Daubechies 4 (db4) Wavelet Function

• Daubechies 4 wavelet is a compactly supported, orthonormal function with two vanishing moments and a four-tap FIR filter foundation for discrete and continuous analysis.
• Analytic approximations using inharmonic sinusoids achieve over 99.99% accuracy, eliminating the need for recursive cascade methods.
• Spline-based quadratic B-spline constructions provide explicit polynomial representations that enhance spectral analysis and enable efficient MATLAB integration.

The Daubechies 4 (db4) wavelet is a compactly supported, orthonormal wavelet constructed for discrete and continuous wavelet analysis. As a member of the Daubechies family, the db4 wavelet exhibits minimal compact support for a given number of vanishing moments and is characterized by four finite impulse response (FIR) filter coefficients. Recent advances provide both explicit analytic approximations for its waveform and a polynomial spline construction route, clarifying its core mathematical structure and facilitating efficient numerical implementation (Vermehren et al., 2015, He et al., 2015).

1. Explicit Time-Domain Approximations of db4 Wavelet and Scaling Functions

Traditional representations of compactly supported wavelets like db4 rely on recursive algorithms or filter-bank constructions. "Close Approximations for Daublets and their Spectra" (Vermehren et al., 2015) introduces closed-form, near-exact analytic representations of the db4 wavelet ($\psi_4$) and scaling function ($\phi_4$) as sums of inharmonic sinusoids: $$\tilde\psi_4(t) = \sum_{k=1}^8 a_k \sin(b_k t + c_k)$$

$$\tilde\phi_4(t) = \sum_{k=1}^8 A_k \sin(B_k t + C_k)$$

where $K=8$ suffices to achieve $>99.99\%$ root-mean-square fit accuracy to the cascade-defined db4 over $[0,7]$.

db4 Wavelet Coefficients $(a_k, b_k, c_k)$

$k$	$a_k$	$b_k$	$c_k$
1	0.3452	4.586	-2.316
2	0.2783	3.460	1.413
3	0.3015	5.770	-0.373
4	0.2129	6.960	-4.943
5	0.1293	2.414	-1.794
6	0.1120	8.161	-3.225
7	0.0295	9.366	-7.567
8	0.0223	1.372	1.102

db4 Scaling Function Coefficients $(A_k, B_k, C_k)$

$k$	$A_k$	$B_k$	$C_k$
1	0.3762	0.672	0.171
2	0.2113	3.226	-2.404
3	0.3900	1.204	0.939
4	0.0770	4.193	2.098
5	0.2661	2.384	-1.379
6	0.0081	5.586	-1.379
7	0.0226	8.537	-1.184
8	0.0205	9.424	3.346

These analytic "near-daublet" forms eliminate the need for recursive cascades and provide explicit, continuous, closed-form expressions (Vermehren et al., 2015).

2. Filter-Bank Foundation: FIR Coefficients and Mask Construction

The orthonormal db4 is derived from a four-tap FIR filter defined by: $$\begin{aligned} h_0 &= \frac{1+\sqrt{3}}{4\sqrt{2}} \approx 0.4829629131 \ h_1 &= \frac{3+\sqrt{3}}{4\sqrt{2}} \approx 0.8365163037 \ h_2 &= \frac{3-\sqrt{3}}{4\sqrt{2}} \approx 0.2241438680 \ h_3 &= \frac{1-\sqrt{3}}{4\sqrt{2}} \approx -0.1294095226 \end{aligned}$$ The low-pass filter mask is explicitly constructed using spline-type masks and Lorentz polynomials (He et al., 2015): $$P_2(z) = \tfrac{1}{4}(1 + 2z + z^2), \quad S_2(z) = \frac{1+\sqrt{3}}{2}z + \frac{1-\sqrt{3}}{2}z^2$$ The mask $P(z) = P_2(z) S_2(z)$, upon shifting and normalization, yields the $h_k$. The associated high-pass coefficients are $g_k = (-1)^k h_{3-k}$. These filters underpin the standard two-scale relations

$$\phi_4(t) = \sqrt{2} \sum_{k=0}^3 h_k \phi_4(2t-k), \quad \psi_4(t) = \sqrt{2} \sum_{k=0}^3 g_k \phi_4(2t-k)$$

ensuring compactly-supported, orthonormal bases in $L^2(\mathbb{R})$ (He et al., 2015).

3. Inharmonic Series Model and Fitting Methodology

The analytic approximation in (Vermehren et al., 2015) leverages an "inharmonic" sum: $$y(t) \approx \sum_{k=1}^K a_k \sin(\omega_k t + \phi_k)$$ The frequencies $\omega_k$ are non-integer multiples selected to enforce exact zero mean over the wavelet's compact support. The phases and amplitudes are determined to minimize mean-square error to the cascade-generated wavelet by nonlinear least-squares (Levenberg–Marquardt algorithm), using the FIR filter output as the regression target. This provides analytic, zero-mean, compactly-supported approximations matching discrete wavelets to within $10^{-4}$ error on $[0,T]$ without repeated filtering or B-spline construction (Vermehren et al., 2015).

4. Spectral Properties and Scalogram Analysis

Fourier Transform and Spectrum

For the periodic extension, the transform is a sum of Dirac deltas: $$\Psi_4^{\rm long}(\omega) = j\pi\sum_{k=1}^8 a_k\left[e^{jc_k}\delta(\omega+b_k) - e^{-jc_k}\delta(\omega-b_k)\right]$$ Compact support ($T=7$ for db4) induces spectral convolution by a $\operatorname{sinc}$, so that the true spectrum is: $$\Psi_4(\omega) = \frac{T}{2} \sum_{k=1}^8 a_k \left[ \mathrm{sinc}\left(\frac{T}{2\pi}(\omega - b_k)\right) - \mathrm{sinc}\left(\frac{T}{2\pi}(\omega + b_k)\right) \right]$$ This sum of shifted $\mathrm{sinc}$ lobes provides a close match to the canonical db4 spectrum (Vermehren et al., 2015).

Scalogram and Frequency Localization

Time–frequency analysis via continuous scalograms shows that, despite visual smearing in 2D projections, the analytic near-daublet accurately preserves db4's known frequency-detection properties. 3D scalogram visualization resolves the correct localization of tone frequencies, confirming fidelity of the analytic approximation for applications in time–scale analysis (Vermehren et al., 2015).

5. Explicit Polynomial Representation via Spline-Type Approach

The alternative construction (He et al., 2015) uses quadratic B-splines: $$B_2(x) = \begin{cases} \frac{1}{2}x^2, & 0\leq x < 1 \ \frac{1}{2}(-2x^2+6x-3), & 1\leq x < 2 \ \frac{1}{2}(3-x)^2, & 2 \leq x < 3 \ 0, & \text{otherwise} \end{cases}$$ The scaling function can be written as a shifted linear combination of these B-splines: $$\phi(x) = a_1 B_2(2x-1) + a_2 B_2(2x-2), \quad a_1 = \frac{1+\sqrt{3}}{2},\,\, a_2 = \frac{1-\sqrt{3}}{2}$$ The wavelet $\psi(x)$ then follows via the high-pass mask applied to the refinement relation. This yields explicit piecewise-cubic forms for both $\phi(x)$ and $\psi(x)$, illustrating the algebraic structure behind the FIR/cascade output (He et al., 2015).

6. Key Properties, Accuracy, and Implementation

• Compact Support: Both $\psi_4$ and $\phi_4$ are supported in $[0,3]$ (filter length minus one).
• Vanishing Moments: $\psi_4$ possesses exactly two vanishing moments: $\int x^m \psi_4(x)\,dx = 0$ for $m=0,1$.
• Orthonormality: Translates and dilates form an orthonormal multiresolution basis for $L^2(\mathbb{R})$.
• Smoothness: Both functions are piecewise-cubic and Hölder-continuous with exponent $\approx 0.55$.
• Approximation Accuracy: The inharmonic sum achieves better than $99.99\%$ accuracy in root-mean-square error relative to the true cascade-based db4.
• MATLAB® Integration: The $\tilde\phi_4$ and $\tilde\psi_4$ near-daublet family (cdb4) is implemented in MATLAB’s Wavelet Toolbox, with analytic time-domain expressions providing seamless integration and display indistinguishable from standard db4 (Vermehren et al., 2015).

7. Theoretical and Practical Significance

The db4 wavelet combines FIR filter efficiency, compact support, explicit orthogonality, and sufficient regularity for signal-processing and analysis tasks. The analytic forms obtained via inharmonic sums and spline masks facilitate hardware synthesis, explicit spectral analysis, and rapid computation. The connection between the FIR/cascade and spline constructions illuminates the algebraic and analytic foundations of Daubechies' original framework. These representations support both rigorous theoretical investigations and practical applications in wavelet-based systems, such as wavelet OFDM or time–frequency analysis (Vermehren et al., 2015, He et al., 2015).