13.1 Two-Channel Filter Banks

Two-channel filter banks split a signal into low-frequency and high-frequency parts, then reconstruct the original from those parts. They form the foundation of subband coding (used in compression) and multiresolution analysis (used in wavelet transforms), making them essential for applications like image compression and denoising.

Two-Channel Filter Bank Structure

Components and Stages

A two-channel filter bank has two stages: analysis (decomposition) and synthesis (reconstruction).

In the analysis stage, the input signal passes through two parallel filters: a lowpass filter $H_0(z)$ and a highpass filter $H_1(z)$. Each filter output is then downsampled by a factor of 2 (keeping every other sample). This cuts the data rate in half for each subband, which is what makes compression possible.

In the synthesis stage, each subband signal is upsampled by a factor of 2 (inserting zeros between samples), then filtered through synthesis filters $G_0(z)$ and $G_1(z)$. The two filtered outputs are summed to produce the reconstructed signal.

Perfect Reconstruction and Aliasing

Perfect reconstruction (PR) means the output is an exact copy of the input, possibly with a delay and a scale factor:

$\hat{X}(z) = c \cdot z^{-d} \cdot X(z)$

where $c$ is a constant gain and $d$ is an integer delay.

Downsampling and upsampling introduce aliasing, because downsampling by 2 causes frequency components to fold on top of each other. The synthesis filters must be designed so that these aliased components cancel exactly when the two subband outputs are summed.

The alias cancellation condition requires:

$G_0(z) H_0(-z) + G_1(z) H_1(-z) = 0$

A common solution is to set $G_0(z) = H_1(-z)$ and $G_1(z) = -H_0(-z)$. Different filter design strategies build on this:

• Quadrature Mirror Filters (QMF): Set $H_1(z) = H_0(-z)$, so the highpass filter is a frequency-mirrored version of the lowpass. Classic QMF designs cancel aliasing but achieve only approximate (not exact) PR.
• Conjugate Quadrature Filters (CQF): Also called power-complementary designs. These achieve exact PR by satisfying both the alias cancellation and the no-distortion conditions simultaneously.

Analysis and Synthesis Filters in Signal Processing

Frequency Subband Splitting

The analysis filters divide the input spectrum into two complementary bands:

• $H_0(z)$ (lowpass) extracts the approximation, capturing the slow-moving, coarse structure of the signal. It passes frequencies from $0$ to roughly $\pi/2$.
• $H_1(z)$ (highpass) extracts the detail, capturing edges, transients, and fine structure. It passes frequencies from roughly $\pi/2$ to $\pi$.

After downsampling by 2, each subband has half the original number of samples. The total number of samples across both subbands equals the original, so no information is gained or lost (assuming PR filters).

Signal Reconstruction and Perfect Reconstruction Conditions

The synthesis filters $G_0(z)$ and $G_1(z)$ must undo the effects of analysis and resampling. Reconstruction works in three steps:

1. Upsample each subband by 2 (insert a zero between every pair of samples).
2. Filter the upsampled lowpass subband with $G_0(z)$ and the upsampled highpass subband with $G_1(z)$.
3. Sum the two filter outputs to get the reconstructed signal.

For perfect reconstruction, the filters must jointly satisfy two conditions:

• Alias cancellation: $G_0(z) H_0(-z) + G_1(z) H_1(-z) = 0$
• No distortion: $G_0(z) H_0(z) + G_1(z) H_1(z) = 2z^{-d}$

The factor of 2 compensates for the energy lost during downsampling, and $z^{-d}$ represents the allowed reconstruction delay.

Applications of Two-Channel Filter Banks

Subband Coding for Compression

Subband coding exploits the fact that signal energy is usually distributed unevenly across frequency bands. The process works as follows:

1. Decompose the input signal into lowpass and highpass subbands using the analysis filter bank.
2. Quantize and encode each subband independently. Allocate more bits to subbands with higher perceptual importance (typically the low-frequency band carries most of the signal energy) and fewer bits to less important subbands.
3. Transmit or store the encoded subbands.
4. At the decoder, reconstruct the signal using the synthesis filter bank.

This approach is more efficient than coding the full-band signal directly because you can tailor the bit allocation to each subband's characteristics. Audio codecs and image compression standards (like JPEG 2000) rely on this principle.

Multiresolution Analysis and Wavelet Transforms

Multiresolution analysis extends the two-channel filter bank by applying it iteratively to the lowpass subband:

1. Decompose the signal into lowpass (approximation) and highpass (detail) subbands.
2. Take the lowpass output and decompose it again with the same filter bank.
3. Repeat for as many levels as needed.

Each level captures detail at a different scale. The highpass subbands from each level represent fine-to-coarse details, while the final lowpass subband is a very coarse approximation of the original signal.

This iterated filter bank structure is exactly what the discrete wavelet transform (DWT) computes. The analysis filters define the wavelet and scaling functions, and the resulting representation is localized in both time and frequency. That dual localization is why wavelets are so effective for denoising (you can threshold small detail coefficients to remove noise) and for compression (most signal energy concentrates in a few large coefficients).