Impulse Response Example

An example impulse response for the first-order recursive filter

$$y(n) = x(n) + 0.9y(n - 1)$$ (6.2)

$$= x(n) + 0.9x(n - 1) + 0.9^2 x(n - 2) + \cdots \protect$$ (6.3)

is shown in Fig.5.2b. The impulse response is a sampled exponential decay, $(1,\, 0.9,\, 0.81,\, 0.73,\,\ldots)$ , or, more formally,

$$h(n) = \left\{\begin{array}{ll} (0.9)^n, & n\geq 0 \\ [5pt] 0, & n<0. \\ \end{array} \right.$$

We can more compactly represent this by means of the unit step function,

$$u(n) \isdef \left\{\begin{array}{ll} 1, & n\geq 0 \\ [5pt] 0, & n<0 \\ \end{array} \right.,$$

so that

$$h(n) = u(n)(0.9)^n, \quad n\in\mathbb{Z}$$

where $n\in\mathbb{Z}$ means $n$ is any integer.