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From: fisico32 on 14 May 2010 08:34
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
From: Clay on 14 May 2010 10:01
On May 14, 8:34 am, "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

If you restrict your implementation to rational numbers, i.e.,
quantized floating point, then yes the result evenually repeats. But a
number system being discrete only says that there is some minimum
positive spacing between the numbers in the system. You don't have to
use equally spaced numbers, so see what you can figure out from
applying this concept?

Clay
From: suren on 14 May 2010 10:38
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
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: Steve Pope on 14 May 2010 10:42
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.

Steve
From: Tim Wescott on 14 May 2010 10:46
fisico32 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?

Perhaps theoretically. But practically speaking you can get signals
that never exactly repeat over any reasonable correlation length of any
filters in the system, so they may as well have a normalized frequency
that's irrational in the discrete-time domain.

The bigger question is -- why care? Signals with irrational normalized
frequencies still go through DSP hardware with no problem -- it's not
like anything breaks. Can you think of a case where it's an issue?

--
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com
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