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Dint
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Hello, I am attempting a past exam paper in preparation for an upcoming exam. The past exam papers do not come with answers and I'm a little unsure as to whether I'm doing all of the questions correctly and would like some feedback if I'm going wrong somewhere.
Any help is greatly appreciated :)
Here are the questions:
Draw a square wave of amplitude 1 and period 1 second whose trigonometric Fourier Series Representation consists of only cosine terms and has no DC component.
Now, I assume they want the FSR to be made up of only cosine terms, there is another question on another past exam that asks for the same thing but in sine terms. A normal square wave that is:
1 for 0<x<.5
0 for -.5<x<0
Consists of only sine terms when you do the FSR (when I did it). It also makes sense because it looks like a sine wave in that it rises at the origin.
2. The attempt at a solution
However, to make it consist of only cosine terms, I'm not quite sure how to draw this, this is what I came up with.
http://img34.imageshack.us/img34/2/squarewave.jpg [Broken]
I did this because I figured it resembles what a cosine wave looks like, in that it peaks at the origin. (I know it's a square wave so it doesn't really peak, but I'm picturing a cosine wave representing the signal in the box)
PART B - 1. Homework Statement
The next part of the problem asks:
For the signal in part A, compute the trigonometrical FSR.
then
Compute the exponential FSR directly from the trigonometric FSR.
[tex]a_{0} = \frac{1}{T} \int\limits_{f_{0}}^{f_{0}+T} f(t) dt[/tex]
[tex]a_{n} = \frac{2}{T} \int\limits_{f_{0}}^{f_{0}+T} f(t) cos\left(\frac{2{\pi}nt}{T}\right) dt[/tex]
[tex]b_{n} = \frac{2}{T} \int\limits_{f_{0}}^{f_{0}+T} f(t) sin\left(\frac{2{\pi}nt}{T}\right) dt[/tex]
I've also been told that for the exponential FSR:
[tex]c_{0} = a_{0}[/tex]
[tex]c_{n} = {\frac{1}{2}}\left(a_{n} - jb_{n}\right)[/tex]
[tex]c_{-n} = {\frac{1}{2}}\left(a_{n} + jb_{n}\right)[/tex]
[tex]a_{0} = \frac{1}{2}[/tex]
By inspection I can see that the average value of one period of the square wave is going to be 0.5.
[tex]a_{n} = \frac{2}{1} \int\limits_{-{\frac{1}{4}}}^{{\frac{1}{4}}} cos\left(\frac{2{\pi}nt}{1}\right) dt[/tex]
Now when I do this integral, I get:
[tex] \frac{1}{{\pi}n} \left[ sin\left(\frac{{\pi}n}{2}\right) - sin\left(-\frac{{\pi}n}{2}\right)\right][/tex]
I'm kinda stuck here. This answer does not give zero, I need the FSR to contain no sine terms, and this answer for my an term does give sine terms...
Any ideas?
Thanks a lot for taking the time to read this.
Any help is greatly appreciated :)
Here are the questions:
Homework Statement
Draw a square wave of amplitude 1 and period 1 second whose trigonometric Fourier Series Representation consists of only cosine terms and has no DC component.
Now, I assume they want the FSR to be made up of only cosine terms, there is another question on another past exam that asks for the same thing but in sine terms. A normal square wave that is:
1 for 0<x<.5
0 for -.5<x<0
Consists of only sine terms when you do the FSR (when I did it). It also makes sense because it looks like a sine wave in that it rises at the origin.
2. The attempt at a solution
However, to make it consist of only cosine terms, I'm not quite sure how to draw this, this is what I came up with.
http://img34.imageshack.us/img34/2/squarewave.jpg [Broken]
I did this because I figured it resembles what a cosine wave looks like, in that it peaks at the origin. (I know it's a square wave so it doesn't really peak, but I'm picturing a cosine wave representing the signal in the box)
PART B - 1. Homework Statement
The next part of the problem asks:
For the signal in part A, compute the trigonometrical FSR.
then
Compute the exponential FSR directly from the trigonometric FSR.
Homework Equations
[tex]a_{0} = \frac{1}{T} \int\limits_{f_{0}}^{f_{0}+T} f(t) dt[/tex]
[tex]a_{n} = \frac{2}{T} \int\limits_{f_{0}}^{f_{0}+T} f(t) cos\left(\frac{2{\pi}nt}{T}\right) dt[/tex]
[tex]b_{n} = \frac{2}{T} \int\limits_{f_{0}}^{f_{0}+T} f(t) sin\left(\frac{2{\pi}nt}{T}\right) dt[/tex]
I've also been told that for the exponential FSR:
[tex]c_{0} = a_{0}[/tex]
[tex]c_{n} = {\frac{1}{2}}\left(a_{n} - jb_{n}\right)[/tex]
[tex]c_{-n} = {\frac{1}{2}}\left(a_{n} + jb_{n}\right)[/tex]
The Attempt at a Solution
[tex]a_{0} = \frac{1}{2}[/tex]
By inspection I can see that the average value of one period of the square wave is going to be 0.5.
[tex]a_{n} = \frac{2}{1} \int\limits_{-{\frac{1}{4}}}^{{\frac{1}{4}}} cos\left(\frac{2{\pi}nt}{1}\right) dt[/tex]
Now when I do this integral, I get:
[tex] \frac{1}{{\pi}n} \left[ sin\left(\frac{{\pi}n}{2}\right) - sin\left(-\frac{{\pi}n}{2}\right)\right][/tex]
I'm kinda stuck here. This answer does not give zero, I need the FSR to contain no sine terms, and this answer for my an term does give sine terms...
Any ideas?
Thanks a lot for taking the time to read this.
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