Hello everyone i would like to find if a sum of periodic systems is periodic, and if it is i would like to find its fundamental period. When i have two signals i think i am capable of doing it lets assume that the period of the first signal is T1=2/3 and the period of the second is T2=3/4 T1/T2=8/9 which is rational so the sum of these two periodic signals is periodic too for its fundamental period i do T12=T1*k=T2*d =>k/d=T2/T1=9/8 =>k=9 and d=8 so T12=6 But what happens when you have the sum of three periodic signals with three periods T1,T2,T3 ? I tried making first the sum of the two signals and then again with the third but the period i was finally doing wasnt the same for every combination, can you help me ? I actually have made some miscalculations, i find the same result no matter the combination But is this way right or is there any other way ? thanks in advance
Before asking about three signals, consider to signals more carefully. You have two signals that have periods T1 and T2. What is the period of the sum of these two signals? Are you even guaranteed that such a period exists? If not, what is required in order for the sum of the two signals to be periodic?
When you add two frequencies, it will result in frequencies of the addition and subtraction of the initial frequencies. Check this site for mathematical expression. http://hyperphysics.phy-astr.gsu.edu/hbase/audio/sumdif.html In acoustics the term "Multiphonics" is used to describe mixing multiple frequencies together.
@PlasmaT : There's something very strange about the claims made in that hyperphysics link you posted. I think the information needs some qualification. The linear physical addition of two sinusoidal electrical voltages (for instance) doesn't lead to the creation of two new frequencies in the composite voltage. The spectrum of such an addition will not reveal any so-called "sum" and "difference" terms, rather only the original frequencies. With respect to modulation, the sum and difference side-bands in RF transmission are created by the physical multiplication of the carrier and modulating signal waveforms. Perhaps I have misinterpreted the information. It will be interesting to see what others think.
@t_n_k : I did not realize until you raised the concern. So i did a check. It think both the terms "multiplication" and "addtion" are correct in how its plugged into the formula. Its because the "sin" multiplication can be deduced to an addition with "cos" values. Please check this link: http://www.radio-electronics.com/in...n/mixers/rf-mixers-mixing-basics-tutorial.php
Thanks @PlasmaT . I'm reasonably familiar with RF modulation & demodulation techniques. For me, the following quote from the (post #3) link is ambiguous, since one might conclude the notion of side-bands arises from the application of the earlier reference to addition of sine terms: "Both the sum and difference frequencies are exploited in radio communication, forming the upper and lower sidebands and determining the transmitted bandwidth." So, the particular issue I have is the apparent implication that the side-bands in RF modulation can be obtained by simple addition of signals - which is clearly not true. Also, the equation as shown is not particularly helpful in relation to understanding the modulation concept with respect to the formation of side-bands. As often is the case in many threads, I have wandered far from the OP's original question .....
When two signals are periodic with periods T1 and T2 then the sum of these two signals will be also a periodic signal if T1/T2 is a rational number, right ?
Yes, with the important understanding that T1/T2 does not have to be rational and, if it's not, then the resulting signal is aperiodic.