Butterworth, Bessel, Linkwitz-Riley... What's the difference, & what's your favorite?

[Bricolo]

I've got some doubts, too


Sallen Key filters are sometibes described with this transfer function:

1/(1+j*1/Q*(w/wc)-(w/wc)²)

with a butterworth filter's equation, I managed to find the same thing (and Q is 0.707, that seems correct)

but with a bessel, whos equation is
3/(3+3*j*(w/wc)+(w/wc)²) , it doesn't seem possible to have something like the first equation

neither with a chebyshev whos equation is

1/(1+e²(2*(w/wc)+1)²)




I tried to solve those equation all day long; now it's time to relax and listen a few cds

[Mr Evil]

Originally posted by Svante


Are you taking about second order filter only? Otherwise, what is your definition of Q?

I don't agree that Q is the usual way of describing the difference, I even think it is inappropriate.

Q = resonant magnification factor, i.e. the ratio of magnitude at resonance to that at infinty or zero (for high-pass and low-pass respecively).

You're right, defining the filter responses by Q only works well at 2nd order. With higher orders, assuming they are constructed from multiple 2nd order sections, each section may have a different Q. Although the Q of each section is still fixed for each filter response, it's no longer the simplest way of describing them.

[Bricolo]

from what I've seen, for orders>2, you decompose them into multiples of 2nd and 1st orders

the total Q is Q1*Q2*Q3*....
and must be equal to the Q you want (0.71 for butterworth)



Someone found a way to write a bessel or chebychev like this?

1/(1+j*1/Q*(w/wc)-(w/wc)²)

[Svante]

Ok, let's go through this.

-A first order filter is described by a f0 only. f0 is the frequency at which the asymtotes for very low and very high frequencies cross.

-A second order filter is described by A f0 and a Q. The definition of f0 is the same as for the first order filter. The definition of Q can be the amplitude at f0. Other definitions exist, but they give the same numerical value for a given filter.

-A third order filter is a first and a second order filter in series. This filter has two f0:s and one Q. This Q is not the amplitude at either f0, but the Q of the 2nd order section.

- A fourth order filter is two 2nd order filters in series. This filter has two f0:s and two Q:s. Neither Q is the amplitude at either f0, but the Q of the 2nd order sections.

-etc, etc...

The bottom line here is that many filters (almost all crossover filters) can be decomposed into a cascade of 1st and 2nd order filters. Each of these sections have a f0, and each 2nd order section also has a Q value. Trying to describe filters of orders >2 with a single Q value is doomed to fail, and will only add confusion. In the light of this:

All first order filters of a given f0 have identical transfer functions.

Filters of any order >1 are commonly described by a f0 (which may be different from those of the subsections) and a "family name", eg butterworth, bessel etc. The f0 can be defined in different ways, two common ways are by the -3 dB point, or the frequency at which the high and low frequency asymptotes cross. If the filter does not fit a well defined family, it is preferrably described by the f0:s and Q:s of all the subsections. Again, a single Q value will not tell the whole story except for the 2nd order filter.

Butterworth filters have sections with identical f0:s, but different Q:s. These filters are 3 dB down at the asymtotic f0.

Linkwitz-Riley filters are two cascaded butterworth filters and are therefore always of even order. These filters are 6 dB down at the asymtotic f0.

Bessel filters filters have cascaded sections with different F0:s and Q:s. These filters are typically more than 3 dB down at the asymtotic f0.

Tjebychev filters have cascaded sections with different F0:s and Q:s. These filters are designed to have a "ripple" of x dB in the pass band, and the f0 is usually defined as when the response goes below x dB. This f0 is lower than the asymtotic f0.

For crossover filters, Butterworth filters produce a flat sum if and only if the filter is odd order, Linkwitz-Riley filters produce a flat sum for even orders.

Bessel and Tjebychev filters do not produce a flat sum, so they are not principally interesting for crossover filters.

PS. When I speak of cascaded filter sections, I think of active filter sections, where there are no interaction effects between from loading the previous section.

HTH

[Mark25]

Thank-you for the useful informative post Svante.

[phase_accurate]

Bessel and Tjebychev filters do not produce a flat sum, so they are not principally interesting for crossover filters.

Just one addition:
2nd order Linkwitz is actually a 2nd order Bessel (with both branches having the same POLE frequency) with one driver wired out of phase.

Regards

Charles

[Ron E]

phase_Accurate wrote:

Just one addition:
2nd order Linkwitz is actually a 2nd order Bessel (with both branches having the same POLE frequency) with one driver wired out of phase.

-------------------------------------------------

I don't think this is accurate. Bessel is a 2nd order TF with a Q of 0.577 and it is essentially a compromise between Butterworth and LR2. It has a compromise peak in amplitude to get a lower dip in power response.

Butterworth Q=0.707 is -3dB at Fc and has a 3dB peak in the vector sum when both sections are at the same frequency, but has flat power response.

LR2 is a Q of 0.5 and is -6dB at Fc and sums flat, but has a dip of 3dB in power response.

People still argue whether on axis frequency or power response is important. There are pros and cons, but on axis frequency response is more important unless the environment is very reverberant.

Getting hung up on alignments is not very useful, though. What we want is a flat sum and good off-axis behavior. Whatever gets you to that target on budget is what works.

[Bricolo]

I think I just have one thing to add to Svante's post


Svante, you say that defining a >2 order filter with only one Q is wrong, but I'm not sure about that.
Imagine a 4th order L-R. It's 2 Butterworth in series
a LR's Q is 0.5
a Butterworth's Q is 0.707
0.707*0.707=0.5

so I think you can also put a Q=1 and Q=0.5 in series, that will end in a LR4


But this is yet to confirm

[Svante]

Originally posted by Ron E
phase_Accurate wrote:

Just one addition:
2nd order Linkwitz is actually a 2nd order Bessel (with both branches having the same POLE frequency) with one driver wired out of phase.

-------------------------------------------------

I don't think this is accurate. Bessel is a 2nd order TF with a Q of 0.577 and it is essentially a compromise between Butterworth and LR2. It has a compromise peak in amplitude to get a lower dip in power response.

Butterworth Q=0.707 is -3dB at Fc and has a 3dB peak in the vector sum when both sections are at the same frequency, but has flat power response.

LR2 is a Q of 0.5 and is -6dB at Fc and sums flat, but has a dip of 3dB in power response.

I was just about to say something similar. I think your Q's are right, Ron, but that you can say nothing about the power response in general. The power response will depend on the distance between the drivers, in relation to the wavelength at the crossover frequency. What you can say, though is that LR with swapped polarity on one driver is the only configuration that produces flat on-axis response (for 2nd order, given ideal drivers)

Originally posted by Bricolo
I think I just have one thing to add to Svante's post


Svante, you say that defining a >2 order filter with only one Q is wrong, but I'm not sure about that.
Imagine a 4th order L-R. It's 2 Butterworth in series
a LR's Q is 0.5
a Butterworth's Q is 0.707
0.707*0.707=0.5

so I think you can also put a Q=1 and Q=0.5 in series, that will end in a LR4


But this is yet to confirm

1+0.5 will not yield the same transfer function as 0.7+0.7. It is true that the amplitude at fc will be the same, but to the sides of fc, the responses will differ. The bottom line is, the definition of Q is not the level at fc, unless the filter is 2nd order. So 1+0.5 is not LR.

[Bricolo]

After looking at some exemple values; it seems you're right.
The total Q is still the same as the 2nd order equivalent, but there's a defined order for the intermediate Qs


LR is made of cascaded butterworth. But I've seen 2nd order LR, how are they made?