## Understanding the Fourier Transform intuitively

Posted by Ed on August 28, 2009

In the following I will explain how the Fourier transform can be understood intuitively. We will start with basis vectors in and make the transition to basis functions.

**The Fourier Transform**

For a function the Fourier transform is given by with (see here and here). In order to understand this formula we consider vectors in :

Every vector can be represented as a linear combination of the basis vectors (1,0,0), (0,1,0) and (0,0,1). Take the vector v=(9,8,4). This vector can be written as a linear combination of the basis vectors in the following way:

Let’s call the numbers in front of the basis vectors coefficients:

where , and are the coefficients.

What if we choose different basis vectors, for example:

Then our vector v=(9,8,4) can be written as:

Thus, for our new basis vectors and the coefficients are 3, 4 and 2.

In general, you can write a vector v as a linear combination of

basis vectors , where in front of the basis vectors you have the coefficients :

In our last example we had together with the coefficients . Just check the formula by plugging in the above values:

———————–

So, every vector can be represented by some basis vectors as .

More general, if we have a vector with n entries instead of 3 we write .

Now, let’s make a step from the discrete to the continuous case.

Say we have a function f(x) and we also want to ask whether it’s possible to represent f(x) as a linear combination of “basis vectors”.

*Question:* Is it possible to write

Let us be more specific and ask: Can we write f(x) as a sum

of ? So, the new basis vectors are .

The question then becomes:

*Question:* Is it possible to write

Indeed, it is possible with a correction for the values of k.

Instead of going from k=1 to n we use infinitely many basis vectors and write

to .

Corrected version:

To finally get to the Fourier integral we replace the sum by an integral:

(The factor is inserted for normalization issues)

One question remains though: How do the coefficients c(k) look like?

We can calculate c(k) by

One can prove that c(k) has this form by plugging c(k) into the formula of f(x) and evaluating the double integral (see here).

**Some remarks**

*1) Summary*

In summary, consider the Fourier integral as a linear combination of basis vectors with the coefficients .

*2) Functions as basis vectors*

You might ask “Why can I consider the functions as basis vectors?”

The functions fulfill some properties similar to those of

basis vectors in .

and are orthonormal to each other:

(i) orthogonal:

with being the inner product of two vectors a and b.

(ii) normal:

In short: where is the Kronecker delta.

For functions f(x) and g(x) we can define an inner product (L2 inner product) as:

.

One can show that for two basis vectors and the inner product is:

(see here). This looks like the Kronecker delta from above.

In particular and are orthogonal to each other since for .

The difference to the Kronecker delta is that the vectors can not be normalized to 1 since .

*3) Interpreting c(k) as projection*

In we represented the vector v=(9,8,4) as a linear combination of the basis vectors :

You can immediately “see” that the coefficients are . But you can also get the coefficients by projecting v onto the basis vectors (Note that we have to project on *normalized* basis vectors):

And as you may know, the projections can be calculated with the dot product:

So, we can write

Now, for a function f(x) the Fourier transform is given by

Similar to above we will interpret the coefficients c(k) as the projection of f(x) onto the basis vectors :

But what is the result of the inner product ?

Recall from above that for two functions f(x) and g(x) we defined the inner product as L2 inner product .

Let’s use this to calculate c(k):

(In the last step I just changed the integration variable from x to t)

*4) Some examples of Fourier transform and coefficients*

On this website the Fourier transform is defined slightly different (compare the factor in front of the integral sign):

with being the coefficient that can be calculated by

Examples for this Fourier transform are given here.

*5) Applications for the Fourier transform*

Have a look at What is a Fourier Transform and what is it used for? and Applications of Fourier Transform in Communications, Astronomy, Geology and Optics.

*6) Java Applets*

Approximation of a function by a Fourier transform

The applet (click on *Start Function FFT*) shows you how you can approximate a function by a Fourier transform (though they talk about the related Fast Fourier Transform). The more coefficients you use the better the approximation becomes.

Applet: Rectangular pulse approximation by Fourier Transform

This applet shows how to approximate a rectangular shaped pulse. If you don’t use enough basis vectors (bandwidth is limited) then the pulse will not look rectangular anymore. This is important in electronics where you want to transmit a signal but the bandwidth is limited.

**References**

[1] U. H. Gerlach, *Linear mathematics in infinite dimensions Signals, Boundary Value Problems and Special Functions*, Febuary 2007, Beta Edition, http://www.math.ohio-state.edu/~gerlach/math/BVtypset/BVtypset.html, “The Fourier Integral Theorem”, http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node31.html

[2] Weisstein, Eric W. *Fourier Transform*, August 2009. From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/FourierTransform.html

[3] Jochen Rau, *Elementary Mathematical Methods for Physics*, Lectures at Dresden University of Technology, winter 1996/97, http://www.mpipks-dresden.mpg.de/~jochen/methoden/intro.html, “Definition” and “Examples” in Lecture 8, http://www.mpipks-dresden.mpg.de/~jochen/methoden/topics/ft_def.html, http://www.mpipks-dresden.mpg.de/~jochen/methoden/topics/ft_ex.html

## Olumide said

Thanks.

If it is true that the Fourier transform

computes the coordinates of a function in an infinite dimensional space spanned by the orthogonal basis vectors . Why is it that is represented as a combination of the basis vectors i.e.

instead of the original set of basis vectors ?

## Ed said

Thank you for your comment. I suppose you mean and . They are different functions, so don’t forget the little hat over .

The minus sign comes from the definition of the L2 inner product: In my post above I tried to derive the coefficients or . I did this by considering a projection, and for projections you use the L2 inner product or scalar product. The L2 inner product involves complex conjugation of which is .

I hope this helps.

Ed