Power Spectral Density Limitations of the Wavelet-OFDM System

2016 24th European Signal Processing Conference (EUSIPCO)

Marwa Chaﬁi ∗, Jacques Palicot ∗, Rémi Gribonval † and Alister G. Burr ‡ ∗ CentraleSupélec, IETR, Campus de Rennes 35576 Cesson - Sévigné Cedex, France Email: {marwa.chaﬁi, jacques.palicot} @supelec.fr †Inria - Bretagne Atlantique, 35042 Rennes Cedex, France Email: [email protected] ‡Dept. of Electronics University of York, York, UK Email: [email protected]

Abstract—Wavelet-OFDM based on the discrete wavelet trans- compared with OFDM, since it does not need a cyclic prefix form is a multicarrier modulation technique of considerable as stated in many references [5], [7], [8], [9], [10]. However, interest, due to its good performance in several respects such as there are other factors which may have a more significant effect the peak-to-average power ratio and the interference cancellation, on bandwidth efficiency, and this has motivated us to conduct as investigated in the literature. More specifically, the Haar a more rigorous study of the power spectral density (PSD) of wavelet has been proposed by various researchers as the most Wavelet-OFDM, which highlights, as we will see, the cost to attractive wavelet for data transmission. In this paper, we address the power spectral density limitations of Wavelet-OFDM, and pay for the advantages enumerated above, especially in the we show analytically and experimentally that the bandwidth case of the Haar wavelet. efficiency of Haar Wavelet-OFDM is significantly poorer than OFDM, having larger main lobe and side lobes compared with In this paper, we study analytically and experimentally OFDM, which reduces the attractiveness of the scheme. the PSD of Wavelet-ODFM, and specifically for the Haar wavelet, since this latter has been promoted in the literature Keywords—Wavelet-OFDM, Orthogonal Frequency Division for its several advantages, but its limitations have rarely been Multiplexing (OFDM), Haar wavelet, Discrete Wavelet Transform investigated. We show that the bandwidth efficiency of Haar (DWT), Power Spectral Density (PSD). Wavelet-OFDM is poorer than conventional OFDM, having large main lobe and side lobes compared with OFDM. We I.INTRODUCTION also addressed the PSD problem as a serious limitation of the Wavelet-OFDM, which should be taken carefully in the study Orthogonal frequency division multiplexing (OFDM) is of this new modulation technique. a very popular modulation technique used in many wireless and wireline communication standards, thanks to its high The paper is organized as follows. Section II defines the spectral efficiency and its ability to overcome the effects Wavelet-OFDM and its variants. The theoretical analysis of the of multipath channels. However, OFDM suffers from some PSD is presented in Section III, while Section IV presents the drawbacks such as high peak-to-average power ratio (PAPR), simulation of the PSD that confirms our analytical result, and and sensitivity to carrier frequency offset and synchronisation further discussions. The conclusions are drawn in Section V errors. To counter these disadvantages, much research has been with some perspectives of the work. conducted in order to design new multi-carrier modulation (MCM) systems as alternatives to OFDM. In this context, Wavelet-OFDM [1], also known as orthogonal wavelet division II.DESCRIPTIONOFTHE WAVELET-OFDM SYSTEM multiplexing (OWDM) [2], has been proposed and promoted Notations: The transmitted MCM signal can be expressed by many authors. Wavelet-OFDM modulation is based on in general as: the inverse discrete wavelet transform (IDWT) instead of the inverse discrete Fourier transform (IDFT) as for OFDM. M−1 In [3], the authors claim that the Haar and the Daubechies x(t) = X X Cm,n gm(t − nT0) . (1) wavelets outperform conventional OFDM in reducing inter- n∈Z m=0 symbol interference and inter-carrier interference in the power | gm{z,n(t) } line communication context. According to [4] and [5], the Haar wavelet outperforms OFDM and the other wavelets in terms M denotes the number of carriers. Cm,n stands for the input of bit error rate. The Haar wavelet has been also presented as complex symbol, time index n, modulated by carrier index m. the wavelet that gives the best PAPR performance [6], [5] and Let us assume that (Cm,n)(m∈[[0,M−1]], n∈Z) are independent and identically distributed, with zero mean and unit variance the lowest computation complexity [7]. 2 σC . T0 is the duration of M input symbols Cm,n (duration of However, the bandwidth efficiency of Wavelet-OFDM has the MCM symbol). The modulation transform and the pulse been rarely addressed in the literature. The common belief shaping filter are jointly modeled by a single function denoted 2 is that Wavelet-OFDM improves the bandwidth efficiency by gm ∈ L (R) (the space of square integrable functions).

A. Expression of the transmitted signal Wavelet-OFDM is an MCM system based on the Wavelet basis instead of the Fourier basis. The modulation system (gm)m∈[[0,M−1]] is represented by the wavelet j functions (ψj,k)j∈[[J0,J−1],k∈[[0,2 −1]] and the scaling functions (φJ0,k)k∈[[0,2J0 −1]] of the ﬁrst scale. The waveforms (gm)m∈[[0,M−1]] can be expressed as:

(ψJ0,k)k [[0,2J0 1]] (φJ0,k)k [[0,2J0 1]], ∈ − ∪ ∈ − if m [[0, 2J0+1 1]] (gm)m [[0,M 1]] :=  ∈ − ∈ − j (ψj,k)j [[J0,J 1],k [[0,2 1]]  ∈ − ∈ −  else.  Figure 1: Some variants of the Wavelet-OFDM. The transmitted Wavelet-OFDM signal is then deﬁned as follows:

J 1 2j 1 − − J x(t) = wj,kψj,k(t nT0) The second 0 complex symbols corresponds to the wavelet − 2 n j=J0 k=0 coefficients w J of the first scale J . First, one X X X ( J0,k)k∈[[0,2 0 −1]] 0 2J0 1 J0+1 − IDWT is performed, which gives in its output 2 scaling J0+1 + aJ0,qφJ0,q(t nT0). (2) coefficients. After that, the next 2 coefficients from the − n q=0 vector Cn are extracted and considered as wavelet coefficients, X X and the second IDWT is performed. The next symbols are • J − 1: last scale considered, with M = 2J , processed in the same way until the last scale j = J − 1 is • J0: first scale considered (J0 ≤ j ≤ J − 1), reached. The vector Cn can be expressed then as: • wj,k: wavelet coefficients located at k-th position from the Cn J scale j, = (aJ0,0, aJ0,1, . . . , aJ0,2 0 1) • a : approximation coefficients located at k-th position − J0,k .(wJ0,0, wJ0,1, . . . , wJ ,2J0 1) 0 from the first scale J , − 0 J +1 j/2 j .(wJ0+1,0, wJ0+1,1, . . . , wJ0+1,2 0 1) • ψj,k = 2 ψ(2 t − kT0): the wavelet orthogonal functions, − J0 . ... .(wj,0, wj,1, . . . , wj,2j 1) 2 J0 − • φJ0,k = 2 φ(2 t−kT0): the scaling orthogonal functions . ... .(wJ 1,0, wJ 1,1, . . . , wJ 1,2J−1 1). (3) at the scale J0. − − − − The symbol in (3) stands for the concatenation operator. Note that the wavelet coefficients wj,k and the approximation . Fig.2 defines the implementation of one decomposition level coefficients aJ0,k represent the complex input symbols Cm,n of (1). The mother wavelet function and the mother scaling j. According to the Mallat algorithm, the IDWT consists of upsampling by a factor of two and filtering the approximation function have a duration of T0, and corresponds to j = 0, k = 0. For each scale j corresponds 2j translated wavelet functions. coefficients (scaling coefficients) and the detail coefficients l From one scale to the next, the number of wavelet functions (wavelet coefficients) respectively by a low-pass f and a high- h is then multiplied by two. pass f filter, whose responses are derived from the wavelet considered. B. Variants and implementation Several variants of the Wavelet-OFDM system can be considered, depending on the first scale J0 selected. Since the scaling functions are considered only for the first scale, J0 then defines the number of the scaling functions φj,k in the modulation system. Fig.1 depicts the wavelet modulation system for different values of J0, for M = 8 carriers. By J convention, when J0 = J, there are 2 scaling functions φj,k and no wavelet function ψj,k considered in the wavelet basis. Note that the position of the functions in Fig.1 is not a coin- Figure 2: IDWT implementation cidence, but it has an importance since it gives an idea about the time frequency localization (∆t, ∆f) of the waveforms, which is studied in Section II-C . In order to implement the Wavelet-OFDM system expressed in (2), we apply the Mallat algorithm [11]. For a C. Time-Frequency analysis Wavelet-OFDM signal based on the wavelets of L = J − J0 scales and the scaling functions of the scale J0, the IDWT To characterize the waveform gm in the time and frequency should be performed L times. L can be also be interpreted as domain, we usually refer to the first and second moments the number of decomposition levels. Let Cn be a vector of in these two dimensions, also known as time mean tgm and M input complex symbols C . The J0 first C symbols m,n 2 m,n frequency mean fGm for the first order, and time localization J0 correspond to the 2 scaling coefficients (aJ0,q)q∈[[0,2J0 −1]]. (TL) and frequency localization (FL) for the second order.

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They are defined as: Definition 2. Auto-correlation mean function + T0 1 ∞ 2 ¯ 1 tgm = t gm(t) dt, (4) Γx(τ) := Γx(t, τ) dt. (16) Egm | | T0 0 −∞ Z Z + 1 ∞ 2 fGm = f Gm(f) df, (5) Let g˜m(t) =g ¯m(−t) Egm | | Z−∞ M 1 + 1/2 1 − T0 4π ∞ 2 2 ¯ ∆tg = (t tg ) gm(t) dt , (6) Γx(τ) = gm(t nT0)¯gm(t nT0 τ) m m T0 − − − Egm − | | m=0 n 0 Z−∞ Z 1 2 X X + / M 1 (n 1)T0 4π ∞ 2 2 1 − − − ∆fGm = (f fgm ) Gm(f) df , (7) = gm(t)¯gm(t τ) Egm − | | T0 nT − −∞ m=0 n 0 Z X X Z− M 1 + + 1 − ∞ ∞ 2 = g (t)¯g (t τ) such that Egm = gm(t) dt. (8) m m T0 − | | m=0 Z−∞ Z−∞ MX1 Egm is the normalized energy of the waveform gm, assumed 1 − to be equal to for all the waveforms g . t and f can = (gm g˜m)(τ). (17) 1 m gm Gm T0 ∗ m=0 be interpreted as the centre of gravity of the waveform gm X in the time domain and frequency domain respectively, or the Definition 3. Power Spectral Density averaged normalized time (frequency resp.) centre. We can also The PSD of a cyclostationary process x is defined as say that the spectrum Gm is localized around the frequency γx(ω) = FT (Γ¯x(τ)), (18) fGm . ∆tgm and ∆fGm can be understood as the duration and the bandwidth of the waveform gm respectively. where FT means the Fourier transform. Without loss of generality, let t = 0. The first and second M 1 ψ 1 − j moment of the wavelet functions (ψj,k)j∈[[J0,J−1],k∈[[0,2 −1]] γx(ω) = FT ( (gm g˜m)(τ)) T0 ∗ for the time and frequency domain, are deduced from the m=0 X first and second moment of the mother wavelet function ψ M 1 1 − as follows: = Gm(ω)G¯m(ω) T0 j j m=0 tψ = 2− (tψ + kT0) = 2− (kT0), (9) j,k MX1 j 1 − 2 fΨj = 2 fΨ, (10) = Gm(ω) . (19) j T0 | | − m=0 ∆tψj,k = 2 ∆tψ, (11) X j In the case of the Wavelet-OFDM system, let ω and ω ∆fΨ = 2 ∆fΨ. (12) Ψ( ) Φ( ) j be the Fourier transform of ψ(t) and φ(t) respectively. From We can observe that, in the time domain, the wavelets of each (2) and (19), we have: scale are translated in time, and have the same TL. In the J 1 2j 1 2J0 1 frequency domain, the wavelets of each scale have the same 1 − − 2 1 − 2 γxwavelet (ω) = Ψj,k(ω) + ΦJ0,q(ω) . FL, and then occupy the same bandwidth. From one scale to T0 | | T0 | | j=J k=0 q=0 the next, the TL is divided by a factor of 2, and the FL is X0 X X multiplied by a factor of 2. j j we have FT (ψj,k) = 2 2 F T (ψ(2 t kT0)) The scaling functions (φ ) ∈ J0 − verify the same rela- J0,k k [[0,2 1]] 1 ω − tionship with the scaling mother function. FT (ψ(2j t)) = Ψ( ) 2j 2j ω j 1 i j kT0 ω III.PSDANALYSIS F T (ψ(2 t kT0)) = e− 2 Ψ( ). (20) − 2j 2j A. Theoretical analysis of the PSD of Wavelet-OFDM Similarly for the scaling function, we have The Fourier transform of the MCM signal x(t) in (1) is ω J0 1 i J kT0 ω FT (φ(2 t kT0)) = e− 2 0 Φ( ). (21) expressed as follows: − 2J0 2J0 M 1 − Hence iωnT0 X(ω) = Cm,nGm(ω)e− . (13) J 1 2j 1 2J0 1 − − − n m=0 1 j ω 2 1 J0 ω 2 − − X X γxwavelet (ω) = 2 Ψ( j ) + 2 Φ( q ) T0 | 2 | T0 | 2 | Gm(ω) is the Fourier transform of gm(t), and ω = 2πf. j=J q=0 X0 Xk=0 X J 1 Definition 1. Auto-correlation function 1 − 2 1 2 = Ψj (ω) + ΦJ0 (ω) . (22) T0 | | T0 | | Γx(t, τ) := E(x(t)¯x(t − τ)), (14) j=J X0 where x¯ is the conjugate of x, and E(.) is the expectation Let us now consider the Haar wavelet, which is the oldest operator. and simplest wavelet, and has a closed form expression in the time and frequency domain. The mother Haar wavelet and the ¯ ′ ′ ′ ′ mother Haar scaling function are expressed as follows Γx(t, τ) = (Cm,nCm ,n )gm,n(t)¯gm ,n (t τ) ′ ′ E − n,n m,m 1 T0 X X √T if 0 t 2 M 1 0 ≤ ≤ − 1 T0 ψ(t) = √ if 2 t T0 (23) Γx(t, τ) = gm(t nT0)¯gm(t nT0 τ). (15) − T0 ≤ ≤ − − −  n m=0 0 else, X X 

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1 if 0 t T0 padding by a factor of 4 is performed on the input signal in φ=(t) √T0 ≤ ≤ (24) (0 else. the frequency domain. The number of carriers considered is M = 16. The experimental curve shows a good ﬁt with the In order to express the PSD of Haar Wavelet-OFDM, let us theoretical one. ﬁrst express |Ψ(ω)| and |Φ(ω)|.

+ ∞ iωt Ψ(ω) = ψ(t)e− dt PSD 0 th Z−∞ 2 ωT0 PSD ωT0 sin exp i 2 4 −10 = i√T0e− , ωT0 4 2 ωT0 −20 sin 4 Ψ(ω) = √T0 . (25)

ωT PSD [dB] | | 0 4 −30

+ −40 1 ∞ iωt Moreover Φ(ω) = φ(t)e− dt √T0 Z−∞ ωT0 −1 −0.5 0 0.5 1 1.5 ωT 2 sin /2 i 0/ Normalized Frequency 6 = √T0e− , x 10 ωT0/2 sin ωT0/2 Figure 3: Theoretical and experimental PSD of Haar Wavelet- Φ(ω) = √T0 . (26) | | ωT0/2 OFDM.

From (22), (25), and (26), we obtain an expression for the PSD of Haar Wavelet-OFDM as follows 2 J 1 2 πfT0 − sin j+1 2 πfT J 2 γx (f) = + sinc( 0/2 0 ) . (27) B. Comparison with OFDM haar πfT0 j+1 | | j=J0 2 X

OFDM B. Width of the main lobe 0 Haar Wavelet

The width of the main lobe of the PSD gives a measure −10 of the bandwidth efﬁciency of the signal. Here we calculate

∆fhaar of the PSD of Haar Wavelet-OFDM expressed in (27). −20

γxhaar (f) = 0 (28) PSD [dB] −30 πfT 2 sin2 0 2j+1 j [J0,J 1] πfT = 0 0 −40 ∀ ∈ − 2j+1 ⇔  2  and sinc(πfT0/ 2J0 ) = 0 | | fT0 −5 0 5 Z∗ Normalized Frequency 5  j [J0,J 1] 2j+1 = kj , kj x 10 ∀ ∈ − fT0 ∈ ⇔ and = kL, kL Z∗ ( 2J0 ∈ J Figure 4: PSD of OFDM and Haar Wavelet-OFDM. fT0 = 2 k, k Z∗. (29) ⇔ ∈ The width of the main lobe is ∆fhaar = f1 − f−1, such that 2J −2J Based on the same parameters and method explained in f1 = and f−1 = . We have then : T0 T0 Section IV-A, the PSD of Haar Wavelet-OFDM and OFDM 2M without cyclic prefix are simulated as displayed in Fig.4. The ∆fhaar = . (30) T0 width of the main lobe of the PSD of Haar Wavelet-OFDM Note here that the width of the main lobe is the same for all is the double that of OFDM. Moreover, Haar Wavelet-OFDM has very large side lobes. Many applications can not tolerate the variants, since it does not depend on the first scale J0. these poor spectrum characteristics. It is true that Haar can be filtered to reduce the side lobes effect, but this will change the IV. SIMULATIONS AND DISCUSSION system performance. Note that OFDM may use a cyclic prefix, In this section, the PSD is obtained by simulation, to con- but it does not normally exceed 25% of the total bandwidth. firm the analysis in Section III, and the bandwidth efficiency of Wavelet-OFDM is discussed and compared with conventional C. Discussion OFDM. Fig.5 displays the sum over each scale of the wavelet in the A. Simulation of the PSD of Haar Wavelet-OFDM frequency domain, and the sum over all the scales which gives the PSD as expressed in (19). As we can observe, the wavelets Fig.3 gives the experimental PSD, and the theoretical PSD of the smallest scale have the largest main lobe width, and this as a function of the normalized frequency based on (27). The defines the main lobe width of the PSD. experimental PSD is simulated using Matlab, and estimated Among the well known wavelets, the Shannon wavelet has via the periodogram method with a rectangular window. Be- the best FL, but it is not orthogonal and has a slow decay in fore applying the wavelet modulation based on IDWT, zero the time domain. In addition, the study investigates the DWT,

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V. CONCLUSION Φ 0 Ψ Ψ The PSD limitations of Wavelet-OFDM have been investi- 1 −10 Ψ gated in this paper, especially for the Haar wavelet. While Haar 2 Ψ Wavelet-OFDM has some advantages, we have shown that it −20 3 PSD has a serious limitation in terms of the bandwidth efﬁciency,

−30 since its spectrum has large main lobe and large side lobes Magnitude [dB] compared with the OFDM system. −40 Even though the Meyer wavelet is not as extensively

−50 promoted in the literature as the Haar wavelet, but has a −5 0 5 Normalized Frequency 5 x 10 better bandwidth efﬁciency, and may potentially compete with OFDM in this respect. Our next study is about the character- Figure 5: Frequency domain wavelets for each scale. istics and performance of Meyer Wavelet-OFDM.

ACKNOWLEDGEMENTS while the Shannon wavelet can be applied only to the complex This work has received a French state support granted to the wavelet transform. The Meyer wavelet is more attractive, as it CominLabs excellence laboratory and managed by the National has a faster decay and satisfies the orthogonality condition. A Research Agency in the "Investing for the Future" program under discrete format approximation of Meyer wavelet noted Dmey reference Nb. ANR-10-LABX-07-01. The authors would also like to is possible, and it can simulate the Meyer wavelet based on a thank the Region Bretagne, France, for its support of this work. finite impulse response (FIR) filter as depicted in Fig.6 1. As a result, the fast wavelet transform can approximate the Meyer REFERENCES wavelet transform, and the DWT can be applied. As simulated [1] Madan Kumar Lakshmanan and Homayoun Nikookar. A review of wavelets for digital wireless communication. Wireless Personal Dmey Wavelet function Dmey Scaling function Communications, 37(3-4):387–420, 2006. 1 1 [2] Scott L Linfoot, Mohammad K Ibrahim, and Marwan M Al-Akaidi. Orthogonal Wavelet Division Multiplex: An Alternative to OFDM. 0.8 Consumer Electronics, IEEE Transactions on, 53(2):278–284, 2007. 0.5 0.6 [3] Youbing Zhang and Shijie Cheng. A Novel Multicarrier Signal Transmission System over Multipath Channel of Low-voltage Power 0 0.4 Line. Power Delivery, IEEE Transactions on, 19(4):1668–1672, 2004. [4] Deepak Gupta, Vipin B Vats, and Kamal K Garg. Performance Analysis 0.2 of DFT-OFDM, DCT-OFDM, and DWT-OFDM Systems in AWGN −0.5 0 Channel. In Wireless and Mobile Communications, 2008. ICWMC’08. The Fourth International Conference on, pages 214–216. IEEE, 2008. −1 −0.2 [5] Jamaluddin Zakaria and Mohd Fadzli Mohd Salleh. Wavelet-based −0.5 0 0.5 −0.5 0 0.5 OFDM Analysis: BER Performance and PAPR Profile for Various Wavelets. In Industrial Electronics and Applications (ISIEA), 2012 IEEE Symposium on, pages 29–33. IEEE, 2012. Figure 6: Discrete Meyer wavelet and scaling function. [6] Waleed Saad, N El-Fishawy, S El-Rabaie, and Mona Shokair. An Efficient Technique for OFDM Dystem using Discrete Wavelet Trans- form. In Advances in Grid and Pervasive Computing, pages 533–541. in Fig.7, Discrete Meyer Wavelet-OFDM has a better spectrum Springer, 2010. efficiency than Haar Wavelet-OFDM, but it is not as good as [7] Marius Oltean and Miranda Naforni¸ta.˘ Efficient Pulse Shaping and Robust Data Transmission Using Wavelets. In Intelligent Signal OFDM without cyclic prefix. Processing, 2007. WISP 2007. IEEE International Symposium on, pages 1–6. IEEE, 2007.

[8] Alaa Ghaith, Rima Hatoum, Hiba Mrad, and Ali Alaeddine. Per- OFDM 0 Dmey formance Analysis of the Wavelet-OFDM New Scheme in AWGN Haar Channel. In Communications and Information Technology (ICCIT), −10 2013 Third International Conference on, pages 225–229. IEEE, 2013. [9] Swati Sharma and Sanjeev Kumar. BER Performance Evaluation of −20 FFT-OFDM and DWT-OFDM. International Journal of Network and

PSD [dB] Mobile Technologies, 2(2):110–116, 2011. −30 [10] Kavita Trivedi, Anshu Khare, and Saurabh Dixit. BER Performance of OFDM with Discrete Wavelet Transform for Time Dispersive Channel. −40 International Journal of Research in Engineering and Technology, 3:2319–1163, 2014. −50 −4 −3 −2 −1 0 1 2 3 4 Normalized Frequency 5 [11] Stéphane Mallat. A Wavelet Tour of Signal Processing. Academic press, x 10 1999. Figure 7: PSD of OFDM and Meyer Wavelet-OFDM.

1Generated using the MATLAB wavefun(’dmey’) command, Wavelet Tool- box.

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