# Minimum Phase Impulse Response

Started by October 29, 2003
```allnor@tele.ntnu.no (Rune Allnor) wrote in message news:<f56893ae.0311032343.312d19a7@posting.google.com>...
> The dispute goes
> over a "simple proof of Fermat's last theorem" (what else...). Harald
> Hanche-Olsen, who made the page referenced above and disputes the
> existence of such a proof,

Ouch! That last sentence can be misunderstood. Hanche-Olsen does not
dispute Fermat's last theorem being proved, he disputes the existence
of a *simple* proof. As far as I know, Wiley, who proved the theorem,
used just about every known aspect of mathemathics to get every piece
in place. Which probably explains why it took 350 years to prove the
theorem.

Rune
```
```Rune Allnor wrote:

> Jerry Avins <jya@ieee.org> wrote in message news:<bo6q8i\$f3k\$1@bob.news.rcn.net>...
>
>>The new poles and zeros yield the same magnitude response as the
>>original, but the phase is minimum. I don't remember why, but I have
>>it on reliable authority (O&S?).
>
>
> With all due respect, Jerry, I get "bad itches" by that sort of argument.
> Please don't misunderstand! I think you are right and I'm not capable
> of doing better myself.
>

Since I gave it the trappings of a proof with "Q.E.D.!", I sympathize
with your itch. I should have written that visualizing my old Spirule, I
can see that zeros in the right-hand s plane introduce less phase shift
than those reflected into the left, and that in the z plane right becomes
outside and left becomes inside. reflections about the s-plane vertical
axis become reciprocal along a radius in the z plane. So much for the
geometry. I used the knowledge that that reflecting a zero about the jw
axis leaves the magnitude response unchanged. I cited O&S as the source,
but it may be Guillemin.

I am happy to use those guys' results without re-deriving them. I know
how to calculate the section modulus of a beam given its shape, but there
are tables for that and I use them.

Jerry

P.S. Digital calculators are valuable tools, but those who have never
become proficient with slide rule or Spirule lack a powerful way to
visualize simple solutions to otherwise complicated problems.

P.P.S. I know a simple way to trisect an angle with ruler, compass, and
pencil. I sometimes use it. It works well. I'm not (for this) a nut.
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;

```
```Jerry Avins <jya@ieee.org> wrote in message news:<bo8mp4\$3vi\$1@bob.news.rcn.net>...
> Rune Allnor wrote:
>
> > Jerry Avins <jya@ieee.org> wrote in message news:<bo6q8i\$f3k\$1@bob.news.rcn.net>...
> >
> >>The new poles and zeros yield the same magnitude response as the
> >>original, but the phase is minimum. I don't remember why, but I have
> >>it on reliable authority (O&S?).
> >
> >
> > With all due respect, Jerry, I get "bad itches" by that sort of argument.
> > Please don't misunderstand! I think you are right and I'm not capable
> > of doing better myself.
> >
>
> Since I gave it the trappings of a proof with "Q.E.D.!", I sympathize
> with your itch. I should have written that visualizing my old Spirule, I
> can see that zeros in the right-hand s plane introduce less phase shift
> than those reflected into the left, and that in the z plane right becomes
> outside and left becomes inside.

Now that you mention it, the 2nd edition of P&M I used when I took that
class did show some figures with some vectors(?) from the zeros to the
unit circle. I never understood why those "vectors" had to be moved
counterclockwise around the zero. (My memory may be failing me now,
I'm not sure if what I remember makes sense at all). If somebody tells
me it had something to do with moving in the complex plane and staying
on some particular sheet of a Riemann surface, things may not be as
total voodoo as they appear at the moment.

> reflections about the s-plane vertical
> axis become reciprocal along a radius in the z plane. So much for the
> geometry. I used the knowledge that that reflecting a zero about the jw
> axis leaves the magnitude response unchanged. I cited O&S as the source,
> but it may be Guillemin.

It was probably O&S. I haven't seen that book, but from what I hear,
those guys did things the mathemathical way.

> I am happy to use those guys' results without re-deriving them. I know
> how to calculate the section modulus of a beam given its shape, but there
> are tables for that and I use them.

Of course I agree with you. My problem is that I like to understand
what's going on. Sure, there are plenty of people out there who do the
maths way better than I will ever be able to. There are thousands of
people who code up those numerical routines faster and more efficient, in
any sense of the word, than me. Still, I like to understand how the stuff
works. By doing that I have a chance of finding out what's easy and what's
not. Which means I can anticipate problems with either coding the routines
or using the routines. And I find out who possess actual knowledge and
skills, and who are merely "politicians" or "imposters", which would be
very useful knowledge whenever I or my projects get under pressure.

> Jerry
>
> P.S. Digital calculators are valuable tools, but those who have never
> become proficient with slide rule or Spirule lack a powerful way to
> visualize simple solutions to otherwise complicated problems.

I have no knowledge of either (I never found out what a spirule is), but
I still trust you on this. Not because of any authority you might have,
but because of the knowledge and skills you consistently demonstrate.

> P.P.S. I know a simple way to trisect an angle with ruler, compass, and
> pencil. I sometimes use it. It works well. I'm not (for this) a nut.

Yeah, right. I suppose the margin of your post was just too small to
describe your trick?[*] Anyway, if the occation ever arises where you can
show me your method face to face, I'll by you a beer.

Rune

[*] Yep, you caught me. According to "authorities" (some maths book, I
can't remember which one) it has been proven that trisecting the angle
by means of the mentioned instruments is impossible. I can't give the
proof myself, I can only refer to "authority". Somehow I suspect you
knew I would respond like that...
```
```Jerry Avins wrote:

> P.P.S. I know a simple way to trisect an angle with ruler, compass, and
> pencil. I sometimes use it. It works well. I'm not (for this) a nut.

(sideways glance)

Using the ruler's scale, or just the straight edge?

```
```"Eric C. Weaver" <weav@sigma.net> wrote in message news:<3fa88fcb\$1@news.announcetech.com>...
> Jerry Avins wrote:
>
> > P.P.S. I know a simple way to trisect an angle with ruler, compass, and
> > pencil. I sometimes use it. It works well. I'm not (for this) a nut.
>
> (sideways glance)
>
> Using the ruler's scale, or just the straight edge?

Right... it took me a couple of hours, but if I can use the scale
I too can trisect the angle. With only the straight edge things can
become difficult.

Rune
```
```Dangerously vocal young programmer wrote:
> > > P.P.S. I know a simple way to trisect an angle with ruler, compass, and
> > > pencil. I sometimes use it. It works well. I'm not (for this) a nut.
> >
> > (sideways glance)
> >
> > Using the ruler's scale, or just the straight edge?
>
> Right... it took me a couple of hours, but if I can use the scale
> I too can trisect the angle. With only the straight edge things can
> become difficult.
>
> Rune

Hey Rune,

Archimedes knew how to do this already, with just a straight edge and
(two I think) pencilmarks. Is that hint enough?

Regards,
Andor
```
```Rune Allnor wrote:
> Jerry Avins <jya@ieee.org> wrote in message news:<bo8mp4\$3vi\$1@bob.news.rcn.net>...
>
...

>>P.P.S. I know a simple way to trisect an angle with ruler, compass, and
>>pencil. I sometimes use it. It works well. I'm not (for this) a nut.
>
>
> Yeah, right. I suppose the margin of your post was just too small to
> describe your trick?[*] Anyway, if the occation ever arises where you can
> show me your method face to face, I'll by you a beer.
>
> Rune
>
> [*] Yep, you caught me. According to "authorities" (some maths book, I
>     can't remember which one) it has been proven that trisecting the angle
>     by means of the mentioned instruments is impossible. I can't give the
>     proof myself, I can only refer to "authority". Somehow I suspect you
>     knew I would respond like that...

Rune,

I wasn't trying to catch you. We needn't be face to face; I'll describe
the method so you can try it yourself. [All geometry problems that are
isomorphic to quadratic or linear equations can be solved with compass
and straightedge. In general, those isomorphic to cubic and higher
equations can not be so solved.]* Trisection is cubic*, so more powerful
tools are needed. Instead of a straightedge, I need a ruler, as I wrote.
A ruler has parallel sides and an end square to them. These extra-
Euclidian features play a part in the construction. I'll give you time
to play with the idea before I thrust it on you. I do not withhold it
now to tease.

The method came to my attention in the late 1940s when it was published
on the front page of the second section of the then two-section New York
Times. It illustrated someone's clam to have solved the trisection
problem; The Times appeared to endorse that claim. Subsequent discussion
was interesting!

Jerry
_______________________________
* I haven't proved that, but accept it from Authority.
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;

```
```Andor wrote:

> Dangerously vocal young programmer wrote:
>
>>>>P.P.S. I know a simple way to trisect an angle with ruler, compass, and
>>>>pencil. I sometimes use it. It works well. I'm not (for this) a nut.
>>>
>>>(sideways glance)
>>>
>>>Using the ruler's scale, or just the straight edge?
>>
>>Right... it took me a couple of hours, but if I can use the scale
>>I too can trisect the angle. With only the straight edge things can
>>become difficult.
>>
>>Rune
>
>
> Hey Rune,
>
> Archimedes knew how to do this already, with just a straight edge and
> (two I think) pencilmarks. Is that hint enough?
>
>
> Regards,
> Andor

That -- and Rune's way -- sounds simpler than my way. Do those ways work
for large angles, say, 90 degrees +/- a little?

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;

```
```an2or@mailcircuit.com (Andor) wrote in message news:<ce45f9ed.0311050704.1fe1b0af@posting.google.com>...
:
> Dangerously vocal young programmer wrote:

Yep, that's what I was getting at... ;)

> > > > P.P.S. I know a simple way to trisect an angle with ruler, compass, and
> > > > pencil. I sometimes use it. It works well. I'm not (for this) a nut.
> > >
> > > (sideways glance)
> > >
> > > Using the ruler's scale, or just the straight edge?
> >
> > Right... it took me a couple of hours, but if I can use the scale
> > I too can trisect the angle. With only the straight edge things can
> > become difficult.
> >
> > Rune
>
> Hey Rune,
>
> Archimedes knew how to do this already, with just a straight edge and
> (two I think) pencilmarks. Is that hint enough?

I am sure he did. Using the scale of a rule or some other length reference,
it's not difficult at all. However, some problems appear to have haunted
maths throughout history:

- Trisecting the angle using only a straight-edge (with no scale or length
refernce) and compass
- Constructing a square of the same area as a circle with given diameter
(or was it vice versa?) using only straight-edge and compass
- Proving one of Euclid's postulates on geometry.

Proving that the first two were impossible was apparently among the first
main contributions of abstract algebra. The disproof of Euclid's 5th(?)
postulate spawned what we now know as "non-Euclidian geometry", and with
it, modern mathematics.

My mistake when I (too sarcastically) flamed Jerry's post was that I
didn't check the basic assumtions of his claim.

Rune
```
```Rune Allnor wrote:

>   ... Using the scale of a rule or some other length reference,
> it's not difficult at all.

I don't know that one, unless trial and error is alowed. The usual way
using trial and error -- called by some "successive refinement" -- is
with dividers. Draftsmen do that regularly.

> However, some problems appear to have haunted maths throughout history:
>
> - Trisecting the angle using only a straight-edge (with no scale or length
>   refernce) and compass
> - Constructing a square of the same area as a circle with given diameter
>   (or was it vice versa?) using only straight-edge and compass
> - Proving one of Euclid's postulates on geometry.
>
> Proving that the first two were impossible was apparently among the first
> main contributions of abstract algebra. The disproof of Euclid's 5th(?)
> postulate spawned what we now know as "non-Euclidian geometry", and with
> it, modern mathematics.
>
> My mistake when I (too sarcastically) flamed Jerry's post was that I
> didn't check the basic assumtions of his claim.

Now _that_ was a trap I did lay!

>
> Rune

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;

```