5.3 Multirate filter banks

Multirate filter banks decompose signals into frequency subbands, each processed at a different sampling rate. This enables efficient analysis, manipulation, and reconstruction of signals for applications ranging from audio/image compression to multicarrier communications and adaptive filtering.

Multirate Filter Bank Fundamentals

A multirate filter bank splits an input signal into subbands using filters, changes the sampling rate within each subband, and then recombines them. The core operations are decimation, interpolation, and the polyphase structures that make everything computationally tractable.

Decimation and Interpolation

Decimation reduces the sampling rate by an integer factor $M$. The process has two stages:

1. Apply an anti-aliasing lowpass filter with cutoff $\pi/M$ to band-limit the signal.
2. Downsample by keeping every $M$-th sample and discarding the rest.

Without the anti-aliasing filter, frequency components above $\pi/M$ fold back into the baseband, causing aliasing distortion.

Interpolation increases the sampling rate by an integer factor $L$. This also has two stages:

1. Upsample by inserting $L-1$ zeros between each original sample.
2. Apply an anti-imaging lowpass filter with cutoff $\pi/L$ and gain $L$ to smooth out the spectral images created by zero-insertion.

These two operations are the building blocks of every multirate filter bank. Getting the filter design right at each stage is what determines whether the overall system can achieve perfect reconstruction.

Analysis and Synthesis Filter Banks

An analysis filter bank takes the input signal and passes it through a bank of bandpass filters, each tuned to a different frequency subband. After filtering, each subband signal is decimated (typically by the number of subbands $M$) so the total number of samples across all subbands equals the original rate.

A synthesis filter bank reverses this process. Each subband signal is interpolated back to the original rate, filtered by a corresponding synthesis filter, and then summed to reconstruct the output. The synthesis filters must be designed to cancel the aliasing introduced during decimation in the analysis stage.

Together, the analysis and synthesis banks form a complete system. If designed properly, the output is a perfect (or near-perfect) delayed copy of the input, meaning all processing can happen in the subband domain without degrading the signal.

Polyphase Representation

Polyphase representation restructures a filter into $M$ parallel branches, each operating at $1/M$ the original rate. For a filter $h[n]$, the $k$-th polyphase component is:

$e_k[n] = h[nM + k], \quad k = 0, 1, \ldots, M-1$

Why does this matter? In a direct-form implementation, you filter at the high rate and then downsample, wasting computation on samples you'll throw away. The polyphase structure moves the filtering after the downsampler, so every multiply-accumulate operation contributes to an output sample. This can reduce computation by a factor of $M$.

Polyphase decomposition also provides a compact matrix formulation of the entire filter bank. The analysis bank can be written as a polyphase matrix $\mathbf{E}(z)$, and perfect reconstruction conditions translate directly into algebraic constraints on this matrix.

Noble Identities

The noble identities let you swap the order of filtering and rate-changing operations, which is essential for deriving polyphase implementations.

• First noble identity: A filter $H(z)$ followed by decimation by $M$ is equivalent to decimation by $M$ followed by $H(z^M)$. In other words, you can move the downsampler before the filter if you replace $z$ with $z^M$ in the transfer function.
• Second noble identity: Interpolation by $L$ followed by a filter $G(z)$ is equivalent to $G(z^L)$ followed by interpolation by $L$.

These identities only hold when the filters are expressed in the appropriate polyphase form. Applying them systematically converts a filter bank from a computationally expensive direct-form structure into an efficient polyphase implementation.

Filter Bank Design Techniques

Several design approaches exist, each trading off between perfect reconstruction guarantees, frequency selectivity, design flexibility, and computational cost. The right choice depends on your application constraints.

Cosine Modulated Filter Banks

Cosine modulated filter banks (CMFBs) generate all $M$ analysis and synthesis filters by cosine-modulating a single prototype lowpass filter. You design one filter well, and the modulation creates the entire bank.

The prototype filter determines the overall performance: its stopband attenuation controls inter-subband leakage, and its transition bandwidth affects frequency resolution. Because all filters share the same prototype, the design problem reduces to optimizing a single filter.

CMFBs can achieve perfect reconstruction and are efficiently implemented using fast DCT algorithms, which reduces the per-sample complexity significantly. The modified discrete cosine transform (MDCT), used in MP3 and AAC audio codecs, and the extended lapped transform (ELT) are both CMFBs.

Quadrature Mirror Filter Banks

Quadrature mirror filter banks (QMFBs) are two-channel filter banks where the highpass analysis filter is a mirror image of the lowpass filter around $\omega = \pi/2$. Specifically, if the lowpass filter is $H_0(z)$, the highpass filter is $H_1(z) = H_0(-z)$.

Classical QMFBs cannot achieve exact perfect reconstruction due to residual amplitude distortion, though the error can be made very small with careful prototype design. The closely related conjugate quadrature filter (CQF) banks do achieve exact PR by imposing a power-complementary condition on the prototype.

QMFBs are widely used as the two-channel building block in tree-structured filter banks for subband coding (e.g., MPEG audio) and as the basis for iterated wavelet decompositions.

Paraunitary Filter Banks

A filter bank is paraunitary if its polyphase analysis matrix $\mathbf{E}(z)$ satisfies:

$\tilde{\mathbf{E}}(z) \mathbf{E}(z) = c \mathbf{I}$

where $\tilde{\mathbf{E}}(z) = \mathbf{E}^H(z^{-1})$ is the paraconjugate. This guarantees perfect reconstruction and orthogonality between subbands.

Orthogonality means the subband signals are decorrelated (for white input), which is ideal for energy compaction in compression. Paraunitary banks can be designed using lattice structures, where the polyphase matrix is factored into a cascade of Givens rotations. Each rotation angle becomes a free design parameter.

The discrete wavelet transform (DWT) with orthogonal wavelets (e.g., Daubechies wavelets) and the lapped orthogonal transform (LOT) are both paraunitary filter banks.

Biorthogonal Filter Banks

Biorthogonal filter banks relax the orthogonality constraint. The analysis and synthesis filters are no longer transposes of each other but instead satisfy a biorthogonality condition: the cross-correlation between analysis filter $k$ and synthesis filter $l$ is a delta function only when $k = l$.

This relaxation provides extra design freedom. You can have symmetric (linear-phase) filters, which paraunitary banks generally cannot achieve for even-length filters. Linear phase is important in image processing because it avoids phase distortion artifacts.

The 9/7 biorthogonal wavelet (used in JPEG 2000 lossy mode) and the 5/3 biorthogonal wavelet (used in JPEG 2000 lossless mode) are the most prominent examples. Both achieve perfect reconstruction while offering linear phase and good frequency selectivity.

M-Channel Filter Banks

M-channel filter banks generalize the two-channel case to $M$ subbands. The input is filtered by $M$ analysis filters, each subband is decimated by $M$, processed, interpolated by $M$, filtered by synthesis filters, and summed.

The design challenge scales with $M$: you need to satisfy PR conditions across all $M$ channels simultaneously. The polyphase matrix becomes $M \times M$, and the aliasing cancellation conditions become a system of $M-1$ equations per frequency.

Design approaches include:

• Cosine modulation of a prototype (CMFBs, as described above)
• Direct optimization of the polyphase matrix entries
• Lattice/cascade factorizations for paraunitary designs
• Tree structures built from cascaded two-channel banks

M-channel banks offer finer frequency resolution than two-channel banks but at higher computational cost and design complexity.

Filter Bank Properties

These properties determine how well a filter bank performs and which applications it suits.

Perfect Reconstruction

A filter bank has perfect reconstruction (PR) if the output equals a scaled, delayed version of the input:

$\hat{x}[n] = c \cdot x[n - n_0]$

with no distortion and no aliasing. PR requires that three conditions are met simultaneously: aliasing is canceled, amplitude distortion is eliminated, and phase distortion is eliminated (or is purely linear).

PR is non-negotiable for lossless compression and any application where the signal passes through the filter bank without subband processing. For lossy compression, near-perfect reconstruction (small but controlled error) is sometimes acceptable if it simplifies the design.

Aliasing Cancellation

When subbands are decimated, spectral components fold over and create aliasing. The synthesis filters must be designed so that these aliased components cancel when the subbands are recombined.

For a two-channel system with analysis filters $H_0(z)$ and $H_1(z)$ and synthesis filters $G_0(z)$ and $G_1(z)$, the aliasing cancellation condition is:

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

This is a necessary (but not sufficient) condition for perfect reconstruction. The remaining condition addresses the transfer function distortion.

Coding Gain

Coding gain quantifies how effectively a filter bank compacts signal energy into fewer subbands. It's defined as:

$G_{TC} = \frac{\frac{1}{M} \sum_{k=0}^{M-1} \sigma_k^2}{\left( \prod_{k=0}^{M-1} \sigma_k^2 \right)^{1/M}}$

where $\sigma_k^2$ is the variance of the $k$-th subband. This is the ratio of the arithmetic mean to the geometric mean of subband variances, and it's always $\geq 1$ (with equality when all variances are identical, meaning no compaction occurred).

Higher coding gain means more energy is concentrated in fewer subbands, which translates to better compression efficiency. For a given signal model (e.g., AR(1) process), the optimal filter bank is the one that maximizes this gain. The KLT (Karhunen-Loève transform) achieves the theoretical maximum, and practical filter banks try to approach it.

Frequency Selectivity

Frequency selectivity measures how well the filters separate adjacent subbands. Poor selectivity means energy leaks between subbands, which degrades compression performance and causes inter-subband interference.

Two key metrics determine selectivity:

• Stopband attenuation: How much the filter suppresses out-of-band frequencies. Higher attenuation means less leakage.
• Transition bandwidth: How quickly the filter rolls off from passband to stopband. Narrower transitions give sharper subband boundaries.

There's a fundamental trade-off: sharper filters require longer impulse responses, which increases computational cost and delay. Cosine modulation and optimization-based methods help achieve good selectivity within practical filter lengths.

Computational Complexity

Complexity depends on three factors: the number of subbands $M$, the filter lengths $N$, and the implementation structure.

• Direct form: Each filter is applied independently. Cost scales as $O(MN)$ per input sample.
• Polyphase form: Filters operate at the decimated rate. Cost drops by a factor of $M$ compared to direct form.
• Fast transform-based: For CMFBs, the modulation step can use an $O(M \log M)$ DCT/FFT, further reducing cost.

For real-time systems, polyphase + fast transform implementations are almost always preferred. The design process often involves explicit trade-offs between complexity and the other properties (PR, selectivity, coding gain).

Applications of Multirate Filter Banks

Subband Coding

Subband coding compresses signals by exploiting the unequal energy distribution across frequency bands. The process works as follows:

1. Decompose the signal into subbands using an analysis filter bank.
2. Quantize each subband independently, allocating more bits to high-energy subbands and fewer to low-energy ones.
3. Encode the quantized subbands using entropy coding.
4. At the decoder, reverse the process: decode, dequantize, and reconstruct using the synthesis filter bank.

MP3 (MPEG-1 Layer III) uses a hybrid structure: a 32-band MDCT-based cosine modulated filter bank. JPEG 2000 uses the 9/7 biorthogonal wavelet (lossy) or 5/3 wavelet (lossless) implemented as iterated two-channel filter banks.

Wavelet Transforms

The discrete wavelet transform (DWT) is implemented by iterating a two-channel filter bank on the lowpass subband at each level. This produces a dyadic frequency decomposition: each successive level halves the bandwidth and doubles the time resolution.

The process for a $J$-level DWT:

1. Apply the two-channel analysis bank (lowpass $H_0$, highpass $H_1$) to the input.
2. Decimate both outputs by 2. Keep the highpass (detail) coefficients.
3. Feed the lowpass (approximation) output back into step 1.
4. Repeat for $J$ levels.

Wavelet packets generalize this by allowing decomposition of both lowpass and highpass subbands at each level, giving a more flexible time-frequency tiling. The lifting scheme provides an efficient way to compute these transforms in-place with reduced memory.

Transmultiplexers

A transmultiplexer converts between time-division multiplexed (TDM) and frequency-division multiplexed (FDM) formats. Structurally, it's the transpose of a standard filter bank: the synthesis bank comes first (at the transmitter), and the analysis bank comes second (at the receiver).

At the transmitter, $M$ input streams are each interpolated, filtered by synthesis filters, and summed to create a composite FDM signal. At the receiver, the analysis filter bank separates the composite signal back into the individual streams, which are then decimated.

The PR conditions for transmultiplexers are the same as for filter banks, just applied in reverse order. This duality means filter bank design techniques transfer directly to transmultiplexer design.

Multicarrier Modulation

In multicarrier modulation, data is transmitted simultaneously on multiple frequency subcarriers. Filter banks provide the modulation/demodulation framework.

OFDM is the most common example. The transmitter uses an inverse FFT (acting as the synthesis bank) to map data symbols onto orthogonal subcarriers. The receiver uses a forward FFT (analysis bank) to recover the symbols. A cyclic prefix handles multipath channel effects.

Filtered multitone (FMT) modulation uses filter banks with better frequency selectivity than the rectangular windows implicit in OFDM. This reduces intercarrier interference and out-of-band emissions, at the cost of higher complexity and the loss of the simple cyclic prefix structure.

Good stopband attenuation in the prototype filter directly translates to lower adjacent-channel interference in these systems.

Adaptive Filtering

Subband adaptive filtering decomposes the input into subbands, runs a separate adaptive filter in each subband, and recombines the outputs. This offers several advantages over fullband adaptive filtering:

• Reduced complexity: Each subband filter is shorter (by a factor of $M$) and operates at a lower rate.
• Faster convergence: The eigenvalue spread of the input correlation matrix is smaller within each subband, so LMS-type algorithms converge faster.
• Better tracking: Subband decomposition allows different adaptation rates for different frequency regions.

Acoustic echo cancellation is a classic application. The echo path can be thousands of taps long at the fullband rate, but subband decomposition breaks this into manageable shorter filters. The trade-off is the delay introduced by the filter bank and potential aliasing artifacts if the bank isn't well-designed.

Advanced Topics in Multirate Filter Banks

Nonuniform Filter Banks

Uniform filter banks split the spectrum into equal-width subbands, but many signals have non-uniform spectral characteristics. Nonuniform filter banks allow different decimation factors across subbands, giving finer resolution where needed and coarser resolution elsewhere.

Common construction methods:

• Tree structures: Cascade two-channel banks in a non-balanced tree. Octave-band decompositions (as in the DWT) are a special case.
• Wavelet packets: Allow arbitrary binary tree decompositions, chosen to match the signal's spectral structure.
• Recombination of uniform subbands: Start with a uniform $M$-channel bank and merge adjacent subbands where fine resolution isn't needed.

Nonuniform banks are particularly useful in audio coding, where perceptual frequency resolution (roughly logarithmic) doesn't match a uniform partition.

Multidimensional Filter Banks

Extending filter banks to 2-D (images) or 3-D (video) signals introduces new design considerations. The simplest approach is separable filtering: apply 1-D filter banks along each dimension independently. This is how most wavelet-based image coders work (row-column processing).

Non-separable filter banks use sampling lattices (e.g., quincunx lattice) that don't align with the Cartesian grid. The quincunx filter bank decimates by a factor of 2 using a checkerboard pattern, splitting the spectrum diagonally rather than horizontally/vertically. This can provide more isotropic frequency decompositions.

Directional filter banks (e.g., contourlets) decompose images along multiple orientations, capturing directional features like edges more efficiently than separable wavelet transforms.

Lifting Scheme

The lifting scheme constructs wavelet filter banks through a sequence of simple operations rather than convolution. A single lifting step consists of:

1. Split: Divide the input samples into even and odd indexed sets.
2. Predict: Use the even samples to predict the odd samples. The prediction error becomes the detail (highpass) coefficients.
3. Update: Use the detail coefficients to update the even samples, producing the approximation (lowpass) coefficients.

Each predict/update step is a simple filtering operation. The key advantages:

• In-place computation: No auxiliary memory needed; results overwrite the input.
• Integer-to-integer transforms: Rounding the predict/update steps gives a perfectly invertible integer transform, critical for lossless compression.
• Modularity: Any FIR wavelet filter bank can be factored into lifting steps (guaranteed by the Euclidean algorithm on the polyphase components).

JPEG 2000 uses lifting-based implementations of both the 9/7 and 5/3 wavelet transforms.

Filter Bank Frames

Standard critically sampled filter banks have exactly as many total subband samples as input samples. Filter bank frames are overcomplete: the number of subbands exceeds the decimation factor, so the representation has built-in redundancy.

A frame satisfies the condition:

$A \|x\|^2 \leq \sum_k |\langle x, \phi_k \rangle|^2 \leq B \|x\|^2$

where $A$ and $B$ are the frame bounds. When $A = B$, the frame is tight, and reconstruction simplifies to applying the adjoint operator scaled by $1/A$.

The redundancy in frames provides robustness: if some subband coefficients are lost or corrupted, the signal can still be reconstructed (approximately or exactly) from the remaining coefficients. This makes frames useful for denoising, sparse representations, and transmission over unreliable channels.

Oversampled Filter Banks

Oversampled filter banks are a specific realization of filter bank frames. In an $M$-channel bank decimated by factor $N < M$, the oversampling ratio is $M/N$. The extra samples per unit time provide redundancy.

Benefits of oversampling:

• Reduced aliasing sensitivity: The less aggressive decimation means aliasing components are smaller, relaxing the filter design requirements.
• Noise robustness: Redundancy allows noise reduction during reconstruction, similar to oversampled A/D conversion.
• Design flexibility: PR conditions become underdetermined (more unknowns than equations), giving more freedom to optimize other properties like frequency selectivity.

The cost is higher data rate in the subband domain (by the oversampling ratio), which matters for compression but is acceptable in applications like subband adaptive filtering or robust communications where the benefits of redundancy outweigh the rate penalty.