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<math>N = 4\,</math> | <math>N = 4\,</math> | ||
| − | In order to find the coefficients, we must first calculate the values of <math>x[n]\,</math> for four consecutive integer values of <math>n\,</math> | + | In order to find the coefficients, we must first calculate the values of <math>x[n]\,</math> for four consecutive integer values of <math>n\,</math>. By plugging values of <math>n\,</math> into the given signal, we find that |
| + | |||
| + | <math>x[0] = 2\,</math> | ||
| + | |||
| + | <math>x[1] = -12\,</math> | ||
| + | |||
| + | <math>x[2] = 2\,</math> | ||
| + | |||
| + | <math>x[3] = -12\,</math> | ||
Revision as of 11:26, 25 September 2008
Define a Periodic DT Signal and Compute the Fourier Series Coefficients
I am going to choose a sine signal, since there have been many cosines done already.
DT signal: $ x[n] = 2\sin(\pi n + \pi) + 4\sin(\frac{\pi}{2} n + \pi)\, $
Now, each sine has its own period, and the fundamental period of the function is the greater of the separate periods.
$ N_2sin = \frac{2\pi}{\pi} k = \frac{2}{1} k $
$ N_4sin = \frac{2\pi}{\frac{\pi}{2}} k = \frac{2}{\frac{1}{2}} k $
Take $ k = 1\, $,
$ N_2sin = 2\, $
$ N_4sin = 4\, $, so the overall fundamental period is
$ N = 4\, $
In order to find the coefficients, we must first calculate the values of $ x[n]\, $ for four consecutive integer values of $ n\, $. By plugging values of $ n\, $ into the given signal, we find that
$ x[0] = 2\, $
$ x[1] = -12\, $
$ x[2] = 2\, $
$ x[3] = -12\, $
