5.5 Wavelet filter banks

Wavelet filter banks overview

Wavelet filter banks decompose signals into frequency subbands using cascaded high-pass and low-pass filters, providing localized time-frequency analysis that traditional Fourier methods can't match. They form the computational backbone of multiresolution analysis and show up everywhere: compression (JPEG 2000), denoising, feature extraction, and beyond.

Key properties of wavelet filter banks

• Time-frequency representation: You get both temporal and spectral information simultaneously, unlike the Fourier transform which gives you one or the other.
• Multiresolution analysis: The signal can be examined at multiple scales, so you can zoom in on fine details or zoom out for coarse structure.
• Filter bank structure: A pair of high-pass and low-pass filters (with downsampling) splits the signal into subbands at each stage.
• Perfect reconstruction: With properly designed filters, the original signal can be recovered exactly from its wavelet coefficients.

Comparison to Fourier transform

The Fourier transform maps a signal into frequency components but discards all time localization. You know which frequencies are present, but not when they occur. The wavelet transform solves this by using a variable-sized analysis window: narrow windows for high frequencies (good time resolution) and wide windows for low frequencies (good frequency resolution).

This makes wavelets far more effective for non-stationary signals and transients. A sudden spike or a chirp that would smear across the entire Fourier spectrum gets captured cleanly in the wavelet domain with both its time location and frequency content preserved.

Wavelet decomposition

Wavelet decomposition breaks a signal into subbands by repeatedly filtering and downsampling. At each stage, a low-pass filter produces approximation coefficients (coarse structure) and a high-pass filter produces detail coefficients (fine structure). The process is then iterated on the approximation coefficients to build a tree of subbands at progressively coarser scales.

Multiresolution analysis

Multiresolution analysis (MRA) is the mathematical framework underpinning wavelet decomposition. The core idea is a nested sequence of subspaces:

$\cdots \subset V_{-1} \subset V_0 \subset V_1 \subset \cdots$

Each subspace $V_j$ represents the signal at resolution $2^j$. The difference between successive resolutions lives in a detail subspace $W_j$, so that $V_{j+1} = V_j \oplus W_j$. The approximation coefficients span $V_j$ and the detail coefficients span $W_j$.

Scaling function

The scaling function $\phi(t)$ generates the approximation subspaces. It satisfies a two-scale relation (also called the refinement equation):

$\phi(t) = \sqrt{2} \sum_k h[k] \, \phi(2t - k)$

where $h[k]$ are the low-pass filter coefficients. This equation says that $\phi(t)$ at one scale can be built from shifted copies of itself at the next finer scale. The Haar scaling function is the simplest example: a unit-height rectangle on $[0,1)$. Daubechies scaling functions are smoother but have wider support.

Wavelet function

The wavelet function $\psi(t)$ generates the detail subspaces. It's derived from the scaling function via the high-pass filter coefficients $g[k]$:

$\psi(t) = \sqrt{2} \sum_k g[k] \, \phi(2t - k)$

For orthogonal wavelets, the high-pass coefficients relate to the low-pass coefficients by the alternating flip: $g[k] = (-1)^k h[1-k]$. The wavelet function captures the high-frequency content that the scaling function misses. The Haar wavelet, for instance, is a step function that takes value $+1$ on $[0, 0.5)$ and $-1$ on $[0.5, 1)$.

Wavelet reconstruction

Reconstruction reverses the decomposition: you upsample the coefficients, filter them with synthesis filters, and sum the results to recover the original signal.

Upsampling in reconstruction

Upsampling by 2 inserts a zero between every pair of coefficient samples, doubling the sequence length. If the detail coefficients at level $j$ are $d_j[n]$, the upsampled version is:

$d_j^{\uparrow}[n] = \begin{cases} d_j[n/2] & n \text{ even} \\ 0 & n \text{ odd} \end{cases}$

This restores the sample rate to match the next finer level before filtering.

Filtering in reconstruction

After upsampling, you apply the synthesis filters: a low-pass reconstruction filter $\tilde{h}[k]$ to the approximation branch and a high-pass reconstruction filter $\tilde{g}[k]$ to the detail branch. The two filtered outputs are then summed to produce the approximation at the next finer scale. For orthogonal wavelets, the synthesis filters are simply the time-reversed versions of the analysis filters.

Perfect reconstruction condition

Perfect reconstruction (PR) means the output equals the input (up to a delay and possible scaling). In the two-channel case, PR requires the analysis filters $H(z), G(z)$ and synthesis filters $\tilde{H}(z), \tilde{G}(z)$ to satisfy two conditions simultaneously:

1. No distortion: The combined transfer function equals a pure delay.
2. Aliasing cancellation: The alias components introduced by downsampling cancel exactly when the two branches are recombined.

In polyphase matrix form, this reduces to requiring the analysis polyphase matrix $\mathbf{E}(z)$ and synthesis polyphase matrix $\mathbf{R}(z)$ to satisfy $\mathbf{R}(z)\mathbf{E}(z) = c \, z^{-d} \mathbf{I}$, where $c$ is a scalar and $d$ is the delay.

Subband coding with wavelets

Subband coding divides a signal into frequency bands, processes each band independently (e.g., quantization, bit allocation), and reconstructs the signal. Wavelet filter banks are a natural fit because their dyadic tree structure provides an octave-band frequency decomposition that matches how many real-world signals distribute their energy.

Subband decomposition

The decomposition follows a recursive structure:

1. Apply the low-pass filter $h[k]$ and high-pass filter $g[k]$ to the input signal.
2. Downsample both outputs by a factor of 2.
3. The high-pass output becomes the detail subband at this level.
4. Repeat steps 1-3 on the low-pass output for the next coarser level.

After $J$ levels, you have $J$ detail subbands plus one coarse approximation subband. This is the dyadic wavelet decomposition tree. The total number of samples across all subbands equals the original signal length (critically sampled).

Subband reconstruction

Reconstruction climbs back up the tree:

1. Upsample the coarsest approximation and the coarsest detail subband by 2.
2. Apply the synthesis low-pass filter $\tilde{h}[k]$ and synthesis high-pass filter $\tilde{g}[k]$ respectively.
3. Sum the two filtered outputs to get the approximation at the next finer level.
4. Repeat until you reach the original resolution.

The synthesis filters are designed so that aliasing from the downsampling step cancels during reconstruction.

Aliasing cancellation in subband coding

Downsampling by 2 introduces aliasing: frequency components above $\pi/2$ fold back into the baseband. In a two-channel filter bank, the output in the $z$-domain contains both the desired signal term and an alias term involving $X(-z)$.

Aliasing cancellation works because the high-pass and low-pass branches produce alias terms with opposite signs. When you design the synthesis filters to satisfy:

$\tilde{H}(z)H(-z) + \tilde{G}(z)G(-z) = 0$

the alias terms cancel upon summation. This is one of the two PR conditions and is automatically satisfied when the filters come from a valid wavelet construction (e.g., conjugate mirror filters or biorthogonal pairs).

Wavelet families

Different wavelet families trade off properties like smoothness, support length, symmetry, and number of vanishing moments. Your choice of wavelet should match the signal characteristics and application requirements.

Haar wavelet

The Haar wavelet is the simplest wavelet: a two-tap filter with coefficients $h = \frac{1}{\sqrt{2}}[1, 1]$ and $g = \frac{1}{\sqrt{2}}[1, -1]$. It has compact support on $[0,1]$, is orthogonal, and has exactly one vanishing moment.

The Haar wavelet is computationally cheap and easy to understand, but its piecewise-constant nature means it produces blocky artifacts and poorly approximates smooth signals. It's useful for pedagogical purposes and for signals with sharp transitions, but rarely the best choice for real applications.

Daubechies wavelets

Daubechies wavelets (denoted dbN, where $N$ is the number of vanishing moments) are orthogonal wavelets with compact support that maximize the number of vanishing moments for a given filter length. The filter length is $2N$ taps.

• db1 is identical to the Haar wavelet.
• db2 has 4 filter taps and 2 vanishing moments.
• db4 has 8 filter taps and can represent polynomials up to degree 3 exactly.

Higher $N$ gives smoother wavelets and better approximation of smooth signals, but at the cost of longer filters (more computation) and reduced time localization. Daubechies wavelets are asymmetric, which can be a drawback for some applications like image processing where symmetric filters avoid phase distortion.

Biorthogonal wavelets

Biorthogonal wavelets use different filters for analysis and synthesis. The analysis and synthesis scaling/wavelet functions form biorthogonal pairs rather than being identical. This relaxation buys you two major advantages:

• Symmetric (linear phase) filters become possible, which is not achievable with orthogonal wavelets (except Haar).
• Independent control over the vanishing moments and regularity of the analysis and synthesis sides.

The CDF 9/7 wavelet (Cohen-Daubechies-Feauveau, with 9-tap and 7-tap filters) is used in the lossy mode of JPEG 2000. The CDF 5/3 wavelet is used in the lossless mode. Biorthogonal wavelets achieve perfect reconstruction despite having non-identical filter pairs, as long as the biorthogonality conditions are satisfied.

Design of wavelet filter banks

Designing a wavelet filter bank means choosing filter coefficients that satisfy PR conditions while optimizing for application-specific criteria. The three main design parameters are vanishing moments, regularity, and support length.

Vanishing moments

A wavelet $\psi(t)$ has $N$ vanishing moments if:

$\int_{-\infty}^{\infty} t^k \psi(t) \, dt = 0 \quad \text{for } k = 0, 1, \ldots, N-1$

This means the wavelet is orthogonal to all polynomials up to degree $N-1$. In practice, if a signal region is well-approximated by a low-degree polynomial, the wavelet coefficients in that region will be very small. This is what makes wavelet compression effective: smooth regions produce near-zero coefficients that can be discarded.

The tradeoff is that $N$ vanishing moments require a minimum filter length of $2N$ (for orthogonal wavelets), which increases computational cost and reduces time localization.

Regularity of wavelets

Regularity measures how smooth the wavelet and scaling functions are. A wavelet with Hölder regularity $\alpha$ is roughly $\alpha$-times differentiable. Higher regularity produces smoother reconstructed signals and reduces visual artifacts in image processing.

Regularity generally increases with the number of vanishing moments, but the relationship isn't one-to-one. For Daubechies wavelets, regularity grows approximately linearly with $N$, at a rate of about $0.2N$. Biorthogonal designs can allocate regularity unevenly: you might want a very smooth synthesis wavelet (for reconstruction quality) while accepting a less smooth analysis wavelet.

Compact support of wavelets

A wavelet has compact support if it's nonzero only on a finite interval. This is essential for practical implementation because it means the corresponding filters are FIR (finite impulse response), which are computationally efficient and numerically stable.

The support width is directly tied to the filter length. For Daubechies dbN wavelets, the support is $[0, 2N-1]$. Longer support gives more vanishing moments and smoother wavelets, but also means each output sample depends on more input samples, increasing computation and reducing the ability to localize events in time.

Applications of wavelet filter banks

Image compression with wavelets

Wavelet-based image compression applies a 2D wavelet transform (typically separable, applying 1D transforms along rows then columns) to produce subbands at multiple scales. Most of the signal energy concentrates in the low-frequency approximation subband, while the detail subbands are sparse.

Compression then proceeds by quantizing the wavelet coefficients (coarser quantization for less perceptually important subbands) and entropy coding the result. JPEG 2000 uses the CDF 9/7 wavelet for lossy compression and achieves notably better quality than DCT-based JPEG at low bit rates, particularly avoiding the blocking artifacts that plague JPEG.

Denoising with wavelets

Wavelet denoising exploits the fact that signal energy tends to concentrate in a few large wavelet coefficients, while noise spreads evenly across all coefficients. The standard procedure:

1. Decompose the noisy signal using a wavelet filter bank.
2. Apply a thresholding rule to the detail coefficients. Hard thresholding sets coefficients below a threshold to zero. Soft thresholding shrinks all coefficients toward zero by the threshold amount.
3. Reconstruct the signal from the modified coefficients.

The threshold is typically set proportional to $\sigma \sqrt{2 \ln N}$ (the universal threshold), where $\sigma$ is the noise standard deviation and $N$ is the signal length. This approach works well because wavelets can distinguish localized signal features from diffuse noise across subbands.

Feature extraction with wavelets

Wavelet coefficients capture localized time-frequency structure, making them natural features for classification and pattern recognition tasks. At each decomposition level, the coefficients reflect activity in a specific frequency band at a specific time location.

Common feature extraction strategies include computing the energy in each subband, statistical moments of the coefficient distributions, or using the coefficients directly as input to a classifier. Applications range from texture analysis in images (where different scales capture different texture granularities) to EEG signal classification (where specific wavelet subbands correspond to clinically relevant brain rhythms like alpha or beta waves).