Impulse Response Example

An example impulse response for the first-order recursive filter

$\displaystyle y(n)$	$\displaystyle =$	$\displaystyle x(n) + 0.9y(n - 1)$	(6.2)
	$\displaystyle =$	$\displaystyle x(n) + 0.9x(n - 1) + 0.9^2 x(n - 2) + \cdots \protect$	(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,

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

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

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

so that

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

where $n\in{\bf Z}$ means

is any integer.

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Implications of Linear-Time-Invariance
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Filter Stability

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