Fourier transforms

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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 Transform[edit]

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.

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