Study and Implementation of Complementary Golay Sequences for PAR reduction in OFDM signals

Transcription

1 Study and Ipleentation of Copleentary Golay Sequences for PAR reduction in OFDM signals Abstract In this paper soe results of PAR reduction in OFDM signals and error correction capabilities by using Copleentary Golay Sequences are presented. This results have been obtained by siulations and later we have ipleented this technique in a DSP. Siulation results show a PAR of db; however, physical results produce a PAR of 6 db. Also, an algorith for generation of Golay Base Sequences on the fly is proposed. Key Words OFDM, Copleentary Golay Sequences, PAR I. INTRODUCTION Today s needs of bandwidth and flexibility are iposing the use of efficient odulations that ay be fit to the characteristics of wireless channels. This is one of the reasons why ulticarrier odulation techniques are finding growing interest for Wireless Local Area Networks (WLAN). Recent WLAN standards, such as Hiperlan type [] and IEEE 8.a [], have adopted the for transission of high bit rates in these networks.the choice of OFDM (Orthogonal Frequency Division Multiplexing) is due to its good perforance in ultipath environents and the nuber of sub-carriers has been chosen so as to itigate the indoor channel effects. Basically, OFDM divides bandwidth into several orthogonal sub-carriers and sends inforation into these sub-carriers. Nevertheless, one of the ain disadvantages of this odulation is its high PAR (Peak-to-Average power Ratio) requiring the use of linear HPAs (High Power Aplifiers) that are very power-inefficient and have an enorous ipact on equipent s autonoy. Matheatically, we can define PAR by: PARfxg = axjx fi j Efjx fi j g This is the relationship between the peaks of the signal and its ean. Since OFDM is a good technique to itigate ultipah effects, it is interesting to try to iprove its disadvantage and reduce the high PAR of this kind of signals. There are a lot of techniques for this purpouse like Copleentary Golay Sequences [], Partial Transit Sequences (PTS) [4], Selective Mapping (SLM)[4], Tone Reservation [5], Clustered Transission [6] and Orthogonal Pilot Sequences (OPS) [7]. Golay Sequences have been chosen for two reasons. First, with this technique, PAR is liited up to db, independently of the nuber of carriers and input data. This is very iportant, because we know in advance how is the dynaic range for HPA. Second is the error correction capability of these codes that allow us to iprove the whole syste. The reaining of the paper is organized as follows. Section II is a brief description of Copleentary Golay Sequences and their characteristics. Then, in section III we show the perforance of this kind of code. Next, section IV analyses the coplexity of the encoder and decoder. In section V we propose an algorith to generate Golay Base Sequences, in this way, we do not need to spend eory to store the, and in section VI we show results obtained in the ipleentation of this encoder. Finally, we draw soe conclusions. II. COMPLEMENTARY GOLAY SEQUENCES Only a few input data sequences produce signals with high PAR. With Golay codes, we generate sequences that once odulated have PAR liited to db []. Two sequences a and b are Copleentary Golay Sequences if the su of their autocorrelations is null except in zero. This is: C a (i) +C b (i) =8 i 6= There are two very interesting relationships between

2 Copleentary Golay Sequences and Reed - Muller codes []. First: each of the!= cosets of RM h (;) in ZRM h (;) having a cosets representative of the for: h X k= x ß(k) x ß(k+) () coprise one of h(+) Golay sequences over Z h of length, where h >, ß is a perutation of f :::g and is the length of the copleentary Golay sequence. In addition, there is another interesting relationship: any sequence of the for X h k= x ß(k) x ß(k+) + X k= c k x k () with c k Z h is a copleentary Golay sequence with respect to the others. With these two results, it is easy to design a block algorith for coding input sequences into Copleentary Golay Sequences (so, PAR of db) and with the error correction capabilities of Reed-Muller codes. We use the first w bits to select the Base Golay Sequence in ZRM h (;), then we take +groups of h bits each one to build the final code. In this way, we have the encoding algorith. We will have therefore w + h ( + )input bits and h output bits, where is the code length, h is the odulation depth and w is the nuber of Golay Base Sequences to use. Thus, the code rate is: w + h ( +)bits R = h bits where, h and w are the paraeters before cited. () In this equation we can see that code rate decreases exponentially as increases, so fro this point of view, we would use sall codes ( sall). And, this is less obvious, but h paraeter does not affect very uch the code rate, so we will have flexibility to odify this paraeter without having uch effect in code rate. These Copleentary Golay Sequences are only valid for phase odulations, this is, M-PSK odulations. It should be noted that they are not valid for M-QAM. III. CHARACTERISTICS AND SIMULATION This section shows soe results obtained by Montecarlo siulations. A. Channel with Aditive White Gaussian Noise First, we begin with soe results in channels with additive White Gaussian Noise (AWGN). This siulations have been carried out with: Sae nuber of sub-carriers as code length (N = ) Without Frequency Guards With Cyclic Prefix Additive White Gaussian Noise (AWGN) Paraeter w at axiu each oent. PAR is always db. We can view in fig. that as we increase the code length ( paraeter), we iprove BER. This is easy to explain. When is larger, the errror correction capability of this code is better so, as we increase, BER decreases. Also, we can see that there are soe E b =N for which uncoded syste is better than coded. This is a block code syste so, when the code can correct all errors, the syste works properly but when the code is not enough to correct the, the syste fails in block and BER increases. This effect is not only for these Golay codes but is very coon in FEC (Forward Error Correction) codes. In this fig. we can also see that the slope icreases as we increase since, as we have said before, it is a block code, and when it fails, it fails in block and the probability of error is larger. And also in fig., when odulation depth (h paraeter) increases, BER increases too and it would be neccesary to increase code length () to copensate for this effect. This is the sae as decreasing E b =N. This effect is better shown in fig.. This figure was obtained with fixed code length and varying odulation depth (h). And finally, also fig. shows that if we decrease code rate for the sae odulation depth, the perforance is better. Obviously the ore redundance, the better perforance.

3 BER BER 4 BER vs SNR in QPSK & 8 PSK QPSK Uncoded = =5 =7 8 PSK Uncoded = =5 = Eb/No Fig.. Code Rate Coparation between QPSK and 8-PSK BER vs SNR with Golay codes length 64 ( = 6) Paraeter w at axiu each oent. With the obtained data we can say that if we use the sae nuber of sub-carriers as code length or a sub-ultiple, PAR obtained is db, except if the nuber of sub-carriers is 8; in this case only with = we can obtain a PAR of db. This is very interesting because we can increase the code length as we want in order to achieve the desired probability of error. But, as we increase the code length (), we increase the coputational load too. It is true that if we use twice the nuber of sub-carriers than the code length we obtain PAR of 6 db, but if we use higher ultiples of the code length, PAR is or db less than in the uncoded syste. Fro this point of view, when the syste has few sub-carriers (64, 8, 56) as in WLAN environents, we will have ore flexibility in order to design the syste (we will be able to use short codes or long codes for very restrictive BER characteristics), but in systes with large nuber of sub-carriers (4, 48 and so on) codes will only be large or very large, and the syste will be ore coplex. QPSK 8 PSK 6 PSK PSK 64 PSK Eb/No Fig.. Coparation M-PSK ( = 6) B. PAR and Nuber of Sub-carriers In [] it is deostrated that if we use twice the nuber of sub-carriers than the code length we obtain PAR of 6 db. We ask ourselves which rates between the nuber of sub-carriers and code length are valid for PAR reduction, and we have siulated soe environents. With these results, we have ore flexibility to design our syste. Siulations have these speciffications: Variable nuber of subcarriers Without Guard Frequencies With Cyclic Prefix With perfect channel (without noise and distortion) C. PAR and Guard Frequencies Another point is how PAR reduction depends on Guard Frequencies. Until this oent, all siulations have been run without guard frequencies. Now: Nuber of sub-carriers (N =, N = + y N = ) Variable Guard Frequencies With Cyclic Prefix Without channel siulation Paraeter w at axiu each oent. In this case, results are not conclusive but only tendencies. When we use guard frequencies, Golay Sequences properties are broken, therefore the PAR obtained is not db but larger; however it is always, or 4 db lower than in the uncoded syste. This diference is larger as we increase odulation depth because PAR is ore or less constant for one, independently of the ratio between Guard Frequencies, the nuber of sub-carriers and h, and in the uncoded syste it is not constant with h paraeter. If we focus our atention in the ratio between the nuber of sub-carriers and guard frequencies, we can

4 conclude that while we hold a sall rate ( %), results are siilar. 9 8 Radix.5 x 5 h= h=4 h=6 Decoder IV. ALGORITHM COMPLEXITY In this section, we will analize the coplexity of the encoder and the decoder. We will use MAC (Multiply Add Carry) operations as a easure, because we will ipleent the in a DSP (Digital Signal Processor). This coputacional load will depend on the code paraeters (, h and w), but ainly on A. Coplexity If we only focus in the encoder algorith and we supose that previously we have generated all Base Golay Sequences (BGS) and we have the into a lookuptable, we will need ( + ) MAC operations and divisions. For the encoder, coplexity will depend exponetially only on Fig.. Nuber cycles in DSP for ipleentation V. GENERATION OF BASE SEQUENCES B. Decoder Coplexity The decoder, does not always do the sae nuber of operations. We will have therefore one lower bound and one higher bound. The higher bound: ( + h +!=) Divisions (+) ( +)Λ + Λ h +(h +)MAC operations. ( +) + w test operations. As in the encoder, the nuber of operations depends expontially on, but also on h and w. And the nuber of operations is larger than in the encoder case. If we copare with the coplexity of the FFT using radix algorit, we can see that: ((N=4 +) log N (5 + 4 (N= )) The axiu nuber of BGS ( w ) is depending on code length ( w»!=). This nuber is very large and increases exponentially with. The eory needed to store all these BGS ay be very large. One way to solve this proble is to liit the nuber of BGS, but the code rate depends on this w. The advantage of this ethod is that we can analyse all the base codes and find the best. Another way is to try to generate these sequences on the fly when it is needed. Looking at equation, we can see that, if we are able to generate a specific perutation, it is easy to build BGS. +N=4 +6 N ) cycles In fig. we can see the evolution of the nuber of cycles depending on paraeters. Actually, DSPs can execute several illions of instructions per second, so this algorith sees viable. Basically, there are two ethods to generate all the perutations: iterative and lexicographic algoriths. Lexicographic algoriths generate all the perutations with a factorial relationship. We can use this in order to design an algorith to generate a specific perutation. This algorith is as follows: 4

5 Algorith to Calculate Perutations. Array Initialization list with all eleents in lexicographic order.. Array initialization perutation vide, aux = n, act = N and i =, where n is nuber of perutation and N total nuber of eleents.. If n od act! =, array perutation is copleted with inverse of list and jup to 7 else jup to res = aux. If integer part of res is and (act )! res >, res = res. 5. Add to perutacion eleent in list in position integer part of res. Delete this eleent fro list. 6. act = act, aux is ultiplied by the fractionary part of res by act!. i = i + and back to. 7. Array perutation is the desired perutation. With this algorith we can generate any perutation without the need to generate the rest. However, not all possible perutations (!) are valid, only half of the generated codes are valid. Syetrical perutation generate the sae code. It is needed therefore to know how the perutations are distributed with respect to their syetrical. This is not easy, P P but first i= ( i)! perutations generate first i= ( i)! codes, and this nuber of codes are enough for our purpouse. When we are encoding, we take first w bits to select BGS, with w»!=. At the begining, this was a proble, but now it plays in our favour, because if we take = 5, wehave5!= = 6 possible codes, but we can only use w =5( 5 =), and P 4 i= (5 i)! =, so we do not loose anything using our algorith. Next table shows soe exaples:!= w ax P i= ( i)! wax Loss TABLE I LOSS COMPARISON TABLE The loss in code rate is not very iportant, because we loose only two or three bits. If we focus in coplexity, this algorith is not very heavy. It needs + MAC operations and test operations. In fig. 4 we can see the diferences between the nuber of cycles if we use the algorith with and without calculating the perutations. In decoding, the diferences are larger with SGB Radix x h= h=4 h=6 h= SGB h=4 SGB h=6 SGB Decoder Fig. 4. Nuber of cycles with Generation Another solution is a hybrid between two ethods, to store =, =4 and so on of all perutations that we want and generate the rest when they are needed. In this way, we reduce coplexity. VI. IMPLEMENTATION We have ipleented this kind of encoder/decoder in a DSP to check whether theoretical characteristics are true or not. We have used TMSC6 DSP fro Texas Instruents and soe converters in order to obtain output physical signals. We have scaled by all the usual values of OFDM paraeters [8], thus N =, BWof 6.5 khz and carrier frequency of 6.5 khz. In fig. 5 we can see the uncoded OFDM output signal. This signal is between and 4.7 V aprox. and in fig. 6 the output coded OFDM signal is shown. We can see clearly that there is a reduction in the peaks of the signal. 5

6 Salida (V) Sequences are not valid for use with M-QAM schees that are ore efficient than M-PSK. However the PAR reduction can copensate the efficiency loss. Also, we have iproved the probability of error of this kind of signals by using the error correction capabilities of Reed-Muller codes. We can fix the probability of error and find Golay encoder paraeters in order to achieve this desired BER. ( and h). We can use Golay codes alone for PAR reduction and error correction, nevertheless it is a good idea to use the with an outer code to iprove the perforance. 5 5 Tiepo (s) Fig. 5. Output OFDM. Uncoded It should be ephasized too that we have designed an algorith to generate BGS on the fly. In this way it is not needed to use eory to store all base sequences, although it is better for sall values of w. Salida (V) Tiepo (s) Fig. 6. Output OFDM. Coded PAR of uncoded signal is nearly 9 db and PAR of coded signal is nearly 6 db. There is a difference between siulations and ipleentations of db [9] because in siulations we use low pass equivalent signal representation and in ipleentation we use the real signal, quantified and with noise. REFERENCES [] ETSI. Broadband radio access networks (bran); hiperlan type ; physical (phy) layer. ( 475 V..),. [] IEEE8. Wireless lan ediu access control (ac) and physical layer (phy) specifications: High-speed physical layer in the 5 ghz band [] J. A. Davis and J. Jedwab. Peak-to-Mean Power Control in OFDM, Golay Copleentary Sequences and Reed-Muller Codes. Dec. 97. [4] L.Ciini Jr and N.R. Sollenberger. Peak-to-Aberage Power Ratio Reduction of an OFDM Signal Using Partial Transit Sequences. Icc, 999. [5] J. Tellado and J. Cioffi. Peak Power Reduction for Multicarrier Transission. Globeco, Nov 998. [6] L.J. Ciini Jr, B. Daneshrad, and N.R. Sollenberger. Clustered OFDM with Transitter Diversity and Coding. Globeco, 996. [7] M. J. Fernández-Getino, J. M. Páez-Borrallo, and O. Edfors. Orthogonal Pilot Sequences for Peak-to-Average Power Reduction in OFDM. Procedings of IEEE VTC,. [8] M. Lobeira, A. García, R. Torres, and J. L. García. Paraeter estiation and indoor channel odelling at 7 GHz for OFDM-based broadband WLAN. Procedings of IST Mobile Suit,. [9] J. Tellado. Peak-to-Average Power Reduction. PhD thesis, Standford University, Septeber 999. VII. CONCLUSIONS The PAR of OFDM signal has been reduced by using Copleentary Golay Sequences. In siulations, PAR is liited to db whereas in physical ipleentation PAR es liited to 6 db. However, this PAR reduction is only valid for OFDM signals without guard frequencies. With guard frequencies, the reduction is not so good. Another constraint is that these Copleentary Golay 6