Impulse Response Example

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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.

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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