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# Discussion Groups | Comp.DSP | interesting insight into impulse definition

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# interesting insight into impulse definition - bharat pathak - 2008-03-11 13:20:00

```Hello All,

When we were deriving ideal HPF equation we came across
interesting definition of impulse,

delta[n] = sin(pi*n)/(pi*n)

Maybe I am getting to understand the sampling theorem....

Regards
Bharat Pathak

Arithos Designs
www.Arithos.com

```
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# Re: interesting insight into impulse definition - Tim Wescott - 2008-03-11 13:40:00

```bharat pathak wrote:
> Hello All,
>
>       When we were deriving ideal HPF equation we came across
>       interesting definition of impulse,
>
>       delta[n] = sin(pi*n)/(pi*n)
>
>       Maybe I am getting to understand the sampling theorem....
>
> Regards
> Bharat Pathak
>
> Arithos Designs
> www.Arithos.com
>
That expression is indeterminate.  The expression

delta[n] =  lim   sin(pi * n + e) / (pi * n + e)
e -> 0

works.

Or, if you use

sinc(x) =   lim   sin(x0) / x0
x0 -> x

then delta[n] = sinc(pi * n)

works also.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html
```
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# Re: interesting insight into impulse definition - robert bristow-johnson - 2008-03-11 15:11:00

```On Mar 11, 1:40 pm, Tim Wescott <t...@seemywebsite.com> wrote:
> bharat pathak wrote:
> > Hello All,
>
> >       When we were deriving ideal HPF equation we came across
> >       interesting definition of impulse,
>
> >       delta[n] = sin(pi*n)/(pi*n)

of course, but with the "removable" singularity explicitly defined as
delta[0]=1.

>
> That expression is indeterminate.

not if you fix the removable singularity.  it says that delta[0]=1 and
for all other integer n, then delta[n]=0.

>  The expression
>
> delta[n] =  lim   sin(pi * n + e) / (pi * n + e)
>             e -> 0
>
> works.

okay, Tim, we're on the same page now.  (another case where i don't
read carefully through the post before starting a response.)

> Or, if you use
>
> sinc(x) =   lim   sin(x0) / x0
>            x0 -> x
>
> then delta[n] = sinc(pi * n)
>
> works also.

that's probably the cleanest way to say it.

r b-j
```
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# Re: interesting insight into impulse definition - Greg Berchin - 2008-03-11 15:24:00

```On Mar 11, 1:20 pm, "bharat pathak" <bha...@arithos.com> wrote:

>       delta[n] = sin(pi*n)/(pi*n)

Maybe I'm reading this too quickly, but I recall all definitions of
delta stating that the amplitude is infinite, with unit area under the
curve.  Doesn't L'Hospital's Rule give an amplitude of one in the
expression above?

Greg
```
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# Re: interesting insight into impulse definition - dbd - 2008-03-11 17:37:00

```On Mar 11, 10:20 am, "bharat pathak" <bha...@arithos.com> wrote:
> Hello All,

>       When we were deriving ideal HPF equation we came across
>       interesting definition of impulse,
>
>       delta[n] = sin(pi*n)/(pi*n)
>
>       Maybe I am getting to understand the sampling theorem....
>
> Regards
> Bharat Pathak
>
> Arithos Designswww.Arithos.com

That equation is not a definition of the delta functional. It is a
relationship that is true of the delta functional acting on the set of
integers as described in common engineering language. In engineering
it is common practice to substitute the values of the delta functional
acting on the real number line for the delta functional itself. See:

http://mathworld.wolfram.com/DeltaFunction.html

There is further confusion added by the use of the expression
'sampling function' for the sync function:

http://mathworld.wolfram.com/SincFunction.html

Dale B. Dalrymple
http://dbdimages.com
```
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# Re: interesting insight into impulse definition - Tim Wescott - 2008-03-11 18:41:00

```Greg Berchin wrote:
> On Mar 11, 1:20 pm, "bharat pathak" <bha...@arithos.com> wrote:
>
>>       delta[n] = sin(pi*n)/(pi*n)
>
> Maybe I'm reading this too quickly, but I recall all definitions of
> delta stating that the amplitude is infinite, with unit area under the
> curve.  Doesn't L'Hospital's Rule give an amplitude of one in the
> expression above?
>
> Greg

The Dirac delta* has an infinite amplitude and an area of one.  It's
very handy for continuous-time analysis.

The Kroneker (SP?) delta is only defined for integer 'times', it has an
amplitude of one for sample zero, and is zero everywhere else.

I certainly hope the OP meant the Kroneker delta, because the above
function (with all due respect for the problem as n approaches zero) is
just a sinc for continuous n.

* An engineer will call it a "function" and scold you for being too
picky if you call it a "functional".  A mathematician will call it a
"functional" and scold you for being too lax if you call it a function.
I'll just avoid either name...

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html
```
______________________________

# Re: interesting insight into impulse definition - Greg Berchin - 2008-03-11 20:15:00

```On Tue, 11 Mar 2008 14:41:18 -0800, Tim Wescott <t...@seemywebsite.com>
wrote:

>I certainly hope the OP meant the Kroneker delta, because the above
>function (with all due respect for the problem as n approaches zero) is
>just a sinc for continuous n.

Exactly the point that I was trying to make.  I, however, interpreted
"delta" to mean "Dirac delta"; hence my confusion.

>* An engineer will call it a "function" and scold you for being too
>picky if you call it a "functional".  A mathematician will call it a
>"functional" and scold you for being too lax if you call it a function.
>  I'll just avoid either name...

Yeah, no sense grabbing that tiger by the tail again.

Greg
```
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# Re: interesting insight into impulse definition - robert bristow-johnson - 2008-03-11 21:52:00

```On Mar 11, 8:15 pm, Greg Berchin <gberc...@comicast.net> wrote:
> On Tue, 11 Mar 2008 14:41:18 -0800, Tim Wescott <t...@seemywebsite.com>
> wrote:
>
> >I certainly hope the OP meant the Kroneker delta, because the above
> >function (with all due respect for the problem as n approaches zero) is
> >just a sinc for continuous n.
>
> Exactly the point that I was trying to make.  I, however, interpreted
> "delta" to mean "Dirac delta"; hence my confusion.
>
> >* An engineer will call it a "function" and scold you for being too
> >picky if you call it a "functional".  A mathematician will call it a
> >"functional" and scold you for being too lax if you call it a function.
> >  I'll just avoid either name...
>
> Yeah, no sense grabbing that tiger by the tail again.

why not?  tigers are cute and fuzzy and warm and cuddly.  especially
when they lay waste to everything they taught up in our continuous-
time signals and systems course.

maybe i'm a glutton for punishment.

r b-j
```
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# Re: interesting insight into impulse definition - Rune Allnor - 2008-03-12 01:30:00

```On 11 Mar, 18:40, Tim Wescott <t...@seemywebsite.com> wrote:
> bharat pathak wrote:
> > Hello All,
>
> >       When we were deriving ideal HPF equation we came across
> >       interesting definition of impulse,
>
> >       delta[n] = sin(pi*n)/(pi*n)
>
> >       Maybe I am getting to understand the sampling theorem....
>
> > Regards
> > Bharat Pathak
>
> > Arithos Designs
> >www.Arithos.com
>
> That expression is indeterminate.  The expression
>
> delta[n] =  lim   sin(pi * n + e) / (pi * n + e)
>             e -> 0
>
> works.
>
> Or, if you use
>
> sinc(x) =   lim   sin(x0) / x0
>            x0 -> x
>
> then delta[n] = sinc(pi * n)
>
> works also.

Or you could use the 'box' function (view with fixed/width
font)

| 1/T     -T/2 <= x <= T/2
Delta(x) = lim |
T->0| 0       elsewhere

As you can see, there onr takes a limit in all these
expressions for Delta(x). This limit is the key, once you
grasp the significance of that you gain some insight.

To give you the big picure, there is a mathematical term
'function space.' There are lots of such spaces, one of
them is 'the collection of functions f(x) which has a
finite integral.' Usually, these definitions specfiy
the functions that belong in a bit more detail, but leave
that for now.

The box function is easily seen to have a finite integral,

T/2                  T/2
integral 1/T dx = [x/T]     = 1
-T/2                 -T/2

so it belongs to this space of integrable functions.
Then there is the distinction between closed and open spaces.
In a 'closed' linear space, all Cauchy points also belong
to the space. A Cauchy point is the limit of a converging
series, e.g. Delta(x) is the Cauchy points of all the
limits above.

Now, this space I am talking about, L2, is closed. That can
be proved without using the Delta(x). Since all these
sequences which belong in L2 converge to Delta(x) and
L2 is a closed space, it follows that Delta(x) belongs
to L2 and therefore can be used as any other function
in L2.

That's the maths part of it. The technical details are
involved, but the big picture is nice and tidy. It seems
that lots of problem occur when people try to assign
'physical' meaning to this (I prefer the term 'intuitive'
rather than 'physical').

In that sense you are approaching a minefield, so handle
with care!

Rune
```
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# Re: interesting insight into impulse definition - Steve Underwood - 2008-03-12 02:40:00

```robert bristow-johnson wrote:
> On Mar 11, 8:15 pm, Greg Berchin <gberc...@comicast.net> wrote:
>> On Tue, 11 Mar 2008 14:41:18 -0800, Tim Wescott <t...@seemywebsite.com>
>> wrote:
>>
>>> I certainly hope the OP meant the Kroneker delta, because the above
>>> function (with all due respect for the problem as n approaches zero) is
>>> just a sinc for continuous n.
>> Exactly the point that I was trying to make.  I, however, interpreted
>> "delta" to mean "Dirac delta"; hence my confusion.
>>
>>> * An engineer will call it a "function" and scold you for being too
>>> picky if you call it a "functional".  A mathematician will call it a
>>> "functional" and scold you for being too lax if you call it a function.
>>>  I'll just avoid either name...
>> Yeah, no sense grabbing that tiger by the tail again.
>
> why not?  tigers are cute and fuzzy and warm and cuddly.  especially
> when they lay waste to everything they taught up in our continuous-
> time signals and systems course.
>
> maybe i'm a glutton for punishment.

I puzzles me why people who have unpopular views about sampled systems
can't be a little bit more discrete about them. :-\

Steve
```
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