From Fourier Series to Fourier Transforms

Recall that where Now let T become large... and so ω becomes small... Fourier Transform of f(x) Inverse Fourier Transform of F(ω).

Example 1 Determine the Fourier Transform of

Note: F(ω) is REAL in this example. These are the graphs of f(t) and F(ω):

Example 2 Determine the Fourier Transform of

Note: F(ω) is COMPLEX in this example. Draw the graph of the modulus of F(ω) (the amplitude spectrum).

Even Functions If f is an even function, then This result arises because cosine is even and so is even...

Example 3 Determine the Fourier Transform of Even function!

Odd Functions If f is an odd function, then This result arises because sine is odd and so is even...

Example 4 Determine the Fourier Transform of Odd function!

Summary: Examplef(t)f(t)F(ω)F(ω) 1EvenReal 2Neither odd nor even Complex 3EvenReal 4OddImaginary

Special Case Use this known result: Substitute Now use: Hence: or

Now look at Tutorial 1