Question

I need help with number 2. I don't know where to get the information?
http://wps.prenhall.com/wps/media/objects/1322/1354712/Ch6_PM%20Digital%20Trans.pdf

Answer 1

the period is the smallest \(T\) such that \(s(t)=s(t+T)\) for all t.

Take your expression of \(s(t)\) (write it here too), write \(s(t+T)\) as well.
This will be respectively something of the form \(\sin(a)\) and \(s(b)\).
then use \(b=a+2\pi\). (using the period of the \(\sin\) function.

Answer 2

i forgot the \(A\) in front of the \(\sin\) but it will not make a difference.

does this make sense to you?

Answer 3

not really. like I have the s(t)= equation but idk after that

Answer 4

\(s(t)=1\sin(2\pi f_0 t+0)=\sin(2\pi f_0 t)\)
and
\(s(t+T)=\sin(2\pi f_0 (t+T))\).

\(s(t)\) is a sinusoidal function. Increase \(t\) and you will come back on the same arguments (mod 2pi).

So the question you want to ask in order to find T is: when is the first time that the argument will have made the "whole angle 2pi"?

Answer 5

\(\sin(t)\) has period \(2\pi\)
\(\sin(2t)\) has period \(\pi\), because after time \(\pi\), \(2t\) has made the complete \(2\pi\) and we repeat the same values.

\(\sin(2\pi t)\) has period 1. Because \(\sin(2\pi(t+T))=\sin(2\pi t+2\pi T)\), and this is true when \(2\pi T=2\pi\) (T=1)

similarly, \(s(t)=s(t+T)\) iff \(\sin(2\pi f_0t) = \sin(2\pi f_0 t + 2\pi f_0 T)\)
iff \(2\pi f_0 T = 2\pi\) (the period of the sine function)

Answer 6

or do you sub 1 in for the frequency?

Answer 7

for \(\sin(2\pi t)\)?

Answer 8

so what is the period for T then? is it 1?

Answer 9

It is \(T=\frac1{f_0}\).

The big lines are :
- write \(s(t)=s(t+T)\)
- write the argument of the functions as \(2\pi f_0 t\) for the first, \(2\pi f_0 t + C\) (T is in C)
- realize that you have a sine function, that means \(C = 2\pi\).
this last equation gives you \(T=1/f_0\).

Answer 10

oh! oh my gosh I didn't see it before silly me! Thank you so much!!

Answer 11

yw (=