The 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
Relation to the Laplace TransformEdit
In 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.
|×||Time Function||Fourier Transform||Property|
|3||, c = constant||Scalar Multiplication|
|4||Differentiation in time domain|
|5||, if||Integration in Time domain|
|6||Differentiation in Frequency Domain|