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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 $ n$ is any integer.


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