Ron N. wrote:
> Jerry Avins wrote:

   ...

>> Why are you busting my chops?
> 
> Not you in particular, just the idea in general that sampling and
> bandlimiting are related by *necessity*. They aren't.  It's just
> that samples are usually worthless (or worth less) unless one
> does some sort of suitable low pass (or bandpass) filtering,
> which a beginner, perhaps such as the OP, doesn't know about
> or has forgotten.

Actually, my point was different. Without prior bandpass filtering, 
aliasing corrupts the signal; agreed. My point was that after sampling, 
there is only this signal (and its images) within the band allowed by 
the sample rate. For a baseband signal, only components up to Fs/2 exist 
in the samples. These are replicated with or without inversion on up the 
spectrum. Whichever image of an aliased signal one chooses, it has all 
of the aliases that the baseband image has. The act of sampling converts 
out-of-band components to in-band aliases. After sampling, there are no 
out-of-band components; they've been converted to in-band components. I 
call that bandlimiting. If you have a better term, I'd like to adopt it.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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Jerry Avins wrote:
> Ron N. wrote:
> > Jerry Avins wrote:
> >> Ron N. wrote:
> >>> Jerry Avins wrote:
> >>>> The spectra are easy to compute *for continuous waveforms*. They are
> >>>> tabulated in many books. You cannot deal with continuous waveforms using
> >>>> a computer, and if you sample the waveform you bandlimit it *whether you
> >>>> intend to or not*.
> >>> Sampling does not automatically bandlimit.  You can undersample
> >>> for some demodulation schemes, for instance.  The sampler does
> >>> not know which band you want.  You have to pick a band filter
> >>> (low pass, or band pass for your selected frequency range).
> >> Sampling aliases all the frequencies above Fs/2 to other frequencies
> >> below Fs/2. Any proper reconstruction procedure will generate a
> >> continuous waveform from those samples all of whose components are below
> >> Fs/2. I call that bandlimiting. What do you call it?
> >
> > Improper reconstruction.  If your signal is above Fs/2 (but narrow-
> > band enough not to be aliased with itself) then you should use a
> > reconstruction formula for that higher frequency band.
> >
> > The reconstruction below Fs/2 is only valid if you have out-of-band
> > information telling you that that is where your signal belongs.
> > Otherwise any reconstruction is ambiguous.
>
> Do you really think the OP was into subband sampling? It was clearly a
> question about baseband and I think you know that too.

Neither.  I wouldn't be surprised if the OP was dealing with a
non-bandlimited signal (which means a baseband reconstruction
will be incorrect, or inexact).  Sampling doesn't fix that.

> Why are you busting my chops?

Not you in particular, just the idea in general that sampling and
bandlimiting are related by *necessity*. They aren't.  It's just
that samples are usually worthless (or worth less) unless one
does some sort of suitable low pass (or bandpass) filtering,
which a beginner, perhaps such as the OP, doesn't know about
or has forgotten.


IMHO. YMMV.
-- 
rhn A.T nicholson d.0.t C-o-M

Ron N. wrote:
> Jerry Avins wrote:
>> Ron N. wrote:
>>> Jerry Avins wrote:
>>>> The spectra are easy to compute *for continuous waveforms*. They are
>>>> tabulated in many books. You cannot deal with continuous waveforms using
>>>> a computer, and if you sample the waveform you bandlimit it *whether you
>>>> intend to or not*.
>>> Sampling does not automatically bandlimit.  You can undersample
>>> for some demodulation schemes, for instance.  The sampler does
>>> not know which band you want.  You have to pick a band filter
>>> (low pass, or band pass for your selected frequency range).
>> Sampling aliases all the frequencies above Fs/2 to other frequencies
>> below Fs/2. Any proper reconstruction procedure will generate a
>> continuous waveform from those samples all of whose components are below
>> Fs/2. I call that bandlimiting. What do you call it?
> 
> Improper reconstruction.  If your signal is above Fs/2 (but narrow-
> band enough not to be aliased with itself) then you should use a
> reconstruction formula for that higher frequency band.
> 
> The reconstruction below Fs/2 is only valid if you have out-of-band
> information telling you that that is where your signal belongs.
> Otherwise any reconstruction is ambiguous.

Do you really think the OP was into subband sampling? It was clearly a 
question about baseband and I think you know that too. Why are you 
busting my chops?

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

Jerry Avins wrote:
> Ron N. wrote:
> > Jerry Avins wrote:
> >> The spectra are easy to compute *for continuous waveforms*. They are
> >> tabulated in many books. You cannot deal with continuous waveforms using
> >> a computer, and if you sample the waveform you bandlimit it *whether you
> >> intend to or not*.
> >
> > Sampling does not automatically bandlimit.  You can undersample
> > for some demodulation schemes, for instance.  The sampler does
> > not know which band you want.  You have to pick a band filter
> > (low pass, or band pass for your selected frequency range).
>
> Sampling aliases all the frequencies above Fs/2 to other frequencies
> below Fs/2. Any proper reconstruction procedure will generate a
> continuous waveform from those samples all of whose components are below
> Fs/2. I call that bandlimiting. What do you call it?

Improper reconstruction.  If your signal is above Fs/2 (but narrow-
band enough not to be aliased with itself) then you should use a
reconstruction formula for that higher frequency band.

The reconstruction below Fs/2 is only valid if you have out-of-band
information telling you that that is where your signal belongs.
Otherwise any reconstruction is ambiguous.


IMHO. YMMV.
-- 
rhn A.T nicholson d.0.t C-o-M

Ron N. wrote:
> Jerry Avins wrote:
>
> > No frequencies above half the sample rate exist in a
> > set of samples.
>
> Of course they exist.  It's just that if they are there *and* the
> baseband spectrum is also there, then they are aliased together
> and you can't tell them apart (without some other external
> information).

dunno if i agree with this.  we have to be specific about what we mean.

by set of samples, i mean

    { x[n]:  n integer, -inf < n < +inf }

in this set there is no concept of inbetween integer n.  x[r] is not
defined unless r is an integer.

this can be represented with a fourier transform (the DTFT)  which is
the Z transform of x[n]

    X(z) = SUM x[n] z^(-n)
            n

with

    z = e^(jw)   for real w .

this X(e^(jw)) is periodic with period 2*pi.  normally we deal with -pi
< w < +pi.  the repeated *images* are not *aliases*.  they are images
and simply from X(z) or from x[n] we don't know whether or not there
was aliasing.  strictly speaking, that information is gone and lacking
any outside info, i think you normally have to assume that a frequency
component that exists was originally between -pi and +pi.  that's why
they're called "aliases", they were originally outside that range and
masquarade as if they were always between -pi and +pi.

in this sense, Jerry is right, i think.  and if you make the hard-core
assumption that x[n] is strictly associated with the bandlimited
reconstruction:

    x(t) = SUM x[n] sinc(t-n)
            n

then there are explicitly no (radian) frequencies above pi in
magnitude.

if you instead say

    x(t) = SUM x[n] delta(t-n)
            n

there are images, but no aliases (unless you have outside information
that tells you something about x(t) before it was sampled).

r b-j

Ron N. wrote:
> Jerry Avins wrote:
>> The spectra are easy to compute *for continuous waveforms*. They are
>> tabulated in many books. You cannot deal with continuous waveforms using
>> a computer,
> 
> You forget about symbolic math programs, which can deal with some
> continuous waveforms if they are functions of a form that the symbolic
> programs can handle (symbolic integration, convolution, etc.)
> A computer can also use the book tabulations.

I didn't forget. If I have to touch all bases and enumerate all possible 
exceptions in every discussion, I won't be able to discuss much.

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

Ron N. wrote:
> Jerry Avins wrote:
>> The spectra are easy to compute *for continuous waveforms*. They are
>> tabulated in many books. You cannot deal with continuous waveforms using
>> a computer, and if you sample the waveform you bandlimit it *whether you
>> intend to or not*.
> 
> Sampling does not automatically bandlimit.  You can undersample
> for some demodulation schemes, for instance.  The sampler does
> not know which band you want.  You have to pick a band filter
> (low pass, or band pass for your selected frequency range).

Sampling aliases all the frequencies above Fs/2 to other frequencies 
below Fs/2. Any proper reconstruction procedure will generate a 
continuous waveform from those samples all of whose components are below 
Fs/2. I call that bandlimiting. What do you call it?

>> No frequencies above half the sample rate exist in a
>> set of samples.
> 
> Of course they exist.  It's just that if they are there *and* the
> baseband spectrum is also there, then they are aliased together
> and you can't tell them apart (without some other external
> information).

In general, they can't be separated even with special information. It's 
hard to unscramble eggs or sounds.

> Actually, in practice, since low pass filters only approximate
> true bandlimiting filters of infinite width, they are always there,
> just hopefully below the desired noise floor.

So is thermal noise. We live with what we have to.

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

Jerry Avins wrote:
> The spectra are easy to compute *for continuous waveforms*. They are
> tabulated in many books. You cannot deal with continuous waveforms using
> a computer,

You forget about symbolic math programs, which can deal with some
continuous waveforms if they are functions of a form that the symbolic
programs can handle (symbolic integration, convolution, etc.)
A computer can also use the book tabulations.


IMHO. YMMV.
-- 
rhn A.T nicholson d.0.t C-o-M

Jerry Avins wrote:
> The spectra are easy to compute *for continuous waveforms*. They are
> tabulated in many books. You cannot deal with continuous waveforms using
> a computer, and if you sample the waveform you bandlimit it *whether you
> intend to or not*.

Sampling does not automatically bandlimit.  You can undersample
for some demodulation schemes, for instance.  The sampler does
not know which band you want.  You have to pick a band filter
(low pass, or band pass for your selected frequency range).

> No frequencies above half the sample rate exist in a
> set of samples.

Of course they exist.  It's just that if they are there *and* the
baseband spectrum is also there, then they are aliased together
and you can't tell them apart (without some other external
information).

Actually, in practice, since low pass filters only approximate
true bandlimiting filters of infinite width, they are always there,
just hopefully below the desired noise floor.


IMHO. YMMV.
-- 
rhn A.T nicholson d.0.t C-o-M

john john wrote:
> Jerry Avins ha scritto:
> 
>> No. The spectra of square and triangular waves are infinite series.
>> Computers can't deal with infinities. If you truncate the series, the
>> waveforms you get might surprise you.
>>
>> What are you trying to see?


> I know about infinite series and I don't cut the series. I'm just
> trying to see the frequency spectrum of a waveform but every frequency
> with its correct phase. I won't the infinite but just the firsts ten
> frequency.

The spectra are easy to compute *for continuous waveforms*. They are 
tabulated in many books. You cannot deal with continuous waveforms using 
a computer, and if you sample the waveform you bandlimit it *whether you 
intend to or not*. No frequencies above half the sample rate exist in a 
set of samples.

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