Understanding Fourier Series for EE Homework

[NHLspl09]

Hey guys, I'm having trouble with a problem assigned for homework in an EE course on Fourier series. We have yet to have a lecture on Fourier series when the homework is due Thursday, and because of the long weekend break we don't have class Tuesday. With little knowledge on Fourier series, from what it seems I have the basic formula (if you can call it that), but am getting kind of confused looking at examples online and was wondering if I could get some help. I only have to complete numbers 2, 4, and 5 of Problem 3-6. Any input or knowledge on the topic/problem at hand would be greatly appreciated!

Homework Statement

Attachment 1 - EE HW P3-6
Attachment 2 - EE HW P3-6 Table

Homework Equations

Attachment 2 - EE HW P3-6 Table

a_o=$\frac{1}{To}$$\int$x(t)dt

a_n=$\frac{2}{To}$$\int$x(t)cos(ηω_ot)dt n≠0

b_n=$\frac{2}{To}$$\int$x(t)sin(ηω_ot)dt


^All with lower bounds of t₁ and an upper bound of t₁+T_o on the integrals

The Attempt at a Solution

All I really know of the Fourier series is what I've found online and what it seems to be primarily are the equations I've posted. Again, any help or advice is greatly appreciated!

 

[MisterX]

You're supposed find the exponential Fourier series coefficients, so the formula you should use is one such as

$c_{n} = \frac{1}{T_{0}} \int ^{t_{1} + T_{0}} _{t_{1}} e^{-int}dt$

where

$i = \sqrt{-1}$

and T₀ is one period

t₁ is any real number. The value of t₁ does not affect the value of any c_n. Choose a value of t₁ that will make the integration simpler. Zero would not be a bad choice for these functions.

You are tasked with finding an expression for c_n for each of the functions on the second sheet, except number three.

 

[NHLspl09]

You're supposed find the exponential Fourier series coefficients, so the formula you should use is one such as

$c_{n} = \frac{1}{2\pi} \int ^{t_{1} + T_{0}} _{t_{1}} e^{int}dt$

where

$i = \sqrt{2}$

and T₀ is one period

t₁ is any real number. The value of t₁ does not affect the value of any c_n. Choose a value of t₁ that will make the integration simpler. Zero would not be a bad choice for these functions.

You are tasked with finding an expression for c_n for each of the functions on the second sheet, except number three.

Interesting, so the equations I posted aren't relevant at all? Cause they seemed to be appearing around multiple sites. Also, when you typed e^int, did you mean eⁱ?? Because you said that i=$\sqrt{2}$

 

[vela]

MisterX meant $i=\sqrt{-1}$.

Using the formula $e^{i\theta} = \cos \theta + i\sin \theta$, you can find how your equations and the one MisterX gave are equivalent. Your equations aren't completely irrelevant, but for this problem, since it asks for the exponential series as opposed to the trig series, you'd be better off just using the formula MisterX gave you.

 

[vela]

By the way, the factor out front should probably be something like 1/T₀, not $1/2\pi$. You should check your textbook for the correct formula.

 

[MisterX]

By the way, the factor out front should probably be something like 1/T₀, not $1/2\pi$. You should check your textbook for the correct formula.

Yes, I changed the limits from another definition and neglected to change the co-effiecient. It should be 1/T₀. I also made a mistake defining i.

Please accept my apologies for this carelessness.

 

[NHLspl09]

MisterX meant $i=\sqrt{-1}$.

Using the formula $e^{i\theta} = \cos \theta + i\sin \theta$, you can find how your equations and the one MisterX gave are equivalent. Your equations aren't completely irrelevant, but for this problem, since it asks for the exponential series as opposed to the trig series, you'd be better off just using the formula MisterX gave you.

Yes, I changed the limits from another definition and neglected to change the co-effiecient. It should be 1/T₀. I also made a mistake defining i.

Please accept my apologies for this carelessness.

No apologies needed! I was just a little fuzzy on what you had mentioned that's all!

Ok, makes a little bit more sense to me now vela and MisterX - yet with this being my first time ever dealing with Fourier series I'm still a little bit confused as to what this problem in general is asking of me.

Also, on a side note/question - so the equations I posted and the equation posted by MisterX are all relevant to Fourier series? It's just that MisterX's equation are more relevant to the problem at hand?

 

[vela]

Right. There are two ways you can write the Fourier series: one as a sum of sines and cosines, the other as a sum of complex exponentials. If you calculate one series, you can actually figure out what the other one is relatively easily, but you might as well just calculate the one asked for directly.

The idea behind the Fourier series is that you can expand a periodic function in terms of a set of basis functions, either sines and cosines or complex exponentials. In other words, you can write
$$f(x) = \sum_{n=-\infty}^\infty c_n e^{in\omega t}$$where $\omega=2\pi/T_0$. The idea is to find what the constants c_n need to equal. That's what the formula tells you how to calculate.

At this point, it's probably best if you just take it on faith this stuff works and simply grind through the integrals. In your lecture, the professor will explain to you why it works.

 

[NHLspl09]

Right. There are two ways you can write the Fourier series: one as a sum of sines and cosines, the other as a sum of complex exponentials. If you calculate one series, you can actually figure out what the other one is relatively easily, but you might as well just calculate the one asked for directly.

The idea behind the Fourier series is that you can expand a periodic function in terms of a set of basis functions, either sines and cosines or complex exponentials. In other words, you can write
$$f(x) = \sum_{n=-\infty}^\infty c_n e^{in\omega t}$$where $\omega=2\pi/T_0$. The idea is to find what the constants c_n need to equal. That's what the formula tells you how to calculate.

At this point, it's probably best if you just take it on faith this stuff works and simply grind through the integrals. In your lecture, the professor will explain to you why it works.

OK I understand that, out of curiosity where do the equations of the sine, square, or triangular wave come in? I understand what you've said, just relating it to some of the problems I have is the challenge now

 

[vela]

Those are the periodic functions f(x).

 

[NHLspl09]

Right, so I would set those equations equal to $$\sum_{n=-\infty}^\infty c_n e^{in\omega t}$$ and solve for c_n?

 

[vela]

Sort of. If you do that — and you will undoubtedly see how to do that eventually — you end up with the formula
$$c_n = \frac{1}{T}\int_0^T f(x)e^{in\omega x}\,dx$$where $\omega=2\pi/T$, so figuring out what the coefficients are is essentially just doing an integral.

 

[NHLspl09]

Sort of. If you do that — and you will undoubtedly see how to do that eventually — you end up with the formula
$$c_n = \frac{1}{T}\int_0^T f(x)e^{in\omega x}\,dx$$where $\omega=2\pi/T$, so figuring out what the coefficients are is essentially just doing an integral.

Which can then be plugged into Matlab, out of curiosity, what is the n that is with iωx?

 

[vela]

It's the same n that's the subscript of c.

 

[NHLspl09]

Having no effect?

 

[vela]

I'm not sure what you mean by that. Have you looked at example 3-5 in your book?

 

[NHLspl09]

I'm not sure what you mean by that. Have you looked at example 3-5 in your book?

I have, (Attachment - EE HW 6 P3-6 Example), but after looking at the graph of the pulse wave (Attachment - EE HW 6 P3-6 Table) and actually reading through the integration steps a few times, it's difficult for me to understand how they found the equation for the pulse wave or if it's even needed to find the end result? It's just difficult for me with zero lecture time on the topic to try and make sense of it that's all, sorry if some questions seem foolish or irrelevant in that matter.

 

[MisterX]

it's difficult for me to understand how they found the equation for the pulse wave or if it's even needed to find the end result?

The right side of the equation for the pulse wave is the sum of infinitely many time-shifted rectangular pulses. $\sqcap$ is a symbol for the rectangle function.

That equation isn't really needed to find the Fourier series coefficients (X₀), because all you have to do is integrate over one period.

 

[NHLspl09]

That equation isn't really needed to find the Fourier series coefficients (X₀), because all you have to do is integrate over one period.

Yeah cause it didn't seem to be used in the integration at all. My next question would be after examining this - if that pulse wave function doesn't effect the integration, I'm not quite sure what would change in looking at the full-rectified sine wave..

 

[MisterX]

It's a different function! An integral expression for Fourier series coefficients would be different!

I didn't mean the function doesn't matter at all, I just meant you don't necessarily have to use the "train," or the actual periodic function. You just need something that is equal to the periodic function for one period, since the integral to get the Fourier series coefficients is over one period.

 

[vela]

Take the first problem, the half-rectified sine wave, for example. First, you need a function that describes what happens in one period. In this case, it would be
$$x(t) = \left\{ \begin{array}{ll} A\sin \omega_0 t & \mathrm{if}~0 \le t \lt T_0/2 \\ 0 & \mathrm{if}~T_0/2 \le t \le T_0 \end{array} \right.$$where $\omega_0 = 2\pi/T_0$. The coefficients X_n are given by
\begin{align*}
X_n &= \frac{1}{T_0}\int_0^{T_0} x(t)e^{-jn\omega_0 t}\,dt \\
&= \frac{1}{T_0}\left[\int_0^{T_0/2} A\sin\omega_0t\,e^{-jn\omega_0 t}\,dt+\int_{T_0/2}^{T_0} 0\,e^{-jn\omega_0 t}\,dt\right] \\
&= \frac{1}{T_0}\int_0^{T_0/2} A\sin\omega_0t\,e^{-jn\omega_0 t}\,dt
\end{align*}
I'll leave it to you to crank out the integral.