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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