Filter Banks and Polyphase Structures

Splitting and Recombining Signals

Up to this point in Module 11, we have treated upsampling and downsampling as tools for changing a single signal's sample rate. Filter banks extend this idea by splitting a wideband signal into multiple narrowband subbands, processing each subband independently, and then recombining them into a reconstructed output. This architecture appears in virtually every modern audio codec (MP3, AAC, Opus), in OFDM wireless systems, and in wavelet-based image compression.

A filter bank has two parts: the analysis bank decomposes the input into subbands, and the synthesis bank reassembles the subbands into the output. When properly designed, the synthesis bank can perfectly undo the analysis bank, recovering the original signal without distortion — a property called perfect reconstruction.

The Analysis Filter Bank

An M-channel analysis filter bank consists of M bandpass filters H₀, H₁, …, H_M−1, each covering a different frequency band. In the simplest case — a uniform DFT filter bank — the filters are modulated versions of a single prototype lowpass filter H₀:

After filtering, each branch is downsampled by M (critical sampling). This keeps the total data rate equal to the input rate — the sum of M subband signals, each at rate f_s/M, equals the original rate f_s. The analysis bank output is M sequences v_k[n], each representing the signal's content near frequency kπ/M.

The Synthesis Filter Bank

The synthesis bank reverses the analysis process. Each subband signal v_k[n] is upsampled by M (inserting M−1 zeros between samples) and then filtered by a synthesis filter G_k. The M filtered branches are summed to produce the reconstructed output x̂[n].

The reconstruction quality depends on how well the filter bank satisfies the perfect reconstruction (PR) conditions: no aliasing from the decimation/interpolation stages, flat overall frequency response, and linear phase. Meeting all three simultaneously requires careful joint design of H_k and G_k.

Polyphase Decomposition: The Efficiency Engine

Naïvely implementing a filter bank requires running M long FIR filters at the full input rate f_s before downsampling. This is computationally wasteful: most of the filter outputs are immediately discarded by the downsampler. The polyphase decomposition reorganizes the computation so that all filtering occurs at the lower subband rate f_s/M.

Any FIR filter H(z) of length N can be written as M shorter filters — called polyphase components E_k(z) — of length ⌈N/M⌉:

In the polyphase implementation of the analysis bank, the input is first passed through a commutator (a serial-to-parallel converter that distributes successive input samples to the M branches), then each branch is filtered by its polyphase component, and finally the M outputs are transformed by an M-point DFT to produce the M subband signals. The result: all arithmetic happens at rate f_s/M, and the DFT can itself be computed efficiently using the FFT.

Quadrature Mirror Filters

The simplest and most widely-used filter bank is the two-channel QMF bank (M = 2). It splits the signal into a lowpass subband (below π/2) and a highpass subband (above π/2). The lowpass analysis filter H₀(z) and the highpass filter H₁(z) are related by:

The QMF design challenge is to make H₀ a good lowpass filter (sharp cutoff near π/2, high stopband attenuation) while simultaneously satisfying the PR conditions. Johnston filters — optimized FIR windows — are a practical solution. Alternatively, conjugate quadrature filters (CQF), also called paraunitary filter banks, achieve exact PR with linear-phase characteristics ideal for audio and image coding.

Applications of Filter Banks

Audio compression (MP3, AAC, Opus): The MPEG psychoacoustic model works in the frequency domain. MP3 uses a 32-channel polyphase filter bank to split audio into 32 subbands, then applies modified discrete cosine transform (MDCT) within each subband. AAC improves on this with a pure MDCT-based filter bank that achieves better frequency resolution. In all cases, the filter bank allows the encoder to allocate more bits to subbands with perceptually important content and fewer bits to masked or less audible bands.

Image compression (JPEG 2000, wavelet coding): The discrete wavelet transform (DWT) is a recursive two-channel QMF bank applied iteratively. The lowpass subband is repeatedly split into lower subbands, creating a multi-resolution representation. JPEG 2000 uses the 9/7 biorthogonal wavelet for lossy coding and the 5/3 LeGall wavelet for lossless coding — both are specific choices of QMF filter pairs.

OFDM communications: An OFDM transmitter can be viewed as a synthesis filter bank where each subcarrier is a separate channel. The IFFT operation is exactly the polyphase synthesis structure: it combines M independent data streams (one per subcarrier) into a wideband time-domain signal. The receiver's FFT is the polyphase analysis bank. This interpretation reveals why guard intervals (cyclic prefix) are needed: they prevent inter-subband aliasing.

Hearing aids and speech processing: Filter banks allow independent processing of different frequency regions, mimicking the cochlea's own frequency decomposition. A hearing aid can apply different amounts of amplification to low, mid, and high frequencies based on the patient's audiogram, all within a single polyphase filter bank implemented on a low-power DSP chip.