View and manipulate signal as vector?

It's said signal may be viewed and manipulated as vetctor. Any
reference or tutorials on this? E-copy is preferred. Thanks!

On Mon, 05 Nov 2007 04:06:08 +0000, zqchen wrote:

> It's said signal may be viewed and manipulated as vetctor. Any
> reference or tutorials on this? E-copy is preferred. Thanks!

No reference or tutorial, but an insight:

In sampled time, a signal s_k can be treated as a vector, possibly
infinitely long, of discrete values (s_0, s_1, s_2, and so on).

If you're intrepid, you can extend this to a continuous-time signal,
although it's usually easier to go back to treating it as a function of
time.

-- 
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

zqchen wrote:
> It's said signal may be viewed and manipulated as vetctor. Any
> reference or tutorials on this? E-copy is preferred. Thanks!

Viewing signals as vectors is very natural - it only causes confusion
if one is blinded by the analogy of vectors and arrows ("a signal is
an arrow?"). One can add signals, distributively multiply them with
scalars, etc. You can easily verfy the vector space axioms for any
signal space (discrete or continuous): http://en.wikipedia.org/wiki/Vector_space

A simple example of viewing signals as vectors is the signal space of
discrete periodic functions with period N (assuming a sampling period
of T = 1). One basis for this vector space are the "time"-vectors

{(1,0,0,..,0),
(0,1,0,...,0),
...,
(0,0,...,1)}

(N of them, as the vector space is N-dimensional). Another basis are
the "frequency"-vectors consisting of the sampled sin() and cos()-
functions with a period that is an integer fraction of N:

{sin(2 pi k /N, cos(2 pi k /N)}, k=0,1,...,N/2 or (N-1)/2, depending
if N is even or odd.

There are also N of them. The basis transform map that takes the
"time"-basis to the "frequency"-basis is the DFT.

Regards,
Andor

I wrote:
...
> {sin(2 pi k /N, cos(2 pi k /N)}, k=0,1,...,N/2 or (N-1)/2,

Forgot the time index. This should be
{sin(2 pi k n /N), cos(2 pi n k /N)}, k=0,1,...,N/2 or (N-1)/2, and
n=0,1,...,N.

>I wrote:
>...
>> {sin(2 pi k /N, cos(2 pi k /N)}, k=0,1,...,N/2 or (N-1)/2,
>
>Forgot the time index. This should be
>{sin(2 pi k n /N), cos(2 pi n k /N)}, k=0,1,...,N/2 or (N-1)/2, and
>n=0,1,...,N.
>
>
passing thoughts..
If I blindly assume all vectors have "direction", can I go ahead and apply
"curl" concept on a sample of 2 dimensional sampled set, (x0, x1) (x2, x3)
etc? 
wiki says curl is "Measures a vector field's tendency to rotate about a
point" . can I take 2 cosecutive samples of sine data and get a good
measure of curl?

By the way my real question was if any one can explain the procedure to
measure gradient in LMS equation . (it takes gradient of the set of tap
weights and tries to minimize error-squared)  

 

zqchen <zhiqun.chen@gmail.com> writes:

> It's said signal may be viewed and manipulated as vetctor. Any
> reference or tutorials on this? E-copy is preferred. Thanks!

Since the concept of "signal as vector" is pretty easy to see for
discrete signals, I'll assume you are asking about continuous-time
signals, or more formally, *functions*.

A function defined on a domain (i.e., a function f:[a,b]-->R, a,b \in R)
is a non-countably infinite-dimensional vector in the sense that it
has a non-countably infinte number of components, namely the values of
the function f(x), a <= x <= b. 

We can define a "dot product" for such functions as 

  f\dot g = \int_{a}^{b} f(t)*g(t) dt

and this can be used very much like the dot product for the more
familiar, finite vectors. For example, two functions are *orthogonal*
if their dot product (as defined above) is zero.

An excellent reference for this (sorry, but you'll have to actually
get up from your computer and go to a library) is [spiegel].

--Randy

@BOOK{spiegel,
  title = "{Applied Differential Equations}",
  author = "{Murray~R.~Spiegel}",
  publisher = "Prentice Hall",
  edition = "third",
  year = "1981"}

-- 
%  Randy Yates                  % "Rollin' and riding and slippin' and
%% Fuquay-Varina, NC            %  sliding, it's magic."
%%% 919-577-9882                %  
%%%% <yates@ieee.org>           % 'Living' Thing', *A New World Record*, ELO
http://www.digitalsignallabs.com

On 5 Nov, 10:19, "pal.debabrata123" <pal.debabrata...@gmail.com>
wrote:
> >I wrote:
> >...
> >> {sin(2 pi k /N, cos(2 pi k /N)}, k=0,1,...,N/2 or (N-1)/2,
>
> >Forgot the time index. This should be
> >{sin(2 pi k n /N), cos(2 pi n k /N)}, k=0,1,...,N/2 or (N-1)/2, and
> >n=0,1,...,N.
>
> passing thoughts..
> If I blindly assume all vectors have "direction",

The have. Try to compute the inner product between x and -x.

> can I go ahead and apply
> "curl" concept on a sample of 2 dimensional sampled set, (x0, x1) (x2, x3)
> etc?

I don't know. Curl includes the cross product between two
vectors, and it is not obvious (to me, at least) how to
compute the vector cross product in dimensons different
from 3.

> By the way my real question was if any one can explain the procedure to
> measure gradient in LMS equation . (it takes gradient of the set of tap
> weights and tries to minimize error-squared)

Read any text on optimizatiom methods. The books I have
are by Luenberger, and Nocedal and Wright.

Rune

On Nov 5, 8:29 am, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 5 Nov, 10:19, "pal.debabrata123" <pal.debabrata...@gmail.com>
> wrote:
>
> > >I wrote:
> > >...
> > >> {sin(2 pi k /N, cos(2 pi k /N)}, k=0,1,...,N/2 or (N-1)/2,
>
> > >Forgot the time index. This should be
> > >{sin(2 pi k n /N), cos(2 pi n k /N)}, k=0,1,...,N/2 or (N-1)/2, and
> > >n=0,1,...,N.
>
> > passing thoughts..
> > If I blindly assume all vectors have "direction",
>
> The have. Try to compute the inner product between x and -x.
>
> > can I go ahead and apply
> > "curl" concept on a sample of 2 dimensional sampled set, (x0, x1) (x2, x3)
> > etc?
>
> I don't know. Curl includes the cross product between two
> vectors, and it is not obvious (to me, at least) how to
> compute the vector cross product in dimensons different
> from 3.
>

Hello Rune et al,

You can do a generalized cross product where you need n-1 basis
vectors when you are working in "n" dimensions. Basically fill in the
bottom n-1 rows of a matrix with the "basis" functions and expand
along the top row by minors. For the curl just fill in the 1st row
with a bunch of partial derivative symbols. I.e., d/da, d/db, d/dc, d/
dd, ... d/dz for the correct number of dimensions.

This is a natural extension to how one finds a vector orthogonal to
two other 3-D vectors. If you recall the technique of filling the the
1st row of a matrix with "i" "j", and "k" and the using the other two
vectors to fill in rows 2 and 3. Then one finds the determinant. But
recalling one may find the determinant by "minors", then write out the
expansion for the determinant of the n by n matrix in terms of the 1st
row. So each component of the curl can be find by the determinant of
its corresponding n-1 by n-1 matrix.

Now the real question becomes what is the physical meaning of such a
curl? Which vector identities still hold? In 3-D if a non zero force
has a zero curl, then the force is conservative.  Stokes and/or
Green's theorems can show this. But what > 3 dimensions?  Special
relativity gives us 4-vectors and their associated idenities. Curl in
83 dimensions? Your guess is as good as mine. I'll have to ponder
this.

IHTH,

Clay

On Nov 5, 12:42 pm, Clay <phys...@bellsouth.net> wrote:
> On Nov 5, 8:29 am, Rune Allnor <all...@tele.ntnu.no> wrote:
>
>
>
>
>
> > On 5 Nov, 10:19, "pal.debabrata123" <pal.debabrata...@gmail.com>
> > wrote:
>
> > > >I wrote:
> > > >...
> > > >> {sin(2 pi k /N, cos(2 pi k /N)}, k=0,1,...,N/2 or (N-1)/2,
>
> > > >Forgot the time index. This should be
> > > >{sin(2 pi k n /N), cos(2 pi n k /N)}, k=0,1,...,N/2 or (N-1)/2, and
> > > >n=0,1,...,N.
>
> > > passing thoughts..
> > > If I blindly assume all vectors have "direction",
>
> > The have. Try to compute the inner product between x and -x.
>
> > > can I go ahead and apply
> > > "curl" concept on a sample of 2 dimensional sampled set, (x0, x1) (x2, x3)
> > > etc?
>
> > I don't know. Curl includes the cross product between two
> > vectors, and it is not obvious (to me, at least) how to
> > compute the vector cross product in dimensons different
> > from 3.
>
> Hello Rune et al,
>
> You can do a generalized cross product where you need n-1 basis
> vectors when you are working in "n" dimensions. Basically fill in the
> bottom n-1 rows of a matrix with the "basis" functions and expand
> along the top row by minors. For the curl just fill in the 1st row
> with a bunch of partial derivative symbols. I.e., d/da, d/db, d/dc, d/
> dd, ... d/dz for the correct number of dimensions.
>
> This is a natural extension to how one finds a vector orthogonal to
> two other 3-D vectors. If you recall the technique of filling the the
> 1st row of a matrix with "i" "j", and "k" and the using the other two
> vectors to fill in rows 2 and 3. Then one finds the determinant. But
> recalling one may find the determinant by "minors", then write out the
> expansion for the determinant of the n by n matrix in terms of the 1st
> row. So each component of the curl can be find by the determinant of
> its corresponding n-1 by n-1 matrix.
>
> Now the real question becomes what is the physical meaning of such a
> curl? Which vector identities still hold? In 3-D if a non zero force
> has a zero curl, then the force is conservative.  Stokes and/or
> Green's theorems can show this. But what > 3 dimensions?  Special
> relativity gives us 4-vectors and their associated idenities. Curl in
> 83 dimensions? Your guess is as good as mine. I'll have to ponder
> this.
>
> IHTH,
>
> Clay- Hide quoted text -
>
> - Show quoted text -

Let correct a slight error.

For finding a generalized cross product, use n-1 basis vectors. For
the curl, use n-2 basis vectors. So for both cases think about "i",
"j","k",... on the 1st row. And for the curl fill in the partial
derivative symbols on the 2nd row. Then fill in the remaining rows
with the basis vectors.

I hope this clarifies things.

Clay

huh, looks like i missed a lot of important things in the past.
Then i can think of signal as a point in the space, right?
Then what convolution does to this vector?