| Line 15: | Line 15: | ||
---- | ---- | ||
---- | ---- | ||
| − | ==Definition== | + | ==Definition of Discrete Time Fourier Transform (DTFT)== |
<math>X(\omega) := \sum_{k=-\infty}^{\infty}x[n]e^{-j\omega k} </math> | <math>X(\omega) := \sum_{k=-\infty}^{\infty}x[n]e^{-j\omega k} </math> | ||
| + | |||
| + | ==Definition of Inverse Discrete Time Fourier Transform (iDTFT)== | ||
| + | <math>x[n] = \frac{1}{2\pi}\int\limits_{0}^{\pi}X(\omega)e^{j\omegan}d\omega </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 | <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 | ||
| Line 31: | Line 34: | ||
so <math>X(\omega + 2\pi) = X(\omega) </math> for all <math>\omega</math> | so <math>X(\omega + 2\pi) = X(\omega) </math> for all <math>\omega</math> | ||
| + | ---- | ||
| + | |||
| + | |||
| + | |||
---- | ---- | ||
---- | ---- | ||
Revision as of 19:58, 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 of Discrete Time Fourier Transform (DTFT)
$ X(\omega) := \sum_{k=-\infty}^{\infty}x[n]e^{-j\omega k} $
Definition of Inverse Discrete Time Fourier Transform (iDTFT)
$ x[n] = \frac{1}{2\pi}\int\limits_{0}^{\pi}X(\omega)e^{j\omegan}d\omega $
$ 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 $
(create a question page and put a link below)
Questions and comments
If you have any questions, comments, etc. please post them on this page.
