The Fourier transform in nature not just time<>frequency domain Most everyone here knows that the FT describes the relationship between a waveform in the time domain and the frequency domain. I was enlightened to read here the other day that the Fourier Transform also describes the relationship in electromagnetics between the current distribution and the far field radiation pattern. Yep, they have sidelobes just like in waveforms... So ... I thought it would be fun and interesting to ask the group to come up with a list of other pairings in nature whose relationship can be described by the FT. Mark
The Fourier transform in nature not just time<>frequency domain
Started by ●March 6, 2009
Reply by ●March 6, 20092009-03-06
makolber@yahoo.com wrote:> The Fourier transform in nature not just time<>frequency domain > > Most everyone here knows that the FT describes the relationship > between a waveform in the time domain and the frequency domain. > > I was enlightened to read here the other day that the Fourier > Transform also describes the relationship in electromagnetics > between the current distribution and the far field radiation > pattern. Yep, they have sidelobes just like in waveforms... > > So ... > > I thought it would be fun and interesting to ask the group to come up > with a list of other pairings in nature whose relationship can be > described by the FT.Optics also involve EM fields. It is nevertheless interesting to note that the image seen in a microscope eyepiece is the FT of another plane in the microscope tube that is plainly visible if you know where to look and have the tool to look there. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●March 6, 20092009-03-06
On Mar 6, 12:37�pm, makol...@yahoo.com wrote:> The Fourier transform in nature �not just time<>frequency domain > > Most everyone here knows that the FT describes the relationship > between a waveform in the time domain and the frequency domain. > > I was enlightened to read here the other day that the Fourier > Transform �also describes the relationship in electromagnetics > between the current distribution and the far field radiation > pattern. � Yep, they have sidelobes just like in waveforms... > > So ... > > I thought it would be fun and interesting to ask the group to come up > with a list of other pairings �in nature whose relationship can be > described by the FT. > > MarkHello Mark, If you want something to really ponder - think about an identity function under Fourier transformation. I.e., which function when FTed is back to itself. (There is more than a single answer - think about making a function as a linear combination of a function and its FT) The Gaussian function, aka Normal distribution or bell curve" is an identity function. And it shows up in many cases where random effects are additive. When they are multiplicative, you will likely end up with a log normal distribution. There are whole books about the ubiquitousness of the Fourier transform. From the EMag stuff, you will find a common beam profile for a laser is essentially Gaussian. The beam is comprised of both a mix of the near field and the far field and all of the fields inbetween. Clay
Reply by ●March 6, 20092009-03-06
On Mar 6, 12:37�pm, makol...@yahoo.com wrote:> The Fourier transform in nature �not just time<>frequency domain > > Most everyone here knows that the FT describes the relationship > between a waveform in the time domain and the frequency domain. > > I was enlightened to read here the other day that the Fourier > Transform �also describes the relationship in electromagnetics > between the current distribution and the far field radiation > pattern. � Yep, they have sidelobes just like in waveforms... > > So ... > > I thought it would be fun and interesting to ask the group to come up > with a list of other pairings �in nature whose relationship can be > described by the FT.1. people doing bitmap image processing use the 2-dim FT where the independent variable is position, not time. 2. when we add two independent random variables, the p.d.f. of the resulting sum (also a random variable) is the convolution of the p.d.f.s of the two random variables being added. then, if you compute the FT of the p.d.f.s, we call those "characteristic functions" and thems get multiplied. you can go one step further and log the characteristic functions. so when we add two independent random variables, we add the logs of their characteristic functions. at the moment those are the only two non-time domain FT applications i can think of. r b-j
Reply by ●March 6, 20092009-03-06
On Mar 6, 1:28�pm, c...@claysturner.com wrote:> On Mar 6, 12:37�pm, makol...@yahoo.com wrote: > > > > > > > The Fourier transform in nature �not just time<>frequency domain > > > Most everyone here knows that the FT describes the relationship > > between a waveform in the time domain and the frequency domain. > > > I was enlightened to read here the other day that the Fourier > > Transform �also describes the relationship in electromagnetics > > between the current distribution and the far field radiation > > pattern. � Yep, they have sidelobes just like in waveforms... > > > So ... > > > I thought it would be fun and interesting to ask the group to come up > > with a list of other pairings �in nature whose relationship can be > > described by the FT. > > > Mark > > Hello Mark, > > If you want something to really ponder �- think about an identity > function under Fourier transformation. I.e., which function when FTed > is back to itself. (There is more than a single answer - think about > making a function as a linear combination of a function and its FT) > The Gaussian function, aka Normal distribution or bell curve" is an > identity function. And it shows up in many cases where random effects > are additive. When they are multiplicative, you will likely end up > with a log normal distribution. There are whole books �about the > ubiquitousness of the Fourier transform. From the EMag stuff, you will > find a common beam profile for a laser is essentially Gaussian. The > beam is comprised of both a mix of the near field and the far field > and all of the fields inbetween. > > Clay- Hide quoted text - > > - Show quoted text -Periodic impulse train transforms to periodic impulse train.
Reply by ●March 6, 20092009-03-06
makolber@yahoo.com wrote:> The Fourier transform in nature not just time<>frequency domain> Most everyone here knows that the FT describes the relationship > between a waveform in the time domain and the frequency domain.> I was enlightened to read here the other day that the Fourier > Transform also describes the relationship in electromagnetics > between the current distribution and the far field radiation > pattern. Yep, they have sidelobes just like in waveforms...> So ...> I thought it would be fun and interesting to ask the group to come up > with a list of other pairings in nature whose relationship can be > described by the FT.Well, in addition to time<-->frequency there is position<-->spatial frequency, often used in optics. Since optics is a subset of electromagntics, maybe you already included that. To make your example a little more interesting, if you place a lens in the wavefront you described, it will focus what would have been at infinity at the focal plane of that lens. You can, for example, take a video camera and arrange it such that it records the Fourier transform of an object. (It works best with a coherent light source, maybe a monochromatic source is good enough after a spatial filter.) There are other transform pairs that come up in physics, such as energy <--> time, and momentum <--> space, but those come about by multiplying one component by hbar of what would otherwise be an ordinary transform pair. -- glen
Reply by ●March 6, 20092009-03-06
On 6 Mar, 18:37, makol...@yahoo.com wrote:> I was enlightened to read here the other day that the Fourier > Transform �also describes the relationship in electromagnetics > between the current distribution and the far field radiation > pattern. �The next obvious step on your path to Nirvana (or the looney bin, whichever is closer) is to investigate why[*] 2+2=4 in such diverse fields as - Maths - Physics - Economy - Medicine - and so on. In fact, it't hard to come up with a single case where 2+2=/=4. Now, *that's* food for tought. Preferably somebody else's, but still. Rune [*] Yes, I'm using the decimal number system.
Reply by ●March 6, 20092009-03-06
makolber@yahoo.com wrote:> The Fourier transform in nature not just time<>frequency domain > > Most everyone here knows that the FT describes the relationship > between a waveform in the time domain and the frequency domain. > > I was enlightened to read here the other day that the Fourier > Transform also describes the relationship in electromagnetics > between the current distribution and the far field radiation > pattern. Yep, they have sidelobes just like in waveforms... > > So ... > > I thought it would be fun and interesting to ask the group to come up > with a list of other pairings in nature whose relationship can be > described by the FT.Incredible observation. How about the application of y = ax + b in the all kinds of human activities? Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
Reply by ●March 6, 20092009-03-06
On Mar 6, 2:10�pm, Rune Allnor <all...@tele.ntnu.no> wrote:> In fact, it's hard to come up with a single case > where 2+2=/=4.I was taught that it is important to keep in mind the possibility that 2 + 2 = 5 for large values of 2 (or small values of 5)....
Reply by ●March 6, 20092009-03-06
Rune Allnor wrote:> On 6 Mar, 18:37, makol...@yahoo.com wrote: > >>I was enlightened to read here the other day that the Fourier >>Transform also describes the relationship in electromagnetics >>between the current distribution and the far field radiation >>pattern. > > > The next obvious step on your path to Nirvana (or the > looney bin, whichever is closer) is to investigate why[*] > 2+2=4 in such diverse fields as > > - Maths > - Physics > - Economy > - Medicine > - and so on. > > In fact, it't hard to come up with a single caseNot hard at all. There is one well known to those who have observed Phd parties for organic chem students who finally got out ;) It's been almost 40 years but IIRC 2 + 2 ~= 3.8> where 2+2=/=4. Now, *that's* food for tought. > Preferably somebody else's, but still. > > Rune > > [*] Yes, I'm using the decimal number system.
