Lightning and Fourier transform of an impulse

Dear list
I would like to ask the group, about a certain doubt I have. In an
examination, the professor asked 'why lighting causes disruptions to
TV and Radio signals?'. The answer supposedly is, since a lighting can
be seen as an electric 'impulse' signal (on the time domain), then in
the frequency domain we have white noise, thus, generating noise in
all frequencies. For the time being, let's ignore the electromagnetics
involved, and let only focus on the Fourier transform part.

The following explanation may be wrong, but the fact that the Fourier
transform(FT) of an impulse is white noise is due to the fact that for
FT 'localizing' a very short and abrupt event in time is very
difficult (we can see this in the Gibbs phenomenon), so all bases
(sines and cosines) are summed up and we end with the continuous
frequency representation, or 'white noise'.

However, a more advanced transform (like a Gabor transform), shows in
a Time frequency plane that the impulse is localized in time and
frequency.

I am highly confused of how to see on a real physical sense the
Fourier transform of an impulse function . Any insights?

regards

Juan Pablo

jnarino wrote:
> Dear list
> I would like to ask the group, about a certain doubt I have. In an
> examination, the professor asked 'why lighting causes disruptions to
> TV and Radio signals?'. The answer supposedly is, since a lighting can
> be seen as an electric 'impulse' signal (on the time domain), then in
> the frequency domain we have white noise, thus, generating noise in
> all frequencies. For the time being, let's ignore the electromagnetics
> involved, and let only focus on the Fourier transform part.
> 
> The following explanation may be wrong, but the fact that the Fourier
> transform(FT) of an impulse is white noise is due to the fact that for
> FT 'localizing' a very short and abrupt event in time is very
> difficult (we can see this in the Gibbs phenomenon), so all bases
> (sines and cosines) are summed up and we end with the continuous
> frequency representation, or 'white noise'.
> 
> However, a more advanced transform (like a Gabor transform), shows in
> a Time frequency plane that the impulse is localized in time and
> frequency.
> 
> I am highly confused of how to see on a real physical sense the
> Fourier transform of an impulse function . Any insights?
> 
> regards
> 
> Juan Pablo

Sometimes when I run into problems like this I start with the math and 
ponder it until the objecting brain cells are beaten into submission by 
their neighbors.

 From a mathematical standpoint this is easy: because the impulse has an 
infinitesimal duration, none of the sine waves that you compare it to 
with the Fourier integral average out, so they all have some non-zero 
value coming out of the integral.  Only because the impulse has infinite 
energy does it have energy at each of the frequencies you try.

To draw this into the physical world, think of the impulse as an 
approximation.  Real infinite-energy events don't exist, nor do real 
infinitesimal time events.  What you simulate with the impulse is a 
really short, really strong event compared to what you're working with 
(like -- a lightning strike near a radio receiver).

So your real signal has some extent in time, and some finite energy. 
But if you tried to do your integrals with it it'd be a pain in the 
behind.  So you just take the area of your real signal, and say that 
it's a Dirac delta functional and proceed.

One way to gain some intuition on this is to consider a signal whose 
time-domain representation is a rectangular signal, so it has width T 
and height 1/T.  Take it's Fourier transform -- you get that familiar 
sinc function.  Now take the limit of the sinc function as T goes to 
zero -- starting to look familiar?

-- 

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

jnarino wrote:
> Dear list
> I would like to ask the group, about a certain doubt I have. In an
> examination, the professor asked 'why lighting causes disruptions to
> TV and Radio signals?'. The answer supposedly is, since a lighting can
> be seen as an electric 'impulse' signal (on the time domain), then in
> the frequency domain we have white noise, thus, generating noise in
> all frequencies. For the time being, let's ignore the electromagnetics
> involved, and let only focus on the Fourier transform part.
> 
> The following explanation may be wrong, but the fact that the Fourier
> transform(FT) of an impulse is white noise is due to the fact that for
> FT 'localizing' a very short and abrupt event in time is very
> difficult (we can see this in the Gibbs phenomenon), so all bases
> (sines and cosines) are summed up and we end with the continuous
> frequency representation, or 'white noise'.
> 
> However, a more advanced transform (like a Gabor transform), shows in
> a Time frequency plane that the impulse is localized in time and
> frequency.
> 
> I am highly confused of how to see on a real physical sense the
> Fourier transform of an impulse function . Any insights?

First of a all, you mean "lightning", the electrical flash that produces 
thunder.

Second, you don't need to think about Fourier transforms, but if you do, 
recognize the while white noise and an impulse have the same power 
spectrum, they do not have the same spectrum. (The inverse transform of 
white noise's spectrum is white noise, but the inverse transform of an 
impulse's spectrum is an impulse. Each to its own kind.)

The spectrum of a lightning strike is very broad because, as you 
recognized, it is quite impulsive. A receiver is affected by any signal 
at the frequency it is tuned to, including any external sources. 
Lightning is one such.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Hi Juan,
Consider a pulse with a finite width.  The shape doesn't matter-- it might
be rectangular, Gaussian, etc.  The frequency spectrum of this pulse will
have a value of 1 at DC, and fall off at higher frequencies.  (There are
more details, but this is a 6 sentence explanation).  

Now, squeeze the pulse and make it narrower. This makes the frequency
spectrum broader.  Here's a reference:  

http://www.dspguide.com/ch10/4.htm

When you have squeezed the pulse into an impulse, the frequency spectrum
has been broadened so much that all is left is a contant value of one.  
Regards,
Steve

But the frequency spectrum exists over all time, not just during the
impulse.

So I don't think this can be used to explain why the lightning
disrupts radio for a short time.

Chris
======================
Chris Bore
BORES Signal Processing
www.bores.com


On Aug 5, 9:39&#4294967295;am, jnarino <jnar...@gmail.com> wrote:
> Dear list
> I would like to ask the group, about a certain doubt I have. In an
> examination, the professor asked 'why lighting causes disruptions to
> TV and Radio signals?'. The answer supposedly is, since a lighting can
> be seen as an electric 'impulse' signal (on the time domain), then in
> the frequency domain we have white noise, thus, generating noise in
> all frequencies. For the time being, let's ignore the electromagnetics
> involved, and let only focus on the Fourier transform part.
>
> The following explanation may be wrong, but the fact that the Fourier
> transform(FT) of an impulse is white noise is due to the fact that for
> FT 'localizing' a very short and abrupt event in time is very
> difficult (we can see this in the Gibbs phenomenon), so all bases
> (sines and cosines) are summed up and we end with the continuous
> frequency representation, or 'white noise'.
>
> However, a more advanced transform (like a Gabor transform), shows in
> a Time frequency plane that the impulse is localized in time and
> frequency.
>
> I am highly confused of how to see on a real physical sense the
> Fourier transform of an impulse function . Any insights?
>
> regards
>
> Juan Pablo

> On Aug 5, 9:39 am, jnarino <jnar...@gmail.com> wrote:

>>I would like to ask the group, about a certain doubt I have. In an
>>examination, the professor asked 'why lighting causes disruptions to
>>TV and Radio signals?'. The answer supposedly is, since a lighting can
>>be seen as an electric 'impulse' signal (on the time domain), then in
>>the frequency domain we have white noise, thus, generating noise in
>>all frequencies. For the time being, let's ignore the electromagnetics
>>involved, and let only focus on the Fourier transform part.
(snip)

>>I am highly confused of how to see on a real physical sense the
>>Fourier transform of an impulse function . Any insights?

Chris Bore wrote:

 > But the frequency spectrum exists over all time, not just during the
 > impulse.

 > So I don't think this can be used to explain why the lightning
 > disrupts radio for a short time.

It is even worse than that.

Note that the Fourier transform of an impulse function
includes both positive and negative time.

But in reality there are no impulse functions,
only approximations to them.  The frequency range of
the Fourier transform depends on how sharp the edge of
the approximate impulse function is.   That is one reason
that lightning affects AM radio much more than FM radio.

-- glen

Chris Bore wrote:
> But the frequency spectrum exists over all time, not just during the
> impulse.
> 
> So I don't think this can be used to explain why the lightning
> disrupts radio for a short time.

Chris,

The lightning isn't really an impulse. Some have durations of 
milliseconds, few less than tens of microseconds. That's more than 
enough to localize the interference in time.

   ...

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Chris Bore wrote:
> But the frequency spectrum exists over all time, not just during the
> impulse.
> 
> So I don't think this can be used to explain why the lightning
> disrupts radio for a short time.

Chris,

The lightning isn't really an impulse. Some have durations of 
milliseconds, few less than tens of microseconds. That's more than 
enough to localize the interference in time.

   ...

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

On Tue, 5 Aug 2008 01:39:47 -0700 (PDT), jnarino <jnarino@gmail.com>
wrote:

>Dear list
>I would like to ask the group, about a certain doubt I have. In an
>examination, the professor asked 'why lighting causes disruptions to
>TV and Radio signals?'. The answer supposedly is, since a lighting can
>be seen as an electric 'impulse' signal (on the time domain), then in
>the frequency domain we have white noise, thus, generating noise in
>all frequencies. For the time being, let's ignore the electromagnetics
>involved, and let only focus on the Fourier transform part.
>
>The following explanation may be wrong, but the fact that the Fourier
>transform(FT) of an impulse is white noise is due to the fact that for
>FT 'localizing' a very short and abrupt event in time is very
>difficult (we can see this in the Gibbs phenomenon), so all bases
>(sines and cosines) are summed up and we end with the continuous
>frequency representation, or 'white noise'.
>
>However, a more advanced transform (like a Gabor transform), shows in
>a Time frequency plane that the impulse is localized in time and
>frequency.
>
>I am highly confused of how to see on a real physical sense the
>Fourier transform of an impulse function . Any insights?
>
>regards
>
>Juan Pablo

I wish I'd seen this yesterday when you posted, but I'll try to catch
up.

As others have pointed out, the FT of an impulsive function isn't
noise, but it generally does have energy spread out over a wide range
of frequencies.    In the case of lightning the power levels involved
are quite high, so it stands to reason that systems sensing spectrum
in the region affected by the lightning would respond to it.

So there are, IMHO, two major things at work 1) Impulsive waveforms
have broad power spectra,   2) There's a LOT of energy in a lightning
strike.

;)

Eric Jacobsen
Minister of Algorithms
Abineau Communications
http://www.ericjacobsen.org

Blog: http://www.dsprelated.com/blogs-1/hf/Eric_Jacobsen.php

Eric Jacobsen <eric.jacobsen@ieee.org> writes:
> [...]
> So there are, IMHO, two major things at work 1) Impulsive waveforms
> have broad power spectra,   2) There's a LOT of energy in a lightning
> strike.

God's impulse radio??? :)
-- 
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