# [sdiy] Fixed filter bank questions

David G Dixon dixon at mail.ubc.ca
Wed May 8 23:04:12 CEST 2019

```Hello SDIY Team!

I'm working on a fixed filter bank design, and I have a few "philosophical"

So, I decided the other day that I wanted to build myself a fixed filter
bank.  The first thing I did was to look at the YuSynth design on the
internet.  I see that he is using pairs of multiple-feedback filters
(exactly as described on pages 150-154 of Don Lancaster's beautiful little
book, "Active-Filter Cookbook" which I have sitting on my lap as I type
this).  That was exactly what I was going to do as well.  Yves says that his
filter sections have a Q of 3.7, and a gain of 1.14.  Actually, his single
sections have a Q of 2.66, and a gain of 1.07.  The overall gain is simply
the square of the individual gains, which is indeed 1.14.  The overall Q is
something that I haven't calculated (I'm presuming I need to derive the
overall transfer function, and I just haven't bothered).  Also, his filter
center frequencies conform to, I'm guessing, the same ones that Moog used in
his 914 or whatever (LP at 88, 125, 175, 250, 350, 500, 750, 1000, 1400,
2000, 2800, 4000, 5600, and HP at 7000 Hz).

So, here is what I'm proposing for my filterbank:

Q of each section of pi (3.14).  This simplifies the math somewhat.

Unity gain across each filter section.

Frequencies every half-octave around 1000 Hz (LP at 88, 125, 177, 250, 354,
500, 707, 1000, 1414, 2000, 2828, 4000, 5657, HP at 8000 Hz)

FIRST QUESTION:  Is there anything wrong with any of these concepts?  I like
the more equally spaced frequencies because I think it will create a
smoother response, plus it just makes more sense mathematically.  (If I'm
doing half octaves, would it be better to do them around 440 Hz rather than
1000 Hz?)  Also, I can beef up the gains of the final output amps a little
bit if unity gain at each filter seems a little anemic.

The multiple-feedback filter requires two capacitors of equal values and
three resistors.  Resistors R1A and R1B form a voltage divider, and their
parallel (Thevenin) resistance is R1.  Resistor R2 is a feedback resistor.
Based on my analysis, once the cap values are chosen, the resistors may be
sized based on the following formulae:

R = 1/(2*pi*F*C)

R2 = R*2Q

R1A = R*Q/G

R2B = R*Q/(2Q^2 - G)

So, as an example, let's take gain G = 1, Q = pi, F = 1000 Hz, and C = 10nF.
Using the formula, R = 15.9k.  Hence, R2 = 100k, R1A = 50k (49.9k) and R2B =
2.67k.  Easy peasy.  Every filter frequency will require different
resistors, but they are easy to calculate.

Yves did it a different way.  He used the same three resistor values in
every filter (R2 = 47k, R1A = 22k, and R1B = 1.8k), and used capacitors in
parallel to find two capacitors which add up to the correct capacitor value
for each frequency.  This means that his resistor choices are very
convenient, but each dual filter requires 8 capacitors at two different
sizes, rather than just 4 capacitors of a single size.  He has restricted
himself to the six most common standard capacitor mantissas (10, 15, 22, 33,
47, 68) in selecting these sizes.  The frequency errors he gets (ignoring
capacitor tolerances) are as high as 3.7%, with an average of 1.4%.

I am proposing to use single standard capacitor values for each filter
(albeit different for each filter, except for two which are the same) - 68,
47, 33, 22, 15, 10, 10, 6.8, 4.7, 3.3, 2.2, 1.5n - but then find the 1%
resistor values closest to the calculated values for each filter.  By this
method, again ignoring capacitor tolerances, my largest frequency error is
only 0.47%, and the average error is only 0.17%.  By using those specific
capacitor values, the values of the resistors are all fairly close, and the
values of R2 are all between 100k and 141k.  This is important, because the
standard 1% resistor value mantissas are more closely spaced in this range,
so it is easier to find resistor values which are very close to the
calculated values.  Of course, the inconvenient aspect of this design is
having to buy a bunch of strange resistor values.  However, I buy oddball
resistor values from Digikey all the time, and if you buy 100 or more, you
pay typically about 1.7 cents per resistor (for Stackpole or Yageo 1/4W
through-hole resistors).  Also, resistors are much cheaper than capacitors,
so cutting the total number of capacitors in half for this build is a
significant cost (and PCB real estate) savings.  So.

SECOND QUESTION:  Is there something fundamentally less desirable about
doing it my way than doing it Yves' way?  Does the minor inconvenience of
having to order and use about 30 different 1% resistor values somehow
outweigh the financial burden of installing twice as many poly film
capacitors as are actually needed in the design?  Also, my PCB will be
smaller and the layout will be simpler.  Note that I may still decide to
decrease my number of capacitor values from 11 to 5 or 6 - just to decrease
the total number of different values I need to buy, plus I will probably
hand-select these capacitors to get the values as close as possible to the
targets.