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[zmix]

I have been searching the net for an explaination of the term
used to describe the Sony Oxford EQ's HF response: "Decramped".

To what does this refer?

I am assuming that it has something to do with aliasing
or possibly rounding errors near the nyquist limit.

[bruno putzeys]

We might need to wait for Paul to chime in to hear the answer, but keep in mind that the wording on the knobs of effects gear doesn't usually accurately describe the process behind it.

[zmix]

Bruno,

It's not a parameter, but it is mentioned in the Oxford sales brochure. It appeared to be a 'marketing' term. I could not find any standard use of the term 'Decramped" in any filter design literature.

I did eventually find some information which indicates that my original guess was correct. "Decramped" HF filters are kept symmetrical around F_c as F_c approaches the nyquist frequency.  
It would seem then that the filters would increase their Q as the frequency is increased.

I wonder how they sound?

[bruno putzeys]

Yezzzzz now I know what it is. I remember asking Paul once if he knew anyone making digital EQ's that compensated for the response alias and him replying that he actually did so. If you make a direct translation from an analogue EQ to a digital one (directly transform poles and zeros), the response ends up aliased, so anything happening above fs/2 has a mirror below fs/2. This is because like analogue filters poles and zeros come in conjugate pairs but unlike analogue the frequency axis wraps round in a circle so the pairs meet eachother at fs/2 (z=-1). If you dial a bell at 20kHz, its mirror image at 24.1kHz spills over into it, distorting the bell shape (I think there's an EQ thread mentioning this). So, you can't reproduce the exact same filter using the exact same number of poles - you need to cancel the aliased response. It's trickier than it sounds because the derivative of the frequency response becomes discontinuous (the curve bends sharply as it it mirrored) at fs/2. Ideally that requires an infinite number of p/z to do, but of course for all practical intents and purposes a few extra will do nicely.

How does it sound: much more like an analogue EQ. Other digital EQ's resort to upsampling to obtain this effect, requiring much more processing power, adding latency and modifying the frequency response even when flat.

[Andy Peters]

zmix wrote on Mon, 09 April 2007 10:02

I have been searching the net for an explaination of the term used to describe the Sony Oxford EQ's HF response: "Decramped". To what does this refer?

Makes you sound less like Lux Interior?  Removes the itch from Poison Ivy?

-a

[Axon]

Bruno Putzeys wrote on Tue, 10 April 2007 13:14

Yezzzzz now I know what it is. I remember asking Paul once if he knew anyone making digital EQ's that compensated for the response alias and him replying that he actually did so. If you make a direct translation from an analogue EQ to a digital one (directly transform poles and zeros), the response ends up aliased, so anything happening above fs/2 has a mirror below fs/2. This is because like analogue filters poles and zeros come in conjugate pairs but unlike analogue the frequency axis wraps round in a circle so the pairs meet eachother at fs/2 (z=-1). If you dial a bell at 20kHz, its mirror image at 24.1kHz spills over into it, distorting the bell shape (I think there's an EQ thread mentioning this). So, you can't reproduce the exact same filter using the exact same number of poles - you need to cancel the aliased response. It's trickier than it sounds because the derivative of the frequency response becomes discontinuous (the curve bends sharply as it it mirrored) at fs/2. Ideally that requires an infinite number of p/z to do, but of course for all practical intents and purposes a few extra will do nicely.

Is this related to the bilinear transform being a first-order approximation? ie, prewarping is supposed to correct frequency response compression in terms of the critical frequencies, but it does nothing for the rest of the behavior?

[bruno putzeys]

Yes. In fact, the problem is independent of the transformation method used. Each of the classic ones (bilinear, matched z, impulse invariant etc) will manage to retain some particular aspect of the filter response but never all of them. One could say that PF invented a new transform method that adds as many poles and zeros as necessary to get a better match.

[Jim Williams]

I would love to see a phase vs frequency plot of this. I would imagine it would look a bit funny around 20k, possibly creating some non linear phase shift or deviation from linear phase.
Such is the problem of EQ curves running into the brick wall.
When analog EQ/filters are run with a 200k bandwidth, the phase response at 20k is quite good, usually within a couple of degrees and a very linear phase curve.

[bruno putzeys]

The maximum phase error using uncorrected EQ's amounts to a maximum of 1 sample. Which is 180

[garret]

I believe Sonoris eq in SAWStudio does something to avoid the eq response problems at nyquist.

http://www.sonoris.nl/en/sneq.php

The free NyquistEq also does something similar...

http://magnus.smartelectronix.com/#NyquistEq

[Axon]

EDIT: I thought what garretg was describing was prewarping, but on second thought, it's not.

Fairly off-topic comment: So I'm more than a little interested in this, because I've been trying to get an IIR RIAA EQ up and running in my spare time, and no matter how I prewarp or upsample, I can't do better than about 0.5db of deviation from the analog filter. I haven't discounted me possibly just being an idiot and writing the filter wrong, but I'm also considering that perhaps direct application of the bilinear transform is just not good enough. I've tried poking around with other transform methods (notably trying to derive a second-order bilinear transform myself... not so successful).

I haven't hit any reference more advanced than my filters textbook yet. Paul's technique wouldn't happen to be published, would it? Otherwise what is the usual literature I should consult on more accurate transform techniques beyond bilinear+prewarping?

[dekkersj]

I would say that oversampling will help. The bilinear transform is only valid for frequencies far off Nyquist. But I am not 100 % positive.

Regards,
Jacco

[bruno putzeys]

I hadn't been looking at it much but I must say I'm a bit baffled to discover that people are using the bilinear transform to build digital parametric EQ's.

It helps to understand what the bilinear transform is for and what it is not. If you have, say, an analogue low-pass or high-pass filter, the bilinear transform will insure that the relative positioning of the poles and zeros produces the same pass-band behaviour. The entire filter response, from DC to infinity, is mapped to the band from DC to fs/2. This is clearly a non-linear mapping.
If you start with a maximally flat butterworth response, the transformed filter will be maximally flat too. The rolloff will be infinitely compressed near fs/2 though. Of course the passband is also frequency-warped, but you don't see that because it's flat... It does become visible if you make a chebychev filter. The transformed filter will have the same ripple amplitude, but the bumps and troughs will be at different frequencies compared to the analogue original.
The only thing pre-warping does is insure that the corner frequency lands where it should. So only the corner frequency is pre-warped!

This makes the bilinear transform quite unsuitable for designing filters that are supposed to modify the entire frequency response (e.g. speaker response correction), because where do you put the corner? Same for an RIAA correction. It has several corners.
There, the direct transform z=log(s/fs) produces markedly better results. The bumps and troughs land at exactly the right frequencies, but as poles and zeros approach fs/2 the amplitude errors grow larger and larger. If you really need to have it as close as you can, I'd suggest a numerical optimisation routine (maybe someone else knows a closed-form alternative but it certainly isn't common knowledge).

[dekkersj]

Right, I understood something similar from my collegeas who are busy in the digital filter design. The bilinear transform is very usefull (and used quite frequently), but only for low frequencies. Let us say fs/10 or so. One can calculate that, by the way.

Another approach is to correct for the anomalies and may be that is the decramping. But I would suggest to oversample a factor of at least 10. Do the prewarp and the bilinear transform after that.

Regards,
Jacco

[bruno putzeys]

The idea of "decramping" is an alternative transform method that (I suspect at the expense of a few extra poles and zeros) is more exact than either direct or bilinear transform, thus rendering oversampling utterly unnecessary. In terms of processing power it is a tremendous reduction compared to upsampling and is obviously the more elegant and effective solution.

<RANT>Using oversampling to get around the issue is like the old hammer conundrum: if all you have in your toolbox is a hammer, every problem starts looking like a nail. As the apparent use of the bilinear transform in some parametric EQ's illustrates, many people leave school knowing (but not understanding) certain basic blocks and spend their lives trying to construct everything out of those. This is how people "invent" solutions like using oversampling just to produce a filter response that the other standard building blocks don't happen to yield readily. The reason why schools don't just show the building blocks but also give the mathematical proof behind them is to show how anyone can formulate their own solutions to their specific problems in the same way that the schoolbook building blocks were once cooked up to address particular problems people were confronted with then. But of course, if schools were subsequently to examine students' ability to design something new instead of reproducing and applying something old, not many diplomas would be handed out.</RANT>