Transcription of Example: the Fourier Transform of a rectangle function ...
{{id}} {{{paragraph}}}
Example: bachelor of science
Example: the Fourier Transform of a rectangle function ...
Example: the Fourier Transform of a rectangle function : rect(t). 1/ 2. 1. F ( ) = . 1/ 2. exp( i t )dt =. i . [exp( i t )]1/ 1/2 2. 1. = [exp( i / 2) exp(i /2)]. i . 1 exp(i / 2) exp( i /2). =. ( /2) 2i sin( /2) F(w) =. ( /2). F ( ) = sinc( /2) Imaginary Component = 0. w Example: the Fourier Transform of a Gaussian, exp(-at2), is itself! . F {exp( at 2 )} = ) exp( i t ) dt 2. exp( at . exp( / 4a). 2. The details are a HW problem! exp( at 2 ) exp( 2 / 4a). 0 t 0 w Fourier Series & The Fourier Transform What is the Fourier Transform ? Fourier Cosine Series for even functions and Sine Series for odd functions The continuous limit: the Fourier Transform (and its inverse).
The Dirac delta function Unlike the Kronecker delta-function, which is a function of two integers, the Dirac delta function is a function of a real variable, t. if 0 0 if 0 t t t δ ⎧∞= ≡ ⎨ ⎩ ≠ t d(t)
Loading..
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: