Which Book Explains the Fourier Transform Clearly?

[pmr]

I need a good book on the Fourier transform, which I know almost noting about.

Some online sources were suggesting Bracewell's "The Fourier Transform & Its Applications." I gave it shot, but it's competely unreadable. On page 1 he throws out an internal expression and says "There, that's the Fourier transform." He gives no reasoning, motivation, or exposition. He then dives into examining the conditions under which the transform exists, how it behaves with even or odd functions, etc...

If I wanted to purposefully confuse a student who was new to mechanics I might throw out the integral expression for the tensor of inertia. I would give no motivation or reasoning. I would state by fiat that it relates to angular momentum somehow. Then I would proceed to give a rigorous proof showing why its eigenvalues are always real. The student would have no idea how to formulate the tensor from first principles, and so they wouldn't really know what it does, why its so useful, or what motivated people to discover it in the first place. They would also have no idea why its symmetric, so they wouldn't really appreciate the proof about its eigenvalues being real.

I need a book on the Fourier transform which is aware of the absurdity of the above approach. Bracewell is definitely not that book.

 

[deskswirl]

Bracewell is definitely not a book for beginners probably more of an intermediate level. I would suggest Linear Systems and Signals by B.P. Lathi which relies on physical insight rather than mathematical definitions. Have you any familiarity with Fourier Series?

 

[pmr]

I'm solid with Fourier series, yep. The integral expression for the Fourier transform feels close enough to a Fourier series that I almost feel like I could figure out what it does on my own (with a blackboard and a lot of free time).

That book by Lathi looks interesting, I'll give it a peek. Though one of the reviews on Amazon is discouraging:

"The students should have already taken several engineering courses, where they have become familiar with circuit analysis, bode plots, Laplace transforms, filters, and several other concepts."

I definitely don't meet that criteria. I'm not an engineer. I'm at the very beginnings of an undergrad physics program. The only circuit analysis I've done was in an intro EM course. It was relatively basic, and Fourier transforms didn't show up. The reason I want to learn about them in the first place is because they're showing up in a quantum book I'm reading (Shankar). So, I'm coming at this from a very different angle. I'm willing to learn about the Fourier transform from an engineering textbook though, provided that I find it comprehensible given my current skill set.

 

[deskswirl]

Sounds like you won't have any difficulty with Lahti as he introduces the Fourier Transform as the infinite period Fourier Series.

The Fourier methods allow one to represent a signal (a physical quantity described by math) in alternate domains like temporal frequency and time. For example the Fourier Series of a periodic signal is a representation of that signal by a set of basis signals (not too dissimilar from unit vectors except they are functions) where the coefficients of the series give the relative amplitude of each frequency component. The square of the amplitude corresponding to energy one can see that the Fourier series is just a decomposition of signal energy among the frequency spectrum. In the limit as the period of the signal you are trying to represent goes to infinity (i.e. the signal is aperiodic) you have the definition of the (inverse) Fourier transform.

How this shows up in QM is that position and momentum (or equivalently wavenumber which is really just spatial frequency) are Fourier transform pairs of each other. Hence whatever you might wish to describe in the spatial domain (like a particle's wavefunction) can be equally represented in terms of momentum.

P.S. Don't believe everything you read on Amazon - Lahti's book is a very gentle introduction.

 

[vanhees71]

A classic is

M. J. Lighthill, Introduction to Fourier analysis and generalized functions, Cambridge University Press 1959

 

[deskswirl]

Lighthill is more advanced than Bracewell and as such is to no benefit to someone teaching themselves without a serious foundation in mathematics.

 

[Dr. Courtney]

There are lots of perspectives on Fourier Analysis and Transforms.

The needs of a mathematician are different from an experimentalist are different from a theorist are different from an engineer.

The wiki pages are a good place to start. But the question is best addressed with more info about the anticipated downstream uses of one's knowledge.

 

[pmr]

At different times I'm an experimentalist, mathematician, theorist, and engineer. Though my skill levels across those domains varies tremednously. I would be interested in an introductory text from any perspective, really. My key requirement is that any introduction should arrive at the transform organically, instead of pulling it out of thin air.

 

[jasonRF]

I don't know if this is what you are looking for, but the free book by Brad Osgood

http://www.e-booksdirectory.com/listing.php?category=392

starts with the Fourier series in chapter 1, and in chapter 2 shows (graphically) what happens to the Fourier series coefficients of a particularly simple signal (square pulse) as the period gets larger and larger. It's free, so you don't have much to lose!

jason

 

[Dr. Courtney]

I don't know if this is what you are looking for, but the free book by Brad Osgood

http://www.e-booksdirectory.com/listing.php?category=392

starts with the Fourier series in chapter 1, and in chapter 2 shows (graphically) what happens to the Fourier series coefficients of a particularly simple signal (square pulse) as the period gets larger and larger. It's free, so you don't have much to lose!

jason

Looks like a great book. Thanks for posting.