Audio Shelving fliters

Hi all!

I'd like to implement 1st and 2nd order shelving filters - which are
generally called 6 db/oct and 12 db/oct filters.

I had a look at the 'Audio EQ Cookbook', but it doesn't say a word
about 1st order filters, and for 2nd order filters it's not clear to me
how the slope parameter S (S <= 1) relates to the filter slope
expressed in db/oct. Matlab simulations showed that RBJ's second order
filter could be a 12dB/oct when S is set to 1 - but in order to achieve
such a rolloff, the amount of boost or cut must be significant (about
30dB) so i keep doubting (Sorry RBJ your document is just great - by
the way thanks for making it available online - it's just that i don't
know how to use it).

I would greatly appreciate if someone could point a useful document -
common analog transfer functions ready for a bilinear transformation or
- even better! - digital transfer functions (yeah i'm a lazy bum ;) )

Thanks! Gus

gusflit@gmx.net wrote:
> Hi all!
>
> I'd like to implement 1st and 2nd order shelving filters - which are
> generally called 6 db/oct and 12 db/oct filters.
>
> I had a look at the 'Audio EQ Cookbook', but it doesn't say a word
> about 1st order filters,

That's because first order shelving filters don't exist.

> and for 2nd order filters it's not clear to me
> how the slope parameter S (S <= 1) relates to the filter slope
> expressed in db/oct.

S is similar to the Q in lowpass filters. Around the 3dB-point, the Q
of lowpass filters significantly affects the slope of the filter. It
reaches the 12dB/oct roll-off only further away from the 3dB frequency.
Similarly, the slope of shelving filters is affected by the S
parameter, even though the order remains constant.

> Matlab simulations showed that RBJ's second order
> filter could be a 12dB/oct when S is set to 1 - but in order to achieve
> such a rolloff, the amount of boost or cut must be significant (about
> 30dB) so i keep doubting (Sorry RBJ your document is just great - by
> the way thanks for making it available online - it's just that i don't
> know how to use it).

Perhaps you don't need a shelving filter at all? Shelving means that at
DC and Nyquist, the filter is flat, with a certain gain difference
between the two flat sections. The slope of the transition band is
controlled with the S parameter, and can be much steeper than 12dB /
oct.

>
> I would greatly appreciate if someone could point a useful document -
> common analog transfer functions ready for a bilinear transformation or
> - even better! - digital transfer functions (yeah i'm a lazy bum ;) )

The cookbook has digital transfer functions. 

Regards,
Andor

Andor a =E9crit :

> gusflit@gmx.net wrote:
> > Hi all!
> >
> > I'd like to implement 1st and 2nd order shelving filters - which are
> > generally called 6 db/oct and 12 db/oct filters.
> >
> > I had a look at the 'Audio EQ Cookbook', but it doesn't say a word
> > about 1st order filters,
>
> That's because first order shelving filters don't exist.
>
> > and for 2nd order filters it's not clear to me
> > how the slope parameter S (S <=3D 1) relates to the filter slope
> > expressed in db/oct.
>
> S is similar to the Q in lowpass filters. Around the 3dB-point, the Q
> of lowpass filters significantly affects the slope of the filter. It
> reaches the 12dB/oct roll-off only further away from the 3dB frequency.
> Similarly, the slope of shelving filters is affected by the S
> parameter, even though the order remains constant.
>
> > Matlab simulations showed that RBJ's second order
> > filter could be a 12dB/oct when S is set to 1 - but in order to achieve
> > such a rolloff, the amount of boost or cut must be significant (about
> > 30dB) so i keep doubting (Sorry RBJ your document is just great - by
> > the way thanks for making it available online - it's just that i don't
> > know how to use it).
>
> Perhaps you don't need a shelving filter at all? Shelving means that at
> DC and Nyquist, the filter is flat, with a certain gain difference
> between the two flat sections. The slope of the transition band is
> controlled with the S parameter, and can be much steeper than 12dB /
> oct.
>
> >
> > I would greatly appreciate if someone could point a useful document -
> > common analog transfer functions ready for a bilinear transformation or
> > - even better! - digital transfer functions (yeah i'm a lazy bum ;) )
>=20
> The cookbook has digital transfer functions.=20
>=20
> Regards,
> Andor

Andor wrote:
> gusflit@gmx.net wrote:
> > Hi all!
> >
> > I'd like to implement 1st and 2nd order shelving filters - which are
> > generally called 6 db/oct and 12 db/oct filters.
> >
> > I had a look at the 'Audio EQ Cookbook', but it doesn't say a word
> > about 1st order filters,
>
> That's because first order shelving filters don't exist.
>
it may be a matter of semantics..

a filter that rises a 6 dB per octave for say two octaves and then goes
flat again I might call a first order shelving filter...

To the OP, a shelving filter is one that has a region of slope but
ultimatly goes back to a flat response.  i.e it may boost all the bass
freqs from <20 Hz to 200 Hz by say 10 dB , then slope down at about 6
dB per octave from 200 Hz to 1 kHz then be flat > 1 kHz...  i.e the
response looks like a shelf...

Mark

Andor wrote:


>>I had a look at the 'Audio EQ Cookbook', but it doesn't say a word
>>about 1st order filters,
> 
> 
> That's because first order shelving filters don't exist.
> 

??????

A standard bass/treble control is the example of the 1st order shelving 
filter. It can be implemented using just one biquad section.

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

Ok i thought i was asking something easy, but looks like it's not. I'll
try to be clearer.

A first order iir filter uses 1 delay element in the feedback loop. A
second order uses 2 of them. That's basically what i meant.
Theoretically, if you want flat regions with no dip/oscillation, a 1st
order filter will yield a max 6dB/oct meanwhile a 2nd will provide
12dB/oct.

1st order shelving filters are available. For example, in Regalia/Mitra
nice approach, if you take a first order lowpass filter (transfer
function Hlp(z) ) and fill in the stopband with the complementary
highpass filter Hhp(z) by a given amount:

G(z) =3D Hlp(z) + K*Hhp(z)

you get a first order High shelving filter G(z) with boost or cut equal
to 20*log10(K).

The problem with Regalia/Mitra cooking recipe is that a pair of filters
with opposite gains (in dB) are not magnitude-complementary - that's
why i'm looking for an other approach.

Maybe that repeating Regalia/Mitra strategy with standard Butterworth
filters would yield to what i'm looking for, but as i said, i'm a lazy
bum - digital shelving filters are ubiquitous and there's no challenge
in deriving their transfer functions from scrap.

Thanks for your input guys!

Gus

Mark a =E9crit :

> Andor wrote:
> > gusflit@gmx.net wrote:
> > > Hi all!
> > >
> > > I'd like to implement 1st and 2nd order shelving filters - which are
> > > generally called 6 db/oct and 12 db/oct filters.
> > >
> > > I had a look at the 'Audio EQ Cookbook', but it doesn't say a word
> > > about 1st order filters,
> >
> > That's because first order shelving filters don't exist.
> >
> it may be a matter of semantics..
>
> a filter that rises a 6 dB per octave for say two octaves and then goes
> flat again I might call a first order shelving filter...
>
> To the OP, a shelving filter is one that has a region of slope but
> ultimatly goes back to a flat response.  i.e it may boost all the bass
> freqs from <20 Hz to 200 Hz by say 10 dB , then slope down at about 6
> dB per octave from 200 Hz to 1 kHz then be flat > 1 kHz...  i.e the
> response looks like a shelf...
>=20
> Mark

Vladimir,

A biquad is a second order section.

Gus

Vladimir Vassilevsky a =E9crit :

> Andor wrote:
>
>
> >>I had a look at the 'Audio EQ Cookbook', but it doesn't say a word
> >>about 1st order filters,
> >
> >
> > That's because first order shelving filters don't exist.
> >
>
> ??????
>
> A standard bass/treble control is the example of the 1st order shelving
> filter. It can be implemented using just one biquad section.
>
> Vladimir Vassilevsky
>=20
> DSP and Mixed Signal Design Consultant
>=20
> http://www.abvolt.com

gusflit@gmx.net wrote:
> Vladimir,
>=20
> A biquad is a second order section.
>=20

You have TWO 1-st order shelf filters combined in one biquad.
Separate controls for bass and for treble.
Is that understood?



>=20
> Vladimir Vassilevsky a =E9crit :
>=20
>=20
>>Andor wrote:
>>
>>
>>
>>>>I had a look at the 'Audio EQ Cookbook', but it doesn't say a word
>>>>about 1st order filters,
>>>
>>>
>>>That's because first order shelving filters don't exist.
>>>
>>
>>??????
>>
>>A standard bass/treble control is the example of the 1st order shelving=

>>filter. It can be implemented using just one biquad section.
>>
>>Vladimir Vassilevsky
>>
>>DSP and Mixed Signal Design Consultant
>>
>>http://www.abvolt.com
>=20
>=20

Vladimir Vassilevsky a =E9crit :

> gusflit@gmx.net wrote:
> > Vladimir,
> >
> > A biquad is a second order section.
> >
>
> You have TWO 1-st order shelf filters combined in one biquad.
> Separate controls for bass and for treble.
> Is that understood?
>

Ok, that was indeed a misunderstanding because i had in mind separate
implementations for hi and lo shelves. Sorry.

Gus wrote:

> 1st order shelving filters are available. For example, in Regalia/Mitra
> nice approach, if you take a first order lowpass filter (transfer
> function Hlp(z) ) and fill in the stopband with the complementary
> highpass filter Hhp(z) by a given amount:
>
> G(z) = Hlp(z) + K*Hhp(z)
>
> you get a first order High shelving filter G(z) with boost or cut equal
> to 20*log10(K).

That's correct. Where was my head.

>
> The problem with Regalia/Mitra cooking recipe is that a pair of filters
> with opposite gains (in dB) are not magnitude-complementary - that's
> why i'm looking for an other approach.

Perhaps wave filters give you what you want:

journals.tubitak.gov.tr/elektrik/issues/elk-03-11-3/elk-11-3-3-0302-1.pdf

I still don't understand why you can't user r b-j's shelving filter ...
?

Regards,
Andor