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Introduction to Digital Filters
    Transfer Function Analysis
       Series and Parallel Transfer Functions
          Series Case

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Series Case

Figure 6.1: Series combination of transfer functions $ H_1(z)$ and $ H_2(z)$ to produce the combined transfer function $ H(z)=H_1(z)H_2(z)$.
\begin{figure}\input fig/series.pstex_t \end{figure}

Figure 6.1 illustrates the series connection of two filters $ H_1(z)=V(z)/X(z)$ and $ H_2(z)=Y(z)/V(z)$. The output $ v(n)$ 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). $

In summary, if the output of filter $ H_1(z)$ is given as input to filter $ H_2(z)$ (a series combination), as shown in Fig.6.1, the overall transfer function is $ H(z)=H_1(z)H_2(z)$--transfer functions of filters connected in series multiply together.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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