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Now that we know <math>a_0\,</math>, we know that <math>x[n] = \frac{2}{3} + \sum_{n=1}^{5}a_k e^{jk\frac{\pi}{3}n}\,</math> | Now that we know <math>a_0\,</math>, we know that <math>x[n] = \frac{2}{3} + \sum_{n=1}^{5}a_k e^{jk\frac{\pi}{3}n}\,</math> | ||
| − | Since <math>\omega_0 = \frac{\pi}{3}\,</math>, let's try and find <math>a_3\,</math> | + | |
| + | Since <math>\omega_0 = \frac{\pi}{3}\,</math>, let's try and find <math>a_3\,</math>, | ||
| + | |||
| + | <math>a_3 = \frac{1}{6}\sum_{n=0}^{5}x[n] e^{-3j\frac{\pi}{3}n}\,</math> | ||
Revision as of 12:32, 25 September 2008
Guess the Periodic Signal
A certain periodic signal has the following properties:
1. N = 6
2. $ \sum_{n=0}^{5}x[n] = 4 $
3. $ \sum_{n=1}^{6}(-1)^nx[n] = 2 $
4. $ a_k = a_{k+3}\, $
Answer
From 1. we know that $ x[n] = \sum_{n=0}^{5}a_k e^{jk\frac{\pi}{3}n}\, $
Using 2., it is apparent that this is the formula for $ a_k\, $. Specifically, for $ a_0\, $, since the only thing under the sum is $ x[n]\, $. So,
Now that we know $ a_0\, $, we know that $ x[n] = \frac{2}{3} + \sum_{n=1}^{5}a_k e^{jk\frac{\pi}{3}n}\, $
Since $ \omega_0 = \frac{\pi}{3}\, $, let's try and find $ a_3\, $,
$ a_3 = \frac{1}{6}\sum_{n=0}^{5}x[n] e^{-3j\frac{\pi}{3}n}\, $
