| Line 4: | Line 4: | ||
1. N = 6 | 1. N = 6 | ||
| − | 2. <math>\sum_{n=0}^{ | + | 2. <math>\sum_{n=0}^{5}x[n] = 4</math> |
| − | 3. <math>\sum_{n=1}^{ | + | 3. <math>\sum_{n=1}^{6}(-1)^nx[n] = 2</math> |
4. <math>a_k = a_{k+3}\,</math> | 4. <math>a_k = a_{k+3}\,</math> | ||
| Line 13: | Line 13: | ||
== Answer == | == Answer == | ||
| − | From 1. we know that <math>x[n] = \frac{1}{6}\sum_{n=0}^{ | + | From 1. we know that <math>x[n] = \frac{1}{6}\sum_{n=0}^{5}a_k e^{jk\frac{\pi}{3}n}\,</math> |
| − | Using 2., it is apparent that this is the formula for <math>a_k\,</math>. Specifically, for <math>a_0\,</math> | + | Using 2., it is apparent that this is the formula for <math>a_k\,</math>. Specifically, for <math>a_0\,</math>, since the only thing under the sum is <math>x[n]\,</math>. So, |
| + | |||
| + | <center><math>\frac{1}{6}\sum_{n=0}^{5}x[n] = 4\,</math> | ||
Revision as of 13:22, 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] = \frac{1}{6}\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,
