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Impulse Response

The impulse response of a digital filter is the output arising from the unit impulse sequence defined as

$$
\delta(n) = \bigg\{{\textstyle{1, \;\; n = 0,}\atop{0, \;\; n\ne0.}}
$$

You can generate an impulse sequence a number of ways; one straightforward way is

imp = [1; zeros(49,1)];

The impulse response of the simple filter with $b = 1$ and $a = [1\, {-0.9}]$ is $h(n) = 0.9^n$, which decays exponentially.

b = 1;
a = [1 -0.9];

h = filter(b,a,imp);

stem(0:49,h)

A simple way to display the impulse response is with the Filter Visualization Tool, fvtool.

fvtool(b,a)

Click the Impulse Response button, $[\uparrow]$, on the toolbar, select Analysis > Impulse Response from the menu, or type the following code to obtain the exponential decay of the single-pole system.

fvtool(b,a,'Analysis','impulse')

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