Fourier Transform

Fact Instead of discrete amplitude for each frequency at intervals of , we have a continuous function giving the amplitude of frequency over infinitesimal frequency intervals

Def Fourier Transform

And the inverse Fourier transform is given by:

Prop

Tophat, DiracDelta and Shah Function

Def Tophat Function Define the tophat function as:

Def DiracDelta Function Delta function is an even function that is zero everywhere except at the origin and has area one. That is:

Prop By the property of delta function, we can get that:

Generally, we can offset the delta function and get the following:

Prop

• The Fourier Transform of the delta function is 1:
• The Inverse Fourier Transform of the delta function is :

Def Shah Function The Shah function is defined by an infinite series of delta functions:

Prop The Fourier Transform of a Shah is a Shah.

Linearity, Symmetry and Duality

Prop The Fourier transform is linear:

Prop The scaling of Fourier Transform:

Corollary

Prop Symmetries

f(t) is real	Real part of F(\omega) is even; Imaginary part is odd
f(t) is real and even	F(\omega) is real and even
f(t) is real and odd	F(\omega) is imaginary and odd

It is equivalent to say that:

where is the complex conjugate of . For a real-valued function , the negative frequency components of the Fourier transform must be complex conjugate of the positive frequency components.

Prop Duality If then

Prop Coordinate Shift

Convolution

Recall that convolution is the integral of the product of two functions after one is reversed and shifted:

The convolution of two functions is defined by

Link to original

It has the following properties:

Convolution is commutative, associative, and distributive over addition.

Link to original

Thrm Convolution Theorem The convolution is function multiplication in Fourier space: where is the convolution operator.

Derivatives of Fourier

Prop If and the first derivative of exists, then . More generally, Proof

Energy and Parseval’s Theorem

Thrm Parseval’s Theorem If is some finite signal, then the energy will be finite. Parseval’s theorem says: