question on meaning of the kernel in Fourier transform

Hi,
  I have been seeking information regarding the meaning of the "e^..."
kernel used in Fourier transform. How does it do the magic of mapping
from time domain to frequency domain. How did it basically originate???
All pointers good exhaustive tutorials are more than welcome.

Hoping for help...

--
TIA..
Jag

jagadeeshbp@gmail.com wrote:
> Hi,
>   I have been seeking information regarding the meaning of the "e^..."
> kernel used in Fourier transform. How does it do the magic of mapping
> from time domain to frequency domain.

The Fourier Transform (FT) is nothing more than a vector basis
representation of a signal. While the theory is valid for both
continuous-time and discrete-time signals, the basic concepts
are easier to follow for the case of finite-length discrete-time
signals.

First a detour to basic calculus in 2D and 3D space.

It is common to define a coordinate system in terms of a set of
orthonormal refernce vectors, and then "navigate" in 2D or 3D
space with respect to these references.

For instance, the point p=(1,3) in 2D space can be represented
as a vector P as

P = 1*i + 3*j

where

   i = [1 0]
   j = [0 1]

But this works regardless of the choise of refernce vectors,
as long as they are orthonormal. For instance the same
point can be represented with respect to a different frame
of reference as

P' = -0.6340*i' + 3.098*j'

where

   i' = [sqrt(3)/2 0.5]^T
   j' = [-0.5 sqrt(3)/2]^T.

The same line of thought apples for vectors of any length N,
they can be represented with respect to whatever orthonormal
frame of reference one finds convenient.

It so happens that the FT can be written as a transform

  X = Wx

where W is an orthonormal matriz that consists only of
the exponential factors.

Rune

Euler's equation can help you see that:

e^ix= cos(x) + i*sin(x)

So the kernel function can also be represented by the basic trigonomic
functions sine and cosine.
The integral can be thought of as a correlation between the kernel
function at a specific frequency and the input signal. To get the whole
spectrum, you perform this 'correlation' across all frequencies.

jagadeeshbp@gmail.com wrote:
> Hi,
>   I have been seeking information regarding the meaning of the "e^..."
> kernel used in Fourier transform. How does it do the magic of mapping
> from time domain to frequency domain. How did it basically originate???
> All pointers good exhaustive tutorials are more than welcome.
> 
> Hoping for help...

I have no time now to write a treatise, but I can probably indicate a 
path you can follow.

Every periodic function of time can be expressed as a sum of sines and 
cosines. If the period is P, the frequencies of the sines and cosines 
are 1/P, 2/P 3/P ... {no limit}.

There is a famous relation between trigonometry and imaginary 
exponentials due to Euler: exp(ix) = cos(x) + i*sin(x). This can be used 
to simplify the mathematics of harmonic analysis, halving the number of 
equations.

I'm sure I omitted something crucial to your understanding, so ask more 
questions.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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jagadeeshbp@gmail.com wrote:

 >   I have been seeking information regarding the meaning of the "e^..."
 > kernel used in Fourier transform. How does it do the magic of mapping
 > from time domain to frequency domain. How did it basically originate???
 > All pointers good exhaustive tutorials are more than welcome.

I am not so sure what you are asking about.  If it is specifically
the properties of the complex exponential others have answered.

Otherwise, look for discussions of orthogonal polynomials, and basis
sets.  The Fourier transform is related to solutions of differential
equations.  Different equations with different boundary conditions
require different basis functions to be used in the corresponding
transform.  Problems involving cylindrical symmetry usually have
Bessel functions as solutions, for example.

Sin and cosine, or the imaginary exponential, are solutions to
the simplest second order differential equation with the simplest
weighting function, and so turn out to be important in the largest
variety of cases.  Otherwise it is just one of a large number
of transforms used in problems of mathematical physics.

-- glen

Like a lot of mathematics, it's to make life easier, and
like a lot of mathematics, you're not really learning
anything new, but just a different way of writing down
something that you already know.....

Sin and Cos are a nuisance in The Calculus - you have to keep
changing the function, and sometimes you have to negate and
sometimes you don't.

Remember that e^(jx) = cos (x) +j(sin(x)), which
then gives us cos (x) = (e^(jx)+e^(-jx))/2, with sin(x)
being something similar?

Well, you know that Fourier splits up time-based functions
into combinations of sines and cosines (still a time based
function) and that the amplitude of those sines and cosines
gives us the frequency response? Well, in order to make
life more simple, what we do is to "split up" those sines
and cosines into their e^(+/-jx) components.

Now all we ahve to deal with are exponentials which are
easy to differentiate and integrate in The Calculus.


jagadeeshbp@gmail.com wrote:
> Hi,
>   I have been seeking information regarding the meaning of the "e^..."
> kernel used in Fourier transform. How does it do the magic of mapping
> from time domain to frequency domain. How did it basically originate???
> All pointers good exhaustive tutorials are more than welcome.
> 
> Hoping for help...
> 
> --
> TIA..
> Jag

Hi all,
  Thank you all for the replies. I had seen the Euler's formula
previously. So that itself made me wonder, how come multiplication of
f(t) can take it from the time domain to frequency domain.

 Again a very basic question......Is there a proof for the statement:
all periodic functions can be split up to a combination of sine and
cosine waves?

 Also the vector/matrix based approach stated: X=Wx is something still
obscure for me. Sorry, I am not an expert in all the math involved. But
still if someone can clarify a little more, I'll be more than
happy......

--
TIA....
Jag.

in article 1125633745.016921.33470@g44g2000cwa.googlegroups.com,
jagadeeshbp@gmail.com at jagadeeshbp@gmail.com wrote on 09/02/2005 00:02:

> I had seen the Euler's formula previously.

that's what makes the kernel e*(j*w*t) have something to do with frequency,
since looking at Euler a different way:

    cos(w*t) = ( e^(+j*w*t) + e^(-j*w*t) )/2

so it's really the case that a real frequency component of frequency, w, is
actually made up of a component at w and another component at -w.  ("w" is
read "omega").

> So that itself made me wonder, how come multiplication of
> f(t) can take it from the time domain to frequency domain.

because when you multiply a component in x(t) with e^(-j*w*t), the component
of x(t) at +w, that is e^(+j*w*t), will become a constant

      (e^(+j*w*t) * e^(-j*w*t)) = e^(j*0) = 1

but all other frequency components will still have an "AC" component to
them.  the component that becomes constant (by multiplying by e^(-j*w*t))
remains after averaging (or integrating) but the other components that,
after multiplying by e^(-j*w*t) remain AC, those components will integrate
to zero.  so after averaging, what you have left is the strength of the
component at e^(+j*w*t).

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

At your level of enquiry, there isn't a proof as  such, but
you can demonstrate to yourself a couple of interesting
examples using Excel. Although it is very good practice to
seek a mathematical proof for each stage of your education,
you won't understand the proof until you have qualified
at degree level in maths! (Indeed, some of those who would
be the authors and experts in this NG don't understand the
mathematical basis for claiming that sampled signals are
a simple multiple of the Diracian Unit Impulse, and yet
they base their whole career on this "fact"!)


Use Excel to plot a sine wave.

Then add to your plot 3 times the frequency at 1/3 the amplitude.

Then add to your plot 5 times the original frequency at 1/5 the
amplitude.

And keep doing this until you are fed up, with all the odd harmonics.

You will see that your final graph is getting very close
to a square wave, thus showing that a square wave is made up of
the odd harmonics.

Now do the same exercise, but with the even harmonics, 2 at 1/2,
4 at 1/4 and so on, and see what you end up with....a triangular
wave.

The proof that you seek lies in using vectors. Just as a single
point can be represented in 3D space by 3 axes nominally at right
angles, so the graph of a complete function can be represented
by adding together a number of simpler graphs from different
axes (Yes! Why not? because every point on the graph can be
represented from the 3 axes, then the whole graph can be
represented by the bringing together of all the points).

Now, it's not so much the "at right angles" which is important,
but "orthogonality", a concept which has the same effect in The
Calculus.

You may have one further stumbling block, which might be
expressed as, "OK, I understand the 3D axes, but an _INFINITE_
series of sines and cosines?"

Yes - you're right to protest, but in practice it's not
infinite - you only have to keep adding in more harmonics
until the difference between two is less than the "space"
betwen two electrons, at which point you stop adding because
the effect is lost in the electrical noise. Nevertheless, you'll
still perceive your square and trinagular waves!

HTH.



jagadeeshbp@gmail.com wrote:
>  Again a very basic question......Is there a proof for the statement:
> all periodic functions can be split up to a combination of sine and
> cosine waves?

That's an interesting way of looking at it, not seen that before,
and much simpler than coming in with the sines and cosines
approach. Perhaps our first introduction to Fourier should
be through the - simpler - exponential approach with sines
and cosines only being introduced through Euler?

While I have your attention, perhaps you could answer the
following question which arises from your oft-repeated
article on sampling and reconstruction (I suspect that you
don't really have a clue which is why you keep ignoring the
question).....

Where did the factor of "T" come from in your opening lines
about sampling where you claim that the comb of Diracian
impulses is mutliplied by the time between the impulses?

Assuming that we were able to generate a Diracian, and then produce
a comb of them by delays and superposition, there wouldn't be a factor
of "T" in such superposition, so where does yours come from?

Consider a 16-bit ADC capable of 100 M Samples per sec. In the first
instance we'll use it to sample a geophysical signal of bandwidth
limited
to 300 HZ and sample at 1 kHz, with suitable analogue instrumentation
to match the input signal to the full range of the ADC. If we now
keep the circuit the same, but now sample at 65.536 MHZ, your claimed
factor of "T" will result in the 16 bit range being compressed down
to one bit. We know this doesn't happen -there will be more samples,
but they'll still be of exactly the same magnitude and exactly
the same 16-bit range, and therefore there isn't a factor of "T"
as you would claim.

Do you know the answer?


robert bristow-johnson wrote:
>
> because when you multiply a component in x(t) with e^(-j*w*t), the component
> of x(t) at +w, that is e^(+j*w*t), will become a constant
>
>       (e^(+j*w*t) * e^(-j*w*t)) = e^(j*0) = 1
>
> but all other frequency components will still have an "AC" component to
> them.  the component that becomes constant (by multiplying by e^(-j*w*t))
> remains after averaging (or integrating) but the other components that,
> after multiplying by e^(-j*w*t) remain AC, those components will integrate
> to zero.  so after averaging, what you have left is the strength of the
> component at e^(+j*w*t).