Origin of Blinear Transform

On Apr 30, 5:53 pm, Oli Charlesworth <c...@olifilth.co.uk> wrote:
> gyansor...@gmail.com said the following on 30/04/2007 20:30:
>
> > Here is a simple derivation
>
> > z=exp(sT)=exp(sT/2)/exp(-sT/2).
>
> > Now expand num and denominator to a 1st order approx
>
> > z:= (1+sT/2+..)/(1-sT/2+...)
>
> I've never been particularly happy with derivations like this; it seems
> we are matching the derivation to suit the desired result.  For example,
> we could equally have factored z as:
>
> z = exp(sT) = exp(3sT/4)/exp(-sT/4)
>
> giving us the bilinear approximation:
>
> z ~= (1 + 3sT/4)/(1 - sT/4)

or it could be z ~= (1 + sT/4)/(1 - 3sT/4)

it's a good point.

probably, the *motivation* (not derivation) i would point to is that
one series for log(z) and keeping the first term.  then once you
define it, you show that it has certain desirable properties and some
undesirable properties that you might be able to pre-warp to
compensate.

 r b-j

On Mon, 30 Apr 2007 11:27:15 -0400, Chris Barrett
<"chrisbarret"@0123456789abcdefghijk113322.none> wrote:

>Does anyone know what the origin is of the bilinear transform?  In other 
>words, how was the bilinear transform derived?

Hello Chris,

  please forgive me for not being certain about 
all of this but I have the recollection 
that the bilinear transform was developed by a team 
of professors at Columbia University in the 1950s.
That team, as I recall, was led by Lofti Zadeh
(the inventor of "fuzzy logic".)

Our pal Julius, here in this newsgroup, has 
met Prof. Zadeh personally.

See Ya',
[-Rick-]

Clay wrote:
> On Apr 30, 2:56 pm, Clay <phys...@bellsouth.net> wrote:
> 
>>On Apr 30, 11:27 am, Chris Barrett
>>
>><"chrisbarret"@0123456789abcdefghijk113322.none> wrote:
>>
>>>Does anyone know what the origin is of the bilinear transform?  In other
>>>words, how was the bilinear transform derived?
>>
>>Hello Chris,
>>
>>I recall using years ago a transform of the form ((1+x^-1)^m)(1-
>>x^-1)^(n) for integration problems where n,m are integers. I believe
>>this is due to Gauss. Anyway, the bilinear is a special case of this,
>>and I'm quite sure this has been around for a long time. Now the
>>question is likely to be who was the one to bring its use into
>>converting S domain equations over to the z domain. I have a book with
>>a lot of z-domain stuff in it from the 1930s. I'll have to hunt around
>>my basement to find the book - I have thousands of them.
>>
>>Clay
> 
> 
> I did a little digging and a transform like the above  y=((1+x)^a)((1-
> x)^b) is due to Jacobi. It is easy to see that the bilinear is a
> special case of this.
> 
> Clay

Is this the "Jacobi method" that you're referring to?  I've found one of 
two ways to discretize the differential equation obtained from taking 
the inverse Laplace of a transfer function H(s).  Neither leads to a 
derivation of the bilinear transform.

Chris Barrett wrote:
> Clay wrote:
> 
>> On Apr 30, 2:56 pm, Clay <phys...@bellsouth.net> wrote:
>>
>>> On Apr 30, 11:27 am, Chris Barrett
>>>
>>> <"chrisbarret"@0123456789abcdefghijk113322.none> wrote:
>>>
>>>> Does anyone know what the origin is of the bilinear transform?  In 
>>>> other
>>>> words, how was the bilinear transform derived?
>>>
>>>
>>> Hello Chris,
>>>
>>> I recall using years ago a transform of the form ((1+x^-1)^m)(1-
>>> x^-1)^(n) for integration problems where n,m are integers. I believe
>>> this is due to Gauss. Anyway, the bilinear is a special case of this,
>>> and I'm quite sure this has been around for a long time. Now the
>>> question is likely to be who was the one to bring its use into
>>> converting S domain equations over to the z domain. I have a book with
>>> a lot of z-domain stuff in it from the 1930s. I'll have to hunt around
>>> my basement to find the book - I have thousands of them.
>>>
>>> Clay
>>
>>
>>
>> I did a little digging and a transform like the above  y=((1+x)^a)((1-
>> x)^b) is due to Jacobi. It is easy to see that the bilinear is a
>> special case of this.
>>
>> Clay
> 
> 
> Is this the "Jacobi method" that you're referring to?  I've found one of 
> two ways to discretize the differential equation obtained from taking 
> the inverse Laplace of a transfer function H(s).  Neither leads to a 
> derivation of the bilinear transform.

Okay.  I almost have it derived.  My equation are too similar to be 
coincidental.

On 30 Apr, 17:27, Chris Barrett
<"chrisbarret"@0123456789abcdefghijk113322.none> wrote:
> Does anyone know what the origin is of the bilinear transform?  In other
> words, how was the bilinear transform derived?

I have no background to state anything with certainty, but the
various posts in this thread, mentioning people like Gauss and
Cauchy, makes me wonder exactly what is meant by "the bilinear
transform."

I don't remember off the top of my head having read anywhere if
he actually developed it, but I wouldn't be the least surprised if
Cauchy developed the concept we know as a "conformal mapping."
As we know, a conformal mapping is a function that transforms one
analytic function into another analytic function. The BLT does
exactly that, so in that sense, it is correct to trace its history
back to at least the times of Cauchy, if not to the man himself.

The concept of the BLT as known in DSP, however, is very
specific to the application. The BLT as we know it requires
both a continuous-time system theory and a discrete-time
system theory to be present, since the BLT describes how
to map system functions between the two domains.

It seems, then, as something of a contradiction if the BLT
should date back to a time before the younger of either
continuous-time or discrete-time systems theory.

Rune