MPEG Filter Banks

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,

$\displaystyle H_1(z) \eqsp H_0(-z),$ (12.97)

was rather severe in that linear-phase FIR implementations only exist in the two-tap case $ H_k(z) = h_{0k}+h_{1k}z^{-1}$ , $ k=0,1$ . In addition to relaxing this constraint, we need to be able to design an $ N$ -channel filter bank for any $ N$ .

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 $ 96$ dB, so the neglected bands (which alias freely) are not significant. An outline of the design procedure is as follows:

  1. Design a lowpass prototype window, $ h(n)$ , with length $ M=LN$ , $ L,M,N \in {\bf Z}.$
  2. The lowpass design is constrained to give aliasing cancellation in neighboring subbands:

    \begin{eqnarray*}
\vert H(e^{j\omega})\vert^2 + \vert H(e^{j(\pi/N)-\omega})\vert^2 &=& 2, \hspace{.5cm}0 < \vert\omega\vert <
\pi/{2N} \\
\vert H(e^{j\omega})\vert^2 &=& 0, \hspace{.5cm}\vert w\vert > \pi/N
\end{eqnarray*}

  3. The filter bank analysis filters $ h_k(n)$ are cosine modulations of $ h(n)$ :

    $\displaystyle h_k(n) \eqsp h(n)\hbox{cos}\left[\left(k+\frac{1}{2}\right)\left(n-\frac{M-1}{2}\right)\frac{\pi}{N} + \phi_k\right],$ (12.98)

    $ k=0,\ldots,N-1$ , where the phases are restricted according to

    $\displaystyle \phi_{k+1} - \phi_k \eqsp (2r+1)\frac{\pi}{2}$ (12.99)

    again for aliasing cancellation.
  4. Since it is an orthogonal filter bank by construction, the synthesis filters are simply the time-reverse of the analysis filters:

    $\displaystyle f_k(n) \eqsp h_k(M-1-n)$ (12.100)

This PQMF filter bank is reportedly used in MPEG audio, layers I and II with $ N=32$ bands and $ M=512$ taps ($ L=8$ ).


Perfect Reconstruction Cosine Modulated Filter Banks

By changing the phases $ \phi_k$ , the pseudo-QMF filter bank can yield perfect reconstruction:

$\displaystyle \phi_k \eqsp \left(k+\frac{1}{2}\right)\left(L+1\right)\frac{\pi}{2}$ (12.101)

where $ L$ is the length of the polyphase filter ($ M=LN$ ).

If $ M=2N$ , then this is the oddly stacked Princen-Bradley filter bank and the analysis filters are related by cosine modulations of the lowpass prototype:

$\displaystyle f_k(n) \eqsp h(n)\hbox{cos}\left[\left(n+\frac{N+1}{2}\right)\left(k+\frac{1}{2}\right)\frac{\pi}{N}\right],\quad k=0,\ldots,N-1$ (12.102)

However, the length of the filters $ M$ can be any even multiple of $ N$ :

$\displaystyle M\eqsp LN, \quad (L/2) \in \cal{Z}$ (12.103)

The parameter $ L$ is called the overlapping factor. These filter banks are also referred to as extended lapped transforms, when $ K \ge 2$ [159].


MPEG Layer III Filter Bank

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 [214] 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 $ 32\times 6=192$ subbands, corresponding to relatively high time resolution and low frequency resolution.
  • Steady-state tones use $ 32\times 18=576$ 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 $ 32\times 6$ -$ 18$ MP3 filter bank, without requiring the ``cross-talk aliasing reduction'' needed by the PQMF/MDCT hierarchical filter bank of MP3 [149, §4.1.6].

The MPEG-4 audio compression standard (there was no MPEG-3), included a new transform coder based on the AAC filter bank [149, §4.1.7].

See, e.g., [16,273] for much more on MPEG coders and related topics. Chapter 4 of [149] contains an excellent summary of MPEG, Sony ATRAC, and Dolby AC-n coders up to 1998.


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Filter Banks Equivalent to STFTs