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Frequency Domain Interpretation

The equation for the inductor, assuming zero initial conditions, transforms to

\begin{eqnarray*}
V(s)=LsI(s)
\end{eqnarray*}

where $ s$ is the complex frequency variable. Taking $ z$ transforms of the sequences $ v$ and $ i$ in the backward-difference scheme yields:

\begin{eqnarray*}
V(z^{-1})=L\frac{1-z^{-1}}{T}I(z^{-1})
\end{eqnarray*}

Thus we can think of our discretized scheme as one obtained under the mapping $ s\to \frac{1-z^{-1}}{T}$ . So here we are mapping from the $ s$ plane to the $ z$ plane. The following figure illustrates where real continuous time frequencies (the $ j\omega$ axis) are mapped:

\epsfbox{eps/freqmap.eps}

dc ($ s=0$ ) mapped to dc ($ z=1$ )

infinite frequency mapped to ($ z=0$ )

We can write the scheme for the trapezoid rule as follows:

\begin{eqnarray*}
V(z^{-1})=\frac{2L}{T}\frac{1-z^{-1}}{1+z^{-1}}I(z^{-1})
\end{eqnarray*}


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``Discrete-Time Lumped Models'', by Stefan Bilbao and Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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