Band limited function zero crossings, how many?

"Ron Hardin" <rhhardin@mindspring.com> wrote in message
news:4033F78F.68ED@mindspring.com...
> And in fact, if my guess is right, any continuous function at all might
appear in a given
> time window and still be band limited (though huge outside that time
window).

Your guess is correct.  There is a family of prolate spheroidal wave
functions, all limited to any given band, that is complete in any given time
interval.

Ron Hardin wrote:

> Is there a maximum number of zero crossings per unit time that a
> band limited
> function can have?  My current feeling is that there isn't.
> 
> I think equivalently, any (continuous) function can be matched by
> a band limited function over a finite interval, if you don't care
> about what happens outside
> that interval?   (``matched'' probably must mean arbitrarily
> closely.)
> 
> What happens, I think, is that you get exponentially large tails
> outside the interval when you do it.
> 
> I'm just going on intuition here.  Does anybody know?

I guess you're not talking about a sampled system, which is discrete 
in time and/or amplitude.
And I guess, that you talk of an ideal band-limiter (ideal low-pass)

However, you're obviously talking of functions like one-dimensional 
signal streams, because I guess that zero-crossings make no sense 
otherwise ? 

Following is required for the signal then:
1. Every second zero crossing (ZC) has opposite direction:
   neg->pos, pos->neg, neg->pos, ...
2. Derivate dA/dt (slope) changes sign between consequent ZCs.
3. Second derivate has ZCs between consequent ZCs
   of the signal.

I'd guess that the second derivate of your function must obey the 
same band-limitation requirements as your original function (except 
near the borders of the interval) .

Thinking of fourier analysis (as far as it is applicable to your 
function), I would think, that your function cannot have more ZCs 
than the cosine function with the maximum frequency which is just 
inside your band.
And, only subharmonics of that maximum frequency are allowed as 
additional components, so that their ZCs coincide with those of 
this fastest cosine component.

Example: If the band-limitation requires frequencies below 500Hz,
you'll end up with ZCs not closer than 1ms.

However, I'm not so familiar with the hidden places of function 
theory. Therefore, you'd better check the other postings, and/or 
reveal which functions you're interested in and why. 
 
Bernhard

Matt Timmermans wrote:

> 
> Your guess is correct.  There is a family of prolate spheroidal wave
> functions, all limited to any given band, that is complete in any given time
> interval.
> 
> 

Er, what does that mean?


Bob

-- 

"Things should be described as simply as possible, but no 
simpler."

                                              A. Einstein

>>>>> "Fred" == Fred Marshall <fmarshallx@remove_the_x.acm.org> writes:

    Fred> If you don't like to use infinite sincs then I bet you will find that
    Fred> suitably windowed sincs will support the same arguments.


Is there not a theorem that says you can't have both a time-limited
and a band-limited signal?  Something about the time-bandwidth product
being constant?

Ray




-- 
--

-- 
Ericsson may automatically add a disclaimer.  Sorry, it's beyond my
control.


On Thu, 19 Feb 2004 08:21:03 +0100, Bernhard Holzmayer
<holzmayer.bernhard@deadspam.com> wrote:

>>Following is required for the signal then:
>>1. Every second zero crossing (ZC) has opposite direction:
>>   neg->pos, pos->neg, neg->pos, ...
>>2. Derivate dA/dt (slope) changes sign between consequent ZCs.
>>3. Second derivate has ZCs between consequent ZCs
>>   of the signal.

Do not forget that the zeroes of a function are not necessarily
real.  For zeroes in the complex plane, the concepts of "positive"
and "negative" are not so well defined.

Quoting from the Kay and Sudhaker paper that I referenced earlier:

"[A] way of representing ... signals is by means of their real and
complex zeros.  A zero-based representation can be shown to be
valid for band-limited signals which is usually the case of
interest.  Often, however, the process of determining the zero
sequence which corresponds to a given waveform may not be trivial.
The real zeros of the waveform are its conventional zero axis
crossings and are easy to determine by clipping.  Usually, the
complex zeros are not easy to identify, except through numerical
factorization of a trigonometric polynomial.  An alternative
procedure is to preprocess the signal prior to the zero extraction
by means of an invertible transform which converts all the complex
zeros into first-order real zeros.  In the case of periodic
band-limited signals, signal recovery to within a scale factor
from the zero crossing sequence is straightforward using a product
expansion.  For aperiodic signals which are represented by an
infinite product, the signal can only ba approximately recovered."

>>Thinking of fourier analysis (as far as it is applicable to your 
>>function), I would think, that your function cannot have more ZCs 
>>than the cosine function with the maximum frequency which is just 
>>inside your band.

It turns out that this is EXACTLY what Kay and Sudhaker did to
"preprocess" the signal -- they added a sine whose frequency was
incrementally above the band limit of the signal.  When the
amplitude of the added sine was sufficiently large, the resulting
"processed" signal became a sine wave with jittered zero
crossings.  This implies that there is an upper limit to the
number of zero crossings of a band limited signal in a finite time
interval -- it is 2(BW)T, where BW is the bandwidth, and T is the
interval.

Greg Berchin

"Bob Cain" <arcane@arcanemethods.com> wrote in message
news:c11ost12qpp@enews2.newsguy.com...
> Matt Timmermans wrote:
>
> >
> > Your guess is correct.  There is a family of prolate spheroidal wave
> > functions, all limited to any given band, that is complete in any given
time
> > interval.
> >
> >
>
> Er, what does that mean?

It means that you can make a sum of these functions that converges to any
function you like within an interval, like Fourier transform can do with
sines and cosines.  In addition, the entire functions are bandlimited to
whatever band you choose, so you are guaranteed to have a bandlimited
result.  There are lots of nifty things you can do with these.

The "prolate spheroidal wave functions" part actually means wave functions
in prolate spheroidal coordinates, which means... bla bla bla... I don't
think you want to go that way.  At least I haven't found any way to relate
this construction to the amazing properties these functions have.

For a lightish introduction to the good stuff, see:
See http://www.math.ucdavis.edu/%7Esaito/courses/ACHA/slepian83.pdf.

"Raymond Toy" <toy@rtp.ericsson.se> wrote in message
news:4n3c97cfbd.fsf@edgedsp4.rtp.ericsson.se...
> >>>>> "Fred" == Fred Marshall <fmarshallx@remove_the_x.acm.org> writes:
>
>     Fred> If you don't like to use infinite sincs then I bet you will find
that
>     Fred> suitably windowed sincs will support the same arguments.
>
>
> Is there not a theorem that says you can't have both a time-limited
> and a band-limited signal?  Something about the time-bandwidth product
> being constant?

Raymond,

Yes.     If one is strictly limited, the other has infinite extent.
         But, sometimes we get hung up on the difference between a signal
and a function
         where one is real and the other is not.
         Infinite-extent functions aren't real signals.  But, windowed ones
can be.
         So, that was my point here about windowed sincs supporting the
argument that there can
         be no more than one zero per sample period over a time span for a
"well behaved"
         function -> which gets closer to a real signal.
         So, allowing the bandwidth to be different than perfectly
bandlimited doesn't hurt the argument.
         (I do believe).

No.      The time bandwidth product from one function or signal to another
isn't constant.
         A "CW" pulse and an "fm sweep" pulse have different TW products.
         Otherwise it wouldn't be very interesting.
         It's really a different aspect of somewhat related subjects.

Fred

"Greg Berchin" <Bank-to-Turn@mchsi.com> wrote in message
news:u5h930lgfso7o04up2lvh0ait9rnnjqck9@4ax.com...
> Do not forget that the zeroes of a function are not necessarily
> real.  For zeroes in the complex plane, the concepts of "positive"
> and "negative" are not so well defined.
>

NB: I was very careful to say that we were talking about zero crossings (of
a real function) and *not* the zeros of a function of a complex variable.
So, something like f(t) and not H(s).  OK?

Fred

"Matt Timmermans" <mt0000@sympatico.nospam-remove.ca> wrote in message
news:oY3Zb.7360$Cd6.652711@news20.bellglobal.com...
> "Bob Cain" <arcane@arcanemethods.com> wrote in message
> news:c11ost12qpp@enews2.newsguy.com...
> > Matt Timmermans wrote:
> >
> > >
> > > Your guess is correct.  There is a family of prolate spheroidal wave
> > > functions, all limited to any given band, that is complete in any
given
> time
> > > interval.
> > >
> > >
> >
> > Er, what does that mean?
>
> It means that you can make a sum of these functions that converges to any
> function you like within an interval, like Fourier transform can do with
> sines and cosines.

Matt,

Meaning that the larger the family or set of them you use, the closer you
get.....  The Fourier Series still has its Gibbs phenomenon.  Do the PSWFs
have some equally nice way to characterise the remaining error?  I probably
should know..... but don't.
(I'm not sure everyone knows what "complete" means.....)

Fred

"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
news:R_-dnZ5R7dKZUqndRVn-vg@centurytel.net...
>
> "Raymond Toy" <toy@rtp.ericsson.se> wrote in message
> news:4n3c97cfbd.fsf@edgedsp4.rtp.ericsson.se...
> > >>>>> "Fred" == Fred Marshall <fmarshallx@remove_the_x.acm.org> writes:
> >
> >     Fred> If you don't like to use infinite sincs then I bet you will
find
> that
> >     Fred> suitably windowed sincs will support the same arguments.
> >
> >
> > Is there not a theorem that says you can't have both a time-limited
> > and a band-limited signal?  Something about the time-bandwidth product
> > being constant?
>
> Raymond,
>
> Yes.     If one is strictly limited, the other has infinite extent.
>          But, sometimes we get hung up on the difference between a signal
> and a function
>          where one is real and the other is not.
>          Infinite-extent functions aren't real signals.

Ooopppss.. I meant "real world" as distinct from "of the real numbers".

Fred