Wavelet-OFDM Vs. OFDM: Performance Comparison Marwa Chafii, Yahya Harbi, Alister Burr

Wavelet-OFDM vs. OFDM: Performance Comparison Marwa Chafii, Yahya Harbi, Alister Burr To cite this version: Marwa Chafii, Yahya Harbi, Alister Burr. Wavelet-OFDM vs. OFDM: Performance Comparison. 23rd International Conference on Telecommunications (ICT), May 2016, Thessaloniki, Greece. hal- 01297214 HAL Id: hal-01297214 https://hal-centralesupelec.archives-ouvertes.fr/hal-01297214 Submitted on 3 Apr 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Wavelet-OFDM vs. OFDM: Performance Comparison Marwa Chafii Yahya J. Harbi, and Alister G. Burr CentraleSupélec, IETR Dept. of Electronics 35576 Cesson - Sévigné Cedex, France University of York, York, UK marwa.chafi[email protected] {yjhh500, alister.burr} @york.ac.uk Abstract—Wavelet-OFDM based on the discrete wavelet trans- conclusion of our study, we propose the Wavelet-OFDM based form, has received a considerable attention in the scientific on the Discrete Meyer (Dmey) wavelet as an alternative to community, because of certain promising characteristics. In this OFDM, since it outperforms OFDM in terms of PAPR and paper, we compare the performance of Wavelet-OFDM based BER at the cost of increased complexity. on Meyer wavelet, and OFDM in terms of peak-to-average power ratio (PAPR), bit error rate (BER) for different channels The remainder of this paper is organized as follows. In and different equalizers, complexity of implementation, and Section II, the Wavelet-OFDM system is described and its power spectral density. The simulation results show that, without different variants with the corresponding implementation are decreasing the bandwidth efficiency, the proposed scheme based presented. The PAPR for different variants, the BER under on Meyer Wavelet-OFDM, outperforms OFDM in terms of PAPR different channel conditions, the implementation complexity by up to 4.5 dB, and in terms of BER by up to 6.5 dB of signal to as well as the PSD are investigated in Section III, IV, V, and noise ratio when using the minimum mean-square error equalizer without channel coding, at the cost of a computational complexity VI respectively. Finally, Section VII concludes the paper and increase. gives the perspectives of the work. Keywords—Wavelet-OFDM, Discrete Wavelet Transform II. INTRODUCTION TO THE WAVELET-OFDM SYSTEM (DWT), Orthogonal Frequency Division Multiplexing (OFDM), Meyer Wavelet, Haar Wavelet. A. Wavelet Basis Let ψ and φ be two functions ∈ L2(R) of a finite support 1 I. INTRODUCTION [0,T0]. such that: 2 2 Several wireline and wireless communication standards kψk = kφk = 1, (1) use orthogonal frequency division multiplexing (OFDM) +∞ and ψ(t) = 0. (2) system as a modulation technique for data transmission. Z−∞ OFDM has proved its worth due to its robustness against 2 R frequency selective channels compared with single carrier L ( ) is the space of square integrable functions. The norm k.k 2 R 2 +∞ 2 modulation, and also to its high spectral efficiency due to of a function g ∈ L ( ) is defined as: kgk = R−∞ |g(t)| dt. its use of orthogonal waveforms. However, OFDM suffers ψ and φ can be named the mother wavelet function and the from some drawbacks such as high peak-to-average power mother scaling function respectively. ratio (PAPR), and synchronisation problems. To counter Let J0 be the first scale selected; we define contracted versions ψ and φ of the functions ψ and φ, for j ≥ J , these disadvantages and to improve its performance in other j j 0 j/2 j respects, new multi-carrier modulation (MCM) systems have ψj (t) = 2 ψ(2 t) (3) been designed, including Wavelet-OFDM [1], [2]. Instead j/2 j φj (t) = 2 φ(2 t). (4) of using the fast Fourier transform (FFT) as in OFDM, T0 Wavelet-OFDM is based on the discrete wavelet transform The contracted functions ψj and φj have a support of [0, 2j ]. (DWT). For every scale j, we define translated versions ψj,k and φj,k of the functions ψj and φj as follows: −j In this paper, the performance of Wavelet-OFDM is com- ψj,k(t) = ψj (t − 2 kT0) j/2 j pared with OFDM, in terms of PAPR, bit error rate (BER) on = 2 ψ(2 t − kT0), (5) −j an additive white gaussian noise (AWGN) channel, a flat fading φj,k(t) = φj (t − 2 kT0) channel and a frequency selective channel. In addition, the j/2 j = 2 φ(2 t − kT0). (6) complexity of implementation and the power spectral density (PSD) are also analysed and compared. We show that the The contracted translated functions ψj,k and φj,k have a sup- kT0 (k+1)T0 j Wavelet-OFDM system performance depends on the number of port of [ 2j , 2j ]. For j ∈ [[J0,J −1]] and k ∈ [[0, 2 −1]], scales considered in the wavelet transform. We also apply the we define the wavelet basis as: frequency domain equalization to the Wavelet-OFDM system k=2J0 −1 J−1 k=2j −1 {φJ0,k}k=0 j=J0 {ψj,k}k=0 , (7) to improve its BER performance. Moreover, based on the study S of the PSD, we show that the selection of the wavelet has a 1The support of a function means here the interval outside which the great impact on the bandwidth efficiency of the system. As a function is equal to zero where J is the last scale considered. B. Expression of the transmitted Wavelet-OFDM signal Wavelets have been used in several wireless communication applications such as data compression, source and channel coding, signal denoising and channel modeling. Moreover, wavelets have been proposed as a modulation basis for mul- ticarrier modulation systems. The resulting system is named Wavelet-OFDM [1], [2] or also known as orthogonal wavelet division multiplexing (OWDM) [3]. In fact, the functions of the wavelet basis defined in (7), can be considered as multi-carrier waveforms for data transmission [4], in the same way as the Fourier basis is considered in a conventional OFDM system. Wavelet-OFDM is the MCM system based on the wavelet basis. Instead of carrying the input symbols upon the exponential Figure 1: Variants of Wavelet-OFDM. functions of Fourier system, the carriers are represented by the wavelet functions (ψj,k)j∈[[J0,J−1],k∈[[0,2j −1]] and the scaling functions (φJ0,k)k∈[[0,2J0 −1]] of the first scale. The transmitted Wavelet-OFDM signal can thus be expressed as follows 2: discrete wavelet transform (IDWT) should be performed L J−1 2j −1 times. L can also be interpreted as the number of decomposition levels. Let Cn be a vector of M input complex x(t) = wj,kψj,k(t − nT0) symbols C . The J0 first C symbols correspond to Xn jX=J0 kX=0 m,n 2 m,n J0 J 2J0 −1 the 2 scaling coefficients (aJ0,k)k∈[[0,2 0 −1]]. The second 2J0 complex symbols correspond to the wavelet coefficients + aJ0,kφJ0,k(t − nT0). (8) Xn kX=0 (wJ0,k)k∈[[0,2J0 −1]] of the first scale J0. First, one IDWT is performed, which gives in its output 2J0+1 scaling coefficients. • wj,k: wavelet coefficients located at k-th position from J0 After that, the next 2 +1 coefficients from the vector Cn scale j, are extracted and considered as wavelet coefficients, and the a : approximation coefficients located at k-th posi- • J0,k second IDWT is performed. The next symbols are processed tion from the first scale J , 0 in the same way until the last scale j = J − 1 is reached. For • ψj,k and φJ0,k are the wavelet and the scaling func- example, let Cn be a vector of M = 8 input symbols, and let tions defined in (5) and (6) respectively. us consider two cases: the first case: The number of carriers M is equal to 2J . The mother wavelet • J0 = 0, which means that we select the maximum and the mother scaling function correspond to ψ0,0 and φ0,0 j number of decomposition levels L = 3. respectively. For each scale j there are 2 corresponding th The vector Cn of input symbols for the n period T0 translated wavelet functions. From one scale to the next, the is expressed as: number of wavelet functions is then multiplied by two. For the J0 first scale J0, there are 2 scaling functions. Cn = {C1,C2,C3,C4,C5,C6,C7,C8} Several variants of Wavelet-OFDM can be constructed, = {a0,0, w0,0, w1,0, w1,1, w2,0, w2,1, w2,2, w2,3.} depending on the first scale J0 considered. Fig. 1 depicts • J0 = 1, which means that we consider L = 2 the different variants of Wavelet-OFDM defined for different decomposition levels. values of J , for a total number of scales J = 3. The position th 0 The vector Cn of input symbols for the n period T0 of the functions in Fig. 1 refers to the time localization and is expressed as: the frequency localization characteristics of the functions. We can observe that these properties change from one variant to Cn = {C1,C2,C3,C4,C5,C6,C7,C8} the other, and the selection of the variant depends on the = {a1,0, a1,1, w1,0, w1,1, w2,0, w2,1, w2,2, w2,3.} application.

Load more