FPGA Central

Hi everyone, this is my first time posting and I hope that its in the righ
place and not violating any rules. I have already tried searching abou
this topic on this website but I was unsuccessful in finding what I wa
seeking for.

This is for my independent study and my goals are to prove tw
relationships.
1. Prove that the Fourier Series (FS) gives the same results as th
Continuous-Time Fourier Transform (CTFT). Using the FS's coefficient
along with the X(jw) of the CTFT.

From my research, I know that we have to use the Dirc Comb and convolve i
with the FS, but from there I'm just lost


2. Prove that the Discrete-Time Fourier Transform is the same as CTFT.

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't kno
what theorem to use to prove this relationship.

So can someone help me out or guide me to the right direction?

Thanks!

On 3 Mrz., 22:31, "dr3amr2" <dnguy...@du.edu> wrote:
> Hi everyone, this is my first time posting and I hope that its in the right
> place 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.

>
> 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't know
> 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

On Mar 4, 12:39 pm, Andor <andor.bari...@gmail.com> wrote:
> On 3 Mrz., 22:31, "dr3amr2" <dnguy...@du.edu> wrote:
>
> > Hi everyone, this is my first time posting and I hope that its in the right
> > place 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.
>
>
>
> > 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't know
> > what theorem to use to prove this relationship.
>

this is essentially the sampling theorem. the CTFT transforms from
one domain (let's call it the "time domain") to a reciprocal domain
(the "frequency domain"). when you sample uniformly in one domain,
that has the effect of copying, shifting, and overlap-adding in the
other domain.

i've posted a reasonably good proof of it (if you can accept the
Neanderthal engineering understanding of the dirac impulse function)
but it has found it's way to Wikipedia

http://en.wikipedia.org/wiki/Nyquist...or_the_theorem

before they kicked me out.

r b-j

On 2008-03-04 13:39:24 -0400, Andor <andor.bariska@gmail.com> said:

> On 3 Mrz., 22:31, "dr3amr2" <dnguy...@du.edu> wrote:
>> Hi everyone, this is my first time posting and I hope that its in the righ
> t
>> place 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.
>
>>
>> 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'
> t know
>> 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".

robert bristow-johnson wrote:\
> i've posted a reasonably good proof of it (if you can accept the
> Neanderthal engineering understanding of the dirac impulse function)
> but it has found it's way to Wikipedia
>
> http://en.wikipedia.org/wiki/Nyquist...or_the_theorem
>
> before they kicked me out.

Hey rb-j,

Since you've been black-balled at Wikipedia, have you considered
directing your efforts to Wikibooks? Though it's a sister project, it
has an independent community which is a) much smaller, and b) not so
anal. Also, they have a book or two on DSP that could benefit from
comp.dsp input.

I just took a brief look at this one, and found it... lacking.
http://en.wikibooks.org/wiki/Digital_Signal_Processing

--
Jim Thomas Principal Applications Engineer Bittware, Inc
jthomas@bittware.com http://www.bittware.com (603) 226-0404 x536
When you have little data, you have lots of freedom. - Olafur Ingolfssonr

Jim Thomas wrote:
> robert bristow-johnson wrote:\
>> i've posted a reasonably good proof of it (if you can accept the
>> Neanderthal engineering understanding of the dirac impulse function)
>> but it has found it's way to Wikipedia
>>
>> http://en.wikipedia.org/wiki/Nyquist...or_the_theorem
>>
>>
>> before they kicked me out.
>
> Hey rb-j,
>
> Since you've been black-balled at Wikipedia, have you considered
> directing your efforts to Wikibooks? Though it's a sister project, it
> has an independent community which is a) much smaller, and b) not so
> anal. Also, they have a book or two on DSP that could benefit from
> comp.dsp input.
>
> I just took a brief look at this one, and found it... lacking.
> http://en.wikibooks.org/wiki/Digital_Signal_Processing
>
You are rights. Its seriously lacking. I can tell just from the second
line of the contents being "Using MATLAB". :-)

Steve

On Mar 5, 9:02 am, Jim Thomas <jtho...@bittware.com> wrote:
> robert bristow-johnson wrote:\
> > i've posted a reasonably good proof of it (if you can accept the
> > Neanderthal engineering understanding of the dirac impulse function)
> > but it has found it's way to Wikipedia
>
> >http://en.wikipedia.org/wiki/Nyquist...heorem#Mathema...
>
> > before they kicked me out.
>
> Hey rb-j,
>
> Since you've been black-balled at Wikipedia, have you considered
> directing your efforts to Wikibooks? Though it's a sister project, it
> has an independent community which is a) much smaller, and b) not so
> anal. Also, they have a book or two on DSP that could benefit from
> comp.dsp input.
>
> I just took a brief look at this one, and found it... lacking.http://en.wikibooks.org/wiki/Digital_Signal_Processing
>

i just took a look, actually registered me as "Rbj" (i dunno why WP
and WB require the first letter to be capitalized). but as i look at
the history of this and similar pages (in http://en.wikibooks.org/wiki/Signals_and_Systems
), there appears to be an admin (Whiteknight) who has done a lot of
the text and i'm in no mood for wasting time fighting with an admin
who wrote some, hmmmm, lacking stuff.

especially if he's an admin, it doesn't matter who's right or who
knows the stuff better. it will eventually be a waste of time (the
important substantive changes that need to be made will eventually get
washed out).

so i think i'll pass on this.

r b-j

robert bristow-johnson wrote:
> On Mar 5, 9:02 am, Jim Thomas <jtho...@bittware.com> wrote:
>> robert bristow-johnson wrote:\
>>> i've posted a reasonably good proof of it (if you can accept the
>>> Neanderthal engineering understanding of the dirac impulse function)
>>> but it has found it's way to Wikipedia
>>> http://en.wikipedia.org/wiki/Nyquist...heorem#Mathema...
>>> before they kicked me out.
>> Hey rb-j,
>>
>> Since you've been black-balled at Wikipedia, have you considered
>> directing your efforts to Wikibooks? Though it's a sister project, it
>> has an independent community which is a) much smaller, and b) not so
>> anal. Also, they have a book or two on DSP that could benefit from
>> comp.dsp input.
>>
>> I just took a brief look at this one, and found it... lacking.http://en.wikibooks.org/wiki/Digital_Signal_Processing
>>
>
> i just took a look, actually registered me as "Rbj" (i dunno why WP
> and WB require the first letter to be capitalized). but as i look at
> the history of this and similar pages (in http://en.wikibooks.org/wiki/Signals_and_Systems
> ), there appears to be an admin (Whiteknight) who has done a lot of
> the text and i'm in no mood for wasting time fighting with an admin
> who wrote some, hmmmm, lacking stuff.
>
> especially if he's an admin, it doesn't matter who's right or who
> knows the stuff better. it will eventually be a waste of time (the
> important substantive changes that need to be made will eventually get
> washed out).
>
> so i think i'll pass on this.
>
> r b-j

I dropped Whiteknight a note, and he says that you'd be more than
welcome to alter "his" book. I asked him to leave a message on your
talk page.

I have interacted with him regularly for the past two years, and he's a
pretty reasonable guy. WB doesn't suffer from the crap that WP does, so
I think your fears are unfounded.

--
Jim Thomas Principal Applications Engineer Bittware, Inc
jthomas@bittware.com http://www.bittware.com (603) 226-0404 x536
If a job's not worth doing, it's not worth doing right.

On Mar 4, 12:39*pm, Andor <andor.bari...@gmail.com> wrote:
> On 3 Mrz., 22:31, "dr3amr2" <dnguy...@du.edu> wrote:
>
> > Hi everyone, this is my first time posting and I hope that its in the right
> > place 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.
>
>
>
> > 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't know
> > 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

"What exactly do you want to do?", Andor?

I think he calls it "homework".

Dirk