Band limited function zero crossings, how many?

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?
-- 
Ron Hardin
rhhardin@mindspring.com

On the internet, nobody knows you're a jerk.

"Ron Hardin" <rhhardin@mindspring.com> wrote in message
news:4033B91D.2151@mindspring.com...
> 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?
> -- 
> Ron Hardin
> rhhardin@mindspring.com
>
> On the internet, nobody knows you're a jerk.

The complete opposite to what I think, but I'm a complete amateur. Out of
interest, are you a professional DSP engineer?
Funky

"Ron Hardin" <rhhardin@mindspring.com> wrote in message
news:4033B91D.2151@mindspring.com...
> 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?

No.

Take a family of sinc functions that are regularly spaced with each
succeeding peak on top of the other sinc's zeros.  That's the typical thing
of course.
Make the family contiguous and N of them in a sequence - to form a "basis
set" of functions.  This set satisfies a condition that will allow you to
adjust the amplitudes of each member of the sequence in order to match N
points of their thus-weighted sum.  It's a simple set of linear equations in
N unknowns to be solved to find the weights.  The set also guarantees that
the sum is a bandlimited result.

I can't say that I've tried exactly the thing you're suggesting but it is
very similar to what's called "supergaining" in antenna design (where the
functions are "space limited" rather than bandlimited). In that case, the
beam pattern goes wild outside of the so-called "visible region" with very
large amplitudes - often larger than the main lobe!

Suppose one of the zeros (I refer to zero-crossings here) lies on one of the
contiguous points that describes the basis set above.  This forces the
aligned sinc to be zero-weighted and satisfies that one condition.  All the
other functions in the basis are zero already on that point.  Then, assume
that you want a zero identically located between the regular zeros.  This
can be realized by adjusting the remaining sum of N-1 variables, etc.

So, the answer to the question is *No* - there is no limit to the number of
zeros in a range as long as there are enough sinc functions.  But, there
will be some very wild amplitudes of those sinc peaks outside of the range
where the zeros exist.  So, it's not clear that it's an "interesting" or
useful functon overall.
A very real example of this type of thing (without supergaining) is
increasing the length of a FIR filter with the objective of reducing the
stopband ripple.  As the ripple is reduced by adding filter coefficients,
the number of zeros in the ripple increases - and that's in a fixed
bandwidth.  OK - timelimited with zeros in frequency - really the same thing
as bandlimited with zeros in time.  If the filter is symmetric, then there
are two coefficients added for each added degree of freedom > each new zero.
Note that each time a single zero is added, the distance between zeros
decreases by M/(M+1) where M is the number of zeros.  So, the reduction in
distance between zeros is less and less distinct.

If you really use N sincs to do this, with the interval of zeros a subset of
their "nonzero span", then there will be the zeros, there will be the sinc
peaks outside of the range of zeros and there will be tails that decay as
1/t outside of the "nonzero span" of sincs.

I think that's all there is to it.

Fred

On Wed, 18 Feb 2004 19:12:25 GMT, Ron Hardin
<rhhardin@mindspring.com> wrote:

>>Is there a maximum number of zero crossings per unit time that a band limited
>>function can have? 

"A Zero Crossing-Based Spectrum Analyzer", Steven M. Kay and 
R. Sudhaker, "IEEE Transactions on Acoustics, Speech, and Signal
Processing", Vol. ASSP-34, No. 1, February 1986.

Greg Berchin wrote:
> 
> On Wed, 18 Feb 2004 19:12:25 GMT, Ron Hardin
> <rhhardin@mindspring.com> wrote:
> 
> >>Is there a maximum number of zero crossings per unit time that a band limited
> >>function can have?
> 
> "A Zero Crossing-Based Spectrum Analyzer", Steven M. Kay and
> R. Sudhaker, "IEEE Transactions on Acoustics, Speech, and Signal
> Processing", Vol. ASSP-34, No. 1, February 1986.

For a stationary gaussian random function, the zero crossing count gives the
rms frequency of the spectrum.  Differentiate the time function and repeat and
get the 2nd, 4th etc moments of the spectrum.

However that doesn't cover the unlikely case proposed.

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).
-- 
Ron Hardin
rhhardin@mindspring.com

On the internet, nobody knows you're a jerk.

Ron Hardin wrote:

> Is there a maximum number of zero crossings per unit time that a band limited
> function can have?  

Doesn't that depend on the frequency? How often does the electric field 
of a blue laser beam cross zero?

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

Jerry Avins wrote:
> 
> Ron Hardin wrote:
> 
> > Is there a maximum number of zero crossings per unit time that a band limited
> > function can have?
> 
> Doesn't that depend on the frequency? How often does the electric field
> of a blue laser beam cross zero?

I think it doesn't matter.  Assume center frequency zero, and any given bandwidth.  Then
there's no limit to the number of zero crossings possible over a given time interval.
(That's the guess.)

So then likewise at any center frequency.
-- 
Ron Hardin
rhhardin@mindspring.com

On the internet, nobody knows you're a jerk.

Ron Hardin wrote:

> Jerry Avins wrote:
> 
>>Ron Hardin wrote:
>>
>>
>>>Is there a maximum number of zero crossings per unit time that a band limited
>>>function can have?
>>
>>Doesn't that depend on the frequency? How often does the electric field
>>of a blue laser beam cross zero?
> 
> 
> I think it doesn't matter.  Assume center frequency zero, and any given bandwidth.  Then
> there's no limit to the number of zero crossings possible over a given time interval.
> (That's the guess.)
> 
> So then likewise at any center frequency.

Do you really mean time interval, like second, nanosecond, etc.? Then 
the number of zero crossings can be made arbitrarily large by choosing 
an arbitrarily high frequency. Or do you mean an arbitrarily large 
number of zero crossings in one period? If you do, the number is limited 
if the bandwidth is.

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

"Jerry Avins" <jya@ieee.org> wrote in message
news:40340cb0$0$3072$61fed72c@news.rcn.com...
> Ron Hardin wrote:
>
> > Jerry Avins wrote:
> >
> >>Ron Hardin wrote:
> >>
> >>
> >>>Is there a maximum number of zero crossings per unit time that a band
limited
> >>>function can have?
> >>
> >>Doesn't that depend on the frequency? How often does the electric field
> >>of a blue laser beam cross zero?
> >
> >
> > I think it doesn't matter.  Assume center frequency zero, and any given
bandwidth.  Then
> > there's no limit to the number of zero crossings possible over a given
time interval.
> > (That's the guess.)
> >
> > So then likewise at any center frequency.
>
> Do you really mean time interval, like second, nanosecond, etc.? Then
> the number of zero crossings can be made arbitrarily large by choosing
> an arbitrarily high frequency. Or do you mean an arbitrarily large
> number of zero crossings in one period? If you do, the number is limited
> if the bandwidth is.

Jerry, how about this: for a given time interval and a given band-limiting
frequency, is the number of zero crossings limited?

My intuition says yes, since band-limiting implies a maximum frequency which
implies a maximum number of zero crossings.

Consider this: if you start with a sampled signal (which is band-limited to
the Nyquist rate by definition), you will have a limited number of zero
crossings in any interval.  The worst case signal goes 1, -1, 1, -1 or
similar, so that you have N zero crossings in an N-sample interval.
Converting this signal to continuous time with a theoretically perfect DAC
yields a band-limited signal again with a maximum of N zero crossings.  I
realize this is more hand-waving then an actual proof, but what do you guys
think?

"Jon Harris" <goldentully@hotmail.com> wrote in message
news:c1134b$1e01up$1@ID-210375.news.uni-berlin.de...
> "Jerry Avins" <jya@ieee.org> wrote in message
> news:40340cb0$0$3072$61fed72c@news.rcn.com...
> > Ron Hardin wrote:
> >
> > > Jerry Avins wrote:
> > >
> > >>Ron Hardin wrote:
> > >>
> > >>
> > >>>Is there a maximum number of zero crossings per unit time that a band
> limited
> > >>>function can have?
> > >>
> > >>Doesn't that depend on the frequency? How often does the electric
field
> > >>of a blue laser beam cross zero?
> > >
> > >
> > > I think it doesn't matter.  Assume center frequency zero, and any
given
> bandwidth.  Then
> > > there's no limit to the number of zero crossings possible over a given
> time interval.
> > > (That's the guess.)
> > >
> > > So then likewise at any center frequency.
> >
> > Do you really mean time interval, like second, nanosecond, etc.? Then
> > the number of zero crossings can be made arbitrarily large by choosing
> > an arbitrarily high frequency. Or do you mean an arbitrarily large
> > number of zero crossings in one period? If you do, the number is limited
> > if the bandwidth is.
>
> Jerry, how about this: for a given time interval and a given band-limiting
> frequency, is the number of zero crossings limited?
>
> My intuition says yes, since band-limiting implies a maximum frequency
which
> implies a maximum number of zero crossings.
>
> Consider this: if you start with a sampled signal (which is band-limited
to
> the Nyquist rate by definition), you will have a limited number of zero
> crossings in any interval.  The worst case signal goes 1, -1, 1, -1 or
> similar, so that you have N zero crossings in an N-sample interval.
> Converting this signal to continuous time with a theoretically perfect DAC
> yields a band-limited signal again with a maximum of N zero crossings.  I
> realize this is more hand-waving then an actual proof, but what do you
guys
> think?

Ah!  You've changed the question from one about a "function" to one about a
"signal"!  A very appropriate thing to do I might add.

So, in contrast to my earlier answer - which did caution about the
practicability of such "functions" - here's another treatment:

Let's assume an (almost) real signal instead of a "function".  The signal is
bandlimited well enough and now let's add that it must be "well-behaved" in
that it is also timelimited well enough AND that it can't have gross
amplitudes away from that section with tightly packed zeros.  In other
words, the signal in the region with the zeros is going to be driven by the
sinc functions whose peaks are in that same region - because we aren't
allowing the sincs outside that region to "blow up" as it were.

Now we can assign a value to every sinc in the interval and one way is to
make them all zero.  That sets a regular interval on the zeros that's the
same as the sinc has and is pi/w where w is the bandwidth in radians per
second or 1/2B seconds where B is the bandwidth limitation.  That is, the
sinc interval is the same as the lower limit of the sampling interval.
Jon is right.  We can also assign values to the sinc peaks of +1, -1, +1,
etc. and will see zeros between the peaks perforce - no matter what the
other sincs are doing because the signal *must* pass through these
alternating sign points.  This roughly sets the same spacing as before - one
zero per sinc peak.
There is no way to get a well-behaved bandlimited *signal* to have more
zeros methinks.

A corollary to this, going back to functions is:
If a function has *more* zeros than this then:
1) they must not be on the points where the sincs cross zero (because all
the other sinc tails cross zero there).
2) therefore, they must be "generated" or dominated by tails of sincs that
are outside the interval.  Start with all zero weights on the sincs inside
the interval .. then any zeros that occur in this interval must be a result
of summing sinc tails from outside the interval.

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

Fred