Difference between revisions of "Discrete-time Fourier transform (DTFT) Slecture by Jacob Holtman" - Rhea

Revision as of 04:17, 29 September 2014

Discrete-time Fourier transform

A slecture by ECE student Jacob Holtman

$X(\omega) := \sum_{k=-\infty}^{\infty}x[n]e^{-j\omega k}$

$X(\omega)$ 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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-The definition of a DTFT is <math>X(w)□(∶=)∑_(n= -∞)^∞▒〖x[n]e^(-jwn) 〗</math>
+==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
+<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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