This section provides some highlights of the history of filter banks used for perceptual audio coding (MPEG audio). For a more complete introduction and discussion of MPEG filter banks, see, e.g., [16,273].
Pseudo-QMF Cosine Modulation Filter Bank
Section 11.3.5 introduced two-channel quadrature mirror filter banks (QMF). QMFs were shown to provide a particular class of perfect reconstruction filter banks. We found, however, that the quadrature mirror constraint on the analysis filters,
was rather severe in that linear-phase FIR implementations only exist in the two-tap case , . In addition to relaxing this constraint, we need to be able to design an -channel filter bank for any .
The Pseudo-QMF (PQMF) filter bank is a ``near perfect reconstruction'' filter bank in which aliasing cancellation occurs only between adjacent bands [194,287]. The PQMF filters commonly used in perceptual audio coders employ bandpass filters with stop-band attenuation near dB, so the neglected bands (which alias freely) are not significant. An outline of the design procedure is as follows:
- Design a lowpass prototype window, , with length ,
- The lowpass design is
constrained to give aliasing cancellation in neighboring subbands:
- The filter bank analysis filters
are cosine modulations of
, where the phases are restricted according to
again for aliasing cancellation.
- Since it is an orthogonal filter bank by construction,
the synthesis filters are simply the time-reverse of the analysis filters:
Perfect Reconstruction Cosine Modulated Filter Banks
By changing the phases , the pseudo-QMF filter bank can yield perfect reconstruction:
where is the length of the polyphase filter ( ).
If , then this is the oddly stacked Princen-Bradley filter bank and the analysis filters are related by cosine modulations of the lowpass prototype:
However, the length of the filters can be any even multiple of :
The parameter is called the overlapping factor. These filter banks are also referred to as extended lapped transforms, when .
MPEG 1 and 2, Layer III is popularly known as ``MP3 format.'' The original MPEG 1 and 2, Layers I and II, based on the MUSICAM coder, contained only 32 subbands, each band approximately 650 Hz wide, implemented using a length 512 lowpass-prototype window, lapped (``time aliased'') by factor of 512/32 = 16, thus yielding 32 real bands with 96 dB of stop-band rejection, and having a hop size of 32 samples [149, §4.1.1]. It was found, however, that a higher coding gain was obtained using a finer frequency resolution. As a result, the MPEG 1&2 Layer III coder (based on the ASPEC coder from AT&T), appended a Princen-Bradley filter bank  having 6 to 18 subbands to the output of each subband of the 32-channel PQMF cosine-modulated analysis filter bank [149, § 4.1.2]. The number of sub-bands and window shape were chosen to be signal-dependent as follows:
- Transients use subbands, corresponding to relatively high time resolution and low frequency resolution.
- Steady-state tones use subbands, corresponding to higher frequency resolution and lower time resolution relative to transients.12.3
- The encoder generates a function called the perceptual entropy (PE) which tells the coder when to switch resolutions.
The MPEG AAC coder is often regarded as providing nearly twice the compression ratio of ``MP3'' (MPEG 1-2 Layer III) coding at the same quality level.12.4 MPEG AAC introduced a new MDCT filter bank that adaptively switches between 128 and 1024 bands (length 256 and 2048 FFT windows, using 50% overlap) [149, §4.1.6]. The nearly doubled number of frequency bands available for coding steady-state signal intervals contributed much to the increased coding gain of AAC over MP3. The 128-1024 MDCT filter bank in AAC is also considerably simpler than the hierarchical - MP3 filter bank, without requiring the ``cross-talk aliasing reduction'' needed by the PQMF/MDCT hierarchical filter bank of MP3 [149, §4.1.6].
Review of STFT Filterbanks
Filter Banks Equivalent to STFTs