104 7. MULTI-VALUED SWITCHING NETWORK SPECTRA
of the complex scalar encoded values of the function f denoted as jf
c
i. Equation 7.3 expresses
this relationship as used in [39] [56].
js
f
i D C
m
jf
c
i (7.3)
Example 7.2 Chrestenson Spectrum of Scalar Encoded Switching Function
Consider the J
2
(literal selection) gate shown in Figure 5.3. Table 7.2 contains the switching
function model in tabular form for this network element with both scalar switching and scalar
complex-values.
Table 7.2: Scalar ternary logic constants
Input Switching Scalar Complex Scalar
Value Output Value Output Value
0 0 a
0
1 0 a
0
2 2 a
2
Since the number of primary inputs is one, n D 1, the appropriate spectral transformation
matrix is C
1
. e spectrum is calculated as the direct product of the Hermitian (conjugate trans-
pose) of C
1
denoted as C
1
with a column vector jJ
2c
i composed of all possible complex scalar
output values for the J
2
gate. e Hermitian is
C
1
D
2
4
a
0
a
0
a
0
a
0
a
1
a
2
a
0
a
2
a
1
3
5
D
2
4
a
0
a
0
a
0
a
0
a
2
a
1
a
0
a
1
a
2
3
5
e scalar Chrestenson spectrum for the J
2
gate is computed as:
jc
J
2
i D C
1
jJ
2c
i D
2
4
a
0
a
0
a
0
a
0
a
2
a
1
a
0
a
1
a
2
3
5
2
4
a
0
a
0
a
2
3
5
D
2
4
2a
0
C a
2
2a
0
C a
2
a
0
C 2a
1
3
5
7.2.2 CHRESTENSON TRANSFORM OF VECTOR-VALUED SWITCHING
FUNCTIONS
Ternary MVSNs are modeled in the vector space by using three-dimensional canonical basis
vectors to represent each switching constant value. To comport with the method of computing
7.2. CHRESTENSON TRANSFORM 105
the Chrestenson spectrum of scalar-valued functions, we define vector-valued complex constants
that serve as mapped counterparts to vector-valued switching constants in Table 7.3. Because we
define these constants as row vectors, it is necessary to represent them as the conjugate transpose
of their more intuitive column vector definition.
Table 7.3: Vector ternary switching constants
Vector Ternary Values
Switching Complex
h0j D
1 0 0
hc
0
j D
a
0
a
0
a
0
h1j D
0 1 0
hc
1
j D
a
0
a
2
a
1
h2j D
0 0 1
hc
2
j D
a
0
a
1
a
2
e vector Chrestenson spectrum S
f
is calculated using the relationship in Equation 7.3
where the column vector of scalar complex conjugate encodings for the function jf
c
i is replaced
with matrix T
s
. Each row of T
s
is the vector complex encoding of f values, hf
c
j, and T is the
switching domain transfer matrix. By the property of truth table isomorphism, the transfer matrix
T can be viewed as a single column of row vectors where each row vector is the switching vector
encoded truth value of function f , hence Equation 7.3 yields T
s
, the complex vector encoded
form of T . is observation leads to Definition 7.3.
Definition 7.3 Chrestenson Spectral Response Matrix
e Chrestenson spectral response matrix T
s
models the functionality of a MVSN in the
Chrestenson spectral domain. e Chrestenson spectral response matrix is defined in Equation
7.3.
T
s
D T C
n
(7.4)
Using Equations 7.3 and 7.4, the vector Chrestenson spectrum S
f
of the n-input, m-output
MVSN modeled by transfer matrix T is given in Equation 7.5.
S
f
D C
m
T
s
(7.5)
Substituting Equation 7.4 in Equation 7.5, the relationship between the switching domain
transfer matrix and the Chrestenson spectrum of an n-input, m-output MVSN becomes
S
f
D C
m
T C
n
:
106 7. MULTI-VALUED SWITCHING NETWORK SPECTRA
Example 7.4 Vector Chrestenson Spectrum To compute the vector Chrestenson spectrum of the
J
2
gate, we first formulate the vector complex encoded values of J
2
as the matrix J
2s
through the
application of the mapping in Equation 7.4.
J
2s
D .J
2
/.C
1
/ D
2
4
1 0 0
1 0 0
0 0 1
3
5
2
4
a
0
a
0
a
0
a
0
a
2
a
1
a
0
a
1
a
2
3
5
D
2
4
a
0
a
0
a
0
a
0
a
0
a
0
a
0
a
1
a
2
3
5
e calculation of the spectral matrix S
J 2
is then accomplished using Equation 7.5.
S
J 2
D .C
1
/.J
2s
/ D
2
4
a
0
a
0
a
0
a
0
a
2
a
1
a
0
a
1
a
2
3
5
2
4
a
0
a
0
a
0
a
0
a
0
a
0
a
0
a
1
a
2
3
5
D
2
4
.3a
0
/ .2a
0
C a
1
/ .2a
0
C a
2
/
.a
0
C a
1
C a
2
/ .a
0
C 2a
2
/ .2a
0
C a
2
/
.a
0
C a
1
C a
2
/ .2a
0
C a
1
/ .a
0
C 2a
1
/
3
5
Each individual Chrestenson spectral coefficient is a row vector within the S
f
spectral
matrix, and several properties of these coefficients are apparent. e 0
t
h-ordered coefficient is
the topmost row vector of S
f
and the first component of this coefficient is always the real value
r
n
where r is the radix or number of distinct function switching values and n is the number of
primary inputs of the MVSN. Furthermore, the first component of the remaining higher-ordered
spectral coefficients is always zero.
e rightmost component of each spectral coefficient vector is identical to the scalar
Chrestenson spectrum when the MVSN is modeled in the switching algebra domain rather than
the vector space domain.
eorem 7.5 Scalar and Vector Spectrum Relation
e scalar Chrestenson spectrum js
f
i and the rightmost column of the vector Chrestenson spectrum S
f
of a function representing the same ternary MVSN are identical.
Proof. For a given switching function f , the scalar Chrestenson spectrum is given as js
f
i D
C
n
jf
c
i and the vector Chrestenson spectrum is given as S
f
D C
n
T
s
. We define a vector jvi of
length 3
n
composed of 3
n
1 zero-valued components and a single unity-valued component of
the form, hvj D
0 0 : : : 0 1
. e rightmost column of the vector Chrestenson spectrum
can be formed by the product S
f
jvi, yielding:

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