fourier transform of impulse train

Hi everybody..
could any one plz explain me how to get the spectra of dirac com
function.my doubt is that what type of signals are fourier
transformable?Can we take fourier transform for a sine/cosine signal.I get
confused vth fourier series and transform while apllying for a signal.
Thanks for all who gonna reply.
Regards,
RAMA

On Feb 22, 12:51 pm, "ramaa" <ramaravi_sh...@yahoo.com> wrote:
> Hi everybody..
> could any one plz explain me how to get the spectra of dirac com
> function.my doubt is that what type of signals are fourier
> transformable?Can we take fourier transform for a sine/cosine signal.I get
> confused vth fourier series and transform while apllying for a signal.
> Thanks for all who gonna reply.


You can view the Fourier transform of a time-domain impulse train as
the frequency spectrum of ideal time-domain sampling of x(t) = 1.  The
sampling theorem demonstrates that the frequency spectrum of a sampled
process must be periodic with period 1/T (T is the sampling period).
However, we also know that the Fourier transform of x(t) = 1 is an
impulse.  Therefore, the result is also a periodic impulse train.

Of course, there are the perennial questions about when it is
allowable to use dirac distributions in Fourier integrals.  I'll leave
that discussion to others.


--
Oli

On Feb 22, 7:51 am, "ramaa" <ramaravi_sh...@yahoo.com> wrote:
> Hi everybody..
> could any one plz explain me how to get the spectra of dirac com
> function.my doubt is that what type of signals are fourier
> transformable?Can we take fourier transform for a sine/cosine signal.I get
> confused vth fourier series and transform while apllying for a signal.
> Thanks for all who gonna reply.
> Regards,
> RAMA

Try finding the book "Dr. Euler's Fabulous Formula" by Paul Nahin. It
will show many wonderful details of how to do what you are asking. The
book is an extension of Nahin's earlier book "An imaginary tale - the
story of the square root of minus one." In fact I would heartily
recommond both of these books to serious DSPers who want to understand
the math behind DSP.

Clay

On Feb 22, 8:36 am, "Oli Charlesworth" <c...@olifilth.co.uk> wrote:
> On Feb 22, 12:51 pm, "ramaa" <ramaravi_sh...@yahoo.com> wrote:
>
> > could any one plz explain me how to get the spectra of dirac com
> > function.my doubt is that what type of signals are fourier
> > transformable?Can we take fourier transform for a sine/cosine signal.I get
> > confused vth fourier series and transform while apllying for a signal.
> > Thanks for all who gonna reply.
>
> You can view the Fourier transform of a time-domain impulse train as
> the frequency spectrum of ideal time-domain sampling of x(t) = 1.  The
> sampling theorem demonstrates that the frequency spectrum of a sampled
> process must be periodic with period 1/T (T is the sampling period).
> However, we also know that the Fourier transform of x(t) = 1 is an
> impulse.  Therefore, the result is also a periodic impulse train.
>
> Of course, there are the perennial questions about when it is
> allowable to use dirac distributions in Fourier integrals.  I'll leave
> that discussion to others.

the Fourier Transform of exp(j*2*pi*f0*t) is  delta(f-f0)

the Fourier series representation of the impulse train

       SUM delta(t - n*T)
        n
is

      SUM 1/T  exp(j*2*pi*(k/T)*t)
        k

so the FT of the impulse train is

      SUM 1/T  delta(f - (k/T))
        k


r b-j

"ramaa" <ramaravi_shree@yahoo.com> wrote in message 
news:Kfmdna3RO6xeDEDYnZ2dnUVZ_hudnZ2d@giganews.com...
> Hi everybody..
> could any one plz explain me how to get the spectra of dirac com
> function.my doubt is that what type of signals are fourier
> transformable?Can we take fourier transform for a sine/cosine signal.I get
> confused vth fourier series and transform while apllying for a signal.
> Thanks for all who gonna reply.
> Regards,
> RAMA

Rama,

I like to think of it this way:

The Fourier Series is a special case of the Fourier Transform as follows:

Any (at all reasonable) time domain signal, sampled or not, finite or of 
infinite extent has a Fourier Transform.  Thereafter, we can qualify the 
nature of this Fourier Transform as follows:

- If the time domain signal is periodic then the resulting Fourier Transform 
will be discrete - that is, made up of Diracs if you will.  They will be at 
f=0 and at the integer multiples of the reciprocal of the period (i.e. the 
fundamental and the harmonic frequencies).

- Correspondingly, if the time domain signal is regularly sampled (that is, 
*it* is discrete) then its Fourier Transform is periodic at multiples of the 
sampling frequency.

- And, if the time domain signal is regularly sampled (it is discrete) , 
then it's Fourier Transform integral can be expressed as a discrete sum.  It 
doesn't *have* to be but it's convenient.  Then it's in a Discrete Fourier 
Transform form.

So, the Fourier Series is a special case and only applies when the function 
being transformed is periodic.

You might take a look at:
http://groups.google.com/group/comp.dsp/browse_thread/thread/cde01fb6543a400b/2656c0f145537e05?lnk=gst&q=discrete+%3C%3E+continuous+marshall&rnum=1#2656c0f145537e05

and similar posts.  It's good to have these mental "cartoons" in mind.

Fred

Fred Marshall wrote:

(snip of question on Fourier transforms)

> I like to think of it this way:

> The Fourier Series is a special case of the Fourier Transform as follows:

> Any (at all reasonable) time domain signal, sampled or not, finite or of 
> infinite extent has a Fourier Transform.  Thereafter, we can qualify the 
> nature of this Fourier Transform as follows:

> - If the time domain signal is periodic then the resulting Fourier Transform 
> will be discrete - that is, made up of Diracs if you will.  They will be at 
> f=0 and at the integer multiples of the reciprocal of the period (i.e. the 
> fundamental and the harmonic frequencies).

Since 0 is an integer, f=0 is included in the general case.

> - Correspondingly, if the time domain signal is regularly sampled (that is, 
> *it* is discrete) then its Fourier Transform is periodic at multiples of the 
> sampling frequency.

In case it isn't so obvious, you can add the case of the regularly 
sampled periodic function with a regularly sampled periodic transform,
the combination of the two previous statements.

You could also add the even functions have real transforms,
(and by symmetry real functions have even transforms), odd
functions have imaginary transforms.

> - And, if the time domain signal is regularly sampled (it is discrete) , 
> then it's Fourier Transform integral can be expressed as a discrete sum.  It 
> doesn't *have* to be but it's convenient.  Then it's in a Discrete Fourier 
> Transform form.

It is also convenient for people who don't believe in delta functions.
(Not that I am one, but there are some out there.)

> So, the Fourier Series is a special case and only applies when the function 
> being transformed is periodic.

-- glen