(New page: Category:slecture Category:ECE438Fall2014Boutin Category:ECE Category:ECE438 Category:signal processing <center><font size= 4> '''Discrete-time Fourier transform'''...) |
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− | + | ==Definition== | |
+ | <math>X(\omega) := \sum_{k=-\infty}^{\infty}x[n]e^{-j\omega k} </math> | ||
+ | <math>X(\omega) </math> is seen to be periodic with a period of <math>2\pi</math> to see this <math>\omega</math> is replaced with <math>\omega + 2k\pi</math> where k is an integer | ||
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+ | <math>X(\omega + 2\pi) = \sum_{n=-\infty}^{\infty}x[n]e^{-j(\omega + 2k\pi)n} </math> | ||
+ | |||
+ | Using the multiplicative rule of exponential the <math>\omega</math> and <math>2k\pi</math> are split into two different exponential | ||
+ | |||
+ | <math>X(\omega + 2\pi) = \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}e^{2k\pi n} </math> | ||
+ | |||
+ | given that n and k are integers k and so <math>e^{-j2k\pi n} = 1 </math> from Euler's identity and so | ||
+ | |||
+ | <math>X(\omega + 2\pi) = \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n} </math> | ||
+ | |||
+ | so <math>X(\omega + 2\pi) = X(\omega) </math> for all <math>\omega</math> | ||
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Revision as of 04:17, 29 September 2014
Discrete-time Fourier transform
A slecture by ECE student Jacob Holtman
Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.
Definition
$ X(\omega) := \sum_{k=-\infty}^{\infty}x[n]e^{-j\omega k} $
$ X(\omega) $ is seen to be periodic with a period of $ 2\pi $ to see this $ \omega $ is replaced with $ \omega + 2k\pi $ where k is an integer
$ X(\omega + 2\pi) = \sum_{n=-\infty}^{\infty}x[n]e^{-j(\omega + 2k\pi)n} $
Using the multiplicative rule of exponential the $ \omega $ and $ 2k\pi $ are split into two different exponential
$ X(\omega + 2\pi) = \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}e^{2k\pi n} $
given that n and k are integers k and so $ e^{-j2k\pi n} = 1 $ from Euler's identity and so
$ X(\omega + 2\pi) = \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n} $
so $ X(\omega + 2\pi) = X(\omega) $ for all $ \omega $
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Questions and comments
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