> On 12 Jan., 19:48, Jerry Avins <j...@ieee.org> wrote:
>> Andor wrote:
>>> On 11 Jan., 16:16, Jerry Avins <j...@ieee.org> wrote:
>>>> Andor wrote:
>>>> ...
>>>>> In the end I had to use 7th order polynomial interpolation FIR filters
>>>>> on my test function to actually catch the zero-crossings using
>>>>> upsampling by factor 4 (5th order was not enough). This can still be
>>>>> implemented efficiently (8*3*1e9 = 24e9 MACS should easily be
>>>>> computable in under a minute on a normal PC).
>>>> I've been puzzling over this for a few days now. Set aside the search
>>>> for actual locations of the zero crossings. If there are at least two
>>>> samples in the period of the highest frequency present in the sampled
>>>> signal, how can any zero crossing fail to be among those counted?
>>> Like this:
>>> http://www.zhaw.ch/~bara/files/zero-crossings_example.png
>>> Plot of the function described above with f=0.98. The function has 80
>>> zero-crossings in the displayed time window, the samples of the
>>> function have exactly one zero-crossing.
>> I see now, thanks. Very nice!
>>
>> What procedures fail if we count zero touchings as zero crossings? Does
>> it come down to the difference between open and closed intervals?
>>
>> Maybe I don't see after all. are the samples close to but not actually
>> on the axis? I see only a grapg, no equation. How does .98 come in?
>
> Yup, the samples don't touch the axis. The equation for this function
> can be found higher up in the thread. For convenience:
>
> x(t) = sin(pi f t) + sin(pi (1-f) t) + eps s((1-f)/2 t)
>
> where s(t) is the square-wave function with period 1 (replace with
> truncated Fourier series to make x(t) bandlimited). x(t) is sampled
> with sampling period T=1. The samples close to the axis have value eps
> (in this example, eps = 0.02) and f=0.98.
Aside from the possibility of Gibbs's phenomenon with the truncated s,
it's clear now. Thanks again.
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������
Reply by Jerry Avins●January 13, 20102010-01-13
dvsarwate wrote:
> On Jan 7, 9:59 am, Andor <andor.bari...@gmail.com> wrote:
>
>> I think the main problem is to find the rough location of the zero-
>> crossing in the sampled data. As my example shows, a zero-crossing of
>> the underlying sampled function will not necessarily show up as a
>> "zero-crossing" in the samples. I would opt for Ron's idea to use
>> quick polynomial interpolation with order >= 3 (can be implemented
>> with very small kernel FIR filters) to oversample the data by 4 or so
>> to find "zero-crossings" in the samples and then use the method of
>> sinc-interpolation outlined in your paper to find the exact location.
>>
>> This reminds me of the old comp.dsp debate on the maximum number of
>> zero-crossings that a bandlimited function may have on any given
>> interval. As it turned out (I think Matt Timmermans came up with the
>> example of the prolate spheroidal wave functions), the number of zero-
>> crossings is not limited ...
>
> But if there could be infinitely many zero-crossings, then no matter
> how
> much we oversample, we could never be sure that we have found them
> all, could we?
>
> If a (low-pass) signal is truly band-limited to W Hz, meaning that
> X(f) = 0 for |f| > W, then the magnitude of its derivative is bounded
> by
> 2\pi W max |x(t)| (cf. Temes, Barcilon, and Marshall, The
> optimization
> of bandlimited systems, Proc. IEEE, Feb. 1973). So, depending on
> the sampling rate and the value of two successive positive samples,
> we can be sure in *some* cases that there is no zero-crossing between
> the two samples: the bound on how quickly x(t) can change does not
> allow for the possibility that the signal could dip down below zero in
> between the two positive samples.
But the derivative grows with both frequency and amplitude. Can a bound
be assigned to the derivative without knowing the pre-sampling amplitude?
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������
Reply by Andor●January 13, 20102010-01-13
On 11 Jan., 19:17, dvsarwate <dvsarw...@gmail.com> wrote:
> On Jan 7, 9:59=A0am, Andor <andor.bari...@gmail.com> wrote:
>
> > I think the main problem is to find the rough location of the zero-
> > crossing in the sampled data. As my example shows, a zero-crossing of
> > the underlying sampled function will not necessarily show up as a
> > "zero-crossing" in the samples. I would opt for Ron's idea to use
> > quick polynomial interpolation with order >=3D 3 (can be implemented
> > with very small kernel FIR filters) to oversample the data by 4 or so
> > to find "zero-crossings" in the samples and then use the method of
> > sinc-interpolation outlined in your paper to find the exact location.
>
> > This reminds me of the old comp.dsp debate on the maximum number of
> > zero-crossings that a bandlimited function may have on any given
> > interval. As it turned out (I think Matt Timmermans came up with the
> > example of the prolate spheroidal wave functions), the number of zero-
> > crossings is not limited ...
>
> But if there could be infinitely many zero-crossings, then no matter
> how
> much we oversample, we could never be sure that we have found them
> all, could we?
>
> If a (low-pass) signal is truly band-limited to W Hz, meaning that
> X(f) =3D 0 for |f| > W, then the magnitude of its derivative is bounded
> by
> 2\pi W max |x(t)| =A0(cf. Temes, Barcilon, and Marshall, The
> optimization
> of bandlimited systems, Proc. IEEE, Feb. 1973). =A0So, depending on
> the sampling rate and the value of two successive positive samples,
> we can be sure in *some* cases that there is no zero-crossing between
> the two samples: the bound on how quickly x(t) can change does not
> allow for the possibility that the signal could dip down below zero in
> between the two positive samples.
On 12 Jan., 19:48, Jerry Avins <j...@ieee.org> wrote:
> Andor wrote:
> > On 11 Jan., 16:16, Jerry Avins <j...@ieee.org> wrote:
> >> Andor wrote:
>
> >> =A0 =A0...
>
> >>> In the end I had to use 7th order polynomial interpolation FIR filter=
s
> >>> on my test function to actually catch the zero-crossings using
> >>> upsampling by factor 4 (5th order was not enough). This can still be
> >>> implemented efficiently (8*3*1e9 =3D 24e9 MACS should easily be
> >>> computable in under a minute on a normal PC).
> >> I've been puzzling over this for a few days now. Set aside the search
> >> for actual locations of the zero crossings. If there are at least two
> >> samples in the period of the highest frequency present in the sampled
> >> signal, how can any zero crossing fail to be among those counted?
>
> > Like this:
>
> >http://www.zhaw.ch/~bara/files/zero-crossings_example.png
>
> > Plot of the function described above with f=3D0.98. The function has 80
> > zero-crossings in the displayed time window, the samples of the
> > function have exactly one zero-crossing.
>
> I see now, thanks. Very nice!
>
> What procedures fail if we count zero touchings as zero crossings? Does
> it come down to the difference between open and closed intervals?
>
> Maybe I don't see after all. are the samples close to but not actually
> on the axis? I see only a grapg, no equation. How does .98 come in?
Yup, the samples don't touch the axis. The equation for this function
can be found higher up in the thread. For convenience:
x(t) =3D sin(pi f t) + sin(pi (1-f) t) + eps s((1-f)/2 t)
where s(t) is the square-wave function with period 1 (replace with
truncated Fourier series to make x(t) bandlimited). x(t) is sampled
with sampling period T=3D1. The samples close to the axis have value eps
(in this example, eps =3D 0.02) and f=3D0.98.
Regards,
Andor
Reply by Jerry Avins●January 12, 20102010-01-12
Andor wrote:
> On 11 Jan., 16:16, Jerry Avins <j...@ieee.org> wrote:
>> Andor wrote:
>>
>> ...
>>
>>> In the end I had to use 7th order polynomial interpolation FIR filters
>>> on my test function to actually catch the zero-crossings using
>>> upsampling by factor 4 (5th order was not enough). This can still be
>>> implemented efficiently (8*3*1e9 = 24e9 MACS should easily be
>>> computable in under a minute on a normal PC).
>> I've been puzzling over this for a few days now. Set aside the search
>> for actual locations of the zero crossings. If there are at least two
>> samples in the period of the highest frequency present in the sampled
>> signal, how can any zero crossing fail to be among those counted?
>
> Like this:
>
> http://www.zhaw.ch/~bara/files/zero-crossings_example.png
>
> Plot of the function described above with f=0.98. The function has 80
> zero-crossings in the displayed time window, the samples of the
> function have exactly one zero-crossing.
I see now, thanks. Very nice!
What procedures fail if we count zero touchings as zero crossings? Does
it come down to the difference between open and closed intervals?
Maybe I don't see after all. are the samples close to but not actually
on the axis? I see only a grapg, no equation. How does .98 come in?
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������
Reply by Andor●January 12, 20102010-01-12
On 11 Jan., 16:16, Jerry Avins <j...@ieee.org> wrote:
> Andor wrote:
>
> =A0 =A0...
>
> > In the end I had to use 7th order polynomial interpolation FIR filters
> > on my test function to actually catch the zero-crossings using
> > upsampling by factor 4 (5th order was not enough). This can still be
> > implemented efficiently (8*3*1e9 =3D 24e9 MACS should easily be
> > computable in under a minute on a normal PC).
>
> I've been puzzling over this for a few days now. Set aside the search
> for actual locations of the zero crossings. If there are at least two
> samples in the period of the highest frequency present in the sampled
> signal, how can any zero crossing fail to be among those counted?
Like this:
http://www.zhaw.ch/~bara/files/zero-crossings_example.png
Plot of the function described above with f=3D0.98. The function has 80
zero-crossings in the displayed time window, the samples of the
function have exactly one zero-crossing.
Regards,
Andor
Reply by dvsarwate●January 11, 20102010-01-11
On Jan 11, 12:37�pm, Jerry Avins <j...@ieee.org> wrote:
> dvsarwate wrote:
> > On Jan 7, 9:59 am, Andor <andor.bari...@gmail.com> wrote:
>
> >> I think the main problem is to find the rough location of the zero-
> >> crossing in the sampled data. As my example shows, a zero-crossing of
> >> the underlying sampled function will not necessarily show up as a
> >> "zero-crossing" in the samples. I would opt for Ron's idea to use
> >> quick polynomial interpolation with order >= 3 (can be implemented
> >> with very small kernel FIR filters) to oversample the data by 4 or so
> >> to find "zero-crossings" in the samples and then use the method of
> >> sinc-interpolation outlined in your paper to find the exact location.
>
> >> This reminds me of the old comp.dsp debate on the maximum number of
> >> zero-crossings that a bandlimited function may have on any given
> >> interval. As it turned out (I think Matt Timmermans came up with the
> >> example of the prolate spheroidal wave functions), the number of zero-
> >> crossings is not limited ...
>
> > But if there could be infinitely many zero-crossings, then no matter
> > how
> > much we oversample, we could never be sure that we have found them
> > all, could we?
>
> > If a (low-pass) signal is truly band-limited to W Hz, meaning that
> > X(f) = 0 for |f| > W, then the magnitude of its derivative is bounded
> > by
> > 2\pi W max |x(t)| �(cf. Temes, Barcilon, and Marshall, The
> > optimization
> > of bandlimited systems, Proc. IEEE, Feb. 1973). �So, depending on
> > the sampling rate and the value of two successive positive samples,
> > we can be sure in *some* cases that there is no zero-crossing between
> > the two samples: the bound on how quickly x(t) can change does not
> > allow for the possibility that the signal could dip down below zero in
> > between the two positive samples.
>
> But doesn't that imply some frequency well above half the sample rate?
>
> Jerry
> --
> Engineering is the art of making what you want from things you can get.
> �����������������������������������������������������������������������
I think Jerry and I are talking at cross-purposes. Suppose we have
two
samples spaced T seconds apart from a signal x(t) that is strictly
band-limited
to have bandwidth W; X(f) = 0 for |f| > W. Suppose also (for
simplicity)
that |x(t)| < 1 and so the magnitude of the derivative is bounded by
2\pi W.
If x(0) = a > 0, then the signal cannot decrease to 0 earlier than t =
a/(2\pi W)
because the rate of change of x(t) is bounded. Similarly, if x(T) = b
> 0 is
the next sample (note that we are not assuming any relation between T
and W),
then the signal must be above zero during the time interval (T - b/
(2\pi W), T)
since otherwise it cannot change fast enough to reach b at t = T.
Thus, if
a/(2\pi W) > T - b/(2\pi W), then there is no zero-crossing between 0
and T.
Note that the stated criterion is equivalent to a + b > 2\pi WT, and
of course
since a + b = x(0) + x(T) < 2, the criterion will never be satisfied
if 2\pi WT > 2.
But, assuming that \pi WT < 1, it is possible that we can deduce for
*some*
adjacent positive samples that x(t) does not cross zero between the
samples. If T is very small (some upsampling may be necessary to get
this),
then maybe we can say that for *most* positive adjacent sample values
a and b, there is no zero-crossing between the samples.
Hope this helps
--Dilip Sarwate
Reply by Jerry Avins●January 11, 20102010-01-11
dvsarwate wrote:
> On Jan 7, 9:59 am, Andor <andor.bari...@gmail.com> wrote:
>
>> I think the main problem is to find the rough location of the zero-
>> crossing in the sampled data. As my example shows, a zero-crossing of
>> the underlying sampled function will not necessarily show up as a
>> "zero-crossing" in the samples. I would opt for Ron's idea to use
>> quick polynomial interpolation with order >= 3 (can be implemented
>> with very small kernel FIR filters) to oversample the data by 4 or so
>> to find "zero-crossings" in the samples and then use the method of
>> sinc-interpolation outlined in your paper to find the exact location.
>>
>> This reminds me of the old comp.dsp debate on the maximum number of
>> zero-crossings that a bandlimited function may have on any given
>> interval. As it turned out (I think Matt Timmermans came up with the
>> example of the prolate spheroidal wave functions), the number of zero-
>> crossings is not limited ...
>
> But if there could be infinitely many zero-crossings, then no matter
> how
> much we oversample, we could never be sure that we have found them
> all, could we?
>
> If a (low-pass) signal is truly band-limited to W Hz, meaning that
> X(f) = 0 for |f| > W, then the magnitude of its derivative is bounded
> by
> 2\pi W max |x(t)| (cf. Temes, Barcilon, and Marshall, The
> optimization
> of bandlimited systems, Proc. IEEE, Feb. 1973). So, depending on
> the sampling rate and the value of two successive positive samples,
> we can be sure in *some* cases that there is no zero-crossing between
> the two samples: the bound on how quickly x(t) can change does not
> allow for the possibility that the signal could dip down below zero in
> between the two positive samples.
But doesn't that imply some frequency well above half the sample rate?
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������
Reply by dvsarwate●January 11, 20102010-01-11
On Jan 7, 9:59�am, Andor <andor.bari...@gmail.com> wrote:
> I think the main problem is to find the rough location of the zero-
> crossing in the sampled data. As my example shows, a zero-crossing of
> the underlying sampled function will not necessarily show up as a
> "zero-crossing" in the samples. I would opt for Ron's idea to use
> quick polynomial interpolation with order >= 3 (can be implemented
> with very small kernel FIR filters) to oversample the data by 4 or so
> to find "zero-crossings" in the samples and then use the method of
> sinc-interpolation outlined in your paper to find the exact location.
>
> This reminds me of the old comp.dsp debate on the maximum number of
> zero-crossings that a bandlimited function may have on any given
> interval. As it turned out (I think Matt Timmermans came up with the
> example of the prolate spheroidal wave functions), the number of zero-
> crossings is not limited ...
But if there could be infinitely many zero-crossings, then no matter
how
much we oversample, we could never be sure that we have found them
all, could we?
If a (low-pass) signal is truly band-limited to W Hz, meaning that
X(f) = 0 for |f| > W, then the magnitude of its derivative is bounded
by
2\pi W max |x(t)| (cf. Temes, Barcilon, and Marshall, The
optimization
of bandlimited systems, Proc. IEEE, Feb. 1973). So, depending on
the sampling rate and the value of two successive positive samples,
we can be sure in *some* cases that there is no zero-crossing between
the two samples: the bound on how quickly x(t) can change does not
allow for the possibility that the signal could dip down below zero in
between the two positive samples.
--Dilip Sarwate
Reply by Jerry Avins●January 11, 20102010-01-11
Andor wrote:
...
> In the end I had to use 7th order polynomial interpolation FIR filters
> on my test function to actually catch the zero-crossings using
> upsampling by factor 4 (5th order was not enough). This can still be
> implemented efficiently (8*3*1e9 = 24e9 MACS should easily be
> computable in under a minute on a normal PC).
I've been puzzling over this for a few days now. Set aside the search
for actual locations of the zero crossings. If there are at least two
samples in the period of the highest frequency present in the sampled
signal, how can any zero crossing fail to be among those counted?
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������