Linear System

Consider the causal, single-input single-output (SISO) system shown in Figure 2. For simplicity, we will take the system to be linear and discrete-time, so that it is characterized by its impulse response $h(n)$ or equivalently its transfer function $H(z)$, which is the $z$ transform of $h(n)$.

$$h(n) \longleftrightarrow H(z)$$ (1)

We will assume that both $h(n)$ and $H(z)$ exist so that we can discuss measuring them interchangeably. We will further assume that $h(n)$ has finite length so that we can measure the response to an input signal in a finite amount of time. The goal of this document is to explain how to excite the system with a signal $s(n)$, measure the response $r(n)$, and use $s(n)$ and $r(n)$ to determine $h(n)$ (and equivalently $H(z)$). In particular, it is useful to pick a signal $s(n)$ that contains a large amount of energy so that measurement noise will not significantly corrupt the measurement results.

Figure 2: Linear system