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This module examines the recursive generation of the impulse response of a second order system. This analysis can be used to determine how to generate a cosine or sine function recursively.
The transfer function of a general second order system is given as:
The Direct Form II structure for the implementation of the second order system is shown in the following figure.
The difference equation for the second order system is
To recursively determine the impulse response of the system start with the following:
Then using Equation 3 the impulse response can be found recursively by:
For the values of n >2 the recursive equation reduces to
because the values of $x(n),x(n-1)$ and $x(n-2)$ will all be zero for n >2.
So, if you know the values of $h(0)$ and $h(1)$ , then the Equation 10 can be used to find future values of $h(n)$ .
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