I'm writing a text about various filterbank structures that have been used for audio coding. In the part about subbanding, I was going to comment that the problem of typical PQMF filterbanks is that you have insufficient resolution in the lower frequencies.

The thing is, typical PQMF filterbanks split in 32 bands. By why can't we use them to split in, say 1024 bands. What makes them worse compared to an MDCT?

My guess would be computation efficiency, but maybe it has to do with critical sampling, too? Properties of the bandpass filters? Honestly, I don't know enough about PQMF filterbanks so any help is welcome here.

Well, basically, the PQMF and the MDCT are both filter banks. PQMF is not PR (Perfect Reconstruction) filter banks whereas the MDCT is PR.

You can design a PQMF filterbanks that split into 1024 bands as with the MDCT, and in my opinion, both the MDCT and the PQMF can be implemented with the FFT. I don't really think that the fast MDCT is very much more efficient than the fast PQMF.

I think the PQMF is also critically sampled - that is it only has X output samples for every X input samples.

I am not really sure about the insufficient resolution in the lower frequencies region issue.

So you're saying that the only relevant difference is the PR?

What about energy compaction? The MDCT has excellent properties coming from the DCT-4 base function, what happens in that regard with a big PQMF?

A typical PQMF bank, as used in MPEG Layer 1, 2 or 3 and Musepack has 32 bands and a resolution of 1380750Hz. At the lower end this a whole bunch of Barks packed together. In fact, Musepack has to employ special noise shaping to work around this problem (ANS).

Sorry.. I went home and did some reading.

It seemed that the PQMF filterbanks cancels aliasing in adjacent channels and assumes that the non-adjacent channels have no aliasing due to the infinite stopband attenuation of the prototype low-pass filter. This works fine when the number of channels are small such as 32 . However, when the number of channels are increased to 1024, then the order of the required prototype lowpass filter is very high due to the sharp cut-off frequency required. As a result, high channels PQMF filterbanks are normally very computationally intensive.

It seemed to me that most practical PQMF filterbanks do not have infinite stopband attenuation at non-adjacent channels. I assumed that must be why Muspack has special noise shaping techniques.

In the case of the QMF filterbanks, aliasing due to non adjacent channels are structurally cancelled as well-unlike the PQMF filterbanks. As a result, the required prototype lowpass filter need not be high in order.

Why use PQMF when there is the QMF?

In the case of the MDCT, aliasing cancellation is done in the overlap and add - time domain samples of the calculated IMDCT and we have control of the aliasing components by choosing an appropriate window shape for the MDCT-IMDCT pair. For the QMF, PQMF there is no such flexibility.


Both the MDCT , QMF, PQMF are filterbanks.. and filterbanks can be implemented using the FFT.

No, because the bands are so wide Bark bands are roughly 80-100Hz in the low regions. A 32 channel PQMF bank gives 1380 750Hz bands. You don't have a possibility to use all the information the psymodel is giving you to do more precise quantization, so you'd have to assume a worst case and lose coding efficiency. ANS is like TNS, but in reverse, so you use time domain noise shaping to control the quantization error in frequency, allowing you to win back some frequency resolution.



Because it's still much more efficient than a stacked/cascaded QMF filterbank?

No. The QMF isn't cascaded. It can be implemented with a single FFT.
I thought the QMF has better properties than the PQMF?

As far as I understand, the QMF splits in 2 bands. If you need to split in more bands, you will need to stack them.

So what you're talking about must be stacking QMF's and replacing that stack by a faster FFT equivalent?

Consider. A stacked QMF has to have longer impulse response for a given overall uniform-band rejection than a PQMF, but it cancels aliasing everywhere in the absense of coding noise.

A PQMF only cancels aliasing in the adjacent band, so the polyphase filter has to be quite long in a per-band form, i.e. the number of taps in each polyphase component stay about constant, regardless of how many bands there are (total filter length of n * p where n is the number of bands, and p is the number of taps in each polyphase component).


An MDCT, on the other hand, is exact reconstruction, and due to its different design cancels aliasing regardless of how many bands away the aliasing is.

Yes, you can argue that it cancels in the time domain. Irrelevant. Duals are duals.

Consider, if I want the same frequency resolution as a 1024 point sin-window MDCT, I have to have a 1024 point QMF first filter (first split), and a PQMF filter much longer (to avoid aliasing).

But, with the QMF, then I need a 512 length second filter, a 256 length third filter ...

So the impulse response of the QMF will always be long, long, long, long, and the PQMF can't be used for a lot of bands without making it long, too.

Life is like that.

No.. I am refering to the polyphase representation of the QMF filterbank. Somehow, I don't remember any stacked structures.

That is the most simple form of the QMF filterbank. But you can generalized the QMF equations directly to any L-channels QMF filterbanks. and I am very sure it is not a cascaded design structure.

Indeed you're talking about a PQMF. Still, you have a fundamental problem with a PQMF, aliasing cancellation only happens in the adjacent bands, not all bands.

Ergo, for good performance, the total filter length for a PQMF has to be LONGER than that from an MDCT, and it's not exact reconstruction, either.

As to QMF, I say with some comfort that a QMF is a two-band split structure. Only.

If you do more bands, it's a PQMF or a Cox's version of a PQMF, which is similar, only different. (Sorry, I don't remember the details very well.)

Layers 1 and 2 use a PQMF. Layer 3 uses an MDCT on top of the PQMF, and then aliasing cancellation (of the wrong magnitude, too, by the way) top of all that to make up for the response of the PQMF filters.

An MDCT of the same length (576) would be better.




This is a notational problem. In the lexicon I'm aware of (Estiban-Galand, Johnston) a QMF is a two-band split. A multiband split is a PQMF.

Both are half-way to a wavelet. Smith's QMF's and IIR QMF's are likewise half-way to a wavelet.

What would be the full way? I got lost in the difference between wavelets and QMF bandsplitting filters somewhere. If you ask me it's just the old and the new name for the same idea. I though the difference was perfect reconstruction, but I'm now seeing "Wavelet audio codecs" with Non-PR filterbanks, so really, I haven't got an idea what would make the distinction.

It is very hard to produce the equations and diagrams here. Unfortunately, I don't have a laptop with me, so I couldn't prepare my arguments in detail.

Like where for example?


Sebi

AES Convention Paper 5301

I suggest a book "Multirate Systems And Filter Banks" by P. P. Vaidyanathan.

And I Malvar's book on lapped transforms.



Wavelets ARE exact-reconstruction. I'll have to look at the non-exact-reconstruction wavelet (cough, hack).

Wavelets provide an orthogonal projection. QMF's don't.

Etc, etc.

N.B. kwong, the QMF was defined as a 2-band filterbank. Cox, Smith, Malvar, Dehery, and others have all produced multiband versions, but they only cancel aliasing in the adjacent band. Collectively they are PQMF's. They are pseudo-orthogonal, and can't be exact reconstruction.

OBT's, etc, are exact-reconstruction, and are very similar, but orthogonal projections...

Umm... these 3/5 and 7/9 wavelets used in image coding don't (biorthogonal).
QMFs come close to orthogonal, though.

Sebi

If you do fetch that paper, I'm sure there'll be more "cough, hack" when you see the proposed method for solving the overlap distortion problem

Well, biorthogonal "wavelets" are not "wavelets" in my book, they are just what you said "biorthogonal wavelets".

They can't be power complimentary among other things, hence I really don't like them a lot.

As to the non-exact-reconstruction "wavelet coder", do you mind if I pass, that way my blood pressure will stay at a more reasonable level...