62 6. BINARY SWITCHING NETWORK SPECTRA

Figure 6.3: Butterﬂy diagram for Example 6.1 calculation.

6.4.2 WALSH TRANSFORM FOR VECTOR-VALUED SWITCHING

FUNCTIONS

When a switching network is modeled in the vector space, the Walsh transform can also be applied

as in the previous section. For the r D 2 binary case, the values h0jand h1j can be mapped to values

along the unit circle in an analogous manner to that of the scalar-valued case. is mapping is

obtained by applying the H

1

transformation to each value so that it may be represented in the

spectral domain.

h0jH

1

D

1 0

1 1

1 1

D

1 1

h1jH

1

D

0 1

1 1

1 1

D

1 1

e vector space model also allows for the values ht j and h¿j. ese values in the Walsh

domain are expressed as follows.

6.4. WALSH TRANSFORM 63

h¿jH

1

D

0 0

1 1

1 1

D

0 0

htjH

1

D

1 1

1 1

1 1

D

2 0

In this manner, the Walsh transform matrix H

1

can be viewed as the collection of row

vectors representing the spectral values corresponding to h0j and h1j. e Hasse diagram of the

vector-valued constants in the switching domain and the Walsh spectral domain are given in

Figure 6.4.

Figure 6.4: Hasse diagram of values in the switching and Walsh spectral domain for r D 2.

e Walsh spectrum of a switching network when modeled in the vector space can analo-

gously be computed as in the case of the more conventional scalar model described in the previous

section. A column vector of all possible output values can be formulated and multiplied with the

Walsh transform matrix. However, each component of the column vector is also a row vector

since a row vector corresponds to each distinct output response. erefore, the column vector

containing all network output responses is itself a matrix, and this matrix is exactly the transfer

matrix T

f

characterizing the switching network. e transfer matrix row vectors are mapped to

corresponding values as given in the Hasse diagram in Figure 6.4 for consistency with the deﬁ-

nition.

eorem 6.2 Walsh Spectrum of Switching Network

e Walsh spectrum of switching network modeled in the vector space is equivalent to the product of the

mapped transfer matrix T

s

and the Walsh transformation matrix H

n

.

Proof. By deﬁnition, the spectrum of a switching function is the product of the Walsh transfor-

mation matrix with a column vector whose components are all possible function values. By the

property of truth table isomorphism, the transfer matrix characterizing a switching network con-

sists of row vectors that are all possible output responses of the network. By the deﬁnition of the

spectrum of network, the spectrum of the network is

64 6. BINARY SWITCHING NETWORK SPECTRA

S

f

D H

n

T

s

:

Example 6.3 Walsh Spectrum of Example Function

Consider the single output switching characterized by the truth table that contains both the scalar

model and vector model values.

x

2

x

1

x

0

f hf j

0 0 0 1

0 1

0 0 1 0

1 0

0 1 0 1

0 1

0 1 1 0

1 0

1 0 0 1

0 1

1 0 1 1

0 1

1 1 0 0

1 0

1 1 1 1

0 1

From the deﬁnition of the Walsh spectrum, a column vector is formed with components

equivalent to the truth table values of the function hf j. is column vector is equivalent to the

transfer matrix characterizing hf j.

T

f

D

2

6

6

6

6

6

6

6

6

6

6

6

4

0 1

1 0

0 1

1 0

0 1

0 1

1 0

0 1

3

7

7

7

7

7

7

7

7

7

7

7

5

e mapped version of the transfer matrix is denoted as T

s

:

6.4. WALSH TRANSFORM 65

T

s

D

2

6

6

6

6

6

6

6

6

6

6

6

4

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

3

7

7

7

7

7

7

7

7

7

7

7

5

e Walsh spectrum is then computed as S

f

D H

f

T

s

:

S

f

D

2

6

6

6

6

6

6

6

6

6

6

6

4

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

3

7

7

7

7

7

7

7

7

7

7

7

5

2

6

6

6

6

6

6

6

6

6

6

6

4

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

3

7

7

7

7

7

7

7

7

7

7

7

5

D

2

6

6

6

6

6

6

6

6

6

6

6

4

8 2

0 2

0 2

0 2

0 C2

0 6

0 C2

0 C2

3

7

7

7

7

7

7

7

7

7

7

7

5

eorem 6.2 leads to the following deﬁnition.

Deﬁnition 6.4 Spectral Response Matrix

e spectral response matrix T

s

characterizes a switching network in the spectral domain and is

equivalent to the transfer matrix T with each row vector mapped to their corresponding spectral

representations.

Each spectral coeﬃcient can be calculated individually through a multiplicative operation

with a speciﬁc row vector. e row vector represents a switching network input stimulus expressed

where each individual primary input value is expressed in the spectral domain. If the input stim-

ulus is represented in the switching domain, the spectral response matrix yields a corresponding

output response with each primary output expressed in the spectral domain. Example 6.5 illus-

trates these two uses of the spectral response matrix.

Example 6.5 Spectral Response Matrix Calculations

Consider the example single-output switching network characterized by the following truth table.

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