# Fourier transform

**Fourier Transform** represents a function as a "linear combination" of complex sinusoids at different frequencies . Fourier proposed that a function may be written in terms of a sum of complex sine and cosine functions with weighted amplitudes.

In Euler notation the complex exponential may be represented as:

Thus, the definition of a Fourier transform is usually represented in complex exponential notation.

The Fourier transform of *s*(*t*) is defined by

Under appropriate conditions original function can be recovered by:

The function is the Fourier transform of . This is often denoted with the operator , in the case above,

The function must satisfy the Dirichlet conditions in order for for the integral defining Fourier transform to converge.

Forward Fourier Transform(FT)/Anaysis Equation

Inverse Fourier Transform(IFT)/Synthesis Equation

## Explanation coming from Linear Algebra

editAccording to linear algebra, for every orthogonal *B* of a vector space *H*, and every element *x* of *H*

holds true.

is an orthonormal basis as can be confirmed by calculating the scalar products. This means that

( denotes the *n'*th component of the vector *x*)
holds true for every *N*-dimensional vector *x*. is exactly the discrete Fourier transform, and we just proved that the inverse discrete Fourier transform of the discrete Fourier transform of a vector *x* is the vector *x*, which is the central theorem of discrete Fourier theory.
Discrete Fourier theory essentially means writing something in the Fourierbasis *B*. This also explains the linearity of the Fourier transformation.

## Relation to the Laplace Transform

editIn fact, the Fourier Transform can be viewed as a special case of the bilateral Laplace Transform. If the complex Laplace variable s were written as , then the Fourier transform is just the bilateral Laplace transform evaluated at . This justification is not mathematically rigorous, but for most applications in engineering the correspondence holds.

## Properties

edit× | Time Function | Fourier Transform | Property |
---|---|---|---|

1 | Linearity | ||

2 | Duality | ||

3 | , c = constant | Scalar Multiplication | |

4 | Differentiation in time domain | ||

5 | , if | Integration in Time domain | |

6 | Differentiation in Frequency Domain | ||

7 | Time reversal | ||

8 | Time Scaling | ||

9 | Time shifting | ||

10 | Modulation | ||

11 | Modulation | ||

12 | Frequency shifting | ||

13 | Convolution |