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

Get Modeling Digital Switching Circuits with Linear Algebra now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.