Discrete Fourier Transforms - Revision history

Wilspa at 21:12, 8 March 2011

2011-03-08T21:12:31Z

← Older revision		Revision as of 14:12, 8 March 2011
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	<math>T_0=\frac{40 samples}{100 samples/second}=0.4 seconds</math>.<br>		<math>T_0=\frac{40 samples}{100 samples/second}=0.4 seconds</math>.<br>
	Therefore the fundamental frequency will be:<br>		Therefore the fundamental frequency will be:<br>
	<math>f_0=1/T_0=2.5 Hz</math>.<br>		<math>f_0=\frac{1}{T_0}=2.5 Hz</math>.<br>
	This is important because if your signal contains a frequency that is not an exact multiple of the fundamental frequency, it will not show up as a single spike like it would in the continuous Fourier Transform.<br>		This is important because if your signal contains a frequency that is not an exact multiple of the fundamental frequency, it will not show up as a single spike like it would in the continuous Fourier Transform.<br>

	Another important thing to know is the range of frequencies that the DFT shows. A DFT shows frequencies in the range <math>[0,2\pi)</math>. However, only 1/2 the frequencies are positive frequencies. In the range <math>[0,\pi]</math> the frequencies are positive. The negative frequencies are in the range <math>(\pi,2\pi)</math>. Why are the negative frequencies above the positive frequencies? That is a good question, and the answer has to do with the unit circle. I need to do more explanation here.		Another important thing to know is the range of frequencies that the DFT shows. A DFT shows frequencies in the range <math>[0,2\pi)</math>. However, only 1/2 the frequencies are positive frequencies. In the range <math>[0,\pi]</math> the frequencies are positive. The negative frequencies are in the range <math>(\pi,2\pi)</math>. Why are the negative frequencies above the positive frequencies? That is a good question, and the answer has to do with the unit circle. I need to do more explanation here.

Wilspa at 21:07, 8 March 2011

2011-03-08T21:07:33Z

← Older revision		Revision as of 14:07, 8 March 2011
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	===What is it used for?===		===What is it used for?===
	The DFT is ~~used~~ in a ~~wide variety of areas~~. ~~Not only~~ is it ~~useful~~ for ~~one dimensional applications such as~~ signal ~~processing~~, it is ~~also used for 2D applications such as CAT scans~~. ~~The DFT allows data to~~ be ~~transformed from one domain into another domain so~~ that ~~minipulation~~ of the ~~data is easier~~, ~~much the same~~ as a Fourier Transform, only ~~in a discrete setting~~. ~~Then~~, ~~when~~ the ~~minipulations~~ are ~~finished,~~ the ~~IDFT~~ (~~Inverse DFT~~) ~~can be used~~ to ~~convert back~~ to ~~the original domain if that is required~~.		The DFT is (as the name suggests) the discrete version of the continuous Fourier Transform. This is important because computers can only truly work in the discrete time domain. You can use it to find out frequency and phase information about a discrete signal.

			===Things to know about a DFT===
			A Discrete Fourier transform has some limitations. The first is that it can only test for discrete frequencies. Each frequency in the output of the DFT is a multiple of the fundamental frequency. The fundamental frequency is based on the number of samples used in the transform and the sampling rate. If the sampling rate of a signal is 100 Hz (100 samples per second) and the number of samples stored is 40 samples, that means that the period of the fundamental frequency is<br>
			<math>T_0=\frac{40 samples}{100 samples/second}=0.4 seconds</math>.<br>
			Therefore the fundamental frequency will be:<br>
			<math>f_0=1/T_0=2.5 Hz</math>.<br>
			This is important because if your signal contains a frequency that is not an exact multiple of the fundamental frequency, it will not show up as a single spike like it would in the continuous Fourier Transform.<br>

			Another important thing to know is the range of frequencies that the DFT shows. A DFT shows frequencies in the range <math>[0,2\pi)</math>. However, only 1/2 the frequencies are positive frequencies. In the range <math>[0,\pi]</math> the frequencies are positive. The negative frequencies are in the range <math>(\pi,2\pi)</math>. Why are the negative frequencies above the positive frequencies? That is a good question, and the answer has to do with the unit circle. I need to do more explanation here.

68.113.10.142: /* What is a DFT? */

2006-10-31T05:42:42Z

What is a DFT?

← Older revision		Revision as of 22:42, 30 October 2006
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	<math>X(f)=\sum _{n=0}^{N-1} x(n) e^{\frac{-j2\pi nm}{N}}		<math>X(f)=\sum _{n=0}^{N-1} x(n) e^{\frac{-j2\pi nm}{N}}
	</math><br>		</math><br>
	Which is using a sum and a ~~noncontinous~~ series of delta functions x(n) instead of the continuous function x(t).		Which is using a sum and a discrete series of delta functions x(n) instead of the continuous function x(t).

	===What is it used for?===		===What is it used for?===
	The DFT is used in a wide variety of areas. Not only is it useful for one dimensional applications such as signal processing, it is also used for 2D applications such as CAT scans. The DFT allows data to be transformed from one domain into another domain so that minipulation of the data is easier, much the same as a Fourier Transform, only in a discrete setting. Then, when the minipulations are finished, the IDFT (Inverse DFT) can be used to convert back to the original domain if that is required.		The DFT is used in a wide variety of areas. Not only is it useful for one dimensional applications such as signal processing, it is also used for 2D applications such as CAT scans. The DFT allows data to be transformed from one domain into another domain so that minipulation of the data is easier, much the same as a Fourier Transform, only in a discrete setting. Then, when the minipulations are finished, the IDFT (Inverse DFT) can be used to convert back to the original domain if that is required.

10.109.3.20 at 03:56, 31 October 2006

2006-10-31T03:56:41Z

← Older revision		Revision as of 20:56, 30 October 2006
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			[[User:wilspa\|Back to my page]]

	==Paul's DFT Page==		==Paul's DFT Page==
	One of the major tools used in signal processing is the DFT, which stands for Discrete Fourier Transform. The reason we need to to a DFT instead of a Fourier Transform is that our computers are limited in their abilites. They use sampling, and they have limited memory, so we have to adapt to the computers.		One of the major tools used in signal processing is the DFT, which stands for Discrete Fourier Transform. The reason we need to to a DFT instead of a Fourier Transform is that our computers are limited in their abilites. They use sampling, and they have limited memory, so we have to adapt to the computers.

10.109.3.20: /* What is it used for? */

2006-10-31T03:28:31Z

What is it used for?

← Older revision		Revision as of 20:28, 30 October 2006
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	Which is using a sum and a noncontinous series of delta functions x(n) instead of the continuous function x(t).		Which is using a sum and a noncontinous series of delta functions x(n) instead of the continuous function x(t).
	===What is it used for?===		===What is it used for?===
	The DFT is used in a wide variety of areas. Not only is it useful for one dimensional applications ~~suc~~ as signal processing, it is also used for 2D applications such as CAT scans. The DFT allows data to be transformed from one domain into another domain so that minipulation of the data is easier, much the same as a Fourier Transform, only in a ~~Discrete~~ setting. Then, when the minipulations are finished, the IDFT (Inverse DFT) can be used to convert back to the original domain if that is required.		The DFT is used in a wide variety of areas. Not only is it useful for one dimensional applications such as signal processing, it is also used for 2D applications such as CAT scans. The DFT allows data to be transformed from one domain into another domain so that minipulation of the data is easier, much the same as a Fourier Transform, only in a discrete setting. Then, when the minipulations are finished, the IDFT (Inverse DFT) can be used to convert back to the original domain if that is required.

Wilspa: /* What is it used for? */

2005-12-06T21:41:01Z

What is it used for?

← Older revision		Revision as of 14:41, 6 December 2005
Line 10:		Line 10:
	Which is using a sum and a noncontinous series of delta functions x(n) instead of the continuous function x(t).		Which is using a sum and a noncontinous series of delta functions x(n) instead of the continuous function x(t).
	===What is it used for?===		===What is it used for?===
			The DFT is used in a wide variety of areas. Not only is it useful for one dimensional applications suc as signal processing, it is also used for 2D applications such as CAT scans. The DFT allows data to be transformed from one domain into another domain so that minipulation of the data is easier, much the same as a Fourier Transform, only in a Discrete setting. Then, when the minipulations are finished, the IDFT (Inverse DFT) can be used to convert back to the original domain if that is required.

Wilspa at 18:31, 6 December 2005

2005-12-06T18:31:30Z

New page

==Paul's DFT Page==
One of the major tools used in signal processing is the DFT, which stands for Discrete Fourier Transform. The reason we need to to a DFT instead of a Fourier Transform is that our computers are limited in their abilites. They use sampling, and they have limited memory, so we have to adapt to the computers.
===What is a DFT?===
A DFT is like doing a Fourier Transform, but instead of doing it with an integral, we do it with discrete values and a sum. A Fourier Transform looks like this:<br>
<math>X(f)=\int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}\, dt
</math><br>
Which uses an integral, while the DFT which looks like this:<br>
<math>X(f)=\sum _{n=0}^{N-1} x(n) e^{\frac{-j2\pi nm}{N}}
</math><br>
Which is using a sum and a noncontinous series of delta functions x(n) instead of the continuous function x(t).
===What is it used for?===