# how to calculate period of composite signal

#### johnvdd

Joined Sep 25, 2012
4
My text book Signals And Systems By Palani says that the fundamental period of sum, product of any signal is the LCM of their periods.

I have x(t) = sin(t) . cos(t) so the fundamental period of x(t) should be = LCM of 2pi and 2pi = 2pi

but as we know sin(t) . cos(t) = sin 2t and its period is pi (not 2pi) ??????

Please help me with finding period of sum, difference, product, division, square, differentiation, integration e.t.c. of periodic signals

#### t_n_k

Joined Mar 6, 2009
5,455
Is it actually possible to find the LCM of several values which are irrational?

In any case, perhaps one should qualify the statement regarding periodicity for the situation of periodic signal products ...

The resultant product of [two] periodic signals has a period equal to the LCM of the original signal periods but the period found by this method may not necessarily be the fundamental period.

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#### johnvdd

Joined Sep 25, 2012
4

OK this "the period found by this method may not necessarily be the fundamental period"

but then what formula/ method should i use to find to get the fundamental period ???????????????

Can u tell me some book, tutorial, site e.t.c. for it ? Please !!

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Please help me with finding period of sum, difference, product, division, square, differentiation, integration e.t.c. of periodic signals

#### t_n_k

Joined Mar 6, 2009
5,455
I would suggest the solution to the problem you raised initially was readily solved - as you indicated.

If

$$f(t)=sin(t)cos(t)= \frac{1}{2}sin(2t)$$

it's not difficult to discern the period T as

$$T=\pi \ seconds$$

The issue apparently arises only with multiplication of periodic signals. I suppose in such matters, one must be competent in the use of trig identities in re-arranging composite trig equations into a form from which it is easier to correctly determine the fundamental period. So if [say] one can re-arrange a function involving products of functions into a simpler form involving only sum (or difference) terms then it becomes easier to find a solution.

Do you have a collection of problems of this type which might be used as practice exercises to tease out those situations where you might experience some difficulty or uncertainty?

Such as ....

Determine the fundamental period of

$$v(t)=\cos{(5t)} \sin{(3t+45^o)}$$

Final Comment: Keep in mind that even when dealing only with addition of periodic functions, the summation may be non-periodic - this applies to cases where the ratio of any two individual composite signal periods is irrational.

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#### WBahn

Joined Mar 31, 2012
25,578
My text book Signals And Systems By Palani says that the fundamental period of sum, product of any signal is the LCM of their periods.

I have x(t) = sin(t) . cos(t) so the fundamental period of x(t) should be = LCM of 2pi and 2pi = 2pi

but as we know sin(t) . cos(t) = sin 2t and its period is pi (not 2pi) ??????

Please help me with finding period of sum, difference, product, division, square, differentiation, integration e.t.c. of periodic signals

Are you sure it says that the fundamental period will be that? It's true that "a" period will be the LCM of the periods, but that period may turn out to be several actual periods. Whether or not that is the case depends on just how the waveforms interact, such as the two you cite in your example. You might look at the context in which the claim was made; the author may say something about that being for unrelated signals or that there are special case exceptions.

An even clearer example is sin^2(wt) (or sin(wt)*sin(w)). In essence, what is happening is that the interactions between the two waveforms is cancelling out the fundamental component of one (or both) leaving you with a composite of two waveforms that effectively have shorter fundamental periods.

#### johnvdd

Joined Sep 25, 2012
4
but can you tell me some books that have a wide, deep and detailed analysis on this topic.

#### Papabravo

Joined Feb 24, 2006
13,550
Doesn't a multiplication of two sinusoids produce sidebands at f1+f2 and f1-f2 when f1 > f2 ?

#### WBahn

Joined Mar 31, 2012
25,578
Yes, but sidebands are different from a fundamental frequency.

The fundamental frequency would be;

f = 1/T

where T is the fundamental period, meaning the smallest period for which:

y(t+T) = y(t)

#### Papabravo

Joined Feb 24, 2006
13,550
Yes, but the original question was to determine the period(frequency) of products of periodic functions. The output of a multiplier ("mixer" in RF terms) is the sum of four components. So did the OP meant to imply that the new period is the LCM of the four components?

#### WBahn

Joined Mar 31, 2012
25,578
The original question specifically asked about the fundamental period. Now, I'm assuming that to mean T as I have defined it -- if the OP meant something else, hopefully he'll speak up. If looking at it as a mixing process, then since both frequencies are the same we have (potentially) something at DC and something at twice the original frequency -- which is pretty much what the OP is seeing.

But what is confusing him (or at least was) is that the rule he is citing basically says that when you multiply to waveforms (or add them, for that matter) the period of the new waveform will be determined by when the waveforms sync up again, which will occur at the least common multiple of their respective periods. What this is missing is that this gives a limit for the longest possible period, but that the waveforms may interact in such a way as to have a shorter period.

In general, knowing the frequencies of the spectral components is not enough to determine the fundamental frequency, just a lower limit on it. Or at least I think that's the case. The more I think about it the less sure I am that it is possible to construct such an example. Clearly we can construct examples in which the LCM rule doesn't yield the actual fundamental ffrequency.

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#### WBahn

Joined Mar 31, 2012
25,578
Is it actually possible to find the LCM of several values which are irrational?
Sure.

The LCM of sqrt(2) and 4 is 4.

The LCM of 4pi and 6pi is 12pi.

The LCM of e, pi, and sqrt(2) is (e*pi*sqrt(2)).