An approach for computing what frequencies are present in a Wave, in particular the Spectrum, entirely from its dependence on time. Allows us to transform a function between time an frequency domains. Allows us to quantitivatively define the Spectrum of a Wave A Gaussian’s transform is another Gaussian.
- Frequency is conjugate to time
- Spatial frequency is conjugate to position Anytime the Fourier Transform is involved, so is the Scale Theorem, and the Uncertainty Principle
Equation
- Where
is called the Fourier Transform of , containing equivalent information to that in lives in the time domain lives in the frequency domain
With respect to space
A spatial fourier transform. Spatial frequency
Derivation from Fourier Series
Let
where these values are defined in Even Function and Odd Function and also Fourier Series Allow t to range from
to , to be continuous Frequency , i.e. for a continuous range of frequencies
Inverse Equation
Derivation from Fourier Series
Note that the
gets factored in for consistency with the above derivation.
With respect to space
Notation
If original function lowercase, use above notation
for the function for the transform If the original function is uppercase: for the function, arbitrarily - For the transform
Examples
Find
See sinc for more info