Series and Parallel Transfer Functions

The transfer function conveniently captures the
*algebraic structure* of a filtering operation with respect to
*series or parallel combination*:

*Transfer functions of filters in series multiply together*

If the output of filter is given as input to filter (series combination), as shown in Fig.6.1, the overall transfer function is . Thus, the transfer functions of two filters connected in series simply*multiply*together.^{7.1}*Transfer functions of parallel filters sum together*

If two filters and are driven by the*same*input signal, and if their outputs are*summed*, as depicted in Fig.6.2, this is called a*parallel combination*of filters and . The transfer function of the parallel combination is then . This result follows immediately from linearity of the*z*transform.

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