Transcription of Example: the Fourier Transform of a rectangle function ...
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Example: tourism industry
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 spectrum Some examples and theorems .. 1. f (t ) = F ( ) exp(i t ) d F ( ) = f (t ) exp( i t ) dt 2 .. Source: Prof. Rick Trebino, Georgia Tech What do we hope to achieve with the Fourier Transform ? We desire a measure of the frequencies present in a wave. This will lead to a definition of the term, the spectrum.
Discrete Fourier Series vs. Continuous Fourier Transform F m vs. m m! Again, we really need two such plots, one for the cosine series and another for the sine series. Let the integer m become a real number and let the coefficients, F m, become a function F(m).! F(m)!
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