any deep thinking on why linear systems are commutative?

Rune Allnor wrote:


> The problem is that DSP is an applied dicipline, with lots of the 
> business being dominated by "hands on problem-solving" Electrical 
> Engineers. I've been trying to attempt the FA-based problem solving 
> strategy with practical problems, and obtained certain encouraging 
> results. However, the reactions from the old-timers with 40+ years 
> experience in EE and who "knew the proper way" of doing things, made 
> me abandon the whole thing. And almost DSP as well.

Do you mean that Hilbert spaces are difficult to get your 
arms around when you are 60ish?  :-)

Hey, if I can do it anyone can.  Actually, the formality of 
F.A. is not all that foreign if you have a strong background 
in conventional DSP.  Things like norms, square integrable 
functions, correlation, and so forth aren't far from what we 
know and love.

Wavelet texts are a really good place to get started with 
it.  They may not call it that but that's what they're 
usually about.


Bob
-- 

"Things should be described as simply as possible, but no 
simpler."

                                              A. Einstein

Bob Cain wrote:


> Wavelet texts are a really good place to get started with it.  They may 
> not call it that but that's what they're usually about.
> 
> 
> Bob

Hey, you almost answered my question before I asked!  Here it is:

Do you have any specific books to recommend for self-directed study on 
the topic?

Thanks.

-- 

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

Tim Wescott wrote:
> Bob Cain wrote:
> 
> 
>> Wavelet texts are a really good place to get started with it.  They 
>> may not call it that but that's what they're usually about.
>>
>>
>> Bob
> 
> 
> Hey, you almost answered my question before I asked!  Here it is:
> 
> Do you have any specific books to recommend for self-directed study on 
> the topic?

For the exposition of functional analysis and the 
prerequisite math in the context of wavelets, my favorite is:

   Wavelets and Subband Coding
   Vetterli and Kovacevic
   Prentice Hall
   ISBN 0-13-097080-8

Chapter 2 will bring you up to speed.  Of course it is not 
nearly as complete as a math course on functional analysis 
would likely be but is adequate to view DSP as "applied 
functional analysis."  (And to dispell the idea that the DFT 
is somehow periodic. :-)


Bob
-- 

"Things should be described as simply as possible, but no 
simpler."

                                              A. Einstein

Tim Wescott <tim@wescottnospamdesign.com> wrote in message news:<10nt7sdsvhd8a65@corp.supernews.com>...
> Bob Cain wrote:
> 
> 
> > Wavelet texts are a really good place to get started with it.  They may 
> > not call it that but that's what they're usually about.
> > 
> > 
> > Bob
> 
> Hey, you almost answered my question before I asked!  Here it is:
> 
> Do you have any specific books to recommend for self-directed study on 
> the topic?

I don't know books for self-study of functional analysis, but you would 
probably find the book 

Strang: "Linear algebra and its applications", 3rd ed., 1988

very useful for understanding the various properties of vector spaces.

Rune

Tim Wescott <tim@wescottnospamdesign.com> wrote in message news:<10nt7sdsvhd8a65@corp.supernews.com>...
> Bob Cain wrote:
> 
> 
> > Wavelet texts are a really good place to get started with it.  They may 
> > not call it that but that's what they're usually about.
> > 
> > 
> > Bob
> 
> Hey, you almost answered my question before I asked!  Here it is:
> 
> Do you have any specific books to recommend for self-directed study on 
> the topic?
> 
> Thanks.

Eh... www.groups.google.com indicates that this post is an answer to 
a post I wrote. I guess it might be that Bob wrote a reply to me, 
that somehow got lost?

(I don't know why, but it has happened in the past that 
www.groups.google.com skips posts.)

Rune

"kiki" <lunaliu3@yahoo.com> wrote in message
news:cljh80$btq$1@news.Stanford.EDU...
> Dear all,
>
> Could you please help me understand better by providing some deep thoughts
> on why linear systems are commutative?
>
> If T1 and T2 are linear systems, then y1=T2(T1(x)) and y2=T1(T2(x))
>
> y1 and y2 are the same.
>
> After seeing some examples of non-linear system and linear systems. I got
> convinced that for non-linear system this property does not hold.
>
> But any deeper thinking? Proof?
>
> Thanks a lot!
>
>

Multivariable systems do not follow commutivity as matrices normally do not
commute.
So this is true for siso systems only.

Tom

"Gordon Sande" <g.sande@worldnet.att.net> wrote in message
news:5fcfd.579$9b.372@edtnps84...
>
>
> kiki wrote:
> > Dear all,
> >
> > Could you please help me understand better by providing some deep
thoughts
> > on why linear systems are commutative?
> >
> > If T1 and T2 are linear systems, then y1=T2(T1(x)) and y2=T1(T2(x))
> >
> > y1 and y2 are the same.
> >
> > After seeing some examples of non-linear system and linear systems. I
got
> > convinced that for non-linear system this property does not hold.
> >
> > But any deeper thinking? Proof?
> >
> > Thanks a lot!
> >
>
> To answer the subject line: Composition of linear (time invariant)
> systems is convolution of their impulse responses. Now you can
> prove that, including the commuting part in passing, various ways
> so it is true for ALL LTI systems.

But not multiple input-output systems.

Tom

"ZZBunker" <zzbunker@netscape.net> wrote in message
news:e4a0829b.0410251545.38c93625@posting.google.com...
>
>   In linear systems it's almost never true that
>   T1T2 = T2T1.
>
For LTI systems T1*T2 = T2*T1 where * is convolution. There are numerous
real world engineering examples of this esp in electronics.I don't want to
split hairs about how linear is linear or how time-invariant is
time-invariant but for us engineers it holds fine most of the time (except
when the systems is multivariable).

Tom