Derivations of transfer functions of common filters?

I've been reading sources that merely give the transfer functions of the filters that are presented, but give no information regarding how have these transfer functions been found.

So are there any sources that would explain how the transfer functions of some common filters have been found/derived? 

On Wed, 29 Jun 2016 02:58:52 -0700, Matti Viljamaa wrote:

> I've been reading sources that merely give the transfer functions of the
> filters that are presented, but give no information regarding how have
> these transfer functions been found.
> 
> So are there any sources that would explain how the transfer functions
> of some common filters have been found/derived?

Do you mean things like Butterworth, Chebychev, etc.?

-- 
Tim Wescott
Control systems, embedded software and circuit design
I'm looking for work!  See my website if you're interested
http://www.wescottdesign.com

On Wednesday, June 29, 2016 at 5:58:57 AM UTC-4, Matti Viljamaa wrote:
> I've been reading sources that merely give the transfer functions of the filters that are presented, but give no information regarding how have these transfer functions been found.
> 
> So are there any sources that would explain how the transfer functions of some common filters have been found/derived?

Does this help for something like the Butterwork filters?

http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/lecture-notes/MITRES_6_007S11_lec24.pdf

Clay

On Wed, 29 Jun 2016 02:58:52 -0700, Matti Viljamaa wrote:

> I've been reading sources that merely give the transfer functions of the
> filters that are presented, but give no information regarding how have
> these transfer functions been found.
> 
> So are there any sources that would explain how the transfer functions
> of some common filters have been found/derived?

I just did a search on "derivation of Butterworth filter" and then 
"derivation of Chebychev filter".  Both got me nice results.

What exactly are you looking for?

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

I'm looking for work -- see my website!

On 29.6.16 12:58, Matti Viljamaa wrote:
> I've been reading sources that merely give the transfer functions of the filters that are presented, but give no information regarding how have these transfer functions been found.
>
> So are there any sources that would explain how the transfer functions of some common filters have been found/derived?


Tim already had a good answer. It could be completed with
'Derivation of Bessel filters'.

-- 

-TV

On Wednesday, June 29, 2016 at 9:58:57 PM UTC+12, Matti Viljamaa wrote:
> I've been reading sources that merely give the transfer functions of the filters that are presented, but give no information regarding how have these transfer functions been found.
> 
> So are there any sources that would explain how the transfer functions of some common filters have been found/derived?

Numerous text books on the derivation of such filters. It usually revolves around a mathematical description of the magnitude squared of the desired transfer function squared. 

|H(f)|^2 = 1/(1+alpha^2 L^2(v))

where L(v) is an nth order polynomial and alpha is passband ripple. v is normalised frequency. We normally consider only the normalised low-pass filter case first and transform the solution for high-pass band pass etc.

L can be defined as a Butterworth polynomial or Chebychev polynomial and so on.
The formula is for magnitude squared so you need to separate the uncausal part from the causal. That is slightly tricky but the end reult is quite simple for at least Butterworth.

See Analog and Digital Signal Processing

Ashok Ambardar

Great book

On Wed, 29 Jun 2016 20:50:54 +0300, Tauno Voipio wrote:

> On 29.6.16 12:58, Matti Viljamaa wrote:
>> I've been reading sources that merely give the transfer functions of
>> the filters that are presented, but give no information regarding how
>> have these transfer functions been found.
>>
>> So are there any sources that would explain how the transfer functions
>> of some common filters have been found/derived?
> 
> 
> Tim already had a good answer. It could be completed with 'Derivation of
> Bessel filters'.

I left that as an exercise to the reader.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

I'm looking for work -- see my website!

On Wednesday, June 29, 2016 at 2:58:57 AM UTC-7, Matti Viljamaa wrote:
> I've been reading sources that merely give the transfer functions of the filters
>  that are presented, but give no information regarding how have these transfer
> functions been found.

As far as I know, it is the other way around.  You desire a transfer function, and then
design a filter to have that function.

> So are there any sources that would explain how the transfer functions of
> some common filters have been found/derived?

https://en.wikipedia.org/wiki/Butterworth_filter

gives a pretty good description of the history of the Butterworth filter, back to 1930.

As well as I know, though, component tolerance was worse in 1930, at least
for consumer grade devices.  Resistors with 20% tolerance might be more usual
than the 5% we usually find now, and 1% for good filters.

I have an RLC bridge with 0.01% resistors in its decade resistor section.

Once the goal is defined, such as maximal flat passband, you then solve the
equations and find the appropriate solution, then name the result.
(Unless someone else did it first.)

-- glen

On Thursday, June 30, 2016 at 2:09:39 AM UTC+3, herrman...@gmail.com wrote:
> On Wednesday, June 29, 2016 at 2:58:57 AM UTC-7, Matti Viljamaa wrote:
> > I've been reading sources that merely give the transfer functions of the filters
> >  that are presented, but give no information regarding how have these transfer
> > functions been found.
> 
> As far as I know, it is the other way around.  You desire a transfer function, and then
> design a filter to have that function.

So you mean that finding a transfer function is "guessing" the nominator and the denominator and then doing plots until the shape is something desired?

On Thursday, June 30, 2016 at 3:00:09 AM UTC-4, Matti Viljamaa wrote:
> On Thursday, June 30, 2016 at 2:09:39 AM UTC+3, herrman...@gmail.com wrote:
> > On Wednesday, June 29, 2016 at 2:58:57 AM UTC-7, Matti Viljamaa wrote:
> > > I've been reading sources that merely give the transfer functions of the filters
> > >  that are presented, but give no information regarding how have these transfer
> > > functions been found.
> > 
> > As far as I know, it is the other way around.  You desire a transfer function, and then
> > design a filter to have that function.
> 
> So you mean that finding a transfer function is "guessing" the numerator and the denominator and then doing plots until the shape is something desired?

yeah, Glen.  i think you mean "desired frequency response" instead of "transfer function".

r b-j