Re: Proving the relationships between FSCTFT and CTFTDTFT
- From: Gordon Sande <g.sande@xxxxxxxxxxxxxxxx>
- Date: Tue, 04 Mar 2008 20:18:48 GMT
On 2008-03-04 13:39:24 -0400, Andor <andor.bariska@xxxxxxxxx> said:
On 3 Mrz., 22:31, "dr3amr2" <dnguy...@xxxxxx> wrote:Hi everyone, this is my first time posting and I hope that its in the rightplace and not violating any rules. I have already tried searching about
this topic on this website but I was unsuccessful in finding what I was
seeking for.
This is for my independent study and my goals are to prove two
relationships.
1. Prove that the Fourier Series (FS) gives the same results as the
Continuous-Time Fourier Transform (CTFT).
You won't find such a proof.
Using the FS's coefficients
along with the X(jw) of the CTFT.
From my research, I know that we have to use the Dirc Comb and convolve it
with the FS, but from there I'm just lost
2. Prove that the Discrete-Time Fourier Transform is the same as CTFT.
Another proof you wont find.
t know
The relation between DTFT and CTFT in sampling is X(e^jwTs)
X(e^jw)|w=wTs = (1/Ts)(sumation)Xc(j(w-((2*pi*k)/Ts))). I just don'
what theorem to use to prove this relationship.
So can someone help me out or guide me to the right direction?
What exactly do you want to do?
Regards,
Andor
If you know how to wave your hands correctly and then fill in the gaps to make
things rigorous you will find some useful heuristics.
1. convolution with an infinite Dirac comb turns the infinite real line into
the cyclic circle. Otherwise known as going from the usual Fourier transform
to Fourier Series for periodic functions.
2. multiplication by an infinite Dirac comb samples the infinite real line to
yield an infinite sequence. You end up with periodic, or aliased, frequencies.
3. multiplcation by and then convolution with an infinite Dirac comb yields
a periodic sequence (provided you slip a bit a scaling in somewhere to get
more than just one point!). You end up with the finite discrete FT.
The trick behind the heuristic is that the infinite Dirac comb has itself as
its Fourier transform.
If you do not get the hand waving and filling in right then all you get is
mysterious nonsense. It is usually straightforward to go from infinite
to bounded but the nonsense often arises when you try to the other way.
This is the heuristic that says:
time and frequency can be described as
1. periodic or unbounded
2. continuous or discrete
so that
periodic time leads to discrete frequencies
unbounded time leads to continuous frequencies
continuous time leads to unbounded frequencies
discrete time leads to periodic frequencies
where the convolution relates the unbounded and periodic while the
multiplication relates the continuous and discrete.
At one of my summer jobs while an undergraduate there was an engineer
who called the combination of hand waving and filling in the gaps as
doing a common sense engineering proof. The problem is that you can either
learn enough measure theory etc or have some common sense. As they say,
common sense is not all that common!
You facts are essentially correct but it is not not clear what you expect
under the title of "proof".
.
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