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Filter pole question

Posted: Wed Jun 25, 2008 7:07 am
by Gorskytron
What does the "pole" of a filter mean?
Such as when a filter is referred to as having a two pole or a four pole.
I understand what dB roll off means but I'm a little unclear on what a pole represents.

Posted: Wed Jun 25, 2008 7:19 am
by Stab Frenzy
1 pole = 6dB/oct rolloff

ie 2 pole = 12dB/oct etc.

I don't know where the term originates, maybe someone will enlighten us. I have a hunch it might be related to the number of inductors used in passive filter design, but that might be wildly wrong.

Posted: Wed Jun 25, 2008 11:14 am
by Mooger5
Stab Frenzy wrote:1 pole = 6dB/oct rolloff

ie 2 pole = 12dB/oct etc.

I don't know where the term originates, maybe someone will enlighten us. I have a hunch it might be related to the number of inductors used in passive filter design, but that might be wildly wrong.
Here´s what I learned so far:

I think the term has a specific origin much like the term "bug" originates from a real bug that got trapped inside a relay back when computers were mostly electro-mechanical.

But yes, the term "pole" refers to one filter section. For a low-pass either one resistor in series to the signal path plus a capacitor to ground or an inductor in series plus resistor to ground are used to attenuate the signal by 6 dB/oct. This is one pole (for high-pass the components positions in the path are reversed). A second pole in series, identical to the first one, will take on the previous rolloff and attenuate the remaining signal by another 6 dB/oct and so on. For a Moog-type 24 dB/oct attenuation, 4 poles in series are needed.

BTW...

To constantly alter the cuttoff frequency one of the components must be variable, and the easiest is the resistance as potentiometers are more available than variable capacitors or inductors of adequate value.
So, since in a four-pole filter there are four resistors, a four-gang potentiometer is required. Add a buffer to the output to compensate for the changes in impedance and there you have a manually controllable four-pole filter.
To make it controllable by several external sources you use voltage-control. Oberheim, Roland and the SSM2040 filters all use the same configuration of operational transconductance amplifiers (OTAs) acting as variable resistors.
These weren´t available in 1966 so Bob Moog used the well-known transistor base-to-emitter inherent resistance. Nobody had thought of it before since this particularity in transistors was regarded as something to avoid rather than an active part in a design. This technique added some distortion to the audio signal so it was impractical in hi-fi terms but in the end it was responsible for the filter´s and the instrument´s musical character. Today it´s known as "Phatness"...

Posted: Wed Jun 25, 2008 12:55 pm
by clubbedtodeath
I'd guess that pole comes from pole point in mathematics.

Posted: Wed Jun 25, 2008 3:34 pm
by GeneralBigbag
That sounds right - pole from complex analysis - not that I know a whole lot about filter implementation, but if you write the transfer function of a filter down, it's of the form p(z)/q(z), (p,q polynomials taking complex arguments z) and the 'poles' are the zeros of the q function - the higher order q is, the more zeros or poles it has.
That said, if you showed me a circuit diagram I would have *no* idea :)

Posted: Wed Jun 25, 2008 4:07 pm
by nathanscribe
Mooger5 wrote:since in a four-pole filter there are four resistors, a four-gang potentiometer is required. Add a buffer to the output to compensate for the changes in impedance and there you have a manually controllable four-pole filter.
To make it controllable by several external sources you use voltage-control.
I'd half-agree with this. You could use a 4-gang pot, but they're not commonly available in more than 2-gang. It's easy to make a 2-pole filter this way, but as good a reason as any for voltage control is to get not just external control, but higher order than 2-pole operation.

Loads of synths use OTAs, they're common and cheap. Transistor ladders are the classic Moog-style ones, diode ladders are around but less common, and Steiner's multi-input filter is also an occasional one. What I like about the modern modulars (and some other synths) is the availability of different filter types. Lowpass gets a bit samey after a while. Nice to have some HP/BP/Notch floating around.

Posted: Wed Jun 25, 2008 7:08 pm
by solderguy
http://electronotes.netfirms.com/free.html


The best info around concerning poles ( and zeroes ) and synths!

Posted: Wed Jun 25, 2008 7:12 pm
by Mooger5
I'd half-agree with this. You could use a 4-gang pot, but they're not commonly available in more than 2-gang. It's easy to make a 2-pole filter this way, but as good a reason as any for voltage control is to get not just external control, but higher order than 2-pole operation.


Alps make 4 and 6-gang potentiometers. They´re used in high-quality Home-Theatre preamps. You can get them easily from Ebay and they´re not so expensive as one might think.
http://cgi.ebay.ie/ALPS-Quad-Potentiome ... m153.l1262
The motorized models even mean they´re voltage-controlled, but the motor wouldn´t be fast enough for synthesizer use.


I think there were full-range filters available in the early days (just not sure about the steepness of the curve), and the difficulty was precisely in making them voltage-controlled. See Till Kopper´s comments about the Moog filter patent here: http://www.till.com/articles/moog/patents.html

Posted: Wed Jun 25, 2008 7:17 pm
by Mooger5
solderguy wrote:http://electronotes.netfirms.com/free.html


The best info around concerning poles ( and zeroes ) and synths!
I was looking for the Electronotes link for ages. Thanks!

Posted: Wed Jun 25, 2008 7:50 pm
by Bitexion
Basically, you can say a 4-pole filter sounds "softer" than a 6dB filter, because the 4-pole has a steeper rolloff curve. A 6dB filter will sound a bit brighter over the whole range. That's basically why the Moog filter always sounds "warm". It's 4-poles.

Posted: Wed Jun 25, 2008 9:00 pm
by nathanscribe
Nice pots, Mooger, but they're a bit OTT for a synth panel 8)

Posted: Wed Jun 25, 2008 9:42 pm
by Solderman
Don't forget the MF-101 and Voyager filters have 2 pole options. I find it gives a more aggressive character. Interesting summary Mooger5. I wonder how it is that the component reversal for multi-mode filters is done without any extra circuits, just alternate wiring and some switching.

Posted: Wed Jun 25, 2008 10:56 pm
by Mooger5
Solderman wrote:Don't forget the MF-101 and Voyager filters have 2 pole options. I find it gives a more aggressive character. Interesting summary Mooger5. I wonder how it is that the component reversal for multi-mode filters is done without any extra circuits, just alternate wiring and some switching.
Yes, 2-pole filters sound buzzier as higher harmonics are less rolled-off per octave.

I only knew about this pole stuff until very recently, since I started researching for an alternative to the 80017A filters in my Juno 106.
The Moog filter requires more discrete components, but it´s not impossible to implement a Memorymoog clone.
An OTA-based 2-pole (LM13700) requires few adjacent components but I think wouldn´t be appropriate to the already thin sound of the Juno´s oscillators...
The SSM2044 plus CA3080 VCA combination is perfect. I made a prototype and it sounds very moogish, only requiring a little more tweaking. When I have the time I´ll finish this project.

The multi-mode operation in the Oberheim filter (and the Steiner-Parker) is still out of my understanding, but for now I find it ingenious since it appears so deceptively simple. Instead of a switch I think a linear pot could be used to continuously select the mode. The wiper would go for the BP tap and the other two pins for LP and HP taps...
I´ve read in the Moog Music forum that a multi-mode 4-pole filter is immensely difficult to create, hence the 2-pole in the Obie.

After I read Kevin Lightner´s "Why a Moog Sounds Like a Moog" article I modded my Rogue to run on 10 Volts like the Mini, and yes it sounds a little more fatter. But curiously (or predictably) I must have miscalculated some resistor values as the filter now has a limited upper-bandwith. My Rogue must be the only Moog synthesizer in the world whose filter has the ARP error. I´ll have to revise the circuit . :lol:

Posted: Wed Jun 25, 2008 11:21 pm
by Bitexion
You can read lots about the filter theory, why 1 pole equals 3dB and rolloff curves around the net.

soundonsound.com has a series of articles called Synth Secrets. One chapter goes deeply into filter theory.

Here's a good wikipedia article:

http://en.wikipedia.org/wiki/Low-pass_filter

Posted: Wed Jun 25, 2008 11:28 pm
by Analog Freak
GeneralBigbag wrote:That sounds right - pole from complex analysis - not that I know a whole lot about filter implementation, but if you write the transfer function of a filter down, it's of the form p(z)/q(z), (p,q polynomials taking complex arguments z) and the 'poles' are the zeros of the q function - the higher order q is, the more zeros or poles it has.
That said, if you showed me a circuit diagram I would have *no* idea :)
You're somewhat on the right track, however, your equation is in the z domain which is for discreet time systems and we're discussing analog systems. You need to be in the s-domain young grasshoppa'. :) Poles are where the transfer function "blows up" and tries to head towards infinity. Zeroes are where the equation will not produce any output. Poles opccur when the denominator equals zero. In other words a divide by zero error, which is mathematically undefined but practically, it is infinity. Zeroes are when the numerator is equal to zero. Obviously zero over anything is zero. I've had to do quite a bit of messing around with FIR and IIR filter design for a class in Digital Signal Processing that I just finished and this stuff get really messy when you get into the higher order systems. 67th order polynomials anyone? We had to do this sort of thing all the time. Anyway, the number of poles determines the order of the transfer function and hence the order of the filter. A two pole filter will have two poles in the transfer function and the equation will be second order. A four pole filter will have four poles and a fourth order transfer equation. Generally speaking, when you're working with an analog filter, a pole is a capacitor. This is because a capacitor blocks DC and as frequency increases reactance (or impedance if you prefer) drops until it is practically zero. When this happens the capacitor is seen as a short and maximum current flows in the circuit. This is only for R-C filters though, which are the most commonly used in small-signal applications such as synthesizers. The short answer to this question is that the term pole comes from (so called) higher mathematics.

As an example here is the simplification process for a fifth order transfer equation in the s (analog complex frequency) domain: Image
Nasty isn't it?