How does Fourier series move analysis from the time domain to the frequency domain?

Conditions

• The discussion concerns periodic functions represented by Fourier series.
• Signal-processing applications are mentioned as motivation.

Reasoning, step by step

1. Start with the time-domain description provided directly by $f(t)$.
2. Express $f(t)$ as an infinite sum of weighted sines and cosines.
3. Interpret the coefficients as measures of how much of each frequency the function contains.
4. Shift the analytical focus from the temporal shape of the wave to its constituent frequencies.

Example

The speaker says this helps us think of a function not just in terms of the time domain, which $f(t)$ does, but also how much we have of each frequency, and that Fourier series and eventually Fourier transforms get us into the frequency domain where we can start doing some signal processing.

Common misconceptions

• Thinking the function is fully understood only by looking at its values over time.
• Assuming Fourier transforms are required to enter the frequency domain, whereas Fourier series already achieves this for periodic functions.

Watch the explanation

Explore next

• What method does the video say will be used later to find the coefficients?
• Why do people use Fourier series instead of just leaving a function as a square wave?