# damping in biquads for butterworth filter?

Started by April 8, 2009
```
Hi, I've been working on trying to implement a 4rth order butterworth from
2 biquads.  I know the coefficients for these two filters will not be the
same because then I get Linkwitz-Riley which I do not want.

I looked through some old threads and someone pointed out that the Fc of
each of the two necessary biquads will be the same, however the Q factor of
each will be different.  Someone else has told me this as well, but how can
this be since the pole locations are different for each filter(because each
biquad has one complex conjugate pair)?  Also, if this is true, then the
a0,a2 coefficients of each biquad should be the only ones that change in
each filter, since they seem to be the only ones that depend on the Q
factor, but in going through the math so far, this doesn't seem to be the
case either?

Obviously I am going astray somewhere but can someone point out where I am
in error?

Thanks,
Ethan

```
```*update*
Okay, I found my mistake in the math, so I can see now that a0 and a2 are
indeed the only coefficients that change.  Which seems to confirm what
others have said, namely that the fc will be the same for each filter and
only the Q will differ.

Is there an easier way to do this?  My design procedure was:

1)figure out what the pre-warped pole locations in the s-domain are
2)take each complex conjugate pole pair to create two second order
sections
3)apply BLT to each biquad, with results that the filters are the same
except for the Q factor.

Since the Q factor is the only thing that changes, is there a way to
figure out what the Q for each of these biquads will need to be to get a
fourth-order butterworth, without going through all the tedium described
above????

Thanks again,
Ethan
```
```Oh, what a lazy idiot.

Enjoy:

Q0 = 0.541196
Q1 = 1.306563

VLV

esfield wrote:

> *update*
> Okay, I found my mistake in the math, so I can see now that a0 and a2 are
> indeed the only coefficients that change.  Which seems to confirm what
> others have said, namely that the fc will be the same for each filter and
> only the Q will differ.
>
> Is there an easier way to do this?  My design procedure was:
>
> 1)figure out what the pre-warped pole locations in the s-domain are
> 2)take each complex conjugate pole pair to create two second order
> sections
> 3)apply BLT to each biquad, with results that the filters are the same
> except for the Q factor.
>
> Since the Q factor is the only thing that changes, is there a way to
> figure out what the Q for each of these biquads will need to be to get a
> fourth-order butterworth, without going through all the tedium described
> above????
>
> Thanks again,
> Ethan
```
```Thanks for the help, Vladimir!

Wow.

So all I had to do was find values of Q such that when multiplied
together, they yeild 1/sqrt(2)??????

It seems so obvious now.

So painfully, painfully obvious.

Ethan
```
```esfield wrote:
> Thanks for the help, Vladimir!
>
> Wow.
>
> So all I had to do was find values of Q such that when multiplied
> together, they yeild 1/sqrt(2)??????

There an infinite number of such Q pairs. Vlad cave you the only ones
that work right.

> It seems so obvious now.
>
>
> So painfully, painfully obvious.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```Yeah, I just realized that.... so what are the other constraints on these
values that force them to be the ones Vlad gave?

Thanks, Ethan
```
```esfield wrote:
> Yeah, I just realized that.... so what are the other constraints on these
> values that force them to be the ones Vlad gave?

Math. All low-pass Butterworth filters have all derivatives zero at f=0.
That gives a 4th-order filter the response 1 + x^4 where x = w/w0. Solve
for the roots.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```On Apr 8, 12:34&#2013266080;pm, "esfield" <ethanstubblefi...@hotmail.com> wrote:
> *update*
> Okay, I found my mistake in the math, so I can see now that a0 and a2 are
> indeed the only coefficients that change. &#2013266080;Which seems to confirm what
> others have said, namely that the fc will be the same for each filter and
> only the Q will differ.
>
> Is there an easier way to do this? &#2013266080;My design procedure was:
>
> 1)figure out what the pre-warped pole locations in the s-domain are
> 2)take each complex conjugate pole pair to create two second order
> sections
> 3)apply BLT to each biquad, with results that the filters are the same
> except for the Q factor.
>
> Since the Q factor is the only thing that changes, is there a way to
> figure out what the Q for each of these biquads will need to be to get a
> fourth-order butterworth, without going through all the tedium described
> above????
>
> Thanks again,
> Ethan

Ethan,

You may like this:

http://www.claysturner.com/dsp/Butterworth%20Filter%20Formulae.pdf

Clay

```
```Thanks again Jerry.  Please bear with me a bit longer!

That gives pole locations at sqrt(j), sqrt(-j), -1/sqrt(j), -1/sqrt(-j).

But, aren't these the pole locations for a second order butterworth?
There are four poles but we only use the left half plane, so this only
gives two poles so this is a second order filter, right?

Also, it's not obvious to me how the Q values come from the pole
locations.  Can you please enlighten me?

Clay, thanks for the link.  I wish I'd had that reference when I started!
Just a quick question: to transform the lpf into hpf, all I do is change
the '1' in each numerator to 's^2', right?
```
```On Apr 8, 3:33&#2013266080;pm, "esfield" <ethanstubblefi...@hotmail.com> wrote:
> Thanks again Jerry. &#2013266080;Please bear with me a bit longer!
>
> That gives pole locations at sqrt(j), sqrt(-j), -1/sqrt(j), -1/sqrt(-j).
>
> But, aren't these the pole locations for a second order butterworth?
> There are four poles but we only use the left half plane, so this only
> gives two poles so this is a second order filter, right?
>
> Also, it's not obvious to me how the Q values come from the pole
> locations. &#2013266080;Can you please enlighten me?
>
> Clay, thanks for the link. &#2013266080;I wish I'd had that reference when I started!
> Just a quick question: to transform the lpf into hpf, all I do is change
> the '1' in each numerator to 's^2', right?

I look at it as replacing each "s" by a 1/s - that inverts the
frewquency response.

For example

1/(1+s)  -> transforms to  1/(1+1/s) now mult top and bot by s to
clean up the expresion and you get s/(s+1) which is now your 1st order
Butterworth highpass filter.

IHTH,

Clay

```