Fourier Transform Pairs

 Some Basic Functions x(t) X(ω) (Synthesis) (Analysis) (impulse) (constant) (unit rectangular pulse, width=1) (sinc) (constant) (impulse) (complex exponential) (shifted impulse) (causal exponential) (same as Laplace w/ s=jω) (Gaussian) (Gaussian)

 Derived Functions (using basic functions and properties) x(t) X(ω) (time scaled rectangular pulse, width=Tp) (rectangular pulse in ω) (triangular pulse, width=2) (scaled triangular pulse, width=2Tp) γ(t) (unit step function)

Using these functions and some Fourier Transform Properties (next page), we can derive the Fourier Transform of many other functions.

© 2015, Erik Cheever

Fourier Transform Properties

 Name Time Domain Frequency Domain Linearity Time Scaling Time Delay (or advance) Complex Shift Time Reversal Convolution Multiplication Differentiation Integration Time multiplication Parseval’s Theorem Duality

Symmetry Properties

 x(t) X(ω) x(t) is real Real part of X(ω) is even, imaginary part is odd x(t) real, even X(ω) is real and even x(t) real, odd X(ω) is imaginary and odd

Relationship between Transform and Series

 If xT(T) is the periodic extension of x(t) then: Where cn are the Fourier Series coefficients of xT(t) and X(ω) is the Fourier Transform of x(t) Furthermore