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

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Parallel Case

Figure 6.2: Parallel combination of transfer functions $ H_1(z)$ and $ H_2(z)$, yielding $ H(z)=H_1(z)+H_2(z)$.
\begin{figure}\input fig/parallel.pstex_t \end{figure}

Figure 6.2 illustrates the parallel combination of two filters. The filters $ H_1(z)$ and $ H_2(z)$ are driven by the same input signal $ x(n)$, and their respective outputs $ y_1(n)$ and $ y_2(n)$ 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). $

where we needed only linearity of the z transform to have that $ {\cal Z}\{y_1+y_2\} = {\cal Z}\{y_1\}+{\cal Z}\{y_2\}$.


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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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