Piecewise Function Homework: Solve Fourier Coefficient b1

[DrOnline]

Homework Statement

Given the following periodic signal:

What is the Fourier coefficient b₁.

I'm not asking for help with the Fourier series. I feel my integration is solid.

What I need help with is creating the correct function!

The Attempt at a Solution

I first declare t=5 as now to be t=0, and T=6 becomes t=2, etc. Am I allowed to do this? It also seems a bit iffy because the period starts at t=4...

f(t) =
1, 0<t<1
t, 1<t<2

This gives me, as far as I can tell, the correct values for 0<t<2, compared to the graph.

Unsure about the "t" function there...

I am doubting whether this is correct, me and a friend have done the series 4-5 times now using this function, and we get the wrong answer, so we think the function we wrote is wrong.

Can somebody help me make sense of this?

 

[dirk_mec1]

f(t) =
1 for t in the interval of [1+4(t-1), 2+4(t-1)]
etc.

 

[LCKurtz]

Homework Statement

Given the following periodic signal:

What is the Fourier coefficient b₁.

I'm not asking for help with the Fourier series. I feel my integration is solid.

What I need help with is creating the correct function!

The Attempt at a Solution

I first declare t=5 as now to be t=0, and T=6 becomes t=2, etc. Am I allowed to do this? It also seems a bit iffy because the period starts at t=4...

f(t) =
1, 0<t<1
t, 1<t<2

Don't forget $f(t)=0,~2<t<4$.

This gives me, as far as I can tell, the correct values for 0<t<2, compared to the graph.

Unsure about the "t" function there...

I am doubting whether this is correct, me and a friend have done the series 4-5 times now using this function, and we get the wrong answer, so we think the function we wrote is wrong.

Can somebody help me make sense of this?

I don't understand the $1.5$ on your picture. But if you calculate the FS for the above function on $(0,4)$ extended periodically, it will converge to the required graph. But the function you are using and the one given are not the same since they disagree for the same value of $t$. One is a translated version of the other and the FS may not look the same.

 

[DrOnline]

The 1.5 is the average value of the interval from "my" 1 to 2, I wrote that to demonstrate how to calculate the average of the function value to a friend.

You tell me to not forget the interval between 2 and 4, but they do not matter, because they will integrate to zero, this is how I see it. I know it is an inaccurate way when I describe the piecewise function, but I only listed the ones relevant for the FS. I do account for them in the period variables when I do the calculations.

My task is only to find the Fourier coefficient b1, so I don't know if that is affected by me messing with the times.

Tomorrow I think I will re-do this and using the times actually on the paper, use the first period, 1 to 4.

f(t)=
0, 0<t<1
1, 1<t<2
-1+t, 2<t<3
0, 3<t<4

I think I understand what you mean about messing with the times, perhaps it make the actual FS not match the actual graph as asked for in the task.