m (→Summary) |
(→Summary) |
||
| Line 41: | Line 41: | ||
From the above math, we can determine all the coefficients: | From the above math, we can determine all the coefficients: | ||
<math> \ \ a_{-2} = 1; \ \ a_{-1} = 2; \ \ a_{0} = 0; \ \ a_1 = 2;\ \ a_{2} = -1 </math> | <math> \ \ a_{-2} = 1; \ \ a_{-1} = 2; \ \ a_{0} = 0; \ \ a_1 = 2;\ \ a_{2} = -1 </math> | ||
| + | |||
| + | The fundamental period of the function is found from: <math>e^{j\omega_0}</math> where he period T = <math>{2\pi \over \omega_o}</math> | ||
| + | |||
| + | Thus, the fundamental period = <math> {2\pi \over 3} </math> | ||
Revision as of 13:48, 25 September 2008
Homework 4 Ben Horst: 4.1 :: 4.2 :: 4.3:: 4.4:: 4.5
Signal
$ x(t) = 2\sin(6t) + 4\cos(3t) $
Fourier Series
$ x(t) = \sum_{k=- \infty }^ \infty a_ke^{jk\omega_0t} $
By Euler's formula, we have: $ x(t)=2({ e^{j 6t} + -e^{-j6t} \over 2j}) + 4({ e^{j3t} + e^{-j3t} \over 2 }) $
$ x(t) = ({ e^{j 6t} + -e^{-j6t} \over j}) + 2e^{j3t} + 2e^{-j3t} $
$ x(t) = -e^{2 j3t} + e^{-2 j3t} + 2e^{1 j3t} + 2e^{-1 j3t} $
Ordering our k's to form a proper series:
$ x(t)= e^{(-2) j3t} + 2e^{(-1)j3t} + 0 + 2e^{(1) j3t} - e^{(2) j3t} $
And making sure we don't forget about $ a_0 $:
$ x(t)= (1)e^{(-2) j3t} + (2)e^{(-1)j3t} + (0)e^{(0)j3t} + (2)e^{(1) j3t} + (-1)e^{(2) j3t} $
Summary
From the above math, we can determine all the coefficients: $ \ \ a_{-2} = 1; \ \ a_{-1} = 2; \ \ a_{0} = 0; \ \ a_1 = 2;\ \ a_{2} = -1 $
The fundamental period of the function is found from: $ e^{j\omega_0} $ where he period T = $ {2\pi \over \omega_o} $
Thus, the fundamental period = $ {2\pi \over 3} $
