CE-OFDM with a Block Estimator Nikolai de Figueiredo and Louis P. Linde Department of Electrical, Electronic and Computer Engineering University of Pretoria Pretoria, South Africa Tel: +27 12 420 2953, Fax: +27 12 362 5000 Email: {Nikolai.DeFigueiredo, Louis.Linde}@up.ac.za Abstract A simple minimum mean-square error channel estimation algorithm was implemented for a constant envelope orthogonal frequency division multiplexing communication platform. The system was shown to have to bit error rate degradation for a range of system parameters. This technology has the potential for application in slow fading wireless environments where power efficiency is paramount. O Index Terms CE-OFDM, channel estimation, PAPR. I. INTRODUCTION RTHOGONAL frequency division multiplexing (OFDM) is a popular modulation technique for wireless digital communications. OFDM owes its popularity as a transmission scheme to the fact that it is resistant to frequency selective fading while offering higher spectral efficiency than single carrier schemes. OFDM does however suffer from two drawbacks. Firstly the peak-to-average power ratio (PAPR) of an OFDM signal is usually large [1], [2]. Secondly OFDM systems are highly susceptible to phenomena that cause intercarrier interference (ICI) and inter-symbol interference (ISI) [3]. There are a number of approaches to the reduction of the PAPR of an OFDM signal. These include clipping, decisionaided reconstruction (DAR) clipping, coding, partial transmission sequence, selective mapping, companding transforms, active constellation extension, tone reservation, and constant envelope OFDM (CE-OFDM) [1], [2]. One of the most desirable of these approaches is CE-OFDM, which reduces the PAPR of a signal to 0 through the process of phase modulation [4], [5]. Another major advantage is the low implementation complexity. The modulation scheme offers the potential for power efficient systems with many mobile applications. Before this technology can be used a number of other building blocks must be developed. These include a channel estimator and synchronisation scheme. Herein a simple block channel estimator, which can be used in conjunction with existing equalisers in packet based systems, is investigated. Although bandwidth efficiency is lost due to the overheads required for channel estimation, the technique justifies the use of the modulation scheme for slow fading environments where The research was made possible via the support from our industry partners Telkom, Unisys, Tellumat, EMC and Alvarion; and the Technology and Human Resources for Industry Program (THRIP) managed by the National Research Foundation (NRF), financed by the Department of Trade and Industry (DTI), South Africa. power efficiency is paramount. The channel estimation scheme also offers the potential for simple implementation of a full synchronisation algorithm without any further data overheads [6]. The system shows favourable BER performance for the four channels that were investigated with between 1 and 3 degradation over the range of values for modulation index and modulation order. In section II the system structure is explained followed by a description of the channel estimation technique in section III. In section IV the results of various simulations are presented and discussed. Finally section V concludes the paper with a summary and ideas for future work. II. SYSTEM OVERVIEW Before the channel estimation scheme could be investigated, the base system was developed. The entire system is depicted in Fig. 1, including transmitter, channel and receiver. The system operates in a block transmission mode assuming the channel coherence time is greater than or equal to the time between training blocks. A. Transmitter The random generator, generating zero mean uniformly distributed binary data, forms the data source. This is followed by M-th order quadrature amplitude modulation (M-QAM) bit-to-symbol mapping and pre-processing. The modulation stage is made up of two operations: firstly the source symbols are inverse fast Fourier transformed (IFFT) generating the OFDM signal; this signal is then used to phase modulate a carrier at an intermediate frequency (IF). Finally a cyclic prefix is added before transmission through the multipath channel. The pre-processing stage is necessary since the phase modulation requires a real input signal. This is achieved by using a conjugate symmetric vector as the input to the IFFT. The conjugate symmetric vector is given by, 0, 1, 2,,,,0,,, 2, 1, where.,. represent the complex conjugate and transpose operators respectively, is the all zero matrix and are N QAM -QAM data symbols. The power of the modulation technique comes from the phase modulator, =
Source Data M-QAM Mapping Pre-processing OFDM Modulation Phase Modulation Mutiplex Insert Cyclic Prefix Multipath Training Data (a) Multipath Cyclic Prefix Removal FDE Phase Demodulation OFDM Demodulation Postprocessing M-QAM Demapping Receive Bits Estimation Fig. 1. System block diagram (a) Transmitter (b) Receiver. (b) where and represent the continuous time phase modulated and OFDM message signals respectively, is the modulation index and is a phase memory term that may be used to achieve continuous phase modulation (CPM). After phase modulation the training block is multiplexed into the transmission vector and a cyclic prefix, with length greater than or equal to the maximum channel delay spread, is inserted at the beginning of each block to mitigate the effects of inter-block interference (IBI) before transmission through the channel. The channel is discussed in detail in Section III. B. Receiver Firstly the cyclic prefix is dropped from the received data, while in parallel the channel is estimated from the demultiplexed training block. This estimate along with the received data is passed to the frequency domain equaliser (FDE). The idea is that the received data is transformed to the frequency domain with a fast Fourier transform (FFT), multiplied with the inverse of the frequency response of the channel and then transformed back to the time domain via the inverse fast Fourier transform (IFFT). The resulting signal, III. BLOCK CHANNEL ESTIMATOR It is assumed that the channel is a multipath Rayleigh fading channel with a coherence time greater than or equal to the time between training blocks. The coherence time of the channel is the time that the channel, which is inherently time varying, may be assumed to be time invariant. One training block is transmitted followed by a number of data blocks. A. Models Four channel power delay spreads (pds) were used for investigation of the efficacy of the channel estimation algorithm. ŝ = =, =0,, 1 is then passed to the phase demodulator consisting of two parts, an arctangent function followed by phase unwrapping. are the received symbols, is the frequency domain correction term,. and. are the inverse- and discrete Fourier transform (IDFT and DFT) operators respectively and is the length of the DFT. The equalised signal is then fast Fourier transformed and post-processed. The post-processing is the recovery of the transmitted data symbols from the estimate of the conjugate symmetric vector. The complex data symbols are then de-mapped and the received bits are detected. This estimate of the transmitted bits is then used to compute the bit error rate (BER) results for various system parameters. Fig. 2. Power delay spread for channels I-IV.
The conventional approach to channel modelling is to sample the channel impulse response thus each tap represents the gain (attenuation due to propagation) of the channel at a discrete delay. In reality this is a composite function of the individual propagation paths, each made up of an attenuation factor and delay. The taps were zero mean Gaussian random variables with variance equal to the square-root of the normalised pds taps. Fig. 2 shows the pds of channels I-IV. s I and II are based on maritime channel models, similar to those in [7], with taps at 0 and 5. I has tap weights of 10/11 and 1/11 and channel II, 2/3 and 1/3. III has an exponentially decaying pds, = /, = 1 exp / 2 10 0< 8.75 0.1188, and channel IV has a uniform pds with taps equal to 1/36. B. Estimation Algorithm A simple channel estimation algorithm was implemented which only requires one matrix multiplication to retrieve the channel estimate. Since a known random binary sequence is transmitted, a pre-calculated matrix is multiplied with the received block of training data. The vector representation of the system signals are, = 0 1 1 = = 0 1 1 0 1 +1 = 1 0 +2, 1 2 where is the correctly timed received vector, h is the channel impulse response and is the AWGN vector. The matrix is the training symbol with equal to the cyclic prefix length. It is a Toeplitz matrix formed from a pseudo random sequence generated with a maximal length linear feedback shift register (LFSR). The pre-calculated matrix is then given by, =, and is used to calculate the channel estimate, =, where is the received training data. The frequency domain representation of the channel is given by, =. The received data may then be equalized by multiplying with the minimum mean-square error (MMSE) correction term, = /, where / is the bit energy to noise density ratio. The MMSE equaliser was chosen for its performance gains in noisy conditions, at a moderate increase in complexity compared to the zero forcing (ZF) solution. Other algorithms such as those that employ singular value decomposition (SVD) were not considered since they incur more computational cost. IV. SIMULATION RESULTS Four experiments were conducted on the system in order to determine the efficacy of the channel estimation algorithm. Each of the first three experiments involved varying one of the system parameters: modulation index, modulation order and number of subcarriers. The fourth experiment, as mentioned before was conducted using one of four different channel models. Each result includes the BER performance of the system in AWGN conditions for comparison. These experiments were chosen in order to gain an understanding of system wide performance, the BER results are an indication of the impact of the channel estimation algorithm on the complex system as a whole. These results give insight into how well the system will perform in a real world context rather than simply quantifying the algorithm s standalone performance. This is significant when moving from the theoretical domain to practical implementation of a communication system. For each of the experiments certain parameters were chosen to remain fixed. The modulation index 2 =0.7 falls in the middle of the range of values, =64 is the least number of subcarriers and =4 is the constellation that is the least sensitive to additive noise. A. Modulation Index The modulation order was fixed at = 4, number of subcarriers = 64 for the channel with exponential delay profile, channel III. Fig. 3 shows the results of the first experiment where the normalised modulation index, 2, was varied from 0.1 to 1.5. Preliminary results indicated that a step size for the modulation index of 0.2 provided enough resolution in the range of values to draw conclusions about the system performance. The solid lines indicate the results where the channel estimation scheme was employed while the dashed lines are for perfect channel knowledge. It can be seen from the figure that the overall system performance, in terms of BER equal to 10, is improved as the modulation index is increased. On the other hand the performance of the channel estimation algorithm decreases. This can been seen in Fig. 4 for the case of 2 =0.7 with a BER degradation of approximately 3 and Fig. 5 for 2 =0.1 with a BER degradation of only 0.1. As the modulation index was increased to 2 =1.5 the channel
estimation again began to induce less error, with degradation of 2. This is an intuitively appealing result since increasing the modulation index results in bandwidth expansion [5]. This is because of the increase in the spread of the constellation points after phase modulation. When demodulating this signal the non-linearity of the arctangent function results in error amplification and the subsequent decrease in system performance. Once the modulation index reaches 2~? = 0.7 the spread of the data is over the entire range of the arctangent function, this is when the degradation is at a maximum. The bandwidth expansion then provides further resistance to error in the channel estimate and improved system performance as the modulation index continues to increase. Fig. 5. BER versus SNR (2~? = 0.1, = 4, = 64) using channel III. Fig. 3. BER versus SNR for various modulation indices ( = 4, = 64) using channel III. Fig. 4. BER versus SNR for various modulation indices ( = 4, = 64) using channel III. B. Modulation Order Again the number of subcarriers was fixed at = 64 and the exponential delay channel was used. The modulation index was fixed at 2~? = 0.7 and the modulation order was varied from = 4 for quadrature phase-shift keying (QPSK) to = 256 for 256-QAM. The BER curves in Fig. 6 indicate that the overall system performance is degraded as the modulation order is increased however the performance of the channel estimator improves. At a BER of 10Fe there was a 2.5 loss in performance for = 4 while for = 256 the loss was reduced to approximately 1. Fig. 6. BER versus SNR for different modulation orders (2~? = 0.7, = 64) using channel III. C. Number of Subcarriers In order to investigate the system sensitivity to number of subcarriers with a fixed length DFT the modulation index was fixed at 2~? = 0.7 and the modulation order = 4. Again channel III was used while the number of subcarriers was increased from = 64, to = 128 and = 256. From
Fig. 7 it can be seen that there is a slight improvement in BER performance and channel estimation performance from =64 to =128. However there is an irreducible error floor when = 256 because the DFT length was fixed at =512. With the use of the conjugate symmetric vector and 512 point DFT carrier orthogonality is lost resulting in the error floor. research. The performance of the system in the presence of time varying fading (Doppler) is another topic for study along with quantification of the system capacity. Lastly multipleinput multiple-output (MIMO) may provide another form of diversity to CE-OFDM. Fig. 7. BER versus SNR for different numbers of sub-carriers (2 =0.7, = 4) using channel III. D. Models Fig. 8 shows the results of the simulation experiments for the four different channel models. The performance of the two maritime channels is worse than the exponential and uniform channels since they have far fewer taps. This results in a loss of multipath diversity. The channel estimator for the maritime channels does however perform better than the others. The system performs comparatively similar for the maritime channels as well as the other two with approximately 1 and 2 degradation respectively. V. CONCLUSION A simple block channel estimator was implemented for CE- OFDM systems. The algorithm provides a low complexity method for estimating the channel in mobile environments where slow fading is present. Furthermore without the need for more overhead the training data may be used for both timing and carrier synchronisation. The system shows favourable BER performance with between 1 and 3 degradation over the range of system parameters investigated. This technology has the potential for the development of wireless communications systems where power efficiency is critical. The development of an entire CE-OFDM based communication system will require a number of other components. Investigation into time and frequency synchronisation algorithms will be necessary as well as forward error correction coding that is applicable to the nature of the modulation scheme. Depending on the application, multiuser or multi-access schemes may also be areas of future Fig. 8. BER versus SNR for channels I-IV (2 =0.7, = 4, = 64). REFERENCES [1] Seung Hee Han and Jae Hong Lee, "An overview of peak-to-average power ratio reduction techniques for multicarrier transmission," Wireless Commun., vol. 12, no. 2, pp. 56-65, Apr. 2005. [2] Tao Jiang and Yiyan Wu, "An Overview: Peak-to-Average Power Ratio Reduction Techniques for OFDM Signals," IEEE Trans. Broadcast., vol. 54, no. 2, pp. 257-268, Jun. 2008. [3] TD Chiueh and PY Tsai, OFDM Baseband Receiver Design for Wireless Communication, Hoboken, Singapore, NJ: John Wiley and Sons (Asia), 2007, Ch. 1-6. [4] S. C. Thompson et al., "Constant Envelope OFDM," IEEE Trans. Commun., vol. 56, no. 8, pp. 1300-1312, Aug. 2008. [5] S.C. Thompson, Constant Envelope OFDM Phase Modulation, Ph.D. dissertation, University of California, San Diego, 2005. [Online]. Available: http://elsteve.com/thesis/thesis.pdf [6] Hlaing Minn et al., "A robust timing and frequency synchronization for OFDM systems," IEEE Trans. Wireless Commun., vol. 2, no. 4, pp. 822-839, Jul. 2003. [7] P. H. Moose et al., "A COFDM-based radio for HDR LOS networked communications," in IEEE International Conf. on Communications, 1999, pp. 187-192. Nikolai de Figueiredo received his undergraduate degree in 2011 from the University of Pretoria and is presently studying towards his Master of Science degree in Electronic Engineering at the same institution. His research interests currently include OFDM, channel estimation, synchronisation, adaptive receivers, and MIMO systems. Professor Louis P. Linde is presently with the Department of Electrical, Electronic and Computer Engineering at the University of Pretoria where he leads the Digital Communications Research Group.