Is This Fourier Series an Odd or Even Function with a Period of 4s?

[Northbysouth]

Homework Statement

For he following Fourier series, which of the answers correctly describes the following function


y(t) = 2 - $\stackrel{1}{π}$∑₁^inf1/nsin(n*πt/2)

a) odd function, period = 2 s
b) Even function, period = 2s
c) Odd function, period = 4s
d) Even functio, period = 4s

Homework Equations

The Attempt at a Solution

From the reading my professor assigned I'm pretty sure that it's odd because it has a sin function, though I still don't understand why.

I'm also confused as to how I find the period. I know that

T =1/f

w=2*π*f

So,

w = (n*π)/2

I'm not quite sure what to make of n.

Any help would be appreciated.

 

[Simon Bridge]

That would be: $$y(t)=2-\frac{1}{\pi}\sum_{n=1}^\infty \frac{1}{n\sin(n\pi t/2)}$$

From the reading my professor assigned I'm pretty sure that it's odd because it has a sin function, though I still don't understand why.

Check y(-t), if it is the same as -y(t) then y(t) is odd, if it is the same as +y(t) then it is even.

I'm also confused as to how I find the period.

y(t+T)=y(t), what is T?

Certainly the period of the sine function is something to do with 2 units.
But, since it is multi-choice, you don't actually have to find the period - just check to see which of the choices is correct.

 

[rude man]

That would be: $$y(t)=2-\frac{1}{\pi}\sum_{n=1}^\infty \frac{1}{n\sin(n\pi t/2)}$$

I think you're misreading what the OP meant, or maybe you're trying to point out that he /she miswrote it.

Your denominator is intended to be the terms in the series, not their inverse. Look at the OP's thumbnail.

 

[rude man]

I'm also confused as to how I find the period. I know that

T =1/f

w=2*π*f

So,

w = (n*π)/2

I'm not quite sure what to make of n.

Any help would be appreciated.

The series is a series of harmonics starting with a dc term ("2") and then a series of sine terms
So the first sine term must be the fundamental frequency where n = 1.
So compare sin(nπt/2) with sin(ωt). In other words, compare π/2 with ω since n = 1.

If you know ω, what is the period T?

BTW by "period" they mean the periodicity of the lowest frequency component, i.e. for which n=1.

 

[Simon Bridge]

I think you're misreading what the OP meant, or maybe you're trying to point out that he /she miswrote it.

... a bit of both by the looks of things - neither thought completed. My excuse is that dinner was just about ready.

Your denominator is intended to be the terms in the series, not their inverse. Look at the OP's thumbnail.

Yep - should have looked at the thumb.
$$y(t)=\frac{1}{\pi}\sum_{n=1}^\infty \frac{1}{n}\sin \left(\frac{n\pi t}{2}\right)$$

Which would make the RHS a Fourier expansion of something - a saw-tooth wave.
... advise still stands.

BTW by "period" they mean the periodicity of the lowest frequency component, i.e. for which n=1.

... or the period of y(t) - let OP work out if this is the same thing.

a pure sine wave would be: $y(t)=\sin(2\pi t/T)$ where T is the period ... comparing with any other sine wave would be how you find T.

But in this case there is a shortcut. Need only use the definition of the period to check to see if either of the options is the correct one.

Care needed though, if $y(t+T)=y(t)$ then $y(t+2T)=y(t)$ also.
But if $T$ is the period of $y(t)$ then $y(t+T/2)\neq y(t)$
So it is a matter of putting the numbers into the equation, and thinking about what "period" means.
Which, I suspect, is the point of the exercise.

 

[LCKurtz]

Homework Statement

For he following Fourier series, which of the answers correctly describes the following function


y(t) = 2 - $\stackrel{1}{π}$∑₁^inf(1/n)sin(n*πt/2)

a) odd function, period = 2 s
b) Even function, period = 2s
c) Odd function, period = 4s
d) Even functio, period = 4s

From the reading my professor assigned I'm pretty sure that it's odd because it has a sin function, though I still don't understand why.

Understandably so. None of the above, even without knowing what "s" is.

 

[Simon Bridge]

@Northbysouth: any of this help?
You should check LCKurtz answer to see why... would y(t) be easier to handle without the "2 -" out the front?
(i.e. the way I wrote it in post #5?)

It is likely that there is a context missing from the problem statement you got.