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

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.

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

DWT (up) and IDWT (down) implementations

Fig. 9

An example of subband coding, J = 3

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

The DWPT implementation scheme

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

Fig. 11

An example of subband coding obtained with DWPT

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.