74 6. BINARY SWITCHING NETWORK SPECTRA

Likewise, the spectral coeﬃcient hs

10f

2

j corresponding to netlist output f

2

is computed

using the spectral output response matrix for f

2

only denoted as T

sf

2

.

hs

10f

2

j D h10jH

2

T

sf

2

D

0 0 1 0

2

6

6

4

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

3

7

7

5

2

6

6

4

1 1

1 1

1 1

1 1

3

7

7

5

D

1 1 1 1

2

6

6

4

1 1

1 1

1 1

1 1

3

7

7

5

D

0 2

6.4.4 COMPUTING WALSH SPECTRAL COEFFICIENTS FROM A NETLIST

e concept of a spectral output response matrix and its use in calculating the Walsh spectral

coeﬃcients and the network output response were developed using monolithic explicit matrices

in the previous section. However, the method of performing the calculation with the spectral

response matrix in distributed factored form is possible and provides a useful technique for deter-

mining spectral coeﬃcients using the vector space model that has not been shown to be possible

with the traditional switching algebra models. e signiﬁcance of the preceding results is that a

method for computing a single spectral coeﬃcient with a simple traversal of an unaltered netlist

is available. ere is no required extraction of a switching function representation, thus all the

problems with intermediate data structures for representation of switching functions and spectra

are avoided. ese problems were the source of one of the most prohibiting factors in utilizing

spectral methods in modern EDA tasks.

e process of computing spectral coeﬃcients through netlist traversals is illustrated using

the example netlist in Figure 6.5. When the netlist transfer matrix is given symbolically in dis-

tributed factored form, it may be graphically represented as shown in Figure 6.7a. Likewise, the

spectral response matrix may be represented in distributed factored form by the diagram in Fig-

ure 6.7b. e spectral response matrix is deﬁned as shown in Figure 6.7b to provide a matrix that

can be used to calculate both the output response due to a stimulus expressed as a canonical basis

vector, or to compute a spectral coeﬃcient by ﬁrst transforming the input stimulus vector to the

Walsh domain. However, if it is desired to compute the spectral coeﬃcients only, the distributed

form in Figure 6.7c may be more convenient since it allows the input stimulus to be represented

as a canonical basis vector and the corresponding output response is then the spectral response.

As an example of computing a spectral coeﬃcient through a traversal of the netlist, we

will utilize the form in Figure 6.7c to compute the spectral coeﬃcients hs

10f

1

j and hs

10f

2

j. ese

6.4. WALSH TRANSFORM 75

Figure 6.7: Distributed factored form a) transfer matrix b) Walsh spectral response matrix.

are the same spectral coeﬃcients that were computed using the explicit monolithic forms of the

matrices in Example 6.11.

Example 6.12 Walsh Coeﬃcient Computation through Netlist Traversal

To compute spectral coeﬃcients through the traversal of a netlist, each netlist gate is described

with its corresponding transfer matrix. Each interconnecting line is then annotated with values

proceeding from the primary inputs to the primary outputs. Initially, a value is assigned to the

primary inputs that correspond to the desired spectral coeﬃcient. Since we are interested in the

spectral coeﬃcients hs

10f

1

j and hs

10f

2

j, we make initial assignments of hx

1

j D h1jand hx

2

j D h0j.

ese values are propagated toward the primary outputs and they are appropriately transformed

as operator transfer matrices are encountered. When a matrix is encountered that has two or more

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