Hi Frank,
>
> 1. FFT filters are strictly speaking no filters (they are not a LTI system),
> so they have some nasty properties, which are more or less audible. The
> audibility depends on the steepness of the filter. So high passes should
> never be made by FFT filters. Never ever.
>
> Filter flanks modulating the signal, a property LTI systems NEVER have.
> May be also low pass filter are a bad idea. For high pass filters I'm
> absolutely sure.
>
Yes, I agree but I've never heard of anyone using an FFT for low pass
filtering. The problem is piecing together the results from the
different windows. LAME does use an FFT filterbank for the psycho
acoustics, where it needs the energies for each frequency. A windowed
FFT gives excellent power spectrum estimation, which is why its use is
suggested by MPEG for the psycho acoustics in all their codecs.
The MDCT is different, it was only invented/discovered in the
1980s to solve the "edge effects" problem. It is a set of overlapped,
modified cosine transformations. The windows have to be specially
constructed to make the transform (forward then backward) lossless.
It is not possible to get a lossless tranform out of any windowed FFT,
nor if the windows overlap less then 50% (for a proof of this, see
Malvar, "Signal Processing with Lapped Tranforms"). I was going
to say Malvar knows what he is doing, except on his web site:
http://www.research.microsoft.com/~malvar/
he has the patently absurd comment that WMA at 64kbs beats
MP3 at 128kbs :-)
The "polyphase filterbank" is similar to the MDCT, but not lossless,
and has since been replaced by the MDCT (for example, in AAC and
Vorbis). Here's what one of the MPEG papers says about the polyhpase
filterbank (maybe this means something to you): "a QMF filter of order
511, with rejection of side lobes better than 96db."
They call the MDCT a QMF filter with perfect reconstruction.
Using your notation (below), LTI filters are what I've seen called FIR
(for when b(i)=0) and IIR (when b(i)<>0). So now I finally understand
what you are talking about :-) I doubt the MDCT can be characterized
as a LTI filter.
Mark
> 2. FFT filters approximating non recursive filter (often called FIR filters,
> which is not correct), but actually they are a mixture of a frequency
> dependent modulator and a filter. Non recursive filters are only a very
> special class of filters. All LTI filtering is done by:
>
> A B
> y(n) := Sum a(i) x(n-i) - Sum b(i) y(n-i)
> i=0 i=1
>
> Every Filter can be characterized by the a(0...A) and b(1...B). For non
> recursive filters is B=0 and A>=0 (also called moving average filters),
> for absolute phase filters is A=0 and B>0 (also called auto regression
> filter). Filters with B>0 and A>0 are mixing both base vectors of filters
> and are also called auto regression moving average filters.
>
> You can divide ever (LTI) filter into two filters, a MA and a AR filter:
>
> A
> v(n) := Sum a(i) x(n-i)
> i=0
>
> B
> y(n) := v(n) - Sum b(i) y(n-i)
> i=1
>
> Now you can set b(0):=1
>
> A
> v(n) := Sum a(i) x(n-i)
> i=0
>
> B
> v(n) := Sum b(i) y(n-i)
> i=0
>
> This gives (x,y,z complex, O is a big omega and omega/fs, j is sqrt(-1) )
>
> A
> v(w)/x(w) = Sum a(i) exp (jOi/fs)
> i=0
>
> B
> v(w)/y(w) = Sum b(i) exp (jOi/fs)
> i=0
>
> Substituting exp(jO/fs) = z gives
>
> A i
> v/x = Sum a(i) z
> i=0
>
> B i
> v/y = Sum b(i) z
> i=0
>
> and
>
> A i
> Sum a(i) z
> i=0
> y/x = -----------
> B i
> Sum b(i) z
> i=0
>
> So you can see: MA = P_A(z), AR = 1/Q_B(z) and ARMA = P_A(z)/Q_B(z).
>
> Example: The easiest AR filter, a integrator (1st order) can only
> be programmed by a infinite long MA filter.
>
>
> Are polyphase filters LTI systems? FFT filters aren't.
> And they are comparable with the subset of MA filters.
>
> --
> Mit freundlichen Grüßen
> Frank Klemm
>
> eMail | [EMAIL PROTECTED] home: [EMAIL PROTECTED]
> phone | +49 (3641) 64-2721 home: +49 (3641) 390545
> sMail | R.-Breitscheid-Str. 43, 07747 Jena, Germany
>
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>
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