Fundamental Theorem of Linear Systems

Theorem: Let a linear time-invariant system $g$ has an $e_f(t)$ denote the complex sinusoid $e^{j 2 \pi ft}$. Then, $g(e_f(.),t)=g(e_f(.),0)e_f(t)=c \, e_f(t)$.

Example: Analog RC Lowpass Filter

Figure 1: A First-Order Analog Lowpass Filter

The impulse response for the circuit in Fig. 1, i.e. the output measured at $y(t)$ when $x(t) = \delta(t)$, is

$$h(t)=\frac{1}{RC}e^{-\frac{1}{RC}t}u(t)$$

For a complex sinusoidal input, $x(t)=e_f(t)=e^{j2\pi ft}$,

$$y(t) = \int_{-\infty}^\infty x(t-\lambda)h(\lambda)\,d\lambda$$

$$= \int_{-\infty}^\infty e^{j2\pi f(t-\lambda)}\frac{1}{RC}e^{-\frac{1}{RC}\lambda}u(\lambda)\,d\lambda$$

$$= e^{j2\pi ft} \left[ \frac{1}{RC}\int_{-\infty}^\infty e^{-j2\pi f\lambda}e^{-\frac{1}{RC}\lambda}\,d\lambda \right]$$

$$= \left[ \frac{\frac{1}{RC}}{{j2\pi f+\frac{1}{RC}}} \right] e^{j2\pi ft}$$

$$= g(e_f(.),0) \, e_f(t)$$

So, $g(e_f(.),0)= H(f)$, which is the transfer function of the system.