Fourier transforms

A Fourier series can sometimes be used to represent a function over an interval. If a function is defined over the entire real line, it may still have a Fourier series representation if it is periodic. If it is not periodic, then it cannot be represented by a Fourier series for all x. In such case we may still be able to represent the function in terms of sines and cosines, except that now the Fourier series becomes a Fourier integral.

The motivation comes from formally considering Fourier series for functions of period 2T and letting T tend to infinity.

Suppose

and

Now, set

and insert the integral formula for the Fourier coefficients:

The summation resembles a Riemann sum for a definite integral, and in the limit ( ) we might get

This informal reasoning suggest the following definition:

Definition 6 (Fourier Transforms)
A function is called the Fourier transform of f(x), if

exist.

is called the inverse Fourier transform of .

The Fourier transform of f is therefore a function of the new variable . This function, evaluated at , is .

Remark 5
The constants 1 and preceding the integral signs in (10) and (11)could be replaced by any two constants whose product is .

Example 7
The Fourier transform of f given by

is

The inverse Fourier transform computes to

Theorem 8 (The Fourier integral)

1.

If f(x) and f'(x) are piecewise continuous in every finite interval

2.

and converges, i.e f(x) is absolutely integrable in

Then

Remark 6
The above conditions are sufficient but not necessary. The similarity with corresponding results for Fourier series is apparent.

We will now develop some properties of the Fourier transform:

Linearity

If , then

provided the Fourier transform of f(t) and g(t) exist.

Scaling

If and , then

 

Time shifting

If and , then

Frequency shifting

If and , then

Symmetry

If , then

Modulation

If and , then

Differentiation in time

Let and suppose that f⁽ⁿ⁾ is piecewise continuous. Assume that . Then

In particular

and

Example 8
Suppose we want

We apply differentiation in time to get

We can also integrate by parts to get

Then we have

Solving this equation, we get

Example 9
Suppose we want to solve

y'-4y=H(t)e^-4t

H(t) is given by

Apply the Fourier transform to the differential equation to get

Setting , we have

Then

From the last equation, we get

Frequency differentiation

Let and suppose that f is piecewise continuous. Then

In particular

and

Convolution

Definition 7 (The convolution (faltung))
If f and g both have Fourier transforms, then the convolution (faltung) f*g of the functions f and g is defined by

Theorem 9 (The convolution theorem)
The Fourier transform of the convolution of f(x) and g(x) is equal to the product of the Fourier transforms of f(x) and g(x).

and

The Fourier transform of the Dirac delta function

Some problems involve the concept of an impulse, which may be intuitively thought of as a force of very large magnitude impacting just for an instant. We can model this idea mathematically as follows:

As is chosen smaller, the duration of this pulse tends to zero while its amplitudes increases without bound. This lead us to define

Strictly speaking, is not really a function in the conventional sense, but it is a quantity called distribution. For historical reasons it is called the Dirac delta function after the physicist P. A. M. Dirac. The delta function has the fundamental property:

Definition 8 (Dirac delta function)

Remark 7
Therefore the Fourier transform of the delta function yields 1.

The sampling theorem

A function f(t) is called band-limited if its Fourier transform is only nonzero on an interval of finite length. This means for some L, if .

Begin with the integral for the inverse Fourier transform.

The complex Fourier series for on [-L,L] is given by

where

Now compare this equations to conclude that

and

Substitute this series for in f(t) to get

Interchanging the summation and the series, we get

Theorem 10 (The sampling theorem)

This means that f(t) is known for all t if just the function values are known for integer values of n. That is, if we sample the signal (function) and determine its values at , then the entire signal can be reconstructed.