The development of the DFT - Revision history

Tsung-lin.yang: /* Inverse DFT */

2008-12-01T00:19:38Z

Inverse DFT

← Older revision		Revision as of 17:19, 30 November 2008
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	So <math>\sum_{m=0}^{N-1}X(m)\, e^{j2\pi\frac{lm}{N}}=\sum_{n=0}^{N-1}x(n)N\delta_{n,l}=Nx(l)</math>		So <math>\sum_{m=0}^{N-1}X(m)\, e^{j2\pi\frac{lm}{N}}=\sum_{n=0}^{N-1}x(n)N\delta_{n,l}=Nx(l)</math>

	<math>x(l) = \frac{1}{N}\sum_{m=0}^{N-1}X(m)\, e^{j2\pi\frac{lm}{N}}\equiv\, IDFT(X(n)) </math>		<math>x(l) = \frac{1}{N}\sum_{m=0}^{N-1}X(m)\, e^{j2\pi\frac{lm}{N}}\equiv\, IDFT(X(m)) </math>

	Key points to note:		Key points to note:

2008-11-20T09:12:19Z

← Older revision		Revision as of 02:12, 20 November 2008
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	How do we put it into computer?		How do we put it into computer?

	We can use A/D converter and a low pass filter to sample the signal that is wanted instead of from <math>t = -\infty to \infty</math>:		We can use A/D converter and a low pass filter to sample the signal that is wanted instead of from <math>t = -\infty\,\, to\,\, \infty</math>:

	[[Image:DFT2.jpg\|1000px]]		[[Image:DFT2.jpg\|1000px]]

2008-11-20T09:12:04Z

← Older revision		Revision as of 02:12, 20 November 2008
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	How do we put it into computer?		How do we put it into computer?

	We can use A/D converter and a low pass filter to sample the signal that is wanted instead of from t = -\infty to \infty:		We can use A/D converter and a low pass filter to sample the signal that is wanted instead of from <math>t = -\infty to \infty</math>:

	[[Image:DFT2.jpg\|1000px]]		[[Image:DFT2.jpg\|1000px]]

2008-11-20T09:11:37Z

← Older revision		Revision as of 02:11, 20 November 2008
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	How do we put it into computer?		How do we put it into computer?
	We can use A/D converter and a low pass filter to sample the signal ~~with~~ wanted instead of from t = -\infty to \infty:
			We can use A/D converter and a low pass filter to sample the signal that is wanted instead of from t = -\infty to \infty:

	[[Image:DFT2.jpg\|1000px]]		[[Image:DFT2.jpg\|1000px]]

2008-11-20T09:09:36Z

@@ Line 1: / Line 1: @@
 If we have a signal, such as following:
 [[Image:DFT1.jpg]]
 How do we put it into computer?
 We can use A/D converter and a low pass filter to sample the signal with wanted instead of from t = -\infty to \infty:
 [[Image:DFT2.jpg]]
 But then we have to make it periodic, so we convolve it with impulse function with NT apart to have impulse function in both time and frequency domain:
 [[Image:DFT3.jpg]]
-From the equations of final signal in both time and frequency domain, we can see that in the computer we have x(n) and in the frequency domain the areas of the impulse functions is:
+[[Image:DFT3.jpg|1000px]]
-<math>\sum_{m=-\infty}^\infty \frac{1} {NT}\sum_{n=0}^{N-1}X(nT)\, e^{-j2\pi\frac{m}{NT}nT} \delta\left(f-\frac{n}{NT}\right)</math>
-Which is the definition of the DFT(x(n)).
+From the equations of final signal in both time and frequency domain, we can see that in the computer we have x(n) and in the frequency domain:

2008-11-20T08:00:09Z

← Older revision		Revision as of 01:00, 20 November 2008
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	<math>\sum_{m=1}^\~~infty~~\delta\left(t-nT\right)<\math>		If we have a signal, such as following:
	~~<math>\alpha\,\!</math>~~		[[Image:DFT1.jpg]]
			How do we put it into computer?
			We can use A/D converter and a low pass filter to sample the signal with wanted instead of from t = -\infty to \infty:
			[[Image:DFT2.jpg]]
			But then we have to make it periodic, so we convolve it with impulse function with NT apart to have impulse function in both time and frequency domain:
			[[Image:DFT3.jpg]]
			From the equations of final signal in both time and frequency domain, we can see that in the computer we have x(n) and in the frequency domain the areas of the impulse functions is:
			<math>\sum_{m=-\infty}^\infty \frac{1} {NT}\sum_{n=0}^{N-1}X(nT)\, e^{-j2\pi\frac{m}{NT}nT} \delta\left(f-\frac{n}{NT}\right)</math>
			Which is the definition of the DFT(x(n)).

2008-11-20T07:35:20Z

New page: <math>\sum_{m=1}^\infty\delta\left(t-nT\right)<\math> <math>\alpha\,\!</math>

New page

<math>\sum_{m=1}^\infty\delta\left(t-nT\right)<\math>
<math>\alpha\,\!</math>