Series and Parallel Transfer Functions
The transfer function conveniently captures the algebraic structure of a filtering operation with respect to series or parallel combination. Specifically, we have the following cases:
- Transfer functions of filters in series multiply together.
- Transfer functions of filters in parallel sum together.
Series Case
![]() |
Figure 6.1 illustrates the series connection of two
filters
and
.
The output
from filter 1 is used as the input to filter 2.
Therefore, the overall transfer function is
![$\displaystyle H(z) \isdefs \frac{Y(z)}{X(z)}
\eqsp \frac{H_2(z)V(z)}{X(z)}
\eqsp H_2(z)H_1(z).
$](http://www.dsprelated.com/josimages_new/filters/img680.png)
![$ H_1(z)$](http://www.dsprelated.com/josimages_new/filters/img34.png)
![$ H_2(z)$](http://www.dsprelated.com/josimages_new/filters/img35.png)
![$ H(z)=H_1(z)H_2(z)$](http://www.dsprelated.com/josimages_new/filters/img36.png)
Parallel Case
Figure 6.2 illustrates the parallel combination of two
filters. The filters and
are driven by the
same input signal
, and their respective outputs
and
are summed. The transfer function of the parallel
combination is therefore
![$\displaystyle H(z) \isdefs \frac{Y(z)}{X(z)} \eqsp \frac{Y_1(z) + Y_2(z)}{X(z)}
\eqsp \frac{Y_1(z)}{X(z)} + \frac{Y_2(z)}{X(z)} \isdefs H_1(z)+H_2(z).
$](http://www.dsprelated.com/josimages_new/filters/img683.png)
![$ {\cal Z}\{y_1+y_2\} = {\cal Z}\{y_1\}+{\cal Z}\{y_2\}$](http://www.dsprelated.com/josimages_new/filters/img684.png)
Series Combination is Commutative
Since multiplication of complex numbers is commutative, we have
![$\displaystyle H_1(z)H_2(z)=H_2(z)H_1(z),
$](http://www.dsprelated.com/josimages_new/filters/img685.png)
By the convolution theorem for z transforms, commutativity of a product of transfer functions implies that convolution is commutative:
![$\displaystyle h_1 \ast h_2
\;\leftrightarrow\;
H_1\cdot H_2
\;=\;
H_2\cdot H_1
\;\leftrightarrow\;
h_2 \ast h_1
$](http://www.dsprelated.com/josimages_new/filters/img686.png)
Next Section:
Partial Fraction Expansion
Previous Section:
Factored Form