Page 157 - DSP Integrated Circuits
P. 157
142 Chapter 4 Digital Filters
Notice the similarity between the bilinear transformation and Richards' vari-
able. Substituting Richards' variable into the chain matrix yields
The chain matrix in Equation (4.31) has element values that are rational
functions in Richards' variable, except for the square-root factor. Fortunately, this
factor can be handled separately during the synthesis. The synthesis procedures
(programs) used for lumped element design can therefore be used with small mod-
ifications in the synthesis of commensurate-length transmission line filters.
The transmission line fil-
ters of interest are, with a
few exceptions, built using
only one-ports. At this stage
it is therefore interesting to
study the input impedance of
the one-port shown in Figure
4.22. From Equation (4.31)
we get the input impedance Figure 422 Terminated transmission line
of a transmission line, with
characteristic impedance ZQ,
loaded with an impedance Z$.
EXAMPLE 4.7
Determine the input impedance of a lossless transmission line with characteristic
impedance ZQ = R that is terminated by an impedance Z%.
Thus, the input impedance is purely resistive and equals ZQ=R.
Hence, an open-ended unit element can be interpreted as a new kind of capacitor
in the ^-domain [10].
A short-circuited unit element can be interpreted as a f-domain inductor.
