Polyphase quadrature filter

A polyphase quadrature filter, or PQF, is a filter bank which splits an input signal into a given number N (mostly a power of 2) of equidistant sub-bands. These sub-bands are subsampled by a factor of N, so they are critically sampled.[1]

This critical sampling introduces aliasing. Similar to the MDCT time domain alias cancellation the aliasing of polyphase quadrature filters is canceled by neighbouring sub-bands, i.e. signals are typically stored in two sub-bands.

Note that signal in odd subbands is stored frequency inverted.

PQF filters are used in MPEG-1 Audio Layer I and II, Musepack (which was based on MPEG-1 layer II), in MPEG-1 Layer III with an additional MDCT, in MPEG-4 AAC-SSR for the 4 band PQF bank, in MPEG-4 V3 SBR for the analysis of the upper spectral replicated band, and in DTS.

PQF has an advantage over the very similar stacked quadrature mirror filter (QMF). Delay and computational effort are much lower.

A PQF filter bank is constructed using a base filter, which is a low-pass at fs/4N. This lowpass is modulated by N cosine functions and converted to N band-passes with a bandwidth of fs/2N.

The base lowpass is typically a FIR filter with a length of 10*N ... 24*N taps. Note that it is also possible to build PQF filters using recursive IIR filters.

Computation

There are different formulas possible. Most of them are based on the MDCT but are slightly modified.


gollark: Ah yes, there was a "rainbow book" series.
gollark: CDs are standardized in what's called the "Red Book" for some reason. I assume they shipped them in a red book or something.
gollark: CDs, of course, store uncompressed PCM audio at 44.1kHz/16 bit sample resolution.
gollark: Because CDs are quite lossy, or something, and have a ton of redundancy.
gollark: In theory you could actually load a ton more data on there if you messed with your CD writer's firmware to ignore the error correction code things.

References
  1. Rothweiler, J. (April 1983). "Polyphase quadrature filters–A new subband coding technique". ICASSP '83. IEEE International Conference on Acoustics, Speech, and Signal Processing. 8: 1280–1283. doi:10.1109/ICASSP.1983.1172005.


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