Abstract
The wavelet transform can be seen as a wavelet-based expansion (decomposition) of a finite-energy signal. In the discrete wavelet transform (DWT), economy in the representation of the signal and possibility of perfect signal reconstruction (PR) are crucial. The simplest formulation of the DWT problem includes two types of basis functions for the expansion: the scaling and wavelet functions. We will see how an ideal, infinite-length but finite-energy signal can be decomposed from the point of view function spaces, and how this decomposition can be obtained using a two-channel digital filter bank. The description of a fast wavelet decomposition/reconstruction algorithm will lead us to the practical implementation of the DWT in the real-life case of a finite-length, sampled input signal, as well as to the properties of PR filters, which are strictly related to the scaling and wavelet functions. After allowing for the necessary conditions that the filters of the bank must satisfy, primarily biorthogonality or orthogonality, a number of degrees of freedom remain available to design different wavelet systems suited for different purposes. A real-world example of signal DWT decomposition will be provided. The chapter ends with an appendix in which the various wavelet systems used for the CWT and/or DWT are reviewed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Note that \(\mathrm {L}^1(\mathbb {R})\) is more restrictive than \(\mathrm {L}^2(\mathbb {R})\), because an absolutely integrable function is also square-integrable, but the converse is not necessarily true.
- 2.
To be precise, we must remark that while the continuous wavelets involved in the CWT are subject to the uncertainty principle of Fourier analysis, the discrete wavelet systems involved in the DWT, often defined through the associated digital filters, do not. However, the DWT does share the MRA property of all forms of wavelet transform: discrete wavelet bases can be shown to possess the MRA property by introducing the concept of nested spanned function spaces (Sect. 14.3).
- 3.
There is an interesting correspondence between wavelet notation and musical notation. In a musical score, each note specifies a frequency and a position in time by its vertical and horizontal placements, respectively, in a way that closely resembles a wavelet signal representation, except that it has fractional jumps in frequency. An article written in 1994 by G. Strang for a nontechnical audience (American Scientist, vol. 82, pp. 250–255) clearly explains this similarity.
- 4.
The signals we are considering are real, and in the DWT they are normally expanded using real functions. For this reason, from now on we will use the notation which is appropriate for real functions, thus writing the inner product without any conjugation sign.
- 5.
The prototype (unscaled) wavelet function is what we called the “mother wavelet” in the previous chapter. The prototype scaling function is sometimes referred to as the “father wavelet”. Note that in order to simplify the notation, here we dropped the subscript adopted in the previous chapter to indicate the mother wavelet: we wrote \(\psi (\theta )\) instead of \(\psi _0(\theta )\).
- 6.
For \(j<0\), the contrary would be true: only coarser features could be represented, and \(V_j\) would be narrower than \(V_0\).
- 7.
Note that in this equation we should actually write \(\eta =\theta /a\) in place of \(\theta \), since the larger scale is unspecified. However, the one presented here is the standard way in which the relation is written in literature.
- 8.
Note that the choice \(j_0=0\) is arbitrary, and we might as well decide to chose a smaller starting scale—a larger degree of detail, e.g. \(j_0=10\), \(a=2^{-10}\)—and write
$$\begin{aligned} \mathrm {L}^2(\mathbb {R})= V_{10} \oplus W_{10} \oplus W_{11} \oplus W_{12} \ldots , \end{aligned}$$or we might prefer a larger starting scale—a smaller degree of detail, e.g. \(j_0=-5\), \(a=2^{5}\)—and write
$$\begin{aligned} \mathrm {L}^2(\mathbb {R})= V_{-5} \oplus W_{-5} \oplus W_{-4} \oplus W_{-3} \ldots . \end{aligned}$$We could even start from \(j_0=-\infty \), i.e., from an infinitely large scale: since \(V_{-\infty } ={0}\), we would then write
$$\begin{aligned} \mathrm {L}^2(\mathbb {R})= \cdots \oplus W_{-2} \oplus W_{-1} \oplus W_0 \oplus W_{1} \oplus W_{2} \oplus \cdots . \end{aligned}$$In this way we would eliminate the scaling function and would get an expansion of the signal on the basis of wavelets solely. In the following discussion we will set \(j_0=0\), unless explicitly stated otherwise.
- 9.
This statement is not in contradiction with what we said in the previous chapter about the frequency response of the filters associated with complex analytic wavelets. Those wavelets are used for CWT only and do not have a corresponding scaling function. They actually act as passband filters.
- 10.
A B-spline is a piecewise polynomial function in one independent variable, exhibiting knots or break-points. The number of internal knots is equal to the degree of the polynomial if there are no knot multiplicities. A B-spline is a continuous function at the knots. For any given set of knots, the B-spline is unique, hence the name, B being short for Basis.
- 11.
Returning for a moment to Fig. 14.10, at this point we may wonder if we would be in a position to proceed further to a more accurate level-8 approximation in \(V_{-1}\). The answer is no; in the example of Fig. 14.10, the signal length is \(N=120\), and after repeated downsampling operations, the coarsest-resolution approximation and detail coefficients (\(c_7[k],\,d_7[k]\)) are made by just one sample each. The decomposition was pushed up to the maximum possible level. The remaining detail coefficients up to \(d_1\) are progressively longer, since they represent less downsampled signals. Now, suppose we had \(N=240\) samples. We would then be able to attain \(J=8\), and (\(c_7[k],\,d_7[k]\)) would have one sample each. However, in Fig. 14.10 we would see level 1 in \(V_7\), since \(J-j=8-1=7\), and so on, up to level 8 in \(V_0\): we would always label the last subspace as \(V_0\).
- 12.
Note that the coefficient vectors \(c_1[k]\) and \(d_1[k]\) cannot be directly combined to reproduce the signal. The coefficients are produced by downsampling and are only half the length of the original signal. It is necessary to reconstruct the approximations and details before combining them.
- 13.
The word “trous” means “holes” in English.
- 14.
Note that that here we are presenting a definition of regularity of the scaling filter, not of the scaling function or of the wavelet.
- 15.
Daubechies (1992) defined two classes of wavelets, via criteria that select a particular scaling filter. One criterion leads to “extremal phase” (minimum phase) Daubechies wavelets, i.e., the ones illustrated here. Another criterion leads to “least asymmetric” Daubechies wavelets, also called “symlets”.
- 16.
Note that in the previous chapter we gave a different and less general definition of the complex Morlet wavelet,
$$\begin{aligned} \psi _0(\theta ) = \frac{1}{^4\sqrt{\pi }} {\mathrm {e}}^{{\mathrm {j}}\omega _0 \theta } {\mathrm {e}}^{-\frac{\theta ^2}{2}}, \end{aligned}$$that corresponds to fixed values of \(T_p\) and \(f_c\). More precisely, it corresponds to \(T_p=2\) and \(f_c=2 \pi /\omega _0\). In the standard expression of the Gaussian probability density function, we would have \(T_p=2\sigma ^2\), where \(\sigma \) is the standard deviation of the Gaussian distribution. Thus, \(T_p=2\) means \(\sigma =1\). Note, however, that the constant factor \(1/\sqrt{\pi T_p}\), that for \(T_p=2\) becomes \(1/\sqrt{2\pi }\), does not coincide with the factor \(1/{^4\sqrt{\pi }}\).
References
Abry, P.: Ondelettes et turbulence, Diderot edn. France, Paris (1997)
Burrus, C.S., Gopinath, R., Guo, H.: Introduction to Wavelets and Wavelet Transforms: A Primer. Prentice-Hall, Upple Saddle River (1998)
Cohen, A., Daubechies, I.: Non-separable bidimensional wavelet bases. Rev. Mat. Iberoam. 9(1), 51–137 (1993)
Cohen, A., Daubechies, I., Feauveau, J.-C.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45, 485–560 (1992)
Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909–996 (1988)
Daubechies, I.: Ten lectures on wavelets. In: CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, USA (1992)
Daubechies, I.: Orthonormal bases of compactly supported wavelets II. Variations on a theme. SIAM J. Math. Anal. 24(2), 499–519 (1993)
Daubechies, I., Lagarias, J.C.: Two-scale difference equations I. Existence and global regularity of solutions. SIAM J. Math. Anal. 22(5), 1388–1410 (1991)
Daubechies, I., Lagarias, J.C.: Two-scale difference equations. II. Local regularity, infinite products of matrices, and fractals. SIAM J. Math. Anal. 23(4), 1031–1079 (1992)
Daubechies, I., Cohen, A., Feauveau, J.-C.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. XLV, 485–560 (1990)
Holschneider, M., Kronland-Martinet, R., Morlet, J., Tchamitchian, P.: A real-time algorithm for signal analysis with the help of the wavelet transform. In: Combes, J.-M., Grossmann, A. (eds.) Wavelets, Time-Frequency Methods and Phase Space (Inverse Problems and Theoretical Imaging). Springer, Berlin (1989)
Kovac̆ević, J., Vetterli, M.: Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R\(^n\). IEEE Trans. Inform. Theory 38(2) 533–555 (1992)
Mallat, S.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Machine Intell. 11(7), 674–693 (1989)
Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, Waltham (2008)
Meyer, Y.: Wavelets and Operators. Cambridge University Press, Cambridge (1993)
Misiti, M., Misiti, Y., Oppenheim, G., Poggi, J.M.: Décomposition en ondelettes et méthodes comparatives: étude d’une courbe de charge électrique. Revue de Statistique Appliquée XLII(2), 57–77 (1994)
Misiti, M., Misiti, Y., Oppenheim, G., Poggi, J.-M. (eds.): Wavelets and Their Applications. Wiley-ISTE, New York (2007)
Nason, G.P., Silverman, B.W.: The stationary wavelet transform and some statistical applications. In: Antoniadis, A., Oppenheim, G.: Wavelets and Statistics. Lecture Notes in Statistics, vol. 103, pp. 281–300. Springer, New York (1995)
Percival, D.B., Walden, A.T.: Wavelet Methods for Time Series Analysis. Cambridge University Press, New York (2000)
Qian, S.: Introduction to Time-Frequency and Wavelet Transforms. Prentice Hall, Upper Saddle River (2001)
Shensa, M.J.: Discrete Wavelet Transforms: The Relationship of the á trous and Mallat algorithms. Treizième Colloque GRETSI—Juan Les Pins, France (1991)
Strang, G., Nguyen, T.: Wavelets and Filter Banks. Wellesley-Cambridge Press, Wellesley (1996)
Tian, J., Wells, R.O. Jr.: Vanishing Moments and Wavelet Approximation. Technical Report CML TR95-01, Computational Mathematics Laboratory, Rice University, Houston, TX, USA (1995)
Teolis, A.: Computational Signal Processing with Wavelets (Applied and Numerical Harmonic Analysis). Birkhauser, Boston (1998)
Vetterli, M.: Filter banks allowing perfect reconstruction. Signal Process. 10, 219–244 (1986)
Vetterli, M., Herley, C.: Wavelets and filter banks: theory and design. IEEE Trans. Acoust. Speech Signal Process. 40(9), 2207–2232 (1992)
Vetterli, M., Kovac̆ević, J.: Wavelets and Subband Coding. Prentice Hall PTR, Englewood Cliffs (1995). Reissued in 2013 by CreateSpace Independent Publishing Platform
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Alessio, S.M. (2016). Discrete Wavelet Transform (DWT). In: Digital Signal Processing and Spectral Analysis for Scientists. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-25468-5_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-25468-5_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25466-1
Online ISBN: 978-3-319-25468-5
eBook Packages: EngineeringEngineering (R0)