The motivation comes from formally considering Fourier series for
functions of period 2*T* 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:

A function is called the

exist.

is called the

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

The Fourier transform of

is

The inverse Fourier transform computes to

- 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

(12) |

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*'-4*y*=*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

(13) **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.

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