Fourier Transform Homework: Determine First 3 Terms

In summary, the homework statement is asking for the first three terms of the Fourier series pictured in the attachment. The first two terms are a_o and a_n, while the third term is undefined.
  • #1
roldy
237
2

Homework Statement


I need to determine the first three terms in the Fourier series pictured in the attachment.
Did I define the peace-wise functions correctly?

I'm re-posting this with the tex code instead of the attached document.

Homework Equations


[tex]
a_o=\frac{1}{2L}\int_{-L}^Lf(t)dt
[/tex]

[tex]
a_n=\frac{1}{L}\int_{-L}^Lf(t)cos\left(\frac{n\pi{t}}{L}\right)dt
[/tex]


The Attempt at a Solution


The plot of the series is symmetric, so therefore I am only going to find what [tex]a_o[/tex] and [tex]a_n[/tex].

[tex]
a_o=\frac{2}{3T} \left[\int_\frac{-3T}{4}^\frac{-T}{4}(-1)dt+\int_\frac{-T}{4}^\frac{T}{4}(1)dt+\int_\frac{T}{4}^\frac{3T}{4}(-1)dt\right]
[/tex]
[tex]
=\frac{2}{3T}\left[-\left(-T/4-(-3T/4)\right)+\left(T/4-(-T/4)\right)+\left(3T/4-T/4\right)\right]

=\frac{2}{3T}\left(T/4-3T/4+T/4+T/4+3T/4-T/4\right)=1/3
[/tex]

[tex]
a_n=\frac{4}{3T}\left[\int_\frac{-3T}{4}^\frac{-T}{4}(-1)cos\left(\frac{n\pi{t}}{3T/4}\right)dt+\int_\frac{-T}{4}^\frac{T}{4}(1)cos\left(\frac{n\pi{t}}{3T/4}\right)dt+\int_\frac{T}{4}^\frac{3T}{4}(-1)cos\left(\frac{n\pi{t}}{3T/4}\right)dt\right]
[/tex]

[tex]
=\frac{4}{3T}\left[\left[-sin\left(\frac{n\pi{\left(-T/4\right)}}{3T/4}\right)+sin\left(\frac{n\pi{\left(-3T/4\right)}}{3T/4}\right)\right]\frac{3T}{4n\pi}+\left[sin\left(\frac{n\pi{\left(T/4\right)}}{3T/4}\right)-sin\left(\frac{n\pi{\left(-T/4\right)}}{3T/4}\right)\right]\frac{3T}{4n\pi}
[/tex]
[tex]
-\left[sin\left(\frac{n\pi{\left(3T/4\right)}}{3T/4}\right)-sin\left(\frac{n\pi{\left(T/4\right)}}{3T/4}\right)\right]\frac{3T}{4n\pi}
[/tex]


[tex]
\frac{1}{n\pi}\left[\left[-sin\left(\frac{-n\pi}{3}\right)+sin\left(-n\pi\right)\right]+\left[sin\left(\frac{n\pi}{3}\right)-sin\left(\frac{-n\pi}{3}\right)\right]-\left[sin\left(n\pi\right)-sin\left(\frac{n\pi}{3}\right)\right]\right]
[/tex]

Distributing the signs through and simplifying:

[tex]
\frac{1}{n\pi}\left[4sin\left(\frac{n\pi}{3}\right)-2sin\left(n\pi\right)\right]
[/tex]

So for n=1,2,3 I get

[tex]
a_1=\frac{2\sqrt{3}}{2\pi}
[/tex]

[tex]
a_2=\frac{\sqrt{3}}{2\pi}
[/tex]

[tex]
a_3=0
[/tex]
 
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  • #2
What is f?
 
  • #3
lanedance said:
What is f?

It's not f. It's f(t), meaning f of t. If you look at the attached picture I came up with f(t) on three intervals. I'm not sure if they are correct. They look correct to me.
 
  • #4
yeah i get its f(t) but can't see any attached pic? pretty important part of the problem
 
  • #5
Sorry about that. For some reason it didn't attach. Here's another try at it.
 

Attachments

  • untitled.JPG
    untitled.JPG
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  • #6
ok, so why integrate over more that a period?

3T/4 - (-3T/4) = 3T/2
 
  • #7
I think that's where I might of made the mistake. I think I should of done it from -T/4 to 3T/4.
 
  • #8
That sounds better... Note that the integral over sin(tnpi/T) would be non zero over that interval as well, you could try your new interval on a sin as a check
 

1. What is a Fourier Transform?

A Fourier Transform is a mathematical tool used to decompose a time-varying signal into its constituent frequencies. It helps in understanding the frequency components present in a given signal.

2. How is the Fourier Transform calculated?

The Fourier Transform is calculated by taking the complex exponential function of the signal and integrating it over a specified range. This process is repeated for different frequencies to obtain the frequency spectrum of the signal.

3. What is the significance of determining the first three terms in a Fourier Transform homework?

The first three terms in a Fourier Transform homework provide important information about the frequency components present in a signal. These terms represent the fundamental frequency, first harmonic, and second harmonic of the signal. They can be used to analyze and understand the behavior of a system or signal in the frequency domain.

4. What are some applications of Fourier Transform?

The Fourier Transform has a wide range of applications in various fields such as signal processing, image processing, audio analysis, and communication systems. It is also used in solving differential equations and in quantum mechanics.

5. Are there any limitations of Fourier Transform?

One of the limitations of Fourier Transform is that it assumes the signal to be infinite and continuous. It also assumes that the signal is periodic, which may not be the case in real-world scenarios. Additionally, the Fourier Transform cannot capture localized information about a signal, which can be important in some applications.

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