DFTJEW - Revision history

Wonoje: /* Discrete Fourier Transform */

2005-12-06T22:36:24Z

Discrete Fourier Transform

← Older revision		Revision as of 15:36, 6 December 2005
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	<math> \mbox{IDFT}[X(m)] \equiv x(k) \equiv \frac{1}{N} \sum_{m=0}^{N-1} X(m) e^{j \frac{2 \pi k m}{N} } </math>		<math> \mbox{IDFT}[X(m)] \equiv x(k) \equiv \frac{1}{N} \sum_{m=0}^{N-1} X(m) e^{j \frac{2 \pi k m}{N} } </math>

			One thing to note about the DFT is that it assumes periodicity. Let us, for example, consider a DFT on the interval of <math>-m </math> to <math> +m</math>. The DFT will expect the data at +<math>m</math> to be equal to the data at -<math>m</math>.


	<small>Principle author: Jeffrey Wonoprabowo</small>		<small>Principle author: Jeffrey Wonoprabowo</small>

2005-12-06T22:30:38Z

Discrete Fourier Transform

← Older revision		Revision as of 15:30, 6 December 2005
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	<math>\mbox{DFT}[x(n)] \equiv X(m) \equiv \sum_{n=0}^{N-1} x(n) e^{-j \frac{2 \pi n m}{N} }</math>		<math>\mbox{DFT}[x(n)] \equiv X(m) \equiv \sum_{n=0}^{N-1} x(n) e^{-j \frac{2 \pi n m}{N} }</math>


			When dealing with a continuous function, an inverse Fourier Transform can be used to go back from the frequency domain into the time domain. The discete analog of the inverse Fourier Transform is the IDFT (Inverse Discrete Fourier Transform) which is defined below:

			<math> \mbox{IDFT}[X(m)] \equiv x(k) \equiv \frac{1}{N} \sum_{m=0}^{N-1} X(m) e^{j \frac{2 \pi k m}{N} } </math>


	<small>Principle author: Jeffrey Wonoprabowo</small>		<small>Principle author: Jeffrey Wonoprabowo</small>

2005-12-06T22:13:19Z

Discrete Fourier Transform

← Older revision		Revision as of 15:13, 6 December 2005
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	The [[FourierTransformsJW\|Fourier Transform]] is a powerful tool to convert a continuous function from the time domain into the frequency domain. The Fourier transform, however, is an integral transform; it is done by integration. This cannot be done with a discrete function. The Discrete Fourier Transform (DFT) allows us to transform a discrete function from the time domain into the frequency domain.		The [[FourierTransformsJW\|Fourier Transform]] is a powerful tool to convert a continuous function from the time domain into the frequency domain. The Fourier transform, however, is an integral transform; it is done by integration. This cannot be done with a discrete function. The Discrete Fourier Transform (DFT) allows us to transform a discrete function from the time domain into the frequency domain.

	~~Let~~ <math>x(n)</math> be a ~~discretized~~ function in time.		If <math>x(n)</math> is a discrete function in time, then the DFT of <math>x(n)</math> would be:

	~~Then~~ the DFT of <math>x(n)</math> would be:

	<math>\mbox{DFT}[x(n)] \equiv X(m) \equiv \sum_{n=0}^{N-1} x(n) e^{-j \frac{2 \pi n m}{N} }</math>		<math>\mbox{DFT}[x(n)] \equiv X(m) \equiv \sum_{n=0}^{N-1} x(n) e^{-j \frac{2 \pi n m}{N} }</math>

2005-12-06T22:11:24Z

Discrete Fourier Transform

← Older revision		Revision as of 15:11, 6 December 2005
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	==Discrete Fourier Transform==		==Discrete Fourier Transform==
	The Fourier Transform is a powerful tool to convert a continuous function from the time domain into the frequency domain. The Fourier transform, however, is an integral transform; it is done by integration. This cannot be done with a discrete function. The Discrete Fourier Transform (DFT) allows us to transform a discrete function from the time domain into the frequency domain.		The [[FourierTransformsJW\|Fourier Transform]] is a powerful tool to convert a continuous function from the time domain into the frequency domain. The Fourier transform, however, is an integral transform; it is done by integration. This cannot be done with a discrete function. The Discrete Fourier Transform (DFT) allows us to transform a discrete function from the time domain into the frequency domain.

			Let <math>x(n)</math> be a discretized function in time.

			Then the DFT of <math>x(n)</math> would be:

			<math>\mbox{DFT}[x(n)] \equiv X(m) \equiv \sum_{n=0}^{N-1} x(n) e^{-j \frac{2 \pi n m}{N} }</math>


	<small>Principle author: Jeffrey Wonoprabowo</small>		<small>Principle author: Jeffrey Wonoprabowo</small>

2005-12-06T22:03:07Z

Discrete Fourier Transform

← Older revision		Revision as of 15:03, 6 December 2005
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	==Discrete Fourier Transform==		==Discrete Fourier Transform==
			The Fourier Transform is a powerful tool to convert a continuous function from the time domain into the frequency domain. The Fourier transform, however, is an integral transform; it is done by integration. This cannot be done with a discrete function. The Discrete Fourier Transform (DFT) allows us to transform a discrete function from the time domain into the frequency domain.


	<small>Principle author: Jeffrey Wonoprabowo</small>		<small>Principle author: Jeffrey Wonoprabowo</small>

2005-11-15T06:14:38Z

New page

==Discrete Fourier Transform==

<small>Principle author: Jeffrey Wonoprabowo</small>