Signals and Systems [Calc the Fundamental Period]

[damasgate]

Homework Statement

Determine the Fundamental Period and Frequency

I got this question wrong on My midterm and I was wondering if you guys could give me some feedback on it

x(t) = cos[50(pi)t] + sin[15(pi)t]

Homework Equations

T(fundamental Period) = 2(pi)/w // w is the frequency

The Attempt at a Solution

I tried finding the peirod of both sine and cosine

for first T = 2(pi)/50(pi) = 1/25
for second T=2(pi)/15(pi) = 2/15

then you have to find the least common Multiple(I think this is where I messed it up)

I got for (1/25) and (2/15) LCM to be 1/75 [ like I said I think this is wrong, I'm not sure how to do LCM with fractions]

I got T=1/75 (fundamental Period)
then I got W=2(pi)/T = 150(pi0
and then I got TWO big Xs on them :(

 

[KataKoniK]

Dear student,

First of all, don't be discouraged by getting a question wrong on your midterm. Making mistakes is a normal part of the learning process, and it's important to take these opportunities to learn and improve.

In this case, your approach was on the right track, but there were a few errors in your calculations. Let's break it down step by step:

1. To find the fundamental period, we need to find the smallest value of t where the function repeats itself. This is also known as the least common multiple (LCM) of the periods of the individual terms in the function.

2. To find the period of a cosine or sine function, we use the formula T = 2(pi)/w, where w is the frequency. In this case, the frequency of the cosine term is 50(pi), and the frequency of the sine term is 15(pi).

3. So, the period of the cosine term is T = 2(pi)/50(pi) = 1/25, and the period of the sine term is T = 2(pi)/15(pi) = 2/15.

4. To find the LCM of these two fractions, we need to find the smallest number that is divisible by both 25 and 15. This number is 75, so the fundamental period is T = 75.

5. Now, to find the frequency, we can use the formula w = 2(pi)/T. Plugging in our value for T, we get w = 2(pi)/75 = 2/75(pi).

So, the fundamental period is T = 75 and the frequency is w = 2/75(pi). I hope this helps clarify the process for you. Keep practicing and don't get discouraged - you'll get it right next time!