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| 1 | +# Discrete Wavelet Transform (Haar / Daubechies) |
| 2 | + |
| 3 | +## Overview & Motivation |
| 4 | + |
| 5 | +The Discrete Wavelet Transform (DWT) decomposes a finite-length signal into a hierarchy of |
| 6 | +approximation and detail coefficients at multiple scales. Unlike the DFT, which projects onto |
| 7 | +infinitely-supported sinusoids, wavelets are compactly supported: they are simultaneously |
| 8 | +localised in time and frequency. This dual localisation makes the DWT the canonical tool for |
| 9 | +detecting transients, edges, and non-stationary features — phenomena that a pure frequency |
| 10 | +representation blurs across the entire spectrum. |
| 11 | + |
| 12 | +In embedded condition-monitoring and edge-ML pipelines, the DWT is valued for three properties: |
| 13 | +its $O(N)$ complexity (a factor of $\log_2 N$ faster than the FFT), its ability to concentrate |
| 14 | +signal energy in very few coefficients (sparsity), and the perfect-reconstruction guarantee that |
| 15 | +lets processed coefficients be inverted exactly. |
| 16 | + |
| 17 | +## Mathematical Theory |
| 18 | + |
| 19 | +### Quadrature-Mirror Filter Bank |
| 20 | + |
| 21 | +A single DWT stage passes a length-$N$ sequence $x[n]$ through two complementary finite-impulse- |
| 22 | +response filters, then downsamples each output by two. |
| 23 | + |
| 24 | +$$ |
| 25 | +c_A[i] = \sum_{k=0}^{P-1} h[k]\, x[2i + k \bmod N], \qquad |
| 26 | +c_D[i] = \sum_{k=0}^{P-1} g[k]\, x[2i + k \bmod N], \qquad i = 0,\ldots,\tfrac{N}{2}-1 |
| 27 | +$$ |
| 28 | + |
| 29 | +where $h$ is the scaling (low-pass) filter with $P$ taps and $g$ is the wavelet (high-pass) filter. |
| 30 | +For an orthogonal wavelet the two filters satisfy the **quadrature-mirror** relation: |
| 31 | + |
| 32 | +$$g[k] = (-1)^k\, h[P - 1 - k]$$ |
| 33 | + |
| 34 | +This guarantees that the two sub-bands together cover the full bandwidth without overlap or gap. |
| 35 | + |
| 36 | +### Multilevel Decomposition (Mallat's Algorithm) |
| 37 | + |
| 38 | +The approximation $c_A$ is fed back into the same filter pair. After $L$ levels the coefficient |
| 39 | +array contains $L$ detail bands and one coarse approximation: |
| 40 | + |
| 41 | +$$ |
| 42 | +\text{coeffs} = [\underbrace{c_D^{(1)}}_{N/2},\; \underbrace{c_D^{(2)}}_{N/4},\; \ldots,\; \underbrace{c_D^{(L)}}_{N/2^L},\; \underbrace{c_A^{(L)}}_{N/2^L}] |
| 43 | +$$ |
| 44 | + |
| 45 | +The layout is critical: level $\ell$ detail starts at offset $\sum_{j=1}^{\ell-1} N/2^j$ within |
| 46 | +the coefficient buffer, and the final approximation occupies the last $N/2^L$ positions. |
| 47 | + |
| 48 | +### Perfect Reconstruction |
| 49 | + |
| 50 | +The synthesis (inverse) stage upsamples each sub-band by two and applies the dual filter pair |
| 51 | +$(h_r, g_r)$. For orthogonal wavelets the synthesis filters are time-reversals of the analysis |
| 52 | +filters: |
| 53 | + |
| 54 | +$$h_r[k] = h[P - 1 - k], \qquad g_r[k] = g[P - 1 - k]$$ |
| 55 | + |
| 56 | +Combined, analysis and synthesis satisfy $H(z)H_r(z^{-1}) + G(z)G_r(z^{-1}) = 2$, the |
| 57 | +alias-cancellation and distortion-free conditions, giving exact reconstruction: |
| 58 | + |
| 59 | +$$\hat{x}[n] = x[n]$$ |
| 60 | + |
| 61 | +to floating-point rounding. |
| 62 | + |
| 63 | +### Parseval Identity (Energy Preservation) |
| 64 | + |
| 65 | +For an orthogonal wavelet the DWT is a unitary transform: |
| 66 | + |
| 67 | +$$\sum_{n=0}^{N-1} x[n]^2 = \sum_{n=0}^{N-1} c[n]^2$$ |
| 68 | + |
| 69 | +where $c$ contains all detail and approximation coefficients. |
| 70 | + |
| 71 | +### Wavelet Families |
| 72 | + |
| 73 | +**Haar** ($P = 2$): $h = [1,\, 1]/\sqrt{2}$. Piecewise-constant basis; discontinuous; simplest |
| 74 | +possible. |
| 75 | + |
| 76 | +**Daubechies-2** ($P = 4$, denoted db2 or D4): coefficients chosen so the wavelet has two |
| 77 | +vanishing moments — it annihilates linear trends. This produces smoother reconstructions and |
| 78 | +better energy compaction for smooth signals. |
| 79 | + |
| 80 | +## Complexity Analysis |
| 81 | + |
| 82 | +| Case | Time | Space | Notes | |
| 83 | +|---------|--------|--------|----------------------------------------------| |
| 84 | +| Forward | $O(N)$ | $O(N)$ | Geometric series: $N + N/2 + \cdots \leq 2N$ | |
| 85 | +| Inverse | $O(N)$ | $O(N)$ | Identical pass through synthesis bank | |
| 86 | + |
| 87 | +Memory is one length-$N$ coefficient buffer, two length-$N/2$ working arrays, and the static |
| 88 | +filter table of size $P$ — no heap allocation. |
| 89 | + |
| 90 | +## Step-by-Step Walkthrough |
| 91 | + |
| 92 | +Input: $x = [1, 2, 3, 4]$, Haar wavelet, $L = 1$. |
| 93 | + |
| 94 | +**Stage 1 (analysis):** |
| 95 | + |
| 96 | +| $i$ | $2i$ | $2i+1$ | $c_A[i] = (x[2i]+x[2i+1])/\sqrt{2}$ | $c_D[i] = (x[2i]-x[2i+1])/\sqrt{2}$ | |
| 97 | +|-----|------|--------|-------------------------------------|-------------------------------------| |
| 98 | +| 0 | 0 | 1 | $(1+2)/\sqrt{2} \approx 2.121$ | $(1-2)/\sqrt{2} \approx -0.707$ | |
| 99 | +| 1 | 2 | 3 | $(3+4)/\sqrt{2} \approx 4.950$ | $(3-4)/\sqrt{2} \approx -0.707$ | |
| 100 | + |
| 101 | +**Coefficient buffer** (layout detail | approx): |
| 102 | + |
| 103 | +$$\text{coeffs} = [-0.707,\; -0.707,\; 2.121,\; 4.950]$$ |
| 104 | + |
| 105 | +**Reconstruction:** upsample $c_A$ and $c_D$, apply synthesis filters, add: |
| 106 | + |
| 107 | +$$\hat{x} = [1.0,\; 2.0,\; 3.0,\; 4.0] \checkmark$$ |
| 108 | + |
| 109 | +## Pitfalls & Edge Cases |
| 110 | + |
| 111 | +- **$N$ must be divisible by $2^L$.** Odd-length or non-aligned signals at deeper levels produce |
| 112 | + fractional half-lengths and are rejected by design. |
| 113 | +- **Boundary handling must match.** Periodic (modulo) extension is used for both analysis and |
| 114 | + synthesis; mixing it with symmetric extension on the other pass destroys perfect reconstruction. |
| 115 | +- **Filter length vs. signal length.** At each level the signal halves; once it equals $P$ the |
| 116 | + periodic convolution wraps completely. Stop decomposition before the signal shorter than the |
| 117 | + filter length to avoid artefacts. |
| 118 | +- **Fast-math reordering.** With `#pragma GCC optimize("fast-math")` floating-point associativity |
| 119 | + relaxes; reconstruction residuals may reach $10^{-5}$ rather than $10^{-7}$ for 32-bit floats. |
| 120 | + |
| 121 | +## Variants & Generalizations |
| 122 | + |
| 123 | +- **Daubechies-$N$ family**: increasing tap count adds vanishing moments, improving energy |
| 124 | + compaction for smooth signals at the cost of longer filters and larger boundary effects. |
| 125 | +- **Biorthogonal wavelets** (e.g. CDF 9/7 used in JPEG 2000): analysis and synthesis filters |
| 126 | + differ but still give perfect reconstruction; not orthogonal, so Parseval does not hold. |
| 127 | +- **Wavelet packet transform**: decomposes both approximation and detail sub-bands at every level, |
| 128 | + forming a full binary tree of sub-bands for adaptive best-basis selection. |
| 129 | +- **Undecimated (stationary) DWT**: omits downsampling; translation-invariant but $O(N \log N)$. |
| 130 | + |
| 131 | +## Applications |
| 132 | + |
| 133 | +- **Denoising**: threshold small detail coefficients (hard or soft thresholding); edges and |
| 134 | + transients survive while broadband noise is suppressed. |
| 135 | +- **Compression**: retain the few large coefficients; the rest encode to zero-runs. |
| 136 | +- **Feature extraction for edge-ML**: wavelet energy per sub-band is a compact descriptor for |
| 137 | + vibration, ECG, or acoustic signals. |
| 138 | +- **Multi-rate filtering**: each level is a critically sampled octave-band filter, useful for |
| 139 | + hearing-aid and audio-codec pipelines. |
| 140 | + |
| 141 | +## Connections to Other Algorithms |
| 142 | + |
| 143 | +- **ConvolutionCorrelation**: the inner analysis/synthesis loops are FIR convolutions; the same |
| 144 | + periodic-indexing trick is used. |
| 145 | +- **FastFourierTransform / RealFastFourierTransform**: alternative time-frequency view; the DWT |
| 146 | + trades uniform frequency resolution for logarithmic-scale (dyadic) resolution. |
| 147 | +- **PowerDensitySpectrum**: Welch's method is an FFT-based power estimator; the DWT gives a |
| 148 | + non-uniform octave-band equivalent. |
| 149 | +- **DiscreteCosineTransform**: a block transform (fixed basis); the DWT is a multi-scale |
| 150 | + transform (variable support). |
| 151 | + |
| 152 | +## References & Further Reading |
| 153 | + |
| 154 | +- S. Mallat, "A Theory for Multiresolution Signal Decomposition: The Wavelet Representation," |
| 155 | + *IEEE Trans. Pattern Analysis and Machine Intelligence*, 11(7), 674–693, 1989. |
| 156 | +- I. Daubechies, *Ten Lectures on Wavelets*, SIAM, 1992. |
| 157 | +- G. Strang and T. Nguyen, *Wavelets and Filter Banks*, Wellesley-Cambridge Press, 1996. |
| 158 | +- A. V. Oppenheim and R. W. Schafer, *Discrete-Time Signal Processing*, 3rd ed., Ch. 11, 2010. |
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