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Non-recirculating filter, second form

Occasionally we will use a variant of the filter above, as shown in Figure 8.10, called the elementary non-recirculating filter, second form. Instead of multiplying the delay output by $Q$ we multiply the direct signal by its complex conjugate $\overline{A}$. (If

\begin{displaymath}
A = a+bi = r{e^{i\alpha}}
\end{displaymath}

is any complex number, its complex conjugate is defined as:

\begin{displaymath}
\overline{A} = a-bi = r{e^{-i\alpha}}
\end{displaymath}

Graphically this flips the entire complex plane across the real axis.) The transfer function of the new filter is

\begin{displaymath}
H(Z) = \vert\overline{Q} - {Z^{-1}}\vert
\end{displaymath}

This gives rise to the same frequency response as the first form since

\begin{displaymath}
\vert\overline{Q} - {Z^{-1}}\vert = \vert Q - \overline{Z^{-1}}\vert = \vert Q- Z\vert
\end{displaymath}

Here we use the fact that $\overline{Q} = {Z^{-1}}$, for all unit complex numbers $Q = {e^{i\omega}}$, since

\begin{displaymath}
\overline{Q} = {e^{-i\omega}} = {Q^{-1}}
\end{displaymath}

Although the two forms of the elementary non-recirculating filter have the same frequency response, their phase responses are different, and we will occasionally use the second form for its phase response.

Figure 8.10: The elementary non-recirculating filter, second form.
\begin{figure}\psfig{file=figs/fig08.10.ps}\end{figure}


next up previous contents index
Next: Elementary recirculating filter Up: Designing filters Previous: Elementary non-recirculating filter   Contents   Index
Miller Puckette 2006-03-03