3.12 Parallel and cascaded filters

In some applications, such as graphic equalizers, it is useful to place filters in parallel as shown in [link] . Can the parallel combination of filters be characterized by a single equivalent filter $h_{eq}\left(t\right)$ ? The answer is yes and results by noting that

Therefore, the last equation in [link] shows that

The equivalent transfer function for the parallel filter structure is given by

Next we wish to find an equivalent filter for the cascaded structure shown in [link] .

This can be done by finding an expression for the intermediate value $y_1\left(t\right)$ :

The output of the cascaded structure is given by

substituting [link] into [link] gives

Reversing the order of integration and rearranging slightly gives

Now let $\xi =\gamma +\tau$ , solving for $\tau$ gives $\tau =\xi -\gamma$ and $d\xi =d\tau$ . Substituting these quantities into [link] leads to

Notice that we can factor $x(t-\xi )$ from the inner integral since $x(t-\xi )$ does not depend on $\gamma$ . The integral in the brackets is recognized as $h_1\left(t\right)*h_2\left(t\right)$ evaluated at $\xi$ . Therefore for the cascaded system, the equivalent impulse response is given by

This can be generalized to any number of cascaded filters giving