w[l] c comb [l] N c comb[n l] g mc m[((l τ n (m) )) N ] N gnc n[((τ n (n) l)) N ] + m n g m c m[l] N c m[n l] (5) m m nm+ n Re g mgnc m[((l τ n (m) )) N ] N c n[((τ (n) n l)) N ] where denotes the conjugate complex. It is evident that the first part (5) of this expression is independent of the cyclic delays and the phases of the gain factors. Thus the SNR can be optimized with respect to these parameters by maximizing the second part (6). Unfortunately it is not possible to optimize these parameters independently, for the following reasons. Between each pair of signals m and n, n m, the cyclic delay leading to maximum SNR is given by the index of the maximum value in the respective summation term of (6), τ n (n) ) opt arg max (m) l cm[((l τ n )) N ] N c n[((τ n (n) l)) N ]. (7) This optimum SNR would be reached by selecting the phase terms according to (τ (m) n (g mg n) j max l ff c m[((l τ n (m) )) N ] N c n[((τ n (n) l)) N ] (6) ff. (8) It becomes visible at this point that an independent optimization of the parameters (g m) and τ n (m) is not possible if >. E.g. if we optimize the delays and phase-rotations for the antenna pairs - and -3, the corresponding parameters of pair -3 will be determined implicitly. We suggest to use the largest terms obtained by (7) for optimizing the cyclic delays. The gain factors will then be optimized using the approach described in the next section. B. Optimum Diversity Weights As we have derived an optimum way to determine the cyclic delays for respective antennas, now the next step should be to determine the diversity branch weight factors. A method to derive optimum diversity weight factors for multiple antenna Pre-DFT processing OFD receiver is presented in [4]. We adopt a similar weight estimation scheme for our Pre-DFT ARC with CDD receiver diversity scheme. The SNR after subchannel diversity combining can be written as [4] SNR Γg H Cg (9) where Γ is the average SNR per diversity branch, C is the covariance matrix of the CIRs of all the diversity branches, and g is the weight vector, defined as g [g, g,..., g ] T. Eigen analysis can be performed on the Hermitian matrix C according to C ZΛZ H, where Λ diag (λ, λ,..., λ ) T is the diagonal matrix whose diagonal elements consist of eigenvalues λ m of C and Z is the unitary matrix whose columns are the eigenvectors corresponding to λ m. It is found that the optimum diversity weight vector g opt is the eigenvector, which corresponds to maximum eigenvalue from diagonal matrix Λ [4].

3 Cyclic delay g g ARC 6 4 agnitudes of Channel Transfer Function Correlation matrix R g Eigen vector [g, g,...,g ] magnitude [db] SNR Channel Channel PreDFT EGC; n 63 PreDFT ARC; n 63 PostDFT RC Fig. 3. Diversity wieght estimation method for Pre-DFT ARC with CDD receiver diversity scheme; for clarity, channel estimation procedure and COFD part of the receiver is not shown in the Figure In the method described above, the covariance matrix C is derived from the CIRs of all diversity branches. The CIR estimators (or equivalently CTF estimators) that are found in literature impose a high computational complexity, thus [4] employed the correlation among the signals of the diversity branches directly in time domain instead of explicitly estimating the CIRs or the CTFs of all the diversity branches. We denote the sampled h received i signal vector at l th sampling T instant as r (l) r (l), r(l),..., r(l). The correlation matrix of the received signal vector at any sampling instant is given by R E{r(l) r (l)h } [ρ p,q] (0) where ρ p,q is the [p, q] th element of R matrix, that represents the correlation among received signal sample for l th instant between p th and q th diversity branches, ρ p,q E[r(l) p r q (l)h ]. Denoting σx as the variance of the transmitted signal and σ n as the variance of the noise component, we find from (0) that R σ xc + σ ni () Above equation shows that the eigenvectors of R and C are the same, hence we can estimate the optimum weight factors based on the correlation matrix. As it is shown in Figure 3, the received signals corresponding to all diversity branches for any sampling instant are put together in a vector (r (l) ) and the autocorrelation of that vector is calculated according to (0). After that the optimum weights for all the receive antenna branches are determined using Eigen analysis as described in (). The principal difference with this method with the method described in [] is that CSI is not required for the above method, thus the combining performance will be improved, as we know that when perfect CSI is never available in a practical situation, and so the performance will always be degraded if weight estimation depends on available CSI. III. ANALYSIS, SIULATIONS AND DISCUSSIONS A. Channel odel A second order stochastic channel model (WSSUS model) suitable for Rayleigh and Ricean fading distributions was used in this work. The frequency-selectivity is described by the spaced-frequency correlation function and by the delay power spectrum (DPS) [3]. In the simulations, realizations of channel transfer functions are generated directly, based on well-defined channel parameters, such as the normalized (or average) received power P 0, the Ricean K- factor K and the RS delay spread τ rms. Indoor WLAN channels subcarrier index Fig. 4. agnitudes of Channel Transfer Functions before and after diversity combining. Pre-DFT ARC, Post-DFT RC, and Pre-DFT EGC are shown in the figure. with τ rms 5ns to 50ns are generated. This corresponds to 0. to sample considering a sampling frequency of 0 Hz as used in IEEE 80.a. Rayleigh fading scenarios with K 0 and Ricean fading scenarios with K 4 were considered. B. Simulation Parameters Simulations are performed with parameters stipulated by the IEEE 80.a WLAN standard: number of OFD subcarriers, N 64, length of cyclic prefix (), N 6 samples, OFD symbol duration, T S 4µs (consists of useful data period of 3.µs and duration of 0.8µs), QPSK symbol mapping with half-rate convolutional coding (corresponds to bps raw bit rate at the receiver), system bandwidth of 0 Hz and operating at the 5 GHz band. Our simulations only considered the dual antenna case (i.e. ). C. Analysis of Channel Responses After Combining The analysis shows that the amount of cyclic shift and the combiner weight factors can be determined effectively in order to achieve optimum SNR in all cases for a Pre-DFT Receiver CDD system, because the CSI can be estimated. Figure 4 shows the magnitude response of the combined channels (equivalent to the SNR per subcarrier) for several receiver diversity schemes, along with the channel responses of the branch channels. It is seen that the Post-DFT RC scheme shows better SNR characteristics, though the responses for Pre-DFT ARC and Pre-DFT EGC are also very close. D. Performance Results and Discussions BER simulations have been performed for dual antenna receiver diversity using Post-DFT RC, Pre-DFT ARC and Pre-DFT EGC. For comparison, pure CDD at the transmitter (Tx-CDD) with fixed cyclic delay of 6 samples [5] and Pre-DFT ARC without cyclic delay (which is equivalent to the technique described in [4]) are also simulated. Figure 5 and 6 show uncoded BER results, which were calculated in a semi-analytical way as follows. For the various receiver concepts compared, the SNR values on the OFD sub-carriers were simulated. Based on these simulated channels, the BERs were determined analytically, using the Q-function, and averaged. The E b shown is the ratio of the average symbol energy per sub-carrier to the noise power density. Coherently detected QPSK with perfect channel estimation is assumed in this analysis.

4 average bit error rate τ rms Rayleigh fading channels 0. samples channel h channel h Pre DFT ARC w/o delay Pre DFT ARC Post DFT RC τ rms sample average E b average bit error rate with coding 0 0 Rate / convolutional coding; constr. length 5 SISO 0 6 Pre DFT ARC w/o delay Pre DFT ARC Post DFT RC average E c Fig. 5. Uncoded BER with and without application of diversity, Rayleigh fading channels with various τ rms. Fig. 7. Performance results in terms of coded BER; Rayleigh channel, τ rms sample; rate convolutional coding with constraint length 5. average bit error rate channel: τ rms 0.5 samples, K 4 channel h channel h Pre DFT ARC w/o delay Pre DFT ARC Post DFT RC average E b Fig. 6. Uncoded BER with and without application of diversity, Ricean channel with K 4. On the Rayleigh channel, Tx-CDD does not give any performance advantage in terms of uncoded BER compared with a single antenna receiver, because the channel gains are added up incoherently just like the noise. On the Ricean channel, the performance with Tx-CDD is even worse, because the combined channel has deeper fades than the component channels. This is one of the main drawbacks of Tx-CDD. The best performance is achieved with Post-DFT RC at the cost of high computational complexity. The Pre-DFT receiver diversity schemes lie in between those results. It is evident that more can be gained over flatter channels (lower τ rms and/or higher K), which is not surprising since in these cases the Pre-DFT combining schemes can add up the channel transfer functions constructively over a wider frequency range. Under the same condition, we observe less performance difference among the Pre-DFT schemes exploiting the CSI. Pre-DFT ARC and Pre-DFT EGC show very similar performance although the average SNR over the subcarriers is significantly higher using ARC []. The gain over Pre-DFT ARC without delay can be significant, but it reduces on channels with a very short channel impulse response. In Figure 7, performance results are given in terms of bit error rate for the coded OFD system. The source data is FEC-coded with a rate convolutional coder, whose constraint length is 5, i.e., the effect of each information bit is spread over roughly 0 FEC coded bits. A block interleaver (with interleaver depth 4) is used to exploit the available frequency diversity. After interleaving the FEC-related bits are spread over subcarriers in an OFD symbol (which consists of 48 data subcarriers). Note that the constraint length was reduced compared with the WLAN standard in order to speed up the computer simulations. Although this affects the absolute results, we expect the general trends and conclusions to be equivalent. BER performance shows that the largest gain is achieved with the traditional Post-DFT RC technique, amounting to almost 7dB at BER. About 3-4 db gain are observed from the Pre-DFT diversity combining techniques, whose performance is remarkably similar. Only ARC without delay is slightly weaker. In particular, after coding, the optimized techniques applied at the receiver, which can use CSI, perform not much better than the unsupervised combining technique using a fixed cyclic delay. It is evident from Figure 5, 6 and 7 that the scheme works better in situation where the RS delay spread is quite small, which means in a typical indoor WLAN environment, Pre-DFT ARC will perform well. In indoor situations, the diversity branches are mostly correlated, so it is a prolific advantage that our scheme works better even if the diversity branches are correlated to each other. This brings another benefit, i.e. usually multiple antennas cannot be used in S due to the fact that the antennas cannot be put with sufficient spatial separation due to space constraint. When the antennas are closely placed to each other, then the diversity branches will experience sufficient correlation which will destroy the benefits that the diversity schemes (such as Post-DFT RC or EGC or SC) bring. In those cases, Pre-DFT ARC scheme can be used to combine the signals efficiently from correlated diversity branches. E. Efficient Implementation We compared the complexity of the schemes in terms of number of multiplications required. Considering that we have a channel which is time-invariant for considerable amount of time, so that N pkt number of OFD symbols can be put in one OFD packet, then the number of multiplications required for one OFD symbol are «( )No N X pre + N pkt logn + N () «N X post N pkt logn + N (3)

5 3.5 3 Relative Processing Cost Vs N; N pkt 50, N o 50 TABLE I COPARISON OF CHANNEL COHERENCE TIE WITH OFD PACKET DURATION FOR IEEE 80.A WLAN STANDARD Relative Processing Cost.5.5 υ, km/h f d, Hz T c, ms max N pkt Number of Subcarriers, N Fig. 8. Relative Processing cost for Pre-DFT ARC and Post-DFT RC in comparison to number of OFD subcarriers; N pkt 50 and N o 50. Relative Processing Cost Relative Processing Cost Vs N pkt ; N 64, N o OFD symbols/packet, N pkt Fig. 9. Relative Processing cost for Pre-DFT ARC and Post-DFT RC in comparison to number of OFD symbols/packet; N 64 and N o 50. where N logn multiplications are required for FFT module per OFD symbol []. The st term of () corresponds to the calculation of covariance matrix shown in Figure 3. N o is the number of time-domain data samples that need to be acquired to obtain the correlation matrix. The computational complexity associated with Eigen analysis for gain factors is not taken into account, as it only required only once for complete OFD packet [4]. Figure 8 and 9 show the relative processing cost between Pre- DFT ARC and Post-DFT RC in comparison to N and N pkt respectively for different values of. In both figures, we can see that the processing cost is drastically reduced in our scheme when N or N pkt or increases. F. Effect of Channel Time-Variance Referring to Figure, our scheme requires channel estimates in time domain for delay insertion and gain factor estimation; and in frequency domain for data detection. If the channel is severely time variant, then the channels need to be estimated very frequently, and thus the savings in complexity due to time-domain combining will be lost in excessive channel estimation burden. Table I summarizes the relationship between channel coherence time with respect to user velocity in the context of IEEE 80.a WLAN system. If the coherence time is defined as the time over which the time correlation function is 0.5, then the coherence time is approximately, T c q 9 6πf d 0.43 f d [7, Section 4.4.3], when f d is maximum doppler shift. In the last column of Table I, we have shown the number of OFD symbols that can be transmitted during T c at corresponding user velocity. In this case, we have taken that one OFD symbol duration is 4µs. The standard specifies a test case of 000 octets in one OFD packet, this corresponds to 000 8/ OFD symbols in a packet with BPSK modulation and rate convolutional coding at the best-case scenario. We can see from the table that a velocity up to 0km/h will have a coherence time that equals to OFD symbols. 0km/h is quite a high velocity for a WLAN user. When the velocity is increased (in the order of tens of km/h), then obviously the complexity will go higher. In general, when the channel is less time-variant, then the coherence time is larger and so the packet duration can be made arbitrarily larger, so the channel estimation frequency will be smaller. IV. CONCLUSION In this paper we have shown that the proposed scheme performs considerably well considering the trade-off between performance and complexity. Because of the low complexity our approach will allow low cost wireless modules targeting the mass market. For less frequency-selective channel, which is very similar to typical indoor or immediate outdoor wireless channel, Pre-DFT ARC with CDD combining is a lucrative option for cost effective, efficient and reliable diversity reception. In situations where diversity branches are correlated to each other (as it is the case in indoor WLANs), Pre- DFT ARC scheme works very well and efficiently combines the diversity branches to increase the transmission quality. It has also been understood that the scheme works well in severely time-variant situations. ACKNOWLEDGEENT This work was partially supported by Samsung, Korea. REFERENCES [] R.V. Nee & R. Prasad, OFD for Wireless ultimedia Communications. Artech House Publishers, 000. [].I. Rahman et al., Performance Comparison between RC Receiver Diversity and Cyclic Delay diversity in OFD WLAN Systems, in Proc. 6 th WPC, vol., October 003, pp [3] K. Witrisal, OFD Air-Interface Design for ultimedia Communications, Ph.D. dissertation, Delft University of Technology, the Netherlands, April 00. [4]. Okada & S. Komaki, Pre-DFT Combining Space Diversity Assisted COFD, IEEE Transactions on Vehicular Technology, vol. 50, no., arch 00. [5]. Bossert et al., On Cyclic Delay Diversity in OFD Based Transmission Schemes, in proc. 7 th International OFD Workshop, Hamburg, Germany, September 00. [6] J.G. Proakis, Digital Communications, 3rd ed. cgraw-hill, New York, 998. [7] T. S. Rappaport, Wireless Communications Principles and Practice. Prentice Hall Inc., 996.