HW12 DFT/Continuous Fourier Transform Relationship - Revision history

Rothsa at 20:14, 4 December 2007

2007-12-04T20:14:33Z

← Older revision		Revision as of 13:14, 4 December 2007
Line 28:		Line 28:
	NOTE: The following Matlab script is not computatively correct -		NOTE: The following Matlab script is not computatively correct -
	it was constructed to only get the graphs that pass visual inspection, not numerical inspection.		it was constructed to only get the graphs that pass visual inspection, not numerical inspection.
	<u>DFT</u>;		::<u>DFT</u>;
	::clear;		::clear;
	::T=.25;		::T=.25;
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	::ylabel('magnitude)');		::ylabel('magnitude)');

	<u>CFT</u>;		::<u>CFT</u>;
	::clear;		::clear;
	::T=.25;		::T=.25;

Rothsa at 20:13, 4 December 2007

2007-12-04T20:13:53Z

← Older revision		Revision as of 13:13, 4 December 2007
Line 28:		Line 28:
	NOTE: The following Matlab script is not computatively correct -		NOTE: The following Matlab script is not computatively correct -
	it was constructed to only get the graphs that pass visual inspection, not numerical inspection.		it was constructed to only get the graphs that pass visual inspection, not numerical inspection.
			<u>DFT</u>;
			::clear;
			::T=.25;
			::t=(0:T:10);
			::N=length(t);
			::f = -1/(2T):1/(NT):1/(2T)-1/(NT);
			::x=(sin(2pit));
			::y=abs(fft(x))
			::figure(1);
			::plot(f,y);
			::title(' DFT')
			::xlabel('frequency');
			::ylabel('magnitude)');

			<u>CFT</u>;
	::clear;		::clear;
	::T=.25;		::T=.25;
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	::figure(1);		::figure(1);
	::plot(f,y);		::plot(f,y);
	::title(' ~~DFT~~')		::title(' CFT')
	::xlabel('~~frequncy~~');		::xlabel('frequency');
	::ylabel('~~dft of x~~)');		::ylabel('magnitude)');

Rothsa at 20:12, 4 December 2007

2007-12-04T20:12:18Z

@@ Line 27: / Line 27: @@
 ::[[Image:hw12b.jpg]]
 NOTE: The following Matlab script is not computatively correct -
-it was constructed to only get the graphs I wanted,
+it was constructed to only get the graphs that pass visual inspection, not numerical inspection.
-not be used for real DFT computations.
+::clear;

Rothsa at 20:10, 4 December 2007

2007-12-04T20:10:11Z

← Older revision		Revision as of 13:10, 4 December 2007
Line 23:		Line 23:
	of the continuous. So, if a simple sine wave is a particular frequency, it's continuous fourier transform (which, by definition, is a graph of the frequencies present in that wave) will spike at that particular frequency. The discrete frequency will also spike, but taper off a bit more gradually.		of the continuous. So, if a simple sine wave is a particular frequency, it's continuous fourier transform (which, by definition, is a graph of the frequencies present in that wave) will spike at that particular frequency. The discrete frequency will also spike, but taper off a bit more gradually.
	Heres the what the DFT of a 1Hz Sine Wave looks like:		Heres the what the DFT of a 1Hz Sine Wave looks like:
	[[Image:hw12.jpg]]		::[[Image:hw12.jpg]]<br>
	Heres the what the CFT of a 1Hz Sine Wave looks like, in comparison:		Heres the what the CFT of a 1Hz Sine Wave looks like, in comparison:
	[[Image:hw12b.jpg]]		::[[Image:hw12b.jpg]]
	NOTE: The following Matlab script is not computatively correct -		NOTE: The following Matlab script is not computatively correct -
	it was constructed to only get the graphs I wanted,		it was constructed to only get the graphs I wanted,
	not be used for real DFT computations.		not be used for real DFT computations.

Rothsa at 20:09, 4 December 2007

2007-12-04T20:09:21Z

← Older revision		Revision as of 13:09, 4 December 2007
Line 22:		Line 22:
	So what does this look like graphically? Well, basically, the Discrete is a bit less accurate on the frequencies it shows since by definition it is the sampled version		So what does this look like graphically? Well, basically, the Discrete is a bit less accurate on the frequencies it shows since by definition it is the sampled version
	of the continuous. So, if a simple sine wave is a particular frequency, it's continuous fourier transform (which, by definition, is a graph of the frequencies present in that wave) will spike at that particular frequency. The discrete frequency will also spike, but taper off a bit more gradually.		of the continuous. So, if a simple sine wave is a particular frequency, it's continuous fourier transform (which, by definition, is a graph of the frequencies present in that wave) will spike at that particular frequency. The discrete frequency will also spike, but taper off a bit more gradually.

	Heres the what the DFT of a 1Hz Sine Wave looks like:		Heres the what the DFT of a 1Hz Sine Wave looks like:
	[[Image:hw12.jpg]]		[[Image:hw12.jpg]]
	Heres the what the CFT of a 1Hz Sine Wave looks like, in comparison:		Heres the what the CFT of a 1Hz Sine Wave looks like, in comparison:
			[[Image:hw12b.jpg]]
	NOTE: The following Matlab script is not computatively correct -		NOTE: The following Matlab script is not computatively correct -
	it was constructed to only get the graphs I wanted,		it was constructed to only get the graphs I wanted,
	not be used for real DFT computations.		not be used for real DFT computations.

Rothsa at 20:06, 4 December 2007

2007-12-04T20:06:52Z

← Older revision		Revision as of 13:06, 4 December 2007
Line 24:		Line 24:

	Heres the what the DFT of a 1Hz Sine Wave looks like:		Heres the what the DFT of a 1Hz Sine Wave looks like:
	~~<math>~~hw12.jpg~~</math>~~		[[Image:hw12.jpg]]
	Heres the what the CFT of a 1Hz Sine Wave looks like, in comparison:		Heres the what the CFT of a 1Hz Sine Wave looks like, in comparison:

Rothsa at 20:06, 4 December 2007

2007-12-04T20:06:11Z

← Older revision		Revision as of 13:06, 4 December 2007
Line 23:		Line 23:
	of the continuous. So, if a simple sine wave is a particular frequency, it's continuous fourier transform (which, by definition, is a graph of the frequencies present in that wave) will spike at that particular frequency. The discrete frequency will also spike, but taper off a bit more gradually.		of the continuous. So, if a simple sine wave is a particular frequency, it's continuous fourier transform (which, by definition, is a graph of the frequencies present in that wave) will spike at that particular frequency. The discrete frequency will also spike, but taper off a bit more gradually.

	Heres the graphs:		Heres the what the DFT of a 1Hz Sine Wave looks like:
			<math>hw12.jpg</math>
			Heres the what the CFT of a 1Hz Sine Wave looks like, in comparison:

			NOTE: The following Matlab script is not computatively correct -
			it was constructed to only get the graphs I wanted,
			not be used for real DFT computations.

Rothsa at 19:36, 4 December 2007

2007-12-04T19:36:49Z

← Older revision		Revision as of 12:36, 4 December 2007
Line 21:		Line 21:

	So what does this look like graphically? Well, basically, the Discrete is a bit less accurate on the frequencies it shows since by definition it is the sampled version		So what does this look like graphically? Well, basically, the Discrete is a bit less accurate on the frequencies it shows since by definition it is the sampled version
	of the continuous. So, if a simple sine wave is a particular frequency, it's continuous fourier transform (which, by definition, is a graph of the frequencies present ~~in a~~ in that wave) will spike at that particular frequency. The discrete frequency will also spike, but taper off a bit more gradually.		of the continuous. So, if a simple sine wave is a particular frequency, it's continuous fourier transform (which, by definition, is a graph of the frequencies present in that wave) will spike at that particular frequency. The discrete frequency will also spike, but taper off a bit more gradually.

	Heres the graphs:		Heres the graphs:

Rothsa at 19:36, 4 December 2007

2007-12-04T19:36:16Z

← Older revision		Revision as of 12:36, 4 December 2007
Line 20:		Line 20:
	The Discrete Transform allows us to chop that bit ol' continuous ones into a bunch of ones and zeros that will fit on your computer along with your Vanilla Ice and Mariah Carey and Marilyn Manson Anthologies.		The Discrete Transform allows us to chop that bit ol' continuous ones into a bunch of ones and zeros that will fit on your computer along with your Vanilla Ice and Mariah Carey and Marilyn Manson Anthologies.

			So what does this look like graphically? Well, basically, the Discrete is a bit less accurate on the frequencies it shows since by definition it is the sampled version
			of the continuous. So, if a simple sine wave is a particular frequency, it's continuous fourier transform (which, by definition, is a graph of the frequencies present in a in that wave) will spike at that particular frequency. The discrete frequency will also spike, but taper off a bit more gradually.

			Heres the graphs:
	~~But, if you're like me, this probably isn't enough description or motivation to truly understand~~ the ~~relationship between DFT's and CFT's. You want pictures, loud and clear. Here they are~~:

Rothsa at 18:12, 3 December 2007

2007-12-03T18:12:43Z

← Older revision		Revision as of 11:12, 3 December 2007
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	~~The Question~~ to ~~answer here is: What~~ is the ~~relationship between the continuous~~ Fourier Transform~~, and the discrete one?~~		===Problem Statement===
			Use Matlab or Octave to show how the DFT is related to the actual Fourier Transform.

			===Solution===
	For those of you who recognize the squiggly codes of mathematical formulas, observe:		For those of you who recognize the squiggly codes of mathematical formulas, observe: