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From: Jerry Avins on 14 May 2010 11:26
On 5/14/2010 10:42 AM, Steve Pope wrote:
> On May 14, 8:34 am, "fisico32"<marcoscipioni1(a)n_o_s_p_a_m.gmail.com>
>
>> In the discrete time domain, in computer simulation, an irrational
>> number(infinite nonrepeating decimal) must be truncated in the number of
>> decimals so it is seen as a rational number too....
>
> Not true; you can implement arbitrary-precision arithmetic
> if you so choose.

Finite time is another important criterion.

Jerry
--
"I view the progress of science as ... the slow erosion of the tendency
to dichotomize." --Barbara Smuts, U. Mich.
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From: Steve Pope on 14 May 2010 11:33
Jerry Avins <jya(a)ieee.org> wrote:

>On 5/14/2010 10:42 AM, Steve Pope wrote:

>> On May 14, 8:34 am, "fisico32"<marcoscipioni1(a)n_o_s_p_a_m.gmail.com>

>>> In the discrete time domain, in computer simulation, an irrational
>>> number(infinite nonrepeating decimal) must be truncated in the number of
>>> decimals so it is seen as a rational number too....

>> Not true; you can implement arbitrary-precision arithmetic
>> if you so choose.

>Finite time is another important criterion.

That's true, but it's valid to discuss computer simulations
that might run indefinitely, with precisions that keep building up
as the simulation goes along.

(I said valid; I didn't say practical... :) )

Steve
From: jim on 14 May 2010 21:19


suren wrote:

> On May 14, 5:34 pm, "fisico32" <marcoscipioni1(a)n_o_s_p_a_m.gmail.com>
> wrote:
> > hello forum,
> >
> > in the continuous time domain, the sum of 2 sinusoids of different
> > frequencies w1, w2, results in a periodic signal only if w1/w2 is a
> > rational number.
> > If w1 and w2 are integers or decimals with finite digits, then their sum is
> > alway periodic (a finite decimal or an infinite periodic decimal can be
> > written as fractions).
> > Only if one of the frequencies is an irrational number then the sum of the
> > two sinusoids will not be a periodic signal.
> >
> > In the discrete time domain, in computer simulation, an irrational
> > number(infinite nonrepeating decimal) must be truncated in the number of
> > decimals so it is seen as a rational number too.... Does that means that,
> > digitally, if we are summing two discrete sinusoids of any frequency (as
> > long as they are discrete periodic sinusoids), because of the impossibility
> > of truly representing an irrational number, we can never get an aperiodic
> > signal from their sum?
> >
> > fisico32
>
> Hi,
> If in discrete time domain, you have a periodic sequence, then the
> digital frequency which is the ratio of the analog frequency you are
> representing to the sampling frequency, i.e. f_a/f_s is a rational

You claim the "analog frequency you are representing" can always be expressed
as a ratio of the sampling frequency. The original question was what if it can
not? In other words, f_a/f_s is a not rational for one of the sampled sinusoids
to be summed.

In pure mathematics that means the sampled sinusoid would go on forever
without repeat. In the real world neither the analog signal period or the
sampling clock can ever be exact enough for anyone to ever be able to determine
that there would or wouldn't be an irrational relation between the two.

-jim






>
> number. So if you have 2 discrete periodic sinusoids, then it means
> that f_a_1/f_s and f_a_2/fs are both rationals. This further implies
> that f_a_1/f_a_2 is also a rational number and hence the sum of these
> 2 periodic sequences will also be periodic.
> Hope this helps.
>
> Regards
> suren

From: Rune Allnor on 18 May 2010 05:58
On 14 Mai, 14:34, "fisico32" <marcoscipioni1(a)n_o_s_p_a_m.gmail.com>
wrote:
> hello forum,
>
> in the continuous time domain, the sum of 2 sinusoids of different
> frequencies w1, w2, results in a periodic signal  only if w1/w2 is a
> rational number.
> If w1 and w2 are integers or decimals with finite digits, then their sum is
> alway periodic (a finite decimal or an infinite periodic decimal can be
> written as fractions).
> Only if one of the frequencies is an irrational number then the sum of the
> two sinusoids will not be a periodic signal.

Don't know what you mean by 'infinite periodic decimal', but apart
from
that I agree.

> In the discrete time domain, in computer simulation, an irrational
> number(infinite nonrepeating decimal) must be truncated in the number of
> decimals so it is seen as a rational number too.... Does that means that,
> digitally, if we are summing two discrete sinusoids of any frequency (as
> long as they are discrete periodic sinusoids), because of the impossibility
> of truly representing an irrational number, we can never get an aperiodic
> signal from their sum?

Correct. Just the same way that a computer simulation never can
produce a signal with arbitrary dynamics, that is infinitely long,
or one that is continuous.

A computer simulation is nothing but a toy, that has only superficial
resemblance to any 'reality' you might think of.

Rune
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