Where can I start learning about Fourier Transforms/Series?

[cmkluza]

I'm trying to learn about Fourier Transforms, specifically how they relate to equalizers, but I can't seem to find any academic guidance. I've asked my maths teacher for help, and I've looked through my school library, but I can't find a single source to start learning about Fourier Transforms.

I've looked online at places like Wikipedia and Wolfram, but those websites don't explain the Fourier Transforms starting from a simple enough position, and I can't start to understand them. Where can I look to start learning? I need to have a fairly good understanding of the derivation of the Fourier Transforms as well as how they work/how to use them.

Any help will be greatly appreciated!

 

[DrClaude]

Try hyperphysics: http://hyperphysics.phy-astr.gsu.edu/hbase/audio/fourier.html

You will basically need to understand Fourier series first. To really understand Fourier transforms, you will need to know calculus, but the basics you get through Fourier series.

 

[jtbell]

those websites don't explain the Fourier Transforms starting from a simple enough position, and I can't start to understand them.

What would be a simple enough position for you? How much mathematics have you studied so far?

 

[cmkluza]

Try hyperphysics: http://hyperphysics.phy-astr.gsu.edu/hbase/audio/fourier.html

You will basically need to understand Fourier series first. To really understand Fourier transforms, you will need to know calculus, but the basics you get through Fourier series.

Thanks, I checked out that link and it seemed to make sense. It showed, like many other websites, the Fourier series:

I've seen this, and I've seen the integrals representing the coefficients, but I can't find where it comes from anywhere. How is this formula derived? Where would I start to derive it?

What would be a simple enough position for you? How much mathematics have you studied so far?

I'm in a Higher Level mathematics course, so what I've studied that I believe is related to the Fourier series/transforms is a fair bit of calculus and trigonometry. From the syllabus I've done algebra, functions and equations, trigonometry, vectors, statistics and probability, and started discrete.

I suppose the simple position I'm looking for just starts at the beginning and explains it from there. What I've seen so far on the Fourier series/transforms hasn't seemed to source much, showing the functions for each and not showing how they were derived.

Thanks for the help up to here, and any further help!

 

[DrClaude]

Thanks, I checked out that link and it seemed to make sense. It showed, like many other websites, the Fourier series:

I've seen this, and I've seen the integrals representing the coefficients, but I can't find where it comes from anywhere. How is this formula derived? Where would I start to derive it?

The short answer is that the Fourier series comes about from the fact that cosine and sine form a complete basis for 2π-periodic functions. The integral (and Fourier transform) arises from the inner product for functions (extension of the dot product for vectors).

 

[cmkluza]

The short answer is that the Fourier series comes about from the fact that cosine and sine form a complete basis for 2π-periodic functions. The integral (and Fourier transform) arises from the inner product for functions (extension of the dot product for vectors).

I'm starting to understand the original equation now, but why is there an $a_0$ but not a $b_0$ in $f(t)$?

Edit: Never mind, there is no $b_0$ because sin(0) = 0. Not sure why I forgot that bit.

 

[cmkluza]

I'm looking online here and it shows the Fourier series "with period of 2L" to be the same as in the picture below:

However in this website it says that it equals $\frac{a_0}{2}$ not just $a_0$ as shown above. Why is this? In fact, what's the purpose of the parts inside of the cosine and sine functions in the above equations? Why is it $cos(\frac{2nπ}{T}t)$ and $sin(\frac{2nπ}{T}t)$?

Edit: Clarification

 

[Incand]

The $a_0$ is divided by two so the general formula for $a_n$ works from the formula $a_n = \frac{1}{L}\int_{-L}^L f(\theta)\cos \frac{n\pi \theta}{L} d\theta$. The first Fourier cosine coefficient is simple the average of the function over the interval, since we integrate from $-L$ to $L$ we have to divide by an extra $2$ beyond the $L$ in the formula.
In complex form you instead have coefficients $c_n$ with $c_0 = \frac{a_0}{2}$ if you really want to get rid of that $2$.

I Believe the $2$ in your $\sin$ and $\cos$ shouldn't be there. I couldn't find them there in your link either. You want a function that repeats itself every interval of $2L$ and that is true for $\sin$ and $\cos$ with the argument $n\pi t/L$.

I think if want to learn the topic properly you really have to lend/buy a book, an internet page just won't give enough information. An excellent book in my opinion is Fourier analysis and it's applications by Folland. It's strikes a middle ground between mathematics and applications which means it can be used by both as a first course in Fourier analysis for engineers,physicists and mathematicians (all you need is calculus + linear algebra).
However if you are mostly looking for application something a book on linear systems or control engineering may be more appropriate.
The other option is just to accept it and follow the formulas, sites like wolfram alpha also are able to calculate Fourier transform and series for you.

 

[cmkluza]

The $a_0$ is divided by two so the general formula for $a_n$ works from the formula $a_n = \frac{1}{L}\int_{-L}^L f(\theta)\cos \frac{n\pi \theta}{L} d\theta$. The first Fourier cosine coefficient is simple the average of the function over the interval, since we integrate from $-L$ to $L$ we have to divide by an extra $2$ beyond the $L$ in the formula.
In complex form you instead have coefficients $c_n$ with $c_0 = \frac{a_0}{2}$ if you really want to get rid of that $2$.

I Believe the $2$ in your $\sin$ and $\cos$ shouldn't be there. I couldn't find them there in your link either. You want a function that repeats itself every interval of $2L$ and that is true for $\sin$ and $\cos$ with the argument $n\pi t/L$.

Sorry, I'm still not quite understanding what the formula is saying. As far as $\frac{a_0}{2}$ goes, I'm still confused. In the general formula $a_n = \frac{1}{L}\int_{-L}^L f(\theta)\cos \frac{n\pi \theta}{L} d\theta$ if you solved for the first term it would be $a_0 = \frac{1}{L}\int_{-L}^L f(\theta)d\theta = \frac{1}{L}[f'(\theta)]^L_{-L}$, so why do you need to divide it by 2?

I've realized now that the $\frac{nπt}{L}$ is really just $\frac{2πnt}{2L}$. So I realize that this shows that it's going through 2π every 2L. $t$ is in there since it is a function of time (t), so then why is there the n in there? I realize that Fourier series probably wouldn't work without the n inside of the trig functions, but what purpose does it serve? Does it have to do with the derivation of the Fourier series?

Apologies if I'm just missing something right in front of my eyes, but thanks for all the help up to here and any future help!

 

[Samy_A]

Sorry, I'm still not quite understanding what the formula is saying. As far as $\frac{a_0}{2}$ goes, I'm still confused. In the general formula $a_n = \frac{1}{L}\int_{-L}^L f(\theta)\cos \frac{n\pi \theta}{L} d\theta$ if you solved for the first term it would be $a_0 = \frac{1}{L}\int_{-L}^L f(\theta)d\theta = \frac{1}{L}[f'(\theta)]^L_{-L}$, so why do you need to divide it by 2?

That $f'(\theta)$ is wrong.

But never mind, and don't bother for now about the $2$ in $\frac{a_0}{2}$.

If you want to learn about Fourier series, mathworld.wolfram.com explains the very basics quite well.

Essentially, it is a way to represent any sufficiently nice periodic function over the interval $[-L,+L]$ as an (infinite) sum of sines and cosines, and more precisely of the functions $\cos(\frac{n\pi t}{L})$ and $\sin(\frac{n\pi t}{L})$ (with $n \in \mathbb N)$.

You need all these functions because together they are sufficient to "represent" all nice periodic functions. Not every Fourier series will contain all these functions, but you need them all to represent all nice periodic functions.

Of course, you get it that when I write "nice", this will get a mathematical definition once you learn the theory in more depth. But for starters, that is not the essence.
The essence is that a periodic function can be decomposed into an (infinite) sum of elementary sine and cosine waves.

On Wikipedia you can also read how this all started:

"The heat equation is a partial differential equation. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was asine or cosine wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series."

You can find a number of introductions to Fourier series online. One example: http://www.stewartcalculus.com/data/CALCULUS Early Transcendentals/upfiles/FourierSeries5ET.pdf

 

[Incand]

Sorry, I'm still not quite understanding what the formula is saying. As far as $\frac{a_0}{2}$ goes, I'm still confused. In the general formula $a_n = \frac{1}{L}\int_{-L}^L f(\theta)\cos \frac{n\pi \theta}{L} d\theta$ if you solved for the first term it would be $a_0 = \frac{1}{L}\int_{-L}^L f(\theta)d\theta = \frac{1}{L}[f'(\theta)]^L_{-L}$, so why do you need to divide it by 2?I've realized now that the $\frac{nπt}{L}$ is really just $\frac{2πnt}{2L}$. So I realize that this shows that it's going through 2π every 2L. $t$ is in there since it is a function of time (t), so then why is there the n in there? I realize that Fourier series probably wouldn't work without the n inside of the trig functions, but what purpose does it serve? Does it have to do with the derivation of the Fourier series?
Apologies if I'm just missing something right in front of my eyes, but thanks for all the help up to here and any future help!

The $n$ is there because it is a series. we sum together $c_0 + a_1cos(\pi \theta/ L) + b_1\sin (\pi \theta/ L) + a_2 \cos(2 \pi x/l) + b_2 \sin (2\pi \theta /L) + \dots$ hoping that it converges to the function we're expanding as a Fourier series.
If you take only the first terms $c_0 + a_1\cos (\pi \theta/ L) + b_1\sin (\pi \theta)$ you get a good approximation to the function (compare it to taylor series!). As to why you take exactly "integers" of cos and sine comes back to that they're form an orthogonal basis in $L^2$-space but you don't need to know that to apply them. (This is usually covered in a linear algebra course and repeated in a course in Fourier analysis/PDE)

The constant is needed to compensate for the average value of the function we're trying to expand. Say we have the function $f(x) = 5+x$ in the interval $-L$ to $L$. The average value of $f(x)$ in the interval is $5$. If we calculate that with the formula we have
$a_0 = \frac{1}{L}\int_{-L}^L (5+x)dx = 10$ so we have to divide by two to get the average. I don't know how to explain it otherwise, maybe try calculate some examples and you see you have to divide the constant by $2$ to get the right function.

In complex form the Fourier series in the interval $-L$ to $L$ would be
$f(x) = \sum_{-\infty}^\infty c_ne^{in\pi x/L}$ and the constant term here is indeed $c_0 = a_0/2$.

Overall I feel you need to get some learning resources with examples and some exercises so you get some experience calculating Fourier series to get a better understanding.

Isn't there an university library where you live? There must be books there covering this subject.

 

[mathwonk]

you might look at An Introduction to Fourier Series and Integrals, by Robert T. Seeley. It is written for undergraduates who know some calculus. It starts from the historical problem of solving the heat equation and shows how the sin and cosine functions arise in this context, goes on to discuss expressing other functions using those, and eventually finishes with an introduction to Fourier transforms. It is less thatn 100 pages long. I have also been told the book by Edwards and Penny gives a nice intro to Fourier series, but I am not sure whether it is the diff eq book or the calculus book, probably the diff eq book.

 

[jack476]

Sophomore-level electrical engineering textbooks tend to go into great detail about Fourier series and transforms. The book I used was Linear Systems and Signals by BP Lathi.

Another book I've used occasionally is Integral Transforms and their Applications (3rd edition) by Debnath and Bhatta. Lots of material on Fourier transforms, but also on Laplace transforms, Z-transforms, and other cool stuff like fractional calculus and a handful of somewhat more obscure transforms.

 

[mathwonk]

Seeley’s book is very good at explaining where all the symbols come from. He starts from the problem of understanding heat flow in a unit disc, and seeks first the relevant differential equation. From the plausible hypothesis that heat flows across a curve proportionally to the normal derivative of temperature, he finds the Laplace equation. Using the natural polar coordinates in the disc he further proposes trying to solve by the method of separation of variables, which reduces partial derivatives to ordinary ones in each variable separately. I.e. he considers first functions which are a product of a function of r and a function of the angle t. Plugging in and imposing the natural conditions of continuity, he finds the only such solutions are of form cr^n.(Ae^int + Be^-int), where t is the angle and n is an integer. More solutions are then obtained as linear combinations of these, or hopefully, infinite such combinations, i.e. series.

The problem of achieving the given boundary values, means for r=1, we want to be able to represent the boundary function as an infinite series in e^int and e^-int, i.e. as a Fourier series. He goes on to discuss when these hopes are realized, proving first Poisson’s theorem giving only ana stmptotic representation, and then discussing also pointwise convergence of Fourier series. This is just chapter one.

Beyond the scope of Seeley’s book, but maybe of interest to you, is the theory of general Fourier transforms. Here we regard a series as a function on the integers, and a summable series as analogous to an integrable function on the circle, i.e. a periodic integrable function. So the theory of Fourier series is viewed as a way to go back and forth between functions on the circle and functions on the integers. The reason this works is those spaces are both abelian groups and they are mutually “dual” somewhat in the sense of dual vector spaces. I.e. notice the circle is a group, and define the dual of a group G, or group of “characters”, as the set of continuous homomorphisms from G to the circle. Note that the integers are the group of continuous homomorphisms from the circle to itself, and the circle is the set of continuous homomorphisms from the integers to the circle.

Further, the real numbers are isomorphic also to to the set of continuous homomorphisms from the reals to the circle. Thus there should be some way to go back and forth between function on the reals and functions on the reals, that is analogous to forming Fourier series; this is the Fourier transform. There are analogous theories for functions on the n torus, a product of n circles, and functions on the n fold product of the integers, as well as between functions on a finite abelian group, and functions on the dual finite abelian group, (discrete Fourier transform). Thus although in general a group and its dual are not usually (and almost never naturally) isomorphic, their algebras of functions are isomorphic, via the Fourier transform. Analogous to finite dimensional vector spaces however a (locally compact abelian) group is isomorphic to its double dual. You can google Pontrjagin duality to pursue this.