Fourier Transforms - Revision history

10.109.3.20: /* Fourier Transform Pairs */

2006-10-31T04:01:50Z

Fourier Transform Pairs

← Older revision		Revision as of 21:01, 30 October 2006
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	This does exactly the opposite of the Fourier Transform. Instead of taking time and converting it to frequency, the inverse transform takes a function in frequency, <math>X(f)</math>, and converts it into a time function, <math>x(t)</math>.		This does exactly the opposite of the Fourier Transform. Instead of taking time and converting it to frequency, the inverse transform takes a function in frequency, <math>X(f)</math>, and converts it into a time function, <math>x(t)</math>.

			*[[User:Wilspa\|Back to my page]]
	==Fourier Transform Pairs==		==Fourier Transform Pairs==
	Most people don't want to do the Fourier Transform integrals all that often, but fortunately it is possible to put general Fourier Transform/Inverse Fourier Transform pairs together, and then from those pairs, perform the particular Fourier Transform you are working on.		Most people don't want to do the Fourier Transform integrals all that often, but fortunately it is possible to put general Fourier Transform/Inverse Fourier Transform pairs together, and then from those pairs, perform the particular Fourier Transform you are working on.
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	<math>\mathcal{F}[X(t)]=x^* (f)</math>		<math>\mathcal{F}[X(t)]=x^* (f)</math>
	<br>		<br>
	[[User:Wilspa\|Back to my page]]
			*[[User:Wilspa\|Back to my page]]
			*[[Fourier transform\|Main Fourier Transform page]]

Frohro: /* Fourier Transforms */

2006-09-29T18:29:45Z

Fourier Transforms

← Older revision		Revision as of 11:29, 29 September 2006
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	This does exactly the opposite of the Fourier Transform. Instead of taking time and converting it to frequency, the inverse transform takes a function in frequency, <math>X(f)</math>, and converts it into a time function, <math>x(t)</math>.		This does exactly the opposite of the Fourier Transform. Instead of taking time and converting it to frequency, the inverse transform takes a function in frequency, <math>X(f)</math>, and converts it into a time function, <math>x(t)</math>.

	~~[[New Page]]~~

	==Fourier Transform Pairs==		==Fourier Transform Pairs==

Frohro: /* Fourier Transforms */

2006-09-29T18:29:23Z

Fourier Transforms

← Older revision		Revision as of 11:29, 29 September 2006
Line 15:		Line 15:
	This does exactly the opposite of the Fourier Transform. Instead of taking time and converting it to frequency, the inverse transform takes a function in frequency, <math>X(f)</math>, and converts it into a time function, <math>x(t)</math>.		This does exactly the opposite of the Fourier Transform. Instead of taking time and converting it to frequency, the inverse transform takes a function in frequency, <math>X(f)</math>, and converts it into a time function, <math>x(t)</math>.

	[New Page]		[[New Page]]

	==Fourier Transform Pairs==		==Fourier Transform Pairs==

Frohro: /* Fourier Transforms */

2006-09-29T18:28:56Z

Fourier Transforms

← Older revision		Revision as of 11:28, 29 September 2006
Line 14:		Line 14:

	This does exactly the opposite of the Fourier Transform. Instead of taking time and converting it to frequency, the inverse transform takes a function in frequency, <math>X(f)</math>, and converts it into a time function, <math>x(t)</math>.		This does exactly the opposite of the Fourier Transform. Instead of taking time and converting it to frequency, the inverse transform takes a function in frequency, <math>X(f)</math>, and converts it into a time function, <math>x(t)</math>.

			[New Page]

	==Fourier Transform Pairs==		==Fourier Transform Pairs==

Wilspa at 21:06, 16 October 2005

2005-10-16T21:06:47Z

← Older revision		Revision as of 14:06, 16 October 2005
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			[[User:Wilspa\|Back to my page]]
	==Fourier Transforms==		==Fourier Transforms==
	A Fourier transform is used to transform a function from the time domain into one from the frequency domain. This can be very useful, because it allows easier minipulation of the function in many cases. The Fourier Transform is often written this way:		A Fourier transform is used to transform a function from the time domain into one from the frequency domain. This can be very useful, because it allows easier minipulation of the function in many cases. The Fourier Transform is often written this way:
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	<math>\mathcal{F}[X(t)]=x^* (f)</math>		<math>\mathcal{F}[X(t)]=x^* (f)</math>
	<br>		<br>
			[[User:Wilspa\|Back to my page]]

Wilspa at 21:01, 16 October 2005

2005-10-16T21:01:08Z

New page

==Fourier Transforms==
A Fourier transform is used to transform a function from the time domain into one from the frequency domain. This can be very useful, because it allows easier minipulation of the function in many cases. The Fourier Transform is often written this way:

<math>
X(f)=\int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}\, dt
</math>

This takes your time function, <math> x(t) </math> and makes it into a frequency function <math> X(f) </math>. This function also has an inverse, namely, the inverse Fourier Transform, and it looks like this:

<math>
x(t)=\int_{-\infty}^{\infty} X(f) e^{j2\pi ft}\, df
</math>

This does exactly the opposite of the Fourier Transform. Instead of taking time and converting it to frequency, the inverse transform takes a function in frequency, <math>X(f)</math>, and converts it into a time function, <math>x(t)</math>.

==Fourier Transform Pairs==
Most people don't want to do the Fourier Transform integrals all that often, but fortunately it is possible to put general Fourier Transform/Inverse Fourier Transform pairs together, and then from those pairs, perform the particular Fourier Transform you are working on.

<math>\mathcal{F}[x(t)]=X(f)</math>
 
<math>\mathcal{F}[dx/dt]=j2\pi fX(f)</math>
 
<math>\mathcal{F}[x(t-t_0)]=e^{-j2\pi ft_0}X(f)</math>
 
<math>\mathcal{F}[e^{j2\pi f_0 t}x(t)]=X(f-f_0)</math>
 
<math>\mathcal{F}[\cos (2\pi f_0 t) x(t)]=\frac{1}{2} X(f-f_0)+\frac{1}{2}X(f+f_0)</math>
 
<math>\mathcal{F}[x(at)]=\frac{1}{\left | a\right \vert} X\left (\frac{f}{a} \right )</math>
 
<math>\mathcal{F}[x(t)*y(t)]=X(f)Y(f)</math>
 
<math>\mathcal{F}[x(t)y(t)]=X(f)*Y(f)</math>
 
<math>\mathcal{F}[\sum_{n=-\infty}^\infty \alpha _n e^{\frac{j2\pi ft}{T}}]=\sum_{n=-\infty}^\infty \alpha _n \delta\left (f-\frac{n}{t}\right )</math>
 
The last transform is actually the transform of a Fourier Series. As you can see, when you take the Fourier Transform of a Fourier Series, you get out the frequency components, <math>\alpha_n \delta\left (f-\frac{n}{t}\right )</math>, of the function.
 
Another thing that you might notice is the symmetry between the forward and inverse transforms. One way to describe this symmetry is:
 
<math>\mathcal{F}[X(t)]=x^* (f)</math>