Re: Fourier Transform Question for Pulsed CW and 802.11b DSSS


<radio913@xxxxxxx> wrote in message
news:1130736886.809206.125170@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>
> Clay S. Turner wrote:
>>
>> 1st show the FT of a pulse is a sinc function. This integration is
>> straight
>> forward in the integral only needs to be done over the support of the
>> pulse.
>>
>
> Could you show us this "straight forward"
> integration?
>

define pulse

Rect(x) = 1 when |x| <a
= 0 otherwise


FT(w) = integral - infty to infty of Rect(x) exp(-jwx) dx

= integral -a to a of exp(-jwx) dx

= exp(-jwx)/(-jw) evaluated at a minus exp(-jwx)/(-jw)
evaluated at -a

recall Euler's theorem exp(jx) = cos(x)+j sin(x)

cos(wa)-j sin(wa) cos(-wa) - j sin(-wa)
= --------------------- - ----------------------
-jw -jw

Now recall symmetry properties cos(-x) = cos(x) and sin(-x) = -
sin(x)


= (cos(wa) - cos(wa) - j sin(wa) - j sin(wa)) / (-jw)

= 2 sin(wa)/w

= 2a * sin(wa)/(wa)


= 2a * sinc(wa)

IHTH,

Clay










.

Relevant Pages

  • Re: Permittivity and the Poynting vector
    ... >>We have a static voltage V on the cable, ... >>pulse in stead of a constant voltage. ... >>propagate from the source to the load at the speed c. ... This integration isn't ...
    (sci.physics.relativity)
  • Re: Mathematica, Axiom or Maxima?
    ... Sometimes using symbolic integration over numerical can result in a much ... coefficient of a rectangular pulse of period say 2T and the pulse width is T ... So one is able to write a small function which will find Fourier coeff. ...
    (sci.math.symbolic)
  • Re: Mathematica, Axiom or Maxima?
    ... Sometimes using symbolic integration over numerical can result in a much ... coefficient of a rectangular pulse of period say 2T and the pulse width is T ... So one is able to write a small function which will find Fourier coeff. ... Most engineers will hardcode SINCfor this one case, ...
    (sci.math.symbolic)
  • Re: Fourier Transform Question for Pulsed CW and 802.11b DSSS
    ... >> Clay S. Turner wrote: ... >>> 1st show the FT of a pulse is a sinc function. ... This integration is ...
    (comp.dsp)
  • Re: integration to do with the error function...
    ... to do with the error function, ... i intend to do a full integration from ... negative infinite to positive infinite. ... from -infty to infty is twice the integral from 0 to +infty. ...
    (sci.math)