The Fourier series is used to express a periodic signal as a ?combination of sinusoidal waves, meaning that we have a resultant signal ?that is made up of a combination of those sine waves.

The Fourier series is used to express a periodic signal as a  combination of sinusoidal waves, meaning that we have a resultant signal  that is made up of a combination of those sine waves. The Fourier  transform is a formula that is used to transform a signal that was  recorded in either time or space to the same signal sampled in the  temporal or spatial frequency. Using this we can help find the frequency  components of these signals. In real applications, signals can be  muddied with noise which can hide their frequency components. The  Fourier transform can process out that extra noise to reveal the  frequencies thus making the Fourier series very good at filtering noise.

The Fourier Transform can either be f(t) for time or f(x) for time.

ω = angular frequency and is related to frequency 

by 

The units for frequency are usually in Hertz(Hz).

k = wavenumber, it has units of inverse length and is related to wavelength 

by 

References:

Washington, A. J., & Evans, R. (2017). Basic Technical Mathematics with Calculus (11th ed.). Pearson Education (US). https://ecpi.vitalsource.com/books/9780134507095 

 Links to an external site.

EngineerYourSound. 2023. What Is The Fourier Series Used For? (Real applications). What Is The Fourier Series Used For? (Real applications) – Loudspeaker & Acoustic Engineering Design (engineeryoursound.com) 

 Links to an external site.

Herman, Russell, 2022, LibreTexts Mathematics, 9.5: Properties of the Fourier Transform. 9.5: Properties of the Fourier Transform – Mathematics