Digital Signal Processing Better wavelet packet tree ... - ScienceDirect

Digital Signal Processing 18 (2008) 885–891

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Digital Signal Processing www.elsevier.com/locate/dsp

Better wavelet packet tree structures for PAPR reduction in WOFDM systems Volkan Kumbasar, O˘guz Kucur ∗ Gebze Institute of Technology, Electronics Engineering Department, Gebze 41400, Kocaeli, Turkey

a r t i c l e

i n f o

a b s t r a c t

Article history: Available online 17 June 2008

This paper presents a proper way to reduce the peak-to-average power ratio (PAPR) in Daubechies (Db) wavelet-based OFDM (WOFDM) by searching better wavelet packet tree (BWPT) structures. These BWPT structures are obtained by using a brute force search algorithm. Numerical and simulation results also show that BWPT structures have lower PAPR values than conventional Mallat structures without any degradation in bit error rate (BER) performance for the same bandwidth occupancy. In addition, this PAPR reduction method does not introduce any additional complexity to WOFDM system since BWPT structures are obtained by selecting the proper ones from wavelet packet tree space. © 2008 Elsevier Inc. All rights reserved.

Keywords: WOFDM Wavelet packets PAPR

1. Introduction Fourier based multi-carrier modulation (MCM) or often called as orthogonal frequency division multiplexing (OFDM) is a strong candidate for the next generation wireless systems which require high data bit rates [1]. The main advantage of OFDM is that it is immune against multi-path fading [1]. Because of this advantage, OFDM is used as standard in several wireline and wireless applications such as HiperLAN/2 and asymmetric digital subscriber line (ADSL) [1]. However, the major disadvantage of OFDM is the peak-to-average power ratio (PAPR) problem [1]. High peaks of the transmitted signal drive the power amplifiers operating near their nonlinear saturation regions, which reduces the power efficiency and causes performance degradation. Therefore, it is necessary to transmit signals with lower PAPR because of operating range of power amplifiers [1]. The PAPR problem is investigated in several works and various techniques have been proposed as reviewed in [2]. Wavelet based multi-carrier modulation, also called wavelet (filter bank) based OFDM (WOFDM) system, which was studied in [3,4], have all the advantages and disadvantages of OFDM systems. In WOFDM systems, the orthogonality is satisfied by orthogonal wavelet filters (filter banks) [5] and no guard interval (cyclic prefix) is needed, thus enhancing the bandwidth efficiency (20%) as compared to conventional OFDM systems [6,7]. Besides, additional bandwidth efficiency (8%) in WOFDM systems is also provided since pilot tones are not necessary [6,7]. In literature, the first work on PAPR analysis of WOFDM systems has been produced in [8], where the authors show that Daubechies family gives the lowest PAPR among other wavelet families and PAPR of Haar wavelet based WOFDM can be reduced by using M-ary signaling. In [9], PAPR reduction of Haar wavelet based WOFDM is examined by applying several spreading codes such as carrier interferometry, Golay complementary and Hadamard. It has been shown that Hadamard code provides the best PAPR reduction; however PAPR reduction by spreading method has the disadvantage of additional complexity for spreading and despreading. Another method is proposed for PAPR reduction in Haar wavelet-based WOFDM in [10]. In this work, data is upsampled, decomposed by discrete wavelet transform (DWT), and decomposed data is mul-

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Corresponding author. Fax: +90 262 605 2205. E-mail address: [email protected] (O. Kucur).

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© 2008

Elsevier Inc. All rights reserved.

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V. Kumbasar, O. Kucur / Digital Signal Processing 18 (2008) 885–891

Fig. 1. Transmitter part.

Fig. 2. Channel and receiver part.

tiplied by the frequency response of root raised cosine roll-off filter before conventional tasks of WOFDM are done at the transmitter. At the receiver site, after conventional tasks of WOFDM, root raised cosine roll-off filter, inverse DWT, and downsampling are used. Therefore, PAPR is reduced at the price of increasing complexity. An approximate PAPR of WOFDM system has been first defined in [11–13], which is similar to the idea of using discrete time approach in OFDM systems. Also, a technique has been proposed to reduce the PAPR of WOFDM systems in [11–13]. This technique improves the PAPR by using a new threshold control method at the cost of some distortion and losing bit error rate (BER) performance [11–13]. It also requires side information from the transmitter to the receiver in order to label the zero padded signals whose energies are below the threshold, therefore uses the bandwidth inefficiently [11–13]. In this paper, a new way is proposed for PAPR reduction, by seeking better wavelet packet tree (BWPT) structures which reduce the PAPR of WOFDM systems. In [14], BWPT structures are searched to reduce inter-symbol interference (ISI) and ICI of WOFDM systems. But the algorithm used in [14] does not search BWPT structures for a fixed sub-carrier or sub-channel number because the algorithm in [14] takes wavelet packet level size as an input. Therefore, to search BWPT structure for a fixed sub-carrier number, a new proper algorithm taking a fixed sub-carrier number as an input is proposed in this paper. In order to seek BWPT structures Haar and Daubechies (Db) wavelet filters are used because they have lower PAPR [8] and better BER performances than the other wavelet families [15]. In this work, BWPT structures with lower PAPRs than conventional Mallat structures are obtained without any performance degradation and any additional complexity in WOFDM systems. The paper is organized as follows. System model and PAPR formulation are given in Section 2. The algorithm is defined in Section 3. Numerical and simulation results are presented in Section 4 and the last section summarizes conclusions. 2. System model and PAPR Figs. 1 and 2 show the transmitter and receiver part of the WOFDM system, respectively. In Fig. 1, the transmitter first uses a digital modulator (bit to symbol mapping) to convert the serial data bit stream d(n) into the serial modulated data stream x(n), then converts x(n) into N parallel data stream xi (n). Here N is defined as the sub-channel number, and R is the symbol rate of x(n) and R /ni is the symbol rate of xi (n). The next step of the transmitter is to perform the orthogonal wavelet modulation. Therefore, each xi (n) is up-sampled by the up-sampling factor ni and filtered by the wavelet modulator filter h i (n). ni and h i (n) are constructed from the wavelet packet (filter bank) tree structure [5].

V. Kumbasar, O. Kucur / Digital Signal Processing 18 (2008) 885–891

887

Fig. 3. Transmitter part of WOFDM system for 5 sub-channels.

For Mallat tree structure, also known as symmetric multistage tree structure, which is used in conventional WOFDM systems [3–5], h i (n) and ni are defined as follows [3,4]

h i (n) =

ni =

⎧ i −1    ⎪ ⎪ ⎪ h ( n )| ⊗ g (n)|↑2k , i ⎪ ↑2 ⎪ ⎨

i = 0, 1, . . . , N − 2,

k=0

N −2 ⎪  ⎪   ⎪ ⎪ ⎪ g (n)|↑2k , ⎩

(1) i = N − 1,

k=0

2i +1 , 2 N −1 ,

i = 0, 1, . . . , N − 2, i = N − 1,

where ⊗ denotes convolution,

M −1 k=0

(2)

indicates the cascade convolution of M filters and |↑2d indicates up-sampling with

a factor of 2d . In (1), g (n) and h(n) are the wavelet’s half-band low-pass filter and half-band high-pass filter impulse responses, respectively. For the quadrature mirror filter (QMF) bank, the relationship between both filters is given as h(n) = (−1)n g ( L − 1 − n), where L is the sequence length of g (n) [5]. In Fig. 1, y (n) denotes the wavelet modulated output signal of the transmitter. For example, the synthesis filter bank of Mallat tree structure with 5 sub-channels, which is used in WOFDM transmitter is depicted in Fig. 3. At the receiver part shown in Fig. 2, the received signal r (n) is filtered by the demodulator wavelet filters with impulse responses h∗i (−n), i = 0, . . . , N − 1 [3,4]. Then, the filtered signal is down-sampled by ni , i = 0, . . . , N − 1 and parallel to serial converted. Finally, the digital demodulator recovers transmitted data bit stream. To ensure the orthogonality of WOFDM, the following has to be satisfied [3,4]





h i (n) ⊗ hk∗ (−n) ↓n = k

PAPR is defined as



PAPR = 10 log10



i = k, i = k.

1, 0,

maxn {| y (n)|2 }

(3)



E {| y (n)|2 }

dB,

(4)

where E {·} is the expectation operator. The complementary cumulative distribution function (CCDF) of PAPR can be defined as follows [3]





P PAPR (dB) > PAPRthr (dB) ,

(5)

where PAPRthr indicates the PAPR threshold in dB. 3. Algorithm The cost function, which is used in the algorithm, is defined as f cos t = arg min v

K 1 

K

i =1

v P PAPR , i

(6)

where v indicates the created wavelet packet tree structure of wavelet packet tree space and K denotes the realized simv denotes the PAPR for the ith random input stream and the vth random ulation number to calculate f cos t . In (6), P PAPR i tree structure. Because of the convolution and up-sampling operators, the analysis of the PAPR for WOFDM systems is very difficult. Therefore, to simplify PAPR calculation, an approximate PAPR expression, which is defined in [11–13], is used. Then, v P PAPR can be defined as i



v P PAPR = 10 log10 i

maxn {| y i (n)|2 } E {| y i (n)|2 }





dB  10 log10

S −1 

i max{| y i (n)|2 }n= 0

1 Si

 S i −1 n=1

| y i (n)|2

dB,

(7)

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V. Kumbasar, O. Kucur / Digital Signal Processing 18 (2008) 885–891

Table 1 PAPRs (dB) for 5, 6 and 7 sub-channels Wavelet

Mallat

BWPT

Improvement

5 sub-channels

Haar Db-4 Db-6

6.280 8.976 10.821

5.882 7.436 7.522

0.398 1.540 3.299

6 sub-channels

Haar Db-4 Db-6

6.706 10.187 12.146

6.418 8.047 8.889

0.288 2.140 3.257

7 sub-channels

Haar Db-4 Db-6

6.995 11.082 13.268

6.788 8.249 9.058

0.207 2.833 4.210

where y i (n) denotes the ith modulated signal (the output of WOFDM transmitter) and S i denotes the length of y i (n). As seen in (7), the term E {| y i (n)|2 } is approximated by a discrete time averaged form. This PAPR equation depends on y i (n), which also depends on the input data bit stream di (n). Because of this relationship, the PAPR can take different values for different input data bit streams. Therefore, to calculate a meaningful PAPR; K random input streams are used to calculate an average PAPR, which is selected as cost function in the algorithm. Seeking BWPT structure K is selected as five million. To create different wavelet packet tree structures, the wavelet modulator filters are now redefined as h iv (n) =

Di  



qdi (n)|↑2d ,

(8)

d=1

qdi (n) =

g (n), h(n),

pdi  0.5,

(9)

pdi < 0.5,

niv = 2 D i .

(10)

In (8) and (10), D i , the wavelet tree level size for the ith filter h iv (n), is a random integer within the interval [1, N − 1], and

in (9) pdi is a random number within the interval [0, 1], which is used to select either g (n) or h(n) for the dth level of the ith filter h iv (n). In other words, in this algorithm, first D i is randomly selected for the ith filter h iv (n). Then, from d = 1 to

D i the dth level of the ith filter h iv (n), which is qdi (n) is selected as g (n) or h(n) according to the random value of p di as seen in (9). From i = 0 to i = N − 1, each wavelet modulator filter h iv (n) is obtained. The pseudo code of the search algorithm is given as follows: 1. Set the cost value and the iteration limit to a large number, and the counter to zero. 2. D i , i = 0, . . . , N − 1, are randomly selected. Then, for each D i , pdi , d = 1, . . . , D i , are randomly selected. Equations (8) and (9) are used to create random wavelet modulator filters, until each wavelet modulator filter, h iv (n) has a different tree structure and satisfy the orthogonality condition in (3). 3. To compare the randomly created  structure to the Mallat structure, the symbol rates of both wavelet packet tree strucN −1 1 tures must be equal. Therefore, if i =0 n v = 1 go to Step 2, else continue. i

4. Calculate the cost value in (6). 5. If the cost value is lower than the current cost value, save this structure as the new tree structure and the cost value as the new cost value. 6. Increase the counter by 1. If the iteration limit is smaller than the counter; go to Step 2, else give out the better tree structure. 4. Numerical and simulation results We implement the previously defined algorithm for Haar (Daubechies-2), Daubechies-4 (Db-4) and Daubechies-6 (Db-6) based binary phase shift keying WOFDM (BPSK-WOFDM) with N = 5, 6 and 7 sub-channels and using R = 2 N −1 symbols/s. Because of the search complexity of the algorithm, we use small sub-channel numbers such as 5, 6 and 7. However, it is possible to implement the algorithm with larger sub-channel numbers, too. Table 1 shows the averaged PAPR in dB for the conventional (Mallat) and BWPT structures of Haar, Db-4 and Db-6 for 5, 6 and 7 sub-channels. As seen in this table, the obtained BWPT structures improve the PAPRs. For example, with BWPT structure we get 3.299 dB improvement for Db-6 BPSK-WOFDM with 5 sub-channels. As seen in Table 1, as the wavelet degree or number of sub-channels increases, the PAPR increases for both conventional and BWPT structures. As the number of sub-channels increases, PAPR improvement decreases for Haar, and increases for Db4 and Db-6 in general. For the same number of sub-channels, PAPR improvement increases as the wavelet degree increases. Figs. 4–6 show the obtained BWPT structures of Haar, Db-4 and Db-6 for the transmitter part with 5, 6 and 7 subchannels, respectively.

V. Kumbasar, O. Kucur / Digital Signal Processing 18 (2008) 885–891

889

(a) (a)

(b)

(b)

Fig. 4. BWPT structures for 5 sub-channels with (a) Haar and (b) Db-4 and Db-6.

Fig. 5. BWPT structures for 6 sub-channels with (a) Haar and (b) Db-4 and Db-6.

(a)

(b) Fig. 6. BWPT structures for 7 sub-channels with (a) Haar and (b) Db-4 and Db-6.

As seen in Figs. 4–6, Db-4 and Db-6 have the same BWPT structures, whereas Haar has different tree structures. Another important result is that the majority of sub-channels of a BWPT structure have the same up-sampling factor. With this fact, we can say that the algorithm seeks for BWPT structures whose majority of sub-channels have the same up-sampling factor. Mathematical expressions for the obtained better wavelet modulator filters for Haar BPSK-WOFDM with 5 sub-channels are given as follows h0 (n) = g (n),

































(11)

h1 (n) = h(n) ⊗ h(n)|↑2 ⊗ h(n)|↑4 ,

(12)

h2 (n) = h(n) ⊗ h(n)|↑2 ⊗ g (n)|↑4 ,

(13)

h3 (n) = h(n) ⊗ g (n)|↑2 ⊗ h(n)|↑4 ,

(14)

h4 (n) = h(n) ⊗ g (n)|↑2 ⊗ g (n)|↑4 .

(15)

Whereas the obtained better wavelet modulator filters for Db-4 and Db-6 BPSK-WOFDM with 5 sub-channels are as follows

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V. Kumbasar, O. Kucur / Digital Signal Processing 18 (2008) 885–891

Fig. 7. CCDF of Db-6 BPSK-WOFDM with 6 sub-channels.

Fig. 8. BER comparison for Db-6 BPSK-WOFDM with 6 sub-channels.

h0 (n) = h(n),

































(16)

h1 (n) = g (n) ⊗ h(n)|↑2 ⊗ h(n)|↑4 ,

(17)

h2 (n) = g (n) ⊗ h(n)|↑2 ⊗ g (n)|↑4 ,

(18)

h3 (n) = g (n) ⊗ g (n)|↑2 ⊗ h(n)|↑4 ,

(19)

h4 (n) = g (n) ⊗ g (n)|↑2 ⊗ g (n)|↑4 .

(20)

Wavelet modulator filters for the systems with 6 and 7 sub-channels can also be expressed similarly in compatible with Figs. 5 and 6, respectively.

V. Kumbasar, O. Kucur / Digital Signal Processing 18 (2008) 885–891

891

To show the improvement of PAPR reduction for BWPT structure, the CCDFs of BWPT structure and Mallat structure for Db-6 BPSK-WOFDM with 6 sub-channels are shown in Fig. 7, as an example. For this result, R is selected as 32 symbols/s. As seen in this figure, the obtained BWPT structure has improved the PAPR. Fig. 8 shows the Monte Carlo simulation results of the BER for the Db-6 based BWPT structured and the Mallat tree structured BPSK-WOFDM systems with 6 sub-channels in an additive white Gaussian noise (AWGN) channel, as an example. In this simulation, R is selected as 32 symbols/s and 160 million symbols are used to calculate each BER value. As seen in this figure, the BWPT structure and the conventional structure have the same BER performance. The same result is obtained by all BWPT structures. Therefore, all obtained BWPT structures improve PAPR without any BER performance degradation with respect to Mallat structures. In addition, BWPT structures have the same bandwidth occupancy as Mallat structures. 5. Conclusions In this paper, the PAPRs of Haar, Db-4 and Db-6 BPSK-WOFDM systems are reduced by BWPT structures as compared to conventional Mallat structures. Simulation results also indicate that the BWPT structures provide the same BER performances as conventional Mallat structures over AWGN channel for the same bandwidth occupancy. We obtain PAPR reduction in WOFDM systems without any lost in BER performance and bandwidth efficiency, any side information and additional complexity. There might be better wavelet packet tree structures than the ones found in this work, i.e., they are not optimal. However, it is hard to define a suitable systematic search algorithm to obtain optimal wavelet packet tree structures which reduce the PAPR in WOFDM systems for fixed number of sub-channels. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

S. Hara, R. Prasad, Multicarrier Techniques for 4G Mobile Communications, Artech House, 2003. Y. Li, G.L. Stüber, Orthogonal Frequency Division Multiplexing for Wireless Communications, Springer, 2006. B.G. Negash, H. Nikookar, Wavelet-based multicarrier transmission over multipath wireless channels, Electron. Lett. 36 (2000) 1787–1788. Y. Zhang, S. Cheng, A novel multicarrier signal transmission system over multipath channel of low-voltage power line, IEEE Trans. Power Deliv. 19 (4) (2004) 1668–1672. G. Strang, T. Nyugen, Wavelets and Filter Banks, Wellesley/Cambridge Press, 1996. Rainmaker Technologies, Inc., RM wavelet based (WOFDM) PHY proposal for 802.16.3, Technical Report IEEE 802.16.3c-01/12, 2001, http://ieee802. org/16. M.K. Laksmanan, H. Nikookar, Review of wavelets for digital wireless communication, Wireless Pers. Commun. 37 (2006) 387–420. A. Ben Aicha, F. Tlili, S. Ben Jebara, PAPR analysis and reduction in WPDM systems, in: IEEE First International Symposium on Control, Communications and Signal Processing, 2004, pp. 315–318. K. Anwar, A.U. Priantoro, M. Saito, T. Hara, M. Okada, H. Yamamoto, On the PAPR reduction for wavelet based transmultiplexer, in: IEEE Int. Symposium on Commun. and Information Technology, 26–29 Oct. 2004, vol. 2, pp. 812–815. H. Sakakibara, E. Okamoto, Y. Iwanami, A wavelet packet modulation method for satellite communications, in: IEEE TENCON 2005, Nov. 2005, pp. 1–5. H. Zhang, D. Yuan, F. Zhao, Threshold method to reduce PAPR in wavelet based multicarrier modulation systems, in: IEEE Int. Symp. on Information Theory, 4–9 Sept. 2005, pp. 1266–1269. H. Zhang, D. Yuan, M. Paetzold, Novel study on PAPRs reduction in wavelet-based multicarrier modulation systems, Digital Signal Process. 17 (2007) 272–279. H. Zhang, D. Yuan, C.-X. Wang, A study on the PAPRs in multicarrier modulation systems with different orthogonal bases, Wireless Commun. Mobile Comput. 7 (3) (2007) 311–318. G. Xingxin, L. Mingquan, F. Zhenming, Optimal wavelet packet based multicarrier modulation over multipath wireless channels, in: IEEE Int. Conf. on Commun., Circuits & Systems and West Sino Expo., 29 June–1 July 2002, pp. 313–317. X. Gao, H. Zhang, D. Yuan, Performance of different wavelets over wavelet packet multicarrier modulation system, in: Proc. ICCC2004, Beijing, China, 15–17 Sept. 2004.

Volkan Kumbasar was born in Berlin, Germany in 1979. He got his B.S. degree in Electronics and Communication Engineering from Kocaeli University, Turkey in 2003. He received his M.S. degree in Electronics Engineering from Gebze Institute of Technology (GIT), Turkey in 2006. From June 2004 to July 2007, he worked as a research assistant at GIT. Currently, he is with Nortel Netas¸ in Turkey and continues his Ph.D. study at GIT. His research interests are wavelets, OFDM, optimization, and MIMO.

Dr. O˘guz Kucur got his B.S. degree in Electronics and Telecommunication Engineering from Istanbul Technical University, Istanbul, Turkey in 1988. He received his M.S. and Ph.D. degrees both in Electrical and Computer Engineering from Illinois Institute of Technology (IIT), Chicago, USA in 1992 and 1998, respectively. From 1996 to 1998 he was a teaching assistant at IIT. He was an assistant professor in the Department of Electrical Engineering at South Dakota State University in the 1998–1999 academical year. He has been working as an assistant professor in the Department of Electronics Engineering at Gebze Institute of Technology, Turkey since October 1999. His research interests are spread spectrum and CDMA systems, OFDM, MIMO and rotation diversity.