Page 163 - DSP Integrated Circuits

148 Chapter 4 Digital Filters

while for #2 = °° we get a = -1 and the incident wave at port 1 is reflected without
a change of sign.

Series Adaptors
Ports are considered to be connected in series if the same current enters all ports
while the ports have the same voltage in a parallel connection. A series connection
of N ports corresponds to an adaptor described by [8,10]

where

and

Figure 4.33 shows the symbol for the two-port
series adaptor. In this case, each port has its own adap-
tor coefficient. However, the coefficients are linearly
dependent since, according to Equation (4.47), we have
Figure 4.33 Two-port
series adaptor
Hence, only one coefficient and
therefore only one multiplication are
required. The other coefficient can be
expressed in terms of the first. The port
for which the adaptor coefficient has
been eliminated is called the dependent
port. The resulting wave-flow graph,
where 0:2 has been eliminated, is shown
in Figure 4.34. In practice, only two-,
three- and four-port series and parallel
adaptors are used. Arbitrarily large
Figure 4.34 Wave-flow graph for a
interconnection networks can be built
two-port series adaptor
using these adaptors only [8, 10]. The
symbol for the three-port series adaptor
is shown in Figure 4.35.
A general three-port series adaptor
At ~l L. A
J1
requires two multiplications and six l * «« f\2
additions. In some cases it is possible to *1 -M- R 2
' !-»&.
'
select one of the adaptor coefficients as 1. Bi^-l ixr
This means that an incident wave to this
port will not be reflected. Hence, this
A 3 B 3
port is called a reflection-free port. The
number of multiplications is reduced to Figure 4.35 Three-port series adaptor