13

C H A P T E R 3

Switching Network Transfer

Functions

3.1 SWITCHING NETWORK MODELS

Switching networks are conventionally modeled using a set of Boolean switching functions where

each of the m primary network outputs are modeled with separate switching functions. We pro-

pose to model such networks as the transformation of a vector hxj 2 H

n

representing the network

input stimulus to a corresponding vector hf j 2 H

m

and the speciﬁc functionality of the network

with a transformation matrix F . us a switching network is modeled as the mathematical map-

ping of a vector from one Hilbert space to another, F W H

n

! H

m

. Figure 3.1 contains a concep-

tual diagram of a switching network modeled conventionally with Boolean switching algebra on

the left and with the proposed vector space model on the right.

Figure 3.1: Conceptual models of switching networks.

e set of switching functions ff

1

; f

2

; : : : ; f

m

g are symbolically denoted using operators

from a Boolean algebra such as hB; C; ; N; 0; 1i where the atomic network elements or “logic

gates” correspond to algebraic operators or expressions involving the operators. For example, an

AND gate corresponds to the multiplicative operator denoted by and the exclusive-OR gate

corresponds to the expression x Ny C Nx y. e primary inputs x

i

are switching variables over the

set B as are the individual network primary outputs denoted as f

i

. A speciﬁc input stimulus vector

may be denoted as an n-dimensional vector whose components are each x

i

2 B, or alternatively,

as a single element from B

n

. Likewise, the output response vector can be considered to be an

element of B

m

.

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