(New page: == Signal == Compute the Fourier series coefficients of the following signal: <font size=4><math>x(t) = 3cos(7t) + 11sin(4t)</math></font> == Fourier series == <font size=4><math>x(...) |
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Compute the Fourier series coefficients of the following signal: | Compute the Fourier series coefficients of the following signal: | ||
| − | + | <math>x(t) = 3cos(7t) + 11sin(4t)\!</math> | |
| − | + | ||
== Fourier series == | == Fourier series == | ||
| − | + | <math>x(t) = 3cos(7t) + 11sin(4t)\!</math> | |
| − | + | ||
<math>x(t) = 3\frac{e^{i7t}+ e^{-i7t}}{2} + 11\frac{e^{i4t}- e^{-i4t}}{2i}</math> | <math>x(t) = 3\frac{e^{i7t}+ e^{-i7t}}{2} + 11\frac{e^{i4t}- e^{-i4t}}{2i}</math> | ||
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== Coefficients == | == Coefficients == | ||
| − | + | <math>w_0=1\!</math> | |
| − | + | ||
<math>a_4= \frac{11}{2i}</math> | <math>a_4= \frac{11}{2i}</math> | ||
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<math>a_7=a_{-7}= \frac{3}{2}</math> | <math>a_7=a_{-7}= \frac{3}{2}</math> | ||
| − | + | <math>a_k = 0\!</math> for all other <math>k \in \mathbb{Z}</math> | |
Revision as of 11:52, 26 September 2008
Signal
Compute the Fourier series coefficients of the following signal: $ x(t) = 3cos(7t) + 11sin(4t)\! $
Fourier series
$ x(t) = 3cos(7t) + 11sin(4t)\! $
$ x(t) = 3\frac{e^{i7t}+ e^{-i7t}}{2} + 11\frac{e^{i4t}- e^{-i4t}}{2i} $
$ x(t) = \frac{3}{2}e^{i7t}+ \frac{3}{2}e^{-i7t} + \frac{11}{2i}e^{i4t}- \frac{11}{2i}e^{-i4t} $
Coefficients
$ w_0=1\! $
$ a_4= \frac{11}{2i} $
$ a_{-4}= -\frac{11}{2i} $
$ a_7=a_{-7}= \frac{3}{2} $
$ a_k = 0\! $ for all other $ k \in \mathbb{Z} $
