Module 1: The Fourier Transform

\[ F(\omega) = \int_{-\infty}^{\infty} f(t)e^{-j\omega t} \, d\tau \]

Example: RC low-pass filter

Here is the circuit we'll analyse:

Figure 1: Series RC low-pass filter used in this module.

Figure 2: Series RL high-pass filter used in this module.

We will derive the transfer function

\[ H(s) = \frac{V_o(s)}{V_s(s)} = \frac{R}{R + \frac{1}{sC}}. \]