On Tue, 22 Jan 2008 23:30:21 +1100, Allan Herriman
<allanherriman@hotmail.com> wrote:
(snipped)
>
>Hi Rick,
>
>Could you please fix the spelling of 'Cartisian' in that document for
>me?
>
>Thanks,
>Allan.
Hi Allan,
I contacted Grant Griffin.
The deed is done. :-)
See Ya',
[-Rick-]
Reply by glen herrmannsfeldt●January 23, 20082008-01-23
Chris Bore wrote:
(snip)
> I think the complex numbers are more intuitive than 1-d sine and
> cosine.
> Sine and cos are 2d ideas. They derive from geometry. A sine is a
> property of an angle. In a 2d word you need two coordinates to define
> a point. (x, y) is OK. So is (cos, sin). But just sin alone, or cos
> alone, is tryint to fit a 2d geometry into a 1d representation.
Or, sine and cosine come from the simplest second order
differential equation. At least that is where they come
from when the subject is DSP. Also, just 1D.
If you say that a second order differential equation is really
two first order equations then it is 2D again.
-- glen
Reply by Chris Bore●January 23, 20082008-01-23
On Jan 21, 7:41�pm, buleg...@columbus.rr.com wrote:
> On Jan 21, 2:03�pm, Darol Klawetter <darol.klawet...@l-3com.com>
> wrote:
>
>
>
>
>
> > On Jan 21, 6:48 am, buleg...@columbus.rr.com wrote:
>
> > > Hi,
>
> > > I have begun a complex number tutorial which shows how e^jwt works. It
> > > is located:
>
> > >http://fourier-series.com/fourierseries2/complex_tutorial.html
>
> > > I have more work to do on this, but come take a look. �I created these
> > > programs for complex numbers in order to explain the complex
> > > representaion of the fourier series.
>
> > > Brent
>
> > Cute, but you should explain why Euler needed "j" in his identity.
> > Explain why it's used as the vertical axis of the phasor plane. Many
> > engineers want a physical intuition that is defied by a reference to
> > the square root of -1. Most introductory texts just state Euler's
> > identity and leave it at that.
>
> I intend to add some more explanation in the near future. �However,
> this whole j thing is difficult because getting that physical
> intuition has not really come to me yet. �I wonder if you must simply
> accept, as an article of faith that Eulers identity works and that
> (e^jwt+e^-jwt)/2 = cos(wt). �That is, the staring point is accepting
> the identity. �The equivelent of an axiom in geometry, then build off
> of the axiom.- Hide quoted text -
>
> - Show quoted text -
I think the complex numbers are more intuitive than 1-d sine and
cosine.
Sine and cos are 2d ideas. They derive from geometry. A sine is a
property of an angle. In a 2d word you need two coordinates to define
a point. (x, y) is OK. So is (cos, sin). But just sin alone, or cos
alone, is tryint to fit a 2d geometry into a 1d representation. To
deal with (cos, sin) pairs is mathematically awkward. A complex number
is just a coordinate pair (x, y) and the maths of complex numbers lets
us deal with those 2d points, their displacements and rotations, with
straightforward maths that we already know. So it simplifies
intuitively as well as mathematically. An Argand diagram, with real
and imaginary axes, makes the hypotenuse be the vector. Then the
complex numbers give a much more intuitive way to deal with this 2d
entity than messing around with 1d.
For me, the trick to finding complex numbers intuitive is to remind
myself that I am dealing with a 2D geometry and always think in
geometric pictures. Then it is the (sin alone) that seems awkward and
loses information.
Chris
============================
Chris Bore
BORES Signal Processing
www.bores.com
chris@bores.com
Chris
Reply by ●January 22, 20082008-01-22
On Jan 22, 6:00�pm, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:
> On Mon, 21 Jan 2008 18:56:34 -0800 (PST), buleg...@columbus.rr.com
> wrote:
>
> � (snipped)
>
>
>
> > To understand rotational/sinusoidal solutions, however, we need to
> >understand complex numbers, which (perhaps) nobody really intuitively
> >understands. �I guess when it all boils down, the only way to get a
> >complete solution to a differential equation (one that includes not
> >just the correct frequeny, but the correct phasing), you need the
> >hidden magic of complex numbers going on "behind the scenes".
>
> >I know this sounds like rambling jibberish, and probably blows my
> >credibility, but, oh well.
>
> >Brent
>
> Hi Brent,
> � � Don't worry.
> Reaching some sort of comfortable understanding
> of the true meaning (whatever that is) of complex numbers
> is not at all easy. �I make no claim that I understand
> their meaning. �I merely understand a little bit
> about their behavior.
>
> Remember, if complex numbers were easy to understand,
> the great mathematician Karl Gauss would *NOT* have
> called the j-operator "the shadow of shadows".
>
> [-Rick-]
Thanks for your nice replies and encouragement. I read ( a lot of )
your book and Steve Smiths book. To some extent those two books
motivated me to do this stuff. Steve is a gifted writer and I feel
that you really struck a wonderful balance between hard math and
visual/descriptive content. In my opinion, you cannot just pick up
an Oppenheim/Shaeffer book and learn DSP. Way too hard. I've tried it
a couple of times over the years and can't get past about 20 pages -
too hard to see what is going on.
Brent
Reply by ●January 22, 20082008-01-22
On Jan 22, 11:00�am, c...@claysturner.com wrote:
> On Jan 21, 10:58�pm, buleg...@columbus.rr.com wrote:
>
>
>
>
>
> > On Jan 21, 10:10�pm, Jerry Avins <j...@ieee.org> wrote:
>
> > > buleg...@columbus.rr.com wrote:
> > > > On Jan 21, 9:37 pm, Jerry Avins <j...@ieee.org> wrote:
> > > >> buleg...@columbus.rr.com wrote:
> > > >>> On Jan 21, 2:03 pm, Darol Klawetter <darol.klawet...@l-3com.com>
> > > >>> wrote:
> > > >>>> On Jan 21, 6:48 am, buleg...@columbus.rr.com wrote:
> > > >>>>> Hi,
> > > >>>>> I have begun a complex number tutorial which shows how e^jwt works. It
> > > >>>>> is located:
> > > >>>>>http://fourier-series.com/fourierseries2/complex_tutorial.html
> > > >>>>> I have more work to do on this, but come take a look. �I created these
> > > >>>>> programs for complex numbers in order to explain the complex
> > > >>>>> representaion of the fourier series.
> > > >>>>> Brent
> > > >>>> Cute, but you should explain why Euler needed "j" in his identity.
> > > >>>> Explain why it's used as the vertical axis of the phasor plane. Many
> > > >>>> engineers want a physical intuition that is defied by a reference to
> > > >>>> the square root of -1. Most introductory texts just state Euler's
> > > >>>> identity and leave it at that.
> > > >>> I intend to add some more explanation in the near future. �However,
> > > >>> this whole j thing is difficult because getting that physical
> > > >>> intuition has not really come to me yet. �I wonder if you must simply
> > > >>> accept, as an article of faith that Eulers identity works and that
> > > >>> (e^jwt+e^-jwt)/2 = cos(wt). �That is, the staring point is accepting
> > > >>> the identity. �The equivelent of an axiom in geometry, then build off
> > > >>> of the axiom.
> > > >> No. At the very least, you can add the Taylor series of cos(x) to the
> > > >> Taylor series of i*sin(x) and observe that the sum is identical to the
> > > >> Taylor series of exp(i*x). A technicality makes that fall short of being
> > > >> a proof, but it sure is a good indication.
>
> > > > That relationship is cool, It points to complex numbers as having more
> > > > meaning than might otherwise have been thought. �It made complex
> > > > numbers a real "head scratcher", but, for me, this relationship does
> > > > not really bring forth any more insight into what imaginary numbers
> > > > are. �To me, imaginary numbers are like a hidden world that works
> > > > behind the scene , but always turns up invisible to us.
>
> > > > For me the real beauty is that the e^jwt is more intuitive as a
> > > > solution to a differential equation than say, sinx and cos x is.
> > > > Sure, it is observed that the fourth derivative of a sine wave is a
> > > > sine wave, but the e^jwt really brings the soultion to a differential
> > > > equation to life.
>
> > > > We can intuitively understand solutions to differential equations that
> > > > have x, x^2, x^3 as solutions.
> > > > We can intuitively understand solutions that have e^-x or e^-t as
> > > > solutions.
>
> > > > �To understand rotational/sinusoidal solutions, however, we need to
> > > > understand complex numbers, which (perhaps) nobody really intuitively
> > > > understands. �I guess when it all boils down, the only way to get a
> > > > complete solution to a differential equation (one that includes not
> > > > just the correct frequeny, but the correct phasing), you need the
> > > > hidden magic of complex numbers going on "behind the scenes".
>
> > > > I know this sounds like rambling jibberish, and probably blows my
> > > > credibility, but, oh well.
>
> > > Imaginary numbers are just a compact bookkeeping scheme for computing
> > > relations that can be arrived at -- albeit with greater effort -- in
> > > other ways.
>
> > > Jerry
> > > --
> > > Engineering is the art of making what you want from things you can get.
> > > �����������������������������������������������������������������������- Hide quoted text -
>
> > > - Show quoted text -
>
> > One of issues about complex numbers and book keeping has to do with my
> > intro elctronics class many years ago. �We were taught to multiply
> > signals by A Le^jwt which was nice, but I don't think I ever fully
> > appreciated that this trick assumes that the other side Le^-jwt
> > automatically comes along for the ride and if you don't understand the
> > implied conjugate, then you don't really get the shorthand notation
> > (which I didn't , and didn't in that class)- Hide quoted text -
>
> > - Show quoted text -
>
> It is more than just a trick. Look up Wronskians. If you are solving
> 2nd order differential equations, you know you will have two linear
> independent solutions to the homogenous part of the equations. The
> initial conditions will determine the appropriate combination.
>
> �A Wronskian is a neat way to show that a set of functions are
> linearly independent. Or course with just two functions all you
> require is for their ratio to be nonconstant to show linear
> independence. �For some types of 2nd order equations, the Wronskian is
> not only nonzero for two linearly independent functions, but sometimes
> it is constant. This shows up in some applications of Schrodinger's
> equation. So if you have a solution in part of a domain, the Wronskian
> can help you find the solution in another part of the domain.
>
> I hope this peaks your interests
>
> Clay
>
> p.s. You will find solutions to Hermitian systems will admit the
> conjugate of a function if the function itself is a solution. In DSP
> you will find a link between functions, their Fourier transforms, and
> Hermitian properties. In Physics, this shows up in Quantum Mechanics,
> where the position and momentum wave representations are Fourier
> transforms of each other.- Hide quoted text -
>
> - Show quoted text -
Thanks for your thoughtful reply. Unfortunately, I can't keep up with
you. I still have to digest Eigenvectors before I can move onto
Wronskians.
Brent
Reply by Randy Yates●January 22, 20082008-01-22
bulegoge@columbus.rr.com writes:
> However, this whole j thing is difficult
I don't see what's so difficult about it. The complex field is simply
a field including the integers that has no extension fields. :)
--
% Randy Yates % "She's sweet on Wagner-I think she'd die for Beethoven.
%% Fuquay-Varina, NC % She love the way Puccini lays down a tune, and
%%% 919-577-9882 % Verdi's always creepin' from her room."
%%%% <yates@ieee.org> % "Rockaria", *A New World Record*, ELO
http://www.digitalsignallabs.com
Reply by Rick Lyons●January 22, 20082008-01-22
On Tue, 22 Jan 2008 23:30:21 +1100, Allan Herriman
<allanherriman@hotmail.com> wrote:
>Hi Rick,
>
>Could you please fix the spelling of 'Cartisian' in that document for
>me?
>
>Thanks,
>Allan.
Hi Allan,
*#$%^&*, that darn typo comes back to haunt
me again! Well, ...Grant Griffin runs the website
where that PDF file is located. I haven't E-mailed
Grant for years. Maybe I'll send Grant an E-mail
to see if he'll post a "corrected" version of that
PDF file.
See Ya',
[-Rick-]
Reply by glen herrmannsfeldt●January 22, 20082008-01-22
bulegoge@columbus.rr.com wrote:
(snip)
> I intend to add some more explanation in the near future. However,
> this whole j thing is difficult because getting that physical
> intuition has not really come to me yet. I wonder if you must simply
> accept, as an article of faith that Eulers identity works and that
> (e^jwt+e^-jwt)/2 = cos(wt). That is, the staring point is accepting
> the identity. The equivelent of an axiom in geometry, then build off
> of the axiom.
Feynman has an interesting way to do it in "Lectures on Physics"
volume 1. He starts out with fractional powers through successive
square roots of 10. (For 10**0.5, 10**0.25, 10**0.125, etc.)
That allows one to compute 10**(x/1024) for integer x.
Then he computes 10**(i/1024) using complex multiplication,
and uses powers of that to compute 10**(i x/1024) for integer x.
The result, surprisingly (if you don't know the answer) is
sine-like curves with an unusual period. He then goes back
to show that if you change the base from 10 such that the
period is 2 pi then the base must be e.
-- glen
Reply by Rick Lyons●January 22, 20082008-01-22
On Mon, 21 Jan 2008 18:56:34 -0800 (PST), bulegoge@columbus.rr.com
wrote:
(snipped)
>
> To understand rotational/sinusoidal solutions, however, we need to
>understand complex numbers, which (perhaps) nobody really intuitively
>understands. I guess when it all boils down, the only way to get a
>complete solution to a differential equation (one that includes not
>just the correct frequeny, but the correct phasing), you need the
>hidden magic of complex numbers going on "behind the scenes".
>
>I know this sounds like rambling jibberish, and probably blows my
>credibility, but, oh well.
>
>Brent
Hi Brent,
Don't worry.
Reaching some sort of comfortable understanding
of the true meaning (whatever that is) of complex numbers
is not at all easy. I make no claim that I understand
their meaning. I merely understand a little bit
about their behavior.
Remember, if complex numbers were easy to understand,
the great mathematician Karl Gauss would *NOT* have
called the j-operator "the shadow of shadows".
[-Rick-]