Lightning and Fourier transform of an impulse

Jerry Avins wrote:

> Chris has a valid objection. The sum of the eternal frequencies present
> is indeed zero at times well removed from the generating impulse.
> However, it is not obvious that a narrow-band subset of them is also
> self canceling. 

Yes well stories involving infinity do tend to have lots of dramatic
plot twists and turns - no doubt about that. 
	What you are saying is that when you attenuate the frequencies that
are being used to cancel other frequencies (and thus make the signal
zero extending into infinity) then the canceled frequencies are going
to come back like Lazurus from the dead. 
	A radio doesn't so much attenuate the frequencies it doesn't need as
amplify the ones it does - No?


-jim



>Nevertheless, if the band is wide enough, the
> cancellation is in fact largely complete. Consider a variable-bandwidth
> filter excited by an impulse. As the bandwidth decreases, the
> cancellation becomes poorer, and the duration of the filtered output
> increases. When the bandwidth becomes very narrow -- the Q is made very
> high -- the output can persist for a long time. We say that the filter
> "rings". It actually does ring in the time domain, but in frequency, it
> is merely selective. As usual, two sides of the same coin.
> 
> >> The answer is, that the common (and your, in this response)
> >> interpretation of 'frequency' is of a sinusoid whose amplitude can
> >> change with time. Hence, a broad spectrum can be of components whose
> >> amplitude is zero outside the 'impulse' of the lightning. But this is
> >> not how the Fourier Transform uses frequency. The amplitude of any
> >> component of a spectrum derived by Fourier Transform is fixed,
> >> constant, for all time. That is not the intuitive, and common,
> >> interpretation. My beef with the Prof's explanation is that he invoked
> >> 'spectrum' as in Fourier Transform - therefore he cannot then abuse
> >> this to introduce components whose amplitude somehow changes (to be
> >> large during, and zero outside of, the impulse).
> >>
> >> Fine, if he abandons Fourier Transforms and talks about frequency
> >> spectra in a non-Fourier way. But if he quotes Fourier then he is
> >> stuck with unchanging spectra.
> 
> I agree that the professor was too glib and oblivious of the subtleties.
> 
> Jerry
> --
> Engineering is the art of making what you want from things you can get.
> �����������������������������������������������������������������������


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jim wrote:
> 
> Jerry Avins wrote:
> 
>> Chris has a valid objection. The sum of the eternal frequencies present
>> is indeed zero at times well removed from the generating impulse.
>> However, it is not obvious that a narrow-band subset of them is also
>> self canceling. 
> 
> Yes well stories involving infinity do tend to have lots of dramatic
> plot twists and turns - no doubt about that. 
> 	What you are saying is that when you attenuate the frequencies that
> are being used to cancel other frequencies (and thus make the signal
> zero extending into infinity) then the canceled frequencies are going
> to come back like Lazurus from the dead. 
> 	A radio doesn't so much attenuate the frequencies it doesn't need as
> amplify the ones it does - No?

Surely, the relative amplitude is all that matters. My point was that 
the frequency spectrum being continuous, effective cancellation appears 
within any sufficiently wide band. The wider the band, the shorter the 
the interval of non-cancellation and conversely. Lightning's spectrum is 
not that of the simple impulse we calculate as undergraduates. It can't 
be; it not only has non-zero duration, but it's causal. Chris is right 
that any one frequency lasts forever. What he missed is that any 
non-zero bandwidth contains components that cancel one another outside 
an interval inversely proportional to that bandwidth.

>> Nevertheless, if the band is wide enough, the
>> cancellation is in fact largely complete. Consider a variable-bandwidth
>> filter excited by an impulse. As the bandwidth decreases, the
>> cancellation becomes poorer, and the duration of the filtered output
>> increases. When the bandwidth becomes very narrow -- the Q is made very
>> high -- the output can persist for a long time. We say that the filter
>> "rings". It actually does ring in the time domain, but in frequency, it
>> is merely selective. As usual, two sides of the same coin.
>>
>>>> The answer is, that the common (and your, in this response)
>>>> interpretation of 'frequency' is of a sinusoid whose amplitude can
>>>> change with time. Hence, a broad spectrum can be of components whose
>>>> amplitude is zero outside the 'impulse' of the lightning. But this is
>>>> not how the Fourier Transform uses frequency. The amplitude of any
>>>> component of a spectrum derived by Fourier Transform is fixed,
>>>> constant, for all time. That is not the intuitive, and common,
>>>> interpretation. My beef with the Prof's explanation is that he invoked
>>>> 'spectrum' as in Fourier Transform - therefore he cannot then abuse
>>>> this to introduce components whose amplitude somehow changes (to be
>>>> large during, and zero outside of, the impulse).
>>>>
>>>> Fine, if he abandons Fourier Transforms and talks about frequency
>>>> spectra in a non-Fourier way. But if he quotes Fourier then he is
>>>> stuck with unchanging spectra.
>> I agree that the professor was too glib and oblivious of the subtleties.
>>
>> Jerry
>> --
>> Engineering is the art of making what you want from things you can get.
>> �����������������������������������������������������������������������
> 
> 
> ----=Posted via Pronews.Com - Unlimited-Unrestricted-Secure Usenet News==----
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-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Chris Bore wrote:
>>Since lightning has a short duration in time we know it has a broad
>>spectrum.  It therefore interferes with many receivers.  Q.E.D.?

> Accepted, but...

> The specrum is broad for ever. So why, then, is the interference not
> also eternal?

> The answer is, that the common (and your, in this response)
> interpretation of 'frequency' is of a sinusoid whose amplitude can
> change with time. Hence, a broad spectrum can be of components whose
> amplitude is zero outside the 'impulse' of the lightning.

As the question was about radio receivers, it is the bandwidth
of the receiver that counts.

> But this is
> not how the Fourier Transform uses frequency. The amplitude of any
> component of a spectrum derived by Fourier Transform is fixed,
> constant, for all time. That is not the intuitive, and common,
> interpretation. My beef with the Prof's explanation is that he invoked
> 'spectrum' as in Fourier Transform - therefore he cannot then abuse
> this to introduce components whose amplitude somehow changes (to be
> large during, and zero outside of, the impulse).

I don't think it is less intuitive than many other parts of
either physics or signal processing.  Real systems have a
finite bandwidth which must be taken into account.

> Fine, if he abandons Fourier Transforms and talks about frequency
> spectra in a non-Fourier way. But if he quotes Fourier then he is
> stuck with unchanging spectra.

Well, you could do it with wavelets.

http://en.wikipedia.org/wiki/Wavelet

-- glen

jim wrote:
(snip)

> 	The story is the frequencies cancel out for most of eternity. If you
> take all the possible cosine functions and sum them they should sum
> coherently at t=0 and cancel at t not 0.
> 	 Another way to look at it is the Fourier pair for a gaussian pulse.
> A very narrow pulse in one domain will produce a very wide pulse in
> the other. So a pulse that is vanishingly small in width in the time
> domain will be so wide it will be virtually flat in the frequency
> domain. This story of course implies that the lightening pulse does
> exist for all time. We perceive it as a short duration event because
> it just rises to a magnitude that is measurable for a very short
> amount of time.  

For those who read the chapter in the Feynman lectures that
I wrote about before, or even for those who didn't, there is
chapter 26:  "Optics: The Principle of Least Time".
He covers Fermat's principle (unrelated to Fermat's
last theorem).   At the end of the chapter, he shows
how all except the light that takes nearly the minimum
time cancels out.  That is, just in the same way as
described above.

Following that are other optics related chapters including
diffraction and interference.

-- glen

Jerry Avins wrote:
(snip)

>>> The specrum is broad for ever. So why, then,
>>> is the interference not also eternal?
(snip)

> I agree that the professor was too glib and
 > oblivious of the subtleties.

I agree, but it is still worth thinking about.

The first one I remember, was considering the sound
from a square wave feeding a speaker, say at 1Hz.
We hear it as individual clicks at the edges.

Since 1Hz is below the minimum frequency that the
ear is sensitive to, we expect that we can't hear a
1Hz sine wave.  For a 1Hz square wave, we should hear
the harmonics within or hearing range, all the odd
frequencies between, say, 21Hz and 19999Hz.

But we don't, we hear clicks every half second.

Now, the signal isn't an impulse, but a periodic
(consider it infinite) square wave with a simple
(but infinite) Fourier transform.  But that
isn't the way we hear it.

-- glen