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Time-frequency localization as mother wavelet selecting criterion for wavelet modulation

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Abstract

Throughout recent years, the discrete wavelet transform (DWT) was used in communications for wavelet modulation (WM). One of its features is the mother wavelets (MW) used. An important number of MW were already proposed in the literature. We investigate in this paper the selection of the MW for the WM, on the basis of its time-frequency localization. We prove, with the aid of Balian-Low theorem, that wavelets have better time-frequency localization than the elements of the bases which are used for signal decomposition in other orthogonal modulation techniques, as, for example, in orthogonal frequency division multiplexing (OFDM). A procedure for the evaluation of the time-frequency localization of MW belonging to Daubechies family is proposed. This procedure is next compared against the WM transmission on different channel types, which reveal the best MW to be used. This advantage of the WM versus OFDM is highlighted by simulation results in fading channels.

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References

  1. Lakshmanan MK, Nikookar H (2006) A review of wavelets for digital wireless communication. Wirel Pers Commun 37(3):387–420

    Article  Google Scholar 

  2. Wu Y, Zou WY (1995) Orthogonal frequency division multiplexing. A multi-carrier modulation scheme. IEEE Trans Consum Electron 41(3):392–399

    Article  Google Scholar 

  3. Jiang T, Xiang W, Chen HH, Ni Q (2007) Multicast broadcasting services support in OFDMA based WiMAX systems. IEEE Commun Mag 45(8):78–86

    Article  Google Scholar 

  4. Le Floch B, Alard M, Berrou C (1995) Coded orthogonal frequency division multiplex. Proc IEEE 83(6):982–996

    Article  Google Scholar 

  5. Baig S, Mughal M (2009) Multirate signal processing techniques for high-speed communication over power lines. IEEE Commun Mag 47(1):70–76

    Article  Google Scholar 

  6. Oltean M, Nafornita M (2007) Efficient Pulse Shaping and Robust Data Transmission Using Wavelets. Proceedings of the third IEEE Internat. Symp. on Intelligent Signal Processing, WISP 2007, Alcala de Henares, Spain, pp 43–48

    Google Scholar 

  7. Oltean M, Isar A (2009) On the Time-Frequency Localization of the Wavelet Signals with Application to Orthogonal Modulation”. Proceedings of IEEE International Symposium SCS’09, Iasi, Romania, pp 173–176

    Google Scholar 

  8. Wornell G, Oppenheim A (1992) Wavelet-based representations for a class of self-similar signals with application to fractal modulation. IEEE Trans Inf Theory 38:785–800

    Article  Google Scholar 

  9. Manglani M, Bell A (2001) Wavelet Modulation Performance in Gaussian and Rayleigh Fading Channels. Proceedings of MILCOM 2001, McLean, VA

    Book  Google Scholar 

  10. Zhang H, Yuan D, Jiong M, Wu D (2004) Research of DFT-OFDM and DWT-OFDM on different transmission scenarios. Proc Int Conf ICITA-2004:31–33

    Google Scholar 

  11. Atzori L, Giusto D, Murroni M (2002) Performance analysis of fractal modulation transmission over fast-fading wireless channels. IEEE Trans Broadcast 48(2):103–110

    Article  Google Scholar 

  12. Silveira LFQ, Silveira LGQ, Assis FM, Pinto EL (2009) Analysis and optimization of wavelet-coded communication systems. IEEE Trans Wirel Commun 8(2):563–567

    Article  Google Scholar 

  13. Bell AE, Manglani MJ (2002) Wavelet modulation in Rayleigh fading channels: Improved performance and channel identification. Proc IEEE Int Conf Acoust Speech Sig Process 3:III-2813–III-2816

    Google Scholar 

  14. Daubechies I (1992) Ten lectures on wavelets, SIAM

  15. Strang G, Shen J (1998) Asymptotic structures of Daubechies scaling functions and wavelets. Appl Comput Harmon Anal 5:312–331

    Article  MATH  MathSciNet  Google Scholar 

  16. Murroni M (2007) A power-based unequal error protection system for digital cinema broadcasting over wireless channels. Signal Process Image Commun 22(3):331–339

    Article  Google Scholar 

  17. Murroni M (2008) Robust transmission of compressed streams over land mobile satellite channel at Ku-Band. IEEE 67th Veh Technol Conf-Spring VTC, Marina Bay, Singapore, pp 2917–2921

    Google Scholar 

  18. Flandrin P (1993) Representations temps-fréquence, Hermes

  19. Mallat S (1999) A wavelet tour of signal processing, 2nd edn. Academic, New York

    MATH  Google Scholar 

  20. Daubechies I (1988) Orthonormal Bases of Compactly Supported Wavelets”. Comm Pure Appl Math 41:909–996

    Article  MATH  MathSciNet  Google Scholar 

  21. Atto A, Pastor D, Isar A (2007) On the statistical decorrelation of the wavelet packet coefficients of a band-limited wide-sense stationary random process. Sig Process Elsevier 87(10):2320–2335

    Article  MATH  Google Scholar 

  22. Zhao F, Zhang H, Yuan D (2004) Performance of COFDM with Different orthogonal Basis on AWGN and frequency Selective Channel. Proc IEEE Internat Sympo on Emerging Technologies: Mobile and Wireless Communications, Shanghai, China, pp 473–475

    Google Scholar 

  23. Gautier M, Lienard J (2007) Applications de la modulation multiporteuse par paquets d’ondelettes aux transmissions sans fil. Ann Telecommun 62(7–8):871–893

    Google Scholar 

  24. Sklar B (1997) Rayleigh fading channels in mobile digital communication systems-part I: characterization. IEEE Commun Mag 35(7):136–146

    Article  Google Scholar 

  25. Wunder G, Junag P et al (2014) 5GNOW: non-orthogonal, asynchronous waveforms for future mobile applications. IEEE Communications Magazine, 97–105

  26. Ramanathan K, Muniraj NJR (2007) DWT-IDWT-based MB-OFDM UWB with digital down converter and digital up converter for power line communication in the frequency band of 50 to 578 MHz. Ann Telecom 70(5–6):181–196

    Google Scholar 

  27. Ahuja N, Lertrattanapanich S, Bose NK (2005) Properties determining choice of mother wavelet. IEE Proc Vis Image Sig Process 152(5):659–664

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. “Politehnica” University of Timisoara, B-dul V. Parvan, 30023, Timisoara, Romania

    Alexandru Isar & Marius Oltean

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  1. Alexandru Isar
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  2. Marius Oltean
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Corresponding author

Correspondence to Alexandru Isar.

Appendix 1

Appendix 1

By concatenating a block computing DWT (named in the following analysis system) with a block which implements the inverse DWT (IDWT) (named in the following synthesis system), an identity system is obtained (Fig. 8). There is a link between these filters and the continuous time-domain MW and SC. Thus, the SC ϕ(t) and the MW ψ(t) can be expressed with the aid of the impulse responses of the low-pass and high-pass filters: \( \phi (t)={\displaystyle \sum_{k=-\infty}^{\infty }h\left[k\right]\phi \left(2t-k\right)} \) and \( \psi (t)={\displaystyle \sum_{k=-\infty}^{\infty }g\left[k\right]\phi \left(2t-k\right)} \), respectively.

As can be seen in Fig. 8, the DWT and IDWT implementations are based on filter banks (FB).

A consequence of the orthogonality of the bases from the MRA and OD refers to the sampled correlations of the SC and MW: R ϕ [n] = R ψ [n] = δ[n], which assures the absence of intersymbol interference (ISI) in WM in conformity with the null ISI Nyquist criterion (A4). So, the use of Nyquist’s filters to shape the pulses is not necessary in WM (A3) [6]. The DWT is a subband-coding scheme, as can be seen in the example in Fig. 9. The DWT has two features: the MW and the number of decomposition levels. Its coefficients take different values for different selections of the features already mentioned when applied to a given discrete in-time signal. For example, for different MW, different sparsity of the wavelet coefficients is obtained.

Fig. 8
figure 8

DWT (up) and IDWT (down) implementations

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Fig. 9
figure 9

An example of subband coding, J = 3

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Consequently, the problem of the selection of the most appropriate features for a given application and for a given signal is of interest. Some criteria for the selection of the most appropriate MW are as follows: the length of its time-domain support, its smoothness or its time or frequency localizations have already been proposed [27]. One of the difficulties encountered in this selection process is given by the diversity of the class of MW. In the case of orthogonal MW, there are the families proposed by Ingrid Daubechies (with compact support and having the highest number of vanishing moments for the considered support length, denoted in the following by Dau), the Coiflets and the Symmlets, the family proposed by Battle and Lemarié, etc.

In case of DWT, at each decomposition level, the sequence of approximation coefficients is split on two branches (Fig. 8), a horizontal one producing the approximation coefficients for the following decomposition levels and a vertical one producing the detail coefficients for the current decomposition level. The detail coefficient sequences are not split further. The discrete wavelet packet transform (DWPT) is obtained from the DWT by splitting the detail coefficient sequences on two branches composed of the low-pass filter h or the high-pass filter g and decimators with factor 2 as well, as is shown in Fig. 10.

Fig. 10
figure 10

The DWPT implementation scheme

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The DWPT is a subband-coding scheme as can be seen in the example in Fig. 11.

Fig. 11
figure 11

An example of subband coding obtained with DWPT

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It can be observed, by comparing Figs. 9 and 11, that the DWPT realizes a subband coding in eight subbands, more than the four subbands obtained by applying the DWT.

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Isar, A., Oltean, M. Time-frequency localization as mother wavelet selecting criterion for wavelet modulation. Ann. Telecommun. 71, 35–46 (2016). https://doi.org/10.1007/s12243-015-0478-3

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