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Calculus/DT signals confusion, searched a lot but no perfect answer

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Expert: Alon Mandes - 6/27/2010


Question
Q1)determine which of them are periodic and find fundamental period.

a)  cos ( (PI*30n)/105 )

b) x(n)=cos(PIn/2)-sin(PIn/8)+3cos(PIn/4  +  PI/3)


how solved them a little, please further, is the working correct?
and also
there is no t, are we supposed to just keep T on the left side?


solution
a)
(30nPI/105)=2PIk
shifting all to right side
left side= 7 * (k/n)
so k is a constant therefore k/n is also a constant,
assuming k and n to be 1
then
left side= 7;
we need fundamental period so,
T=7 seconds.


solution
b)
taking all the values in the brackets of the functions of cos sin and 3cos and equating them to 2PIk separately.
getting
T=4, T=16 AND T =8.
now what is do be done?
on net some sources have done division, some say take LCM.
what to do and why?.


and if possible please illustrate the graph these signals would make

thank you

Answer

the 2nd graph
Hello Wallaq,
In section (a) what you did was right . The period is 7 .
In section (b) the signal is composed of 3 periodec sub-signals. In this case, it's called
group velocity, & this velocity is dictated by slowest component . Therefore T=16 is the
domenant period. ("Immagine 2 people running together, one runs slow & the other fast , in
this case , as a group , the slower person will determine the speed ")

I attaced an image of the second graph .

Alon.  

Alon Mandes

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