# [PD] unit impulse without [dirac~]

Charles Henry czhenry at gmail.com
Sat Nov 17 20:06:36 CET 2007

```Personally, I like to use the sinc/rectangular impulse functions to
define the dirac delta, because it has a handy symmetry with fourier
analysis.

our rectangular function, in the time domain is
g(t)={1/(2T) , -T<t<T
0     , elswhere
which has fourier transform,
G(f)=sinc(f*T)

in the lim T->0
g(t)->dirac(t)
and lim T->0  G(f)=lim T->0 sinc(f*T) -> 1 for all f

likewise for
h(t)=fs*sinc(fs*t)
H(f)={ 1,  -fs/2 < f < fs/2
0,   elsewhere

and in the lim fs->infinity
g(t)->dirac(t)
G(f)-> 1  for all f

Chuck

On Nov 16, 2007 6:03 PM, Uğur Güney <ugurguney at gmail.com> wrote:
> # What is preserved is the area under the function, it is equal to one (its
> just 1, with no physical dimensions like energy. So, if x axis has the
> dimensions of L(ength), y-axis has the dimension of 1/L). When you let the
> bottom side's length of a triangle shaped function to go to zero, for
> preserving its area, its height goes to infinity. Dirac Delta Function is
> defined as this limiting case, other limiting case where the height goes to
> zero and so length goes to infinity is unrelated to Dirac Delta Function.
> (actually it is not a function, but a distribution. :-) Its behavoir is very
> pathologic for a function. Distributions are more general.)
> # The idea behind all of this cumbersome things become useful when you
> multiply a function with Dirac Delta and take the integral:
>
> integral{f(t)*delta(t-a)*dt} = f(a)
>
> # So, Delta function takes a sample (a snapshot) from the function at time
> a.
> # It can be used as a mathematical approximations of some real signals
> (there can't be any physical quantity of which amplitude is infinity) or a
> tool for solving problems.
> -uğur-
>
>
>
>
>  On Nov 17, 2007 11:01 AM, Andy Farnell <padawan12 at obiwannabe.co.uk> wrote:
> > Is it correct to say that the Dirac impulse preserves
> > energy, as it tends towards zero time length the amplitude goes to
> infinity and if we
> > squashed its amplitude to zero it would be infinitely long? In which case
> Dirac impulses
> > are theoretical and not practical digital signals?
> >
>
>
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```