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| − | 8 + sin<math>( \frac{2 pi n}{N} )</math> + 8cos<math>( \frac{4 pi n}{N} )</math> | + | 8 + sin<math>( \frac{2 pi n}{N} )</math> + 8cos<math>( \frac{4 pi n}{N} )</math> |
| − | = 8 + <math>( \frac{1}{2j})</math> { | + | = 8 + <math>( \frac{1}{2j})</math> {<math>( e^( \frac{j2 pi n}{N} ) </math> - <math>e^( \frac{-j2 pi n}{N} )</math>} + 4 {<math>e^( \frac{j4 pi n}{N} </math> + <math>e^( \frac{-j4 pi n}{N} </math>} |
| − | = 8 + <math>( \frac{-1j}{2})</math> | + | = 8 + <math>( \frac{-1j}{2})</math> <math>( e^( \frac{j2 pi n}{N} ) </math> + <math>( \frac{1j}{2})</math> <math>( e^( \frac{-j2 pi n}{N} ) </math> +4 <math>( e^( \frac{j4 pi n}{N} ) </math> +4 <math>( e^( \frac{-j4 pi n}{N} ) </math> |
| − | Therfore, we have | + | Therfore, we have the coefficients as |
<math>a_0</math> = 8 | <math>a_0</math> = 8 | ||
| + | |||
| + | <math>a_1</math> = <math>( \frac{-1 j }{2} )</math> | ||
| + | |||
| + | <math>a_-1</math> = <math>( \frac{1 j }{2} )</math> | ||
| + | |||
| + | |||
| + | <math>a_2</math> = 4 | ||
| + | |||
| + | |||
| + | <math>a_-2</math> = 4 | ||
Latest revision as of 17:11, 26 September 2008
Let the DT siganl be
8 + sin$ ( \frac{2 pi n}{N} ) $ + 8cos$ ( \frac{4 pi n}{N} ) $
= 8 + $ ( \frac{1}{2j}) $ {$ ( e^( \frac{j2 pi n}{N} ) $ - $ e^( \frac{-j2 pi n}{N} ) $} + 4 {$ e^( \frac{j4 pi n}{N} $ + $ e^( \frac{-j4 pi n}{N} $}
= 8 + $ ( \frac{-1j}{2}) $ $ ( e^( \frac{j2 pi n}{N} ) $ + $ ( \frac{1j}{2}) $ $ ( e^( \frac{-j2 pi n}{N} ) $ +4 $ ( e^( \frac{j4 pi n}{N} ) $ +4 $ ( e^( \frac{-j4 pi n}{N} ) $
Therfore, we have the coefficients as
$ a_0 $ = 8
$ a_1 $ = $ ( \frac{-1 j }{2} ) $
$ a_-1 $ = $ ( \frac{1 j }{2} ) $
$ a_2 $ = 4
$ a_-2 $ = 4
