2010 Second International Conference on Computational Intelligence, Communication Systems and etworks PPR Reduction in OFDM Systems: Zadoff-Chu Based Pre/Post-Coding Techniques Imran Baig and Varun Jeoti Electrical and Electronic Engineering Department, Universiti Teknologi PETROS, Tronoh, 31750, Perak, Malaysia Imran_baig_mirza@yahoo.com and varun_jeoti@petronas.com.my bstract High Peak to verage Power Ratio (PPR is one of the major drawbacks in Orthogonal Frequency Division Multiplexing (OFDM. The high PPR increases the complexity of nalogue to Digital (/D and Digital to nalogue (D/ converters and also reduces the efficiency of RF High Power mplifier (HP. In this paper, we propose two novel PPR reduction techniques for OFDM systems, Zadoff-Chu matrix (ZCT precoding based PPR reduction technique and ZCT postcoding based PPR reduction technique. Simulation results show that, at clip rate of, the PPR of our both proposed systems reduced to 0dB, 3dB, 4.1dB and 4.3dB for M-QM (where M = 4, 16, 64, 256. Both ZCT precoding and postcoding based OFDM systems are shown better PPR gain and Bit Error Rate (BER performance than alsh Hadamard precoded OFDM (HT-OFDM and OFDM conventional. Keywords- Zadoff-Chu matrix ; OFDM conventional; HT-OFDM; Peak to verage Power Ratio; BER I. ITRODUCTIO Orthogonal Frequency Division Multiplexing (OFDM is a multicarrier transmission scheme that has become the technology of choice for next generation wireless and wireline digital communication systems because of its high speed data rates, high spectral efficiency, high quality service and robustness against narrow band interference and frequency selective fading [1]. OFDM thwarts Inter Symbol Interference (ISI by inserting a uard Interval (I using a Cyclic Prefix (CP and moderates the frequency selectivity of the Multi Path (MP channel with a simple equalizer [2]. OFDM is widely adopted in various communication standards like Digital udio Broadcasting (DB, Digital Video Broadcasting (DVB, ireless Local rea etworks (L, ireless Metropolitan rea etworks (M, ireless Personal rea etworks (P and even in the beyond 3 ide rea etworks ( etc. However, among others, the Peak to verage Power Ratio (PPR is still one of the major drawbacks in the transmitted OFDM signal [3]. For zero distortion of the OFDM signal, the RF High Power mplifier (HP must not only operate in its linear region but also with sufficient back-off. Thus, HP with a large dynamic range are required for OFDM systems. These amplifiers are very expensive and are major cost components. Thus, if we reduce the PPR it not only means that we are reducing the cost of OFDM system and reducing the complexity of /D and D/ converters, but also increasing the transmit power, thus, for same range improving received SR, or for the same SR improving range. The literature is replete with a large number of PPR reduction techniques. mong them, schemes like constellation shaping [4], phase optimization [5], nonlinear companding transforms [6], Tone Reservation (TR and Tone Injection (TI [7]-[8], clipping and filtering [9], Partial Transmit Sequence (PTS [10], precoding based techniques [11] and Precoding based Selected Mapping (PSLM [12] are popular. The precoding based techniques, however, show great promise as they are simple linear techniques to implement without the need of any side information. This paper presents two novel PPR reduction techniques, namely, Zadoff-Chu matrix (ZCT precoding based PPR reduction technique and ZCT postcoding based PPR reduction technique for OFDM systems. In the proposed schemes, the reshaping of the ZCT is carried out one way for precoding and another way for postcoding. For precoding we reshape the ZCT row wise and precode the constellation symbols before the I with ZCT and for postcoding, we reshape the ZCT column wise and implement ZCT after the I. The rest of the paper is organized as follows: Section II describes the basics of the OFDM system and PPR, In Section III we present the proposed ZCT precoding and postcoding based OFDM system models, and Section IV presents performance evaluation and computer simulation results and section V concludes the paper. II. OFDM SYSTEM D PPR Fig. 1 illustrates the block diagram of an OFDM system. Baseband modulated symbols are passed through serial to parallel converter which generates complex vector of size. e can write the complex vector of size as X = [X 0, X 1, X 2 X -1 ] T. X is then passed through the I block. The complex baseband OFDM signal with subcarriers can be written as 978-0-7695-4158-7/10 $26.00 2010 IEEE DOI 10.1109/CICSy.2010.34 374 373
= Figure 1. Block diagram of eneral OFDM system., n=0, 1, 2... -1 (1 The PPR of OFDM signal in (1 can be written as PPR = [ ] where E [.] denotes expectation and the Complementary Cumulative Distribution Function (CCDF for an OFDM signal can be written as P (PPR > = 1 (1 e PPRo (3 where is the clipping level. This equation can be read as the probability that the PPR of a symbol block exceeds some clip level. III. I PROPOSED TECHIQUES D SYSTEM MODEL. Zadoff-Chu Sequences Zadoff-Chu sequences are class of poly phase sequences having optimum correlation properties. Zadoff- Chu sequences have an ideal periodic autocorrelation and constant magnitude. ccording to [13], the Zadoff-Chu sequences of length L can be defined as: =. where k = 0, 1, 2 L-1, q is any integer, r is any integer relatively prime to L and j= 1. B. ZCT Precoding Based OFDM system (2 (4 Fig.2. shows the block diagram of ZCT precoding based OFDM system. In the ZCT precoding based OFDM system baseband modulated data is passed through convertor which generates a complex vector of size that can be written as X = [X 0, X 1, X 2 X -1 ] T. Figure 2. Block diagram of ZCT precoding based OFDM system Then ZCT precoding is applied to this complex vector which transforms this complex vector into new vector of length that can be written as Y=PX= [Y 0, Y1, Y2 Y -1 ] T, here R is a ZCT based row-wise precoding matrix of size L =. ith the use of reordering as given in equation (5 = (5 matrix R with row wise reshaping can be written as = In other words, the 2 point long Zadoff-Chu sequence fills the precoding matrix row-wise. R is x, ZCT complex orthogonal matrix with length L 2 =. By letting, q = 1 and r = 1, the ZCT for Even L can be written as r k = exp [(j*pi*k 2 / L 2 ]. ccordingly, precoding X gives rise to Y as follows: Y = (7 =, =0,1, 1 (8, means m th row and l th column of precoder matrix. The complex baseband OFDM signal with subcarriers without precoding is given by = Zadoff-Chu I (6., n=0,1,2,...,-1 (9 However, expanding (9 while using q = 1 and r = 1 in (4, gives complex baseband ZCT precoding based OFDM signal with subcarriers as Zadoff-Chu -1 / FDE 375 374
=. (10 The expression in (10 suggests that x n are I of constellation data X l premultiplied with quadratic phase and I precoded, and then alternated with ±1. The PPR of ZCT-OFDM signal in (10 can be written as PPR = [ ] (11 where E [.] denotes expectation and the CCDF for an ZCT based OFDM signal can be written as P (PPR > = 1 (1 e PPRo (12 where is the clipping level. C. ZCT Postcoding Based OFDM System In other words, the 2 point long Zadoff-Chu sequence fills the precoding matrix column-wise. C is x, ZCT complex orthogonal matrix with length L 2 =. By letting, q = 1and r = 1, the ZCT for Even L can be written as c k = exp [j*pi*k 2 / L 2 ]. ccordingly, postcoding y gives rise to w as follows: w =C.x (15 C is a ZCT column wise postcoding of size L =. Expanding (13 while using q = 1 and r=1 in (4, gives complex baseband ZCT postcoding based OFDM signal with subcarriers can be written as =.... =0,1, 1 (16 The expression in (16 suggests that w m are I of constellation data X l premultiplied with quadratic phase and I postcoded, and then alternated with ±1. The PPR of ZCT postcoding based OFDM signal in (16 can be written as PPR = [ ] (17 I Zadoff-Chu Zadoff-Chu -1 / FDE where E [.] denotes expectation and the CCDF for an ZCT postcoding based OFDM signal can be written as P (PPR > = 1 (1 e PPRo (18 where is the clipping level. This equation can be read as the probability that the PPR of a symbol block exceeds some clip level. Figure 3. Block diagram of ZCT postcoding based OFDM system Fig.3. shows the block diagram of ZCT postcoding based OFDM system. In the ZCT postcoding based OFDM system baseband modulated data is passed through convertor which generates a complex vector of size that can be written as X = [X 0, X 1, X 2 X -1 ] T. Then I is performed to this complex vector which transforms this complex vector into new vector of length that can be written as y = I{X} = [y 0, y1, y2 y -1 ] T. ith the use of reordering as given in equation (13 = (13 matrix C with column wise reshaping is written as = (14 IV. PERFORMCE EVLUTIO D SIMULTIO RESULTS e performed extensive simulations in MTLB in order to evaluate performance of the ZCT precoding and postcoding based OFDM systems. To show the PPR analysis of ZCT precoding and postcoding based OFDM, data is generated randomly then modulated by M-QM (where M=4, 16, 64, 256. To show overall performance of the ZCT precoding and postcoding based OFDM systems for PPR and BER in MTLB we considered 64 point I for M-QM. e also compared our simulation results with HT-Precoded OFDM system and OFDM conventional. Fig.4 shows the CCDF comparisons of ZCT HT-OFDM system, and OFDM conventional for =64. t clip rate of 10, the PPR gain of 7dB and 8dB is system and OFDM respectively for 4-QM modulation. 376 375
ZCT Pre/Post-Coding Based OFDM Systems with =64 for 4QM ZCT Pre/Post-Coding Based OFDM Systems with =64 for 64QM Figure 4. CCDF of ZCT precoded OFDM, ZCT postcoded OFDM, HT-OFDM and OFDM conventional with =64 for 4-QM Figure 6. CCDF of ZCT precoded OFDM, ZCT postcoded OFDM, HT-OFDM and OFDM conventional with =64 for 64-QM ZCT Pre/Post-Coding Based OFDM Systems with =64 for 16QM ZCT Pre/Post-Coding Based OFDM Systems with =64 for 256QM Figure 5. CCDF of ZCT precoded OFDM, ZCT postcoded OFDM, HT-OFDM and OFDM conventional with =64 for 16-QM Figure 7. CCDF of ZCT precoded OFDM, ZCT postcoded OFDM, HT-OFDM and OFDM conventional with =64 for 256-QM Fig.5 shows the CCDF comparisons of ZCT HT-OFDM system, and OFDM conventional for =64. t clip rate of 10, the PPR gain of 4.2dB and 5dB is system and OFDM respectively for 16-QM modulation. Fig.6 shows the CCDF comparisons of ZCT precoding and system, and OFDM conventional for =64. t clip rate of 10, the PPR gain of 3.3dB and 4dB is achieved when we compare ZCT precoding and postcoding based OFDM systems with HT-OFDM system and OFDM respectively for 64-QM modulation. Fig.7 shows the CCDF comparisons of ZCT HT-OFDM system, and OFDM conventional for =64. PPR of M-QM( M=4,16,64,256 0 1 2 3 4 5 Figure 8. CCDF of M-QM (M=4, 16, 64, 256 4-QM 16-QM 64-QM 256-QM 377 376
Bit Error Rate 10-4 10-5 Bit Error Probability Curve for 4-QM using HP(Rapp model ZCT Precoded ZP-OFDM ZCT Postcoded ZP-OFDM -2 Eb/o, db Figure 9. BER vs. Eb=0 for ZCT precoded OFDM, ZCT postcoded OFDM, HT-OFDM and OFDM conventional for =64, HP model: Rapp model with smoothness factor = 2 t clip rate of 10, the PPR gain of 2.5dB and 2.8dB is system and OFDM respectively for 256-QM modulation. It is to be noted that M-QM has itself PPR. Fig.8 shows the PPR of M-QM (M=4, 16, 64, 256. t clip rate of 10, the PPR is 0dB, 3dB, 4.1dB and 4.3dB for 4QM, 16QM, 64QM and 256QM respectively. Fig.9 shows the BER performance in while using HP. It can be seen that ZCT precoding and postcoding based OFDM systems has no BER degradation. The effect of Rapp s solid state power amplifier is more detrimental on HT-OFDM system and OFDM conventional as compared to that on ZCT precoding and postcoding based OFDM systems. Table 1 summarizes the PPR of, our both proposed ZCT precoding and postcoding based OFDM systems and M-QM (M=4, 16, 64, 256, it is to be noted that M- QM has itself PPR. Hence, it is concluded from table 1 that our both proposed ZCT precoding and postcoding based OFDM systems has zero PPR. TBLE 1 T CLIP RTE OF, PPR OF M-QM and ZCT Pre/Postcoding based OFDM Type Of Modulation (M-QM M=4, 16, 64, 256 Theoretical BER PPR Of M-QM M=4, 16, 64, 256 HT-OFDM OFDM Conventional PPR of ZCT Precoded OFDM PPR of ZCT Postcoded OFDM 4-QM 0 0 0 16-QM 3 3 3 64-QM 4.1 4.1 4.1 256-QM 4.3 4.3 4.3 V. COCLUSIO In this paper, two PPR reduction schemes based on ZCT precoding and postcoding in OFDM systems are proposed. Table 1 show, that the PPR of our both proposed systems is equal to the PPR of M-QM itself. Hence, it is concluded that the proposed pre/postcoded OFDM systems introduce no PPR or BER degradation. In addition, the ZCT precoding and postcoding based OFDM systems for PPR reduction do not require any power increment, complex optimization and side information to be sent for the receiver. dditionally, ZCT-OFDM systems also take advantage of frequency variations of the communication channel and can also offer substantial performance gain in fading multipath channels. Thus, it is concluded that both ZCT pre/postcoding based OFDM systems are more favourable than the HT-OFDM system and OFDM conventional. REFERECES [1] Y. u and Z. Y. illiam, Orthogonal frequency division multiplexing: multi-carrier modulation scheme, IEEE Trans. Consumer Electronics, vol. 41, no. 3, pp. 392 399, ug. 1995. [2] B. Muquet, Z. ang,. B. iannakis, M. Courville, and P. Duhamel, Cyclic prefixing or zero padding for wireless multicarrier transmissions,'' IEEE Trans. Comm., vol. 50, pp. 2136-2148, Dec. 2002. [3] R. V ee and. ild, Reducing the Peak-To-verage Power Ratio of OFDM, Vehicular Technology Conference, 1998. VTC98. 48 th IEEE, Volume.3, 18-21 May 1998. [4] Y. Kou,. S. Lu and. ntoniou, new peak-to-average power-ratio reduction algorithm for OFDM systems via constellation extension, IEEE Trans. ireless Communications, vol. 6, no. 5, pp. 1823 1832, May 2007. [5] H. ikookar and K. S. Lidsheim, Random phase updating algorithm for OFDM transmission with low PPR, IEEE Trans. Broadcasting, vol. 48, no. 2, pp. 123 128, Jun. 2002. [6] T. Jiang,. Yao, P. uo, Y. Song and D. Qu, Two novel nonlinear Companding schemes with iterative receiver to reduce PPR in multicarrier modulation systems, IEEE Trans. Broadcasting, vol. 52, no. 2, pp. 268 273, Mar. 2006. [7] J. T. Mourelo, PPR Reduction for Multicarrier Modulation, PhD thesis, University of Stanford, 1999. [8] S. Yoo, S. Yoon, S. Y. Kim, and I. Song, novel PPR reduction scheme for OFDM systems: Selective mapping of partial tones (SMOPT, IEEE Trans. Consumer Electronics, vol. 52, no. 1, pp.40 43, Feb. 2006. [9] L. ang and C. Tellambura, Simplified Clipping and Filtering Technique for PR Reduction in OFDM Systems, Signal Processing Letters, IEEE, vol.12, no.6, pp. 453-456, June 2005. [10] S. H. Han and J. H. Lee, "PPR Reduction of OFDM Signals Using a Reduced Complexity PTS Technique", Signal Processing Letters, IEEE, Vol.11, Iss.11, ov. 2004, Pages: 887-890. [11] I. Baig and V. Jeoti, PPR nalysis of DHT-Precoded OFDM System for M-QM, the 3rd International Conference on Intelligent and dvanced Systems (ICIS2010 June.2010, Kuala Lumpur, Malaysia. [12] I. Baig and V. Jeoti, DCT Precoded SLM Technique for PPR Reduction in OFDM Systems, the 3rd International Conference on Intelligent and dvanced Systems (ICIS2010 June.2010, Kuala Lumpur, Malaysia. [13] S. obilet, J. F. Helard and D. Mottier, Spreading sequences for uplink and downlink MC-CDM Systems: PPR an MI minimization, European Transactions on Telecommunications, 2002. 378 377