3.3. TRANSFER MATRICES OF COMMON SWITCHING CIRCUITS 17
jiihi j is a square Dirac-delta matrix whose i
th
row vector is hij allowing for the previously noted
solution of P
i
to be defined as consisting of null row vectors for all but the i
th
row vector which is
equivalent to hf
i
j. erefore, the projection matrix P
i
D jiihf
i
j. Substituting this result into the
result of eorem 3.5 yields Equation 3.5.
3.3 TRANSFER MATRICES OF COMMON SWITCHING
CIRCUITS
Equation 3.3 may be used to determine the transfer matrices for common switching network
components serving as atomic operators or, logic gates.” Example 3.7 illustrates the calculation
of the transfer matrix A for a two-input AND gate.
Example 3.7 Transfer Matrix for 2-input AND Gate
Consider a 2-input AND gate whose symbol and switching algebra truth table are shown in
Figure 3.2. e four possible input stimuli are h00j, h01j, h10j, and h11j. e projection ma-
trix relationships for the AND gate then become h00jP
0
D h0j, h01jP
1
D h0j, h10jP
2
D h0j, and
h11jP
3
D h1j. From Lemma 3.3, the projection matrices are of the form:
Figure 3.2: Circuit symbol and truth table for binary AND gate.
P
0
D
2
6
6
4
1 0
0 0
0 0
0 0
3
7
7
5
P
1
D
2
6
6
4
0 0
1 0
0 0
0 0
3
7
7
5
P
2
D
2
6
6
4
0 0
0 0
1 0
0 0
3
7
7
5
P
3
D
2
6
6
4
0 0
0 0
0 0
0 1
3
7
7
5
e overall transfer matrix for a two-input AND gate A is given by Equation 3.3.

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