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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