Personal reminder (or CheetSheet) about Fourier Transform

Recently, I’m studying Fourier Transform by watching the lectures from Stanford University. I felt that I already forget the math basics that I’ve learnt in college. So, to set up a quick lookup table for myself, I decide to write something to memorize it.

 

Here I listed some formulas that I’m encountered, And maybe more formulas to come in the future. Hope that will be useful for somebody else too. 

 

 

Trigonometric Functions and Complex Exponentials:

 

 

Fourier Coefficients:

 

Fourier Series:

Sinc Function:

Duality Properties of FT

 

 

Here is the definition of Reversed Signal

 

Linearity of FT

 

 

Shift Theorem of FT

 

 

Stretch Theorem of FT

 

 

Combine Shift and Stretch of FT

 

 

 

Definition of Convolution:

 

Convolution Theorem, Which means FT of Convolution is multiplication of FT

 

Definition of Rapidly decreasing functions

 

 

The Delta distribution

 

The Delta distribution property

 

Delta with convolution

 

 

How a Distribution works on a function

 

 

The derivation of How Fourier Transform move from distribution to function

 

 

 

Shift and Stretch of FT

 

Derivation of FT of delta function

 

Different notations for delta function

 

FT of sine and cosine functions

 

 

how Derivation shifts from T to phi.

 

 

 

derivation of FT of H

 

 

function g times distribution T means distribution T works on function g times phi

 

 

Derivation Theorem for distributions

 

 

 

shift one way in distribution means shift in the other way of function

 

 

psi Convolve with f works on phi means f works on reversed psi convolve with phi

 

 

Shah function definition

 

 

Derivation of the Sampling Theorem

 

 

 

Definition of Discrete Fourier Transform 

 

 

 

DFT operation as a matrix multiplication

 

 

Reference:

 

Lecture Notes for EE 261

The Fourier Transform and Its Applications by Prof. Brad Osgood.