Page 211 - A Practical Guide from Design Planning to Manufacturing

184 Chapter Six

Y Y
A B
Z Z

Y Y
A B
Z Z
Figure 6-5 DeMorgan’s theorem.
A = YZ = Y + Z B = Y + Z = Y&Z

All the two-input functions shown in Fig. 6-4 can be defined for any
number of inputs. It is possible to perform logic design using only some
of the types of gates, but it is often far more intuitive to use all the
types. In fact, DeMorgan’s theorem shows that using inverters, AND
logic can be changed to OR logic or vice versa (Fig. 6-5).
DeMorgan’s theorem points out that inverting any function is accom-
plished by inverting all the terms of the function while changing ANDs
to ORs and ORs to ANDs. This means that any function could be created
using only NAND gates or NOR gates. While performing logic design it
is typically easiest to use more types than this, but during circuit design
DeMorgan’s theorem is often used to pick the most convenient type of
gate. The equation given in Fig. 6-6 shows the effect of repeated appli-
cation of DeMorgan’s theorem.
Implementing any complex logical function begins with writing out
the truth table, the table listing the desired outputs for each possible
combination of inputs. The goal of logic design is to find the logic gates
and connections needed to implement the function. ANDing together dif-
ferent combinations of the inputs and their complements can create func-
tions that are true for only one possible combination of inputs. These are
called fundamental products. By ORing together the needed fundamental
products, any desired function can be realized. A logical function written
in this fashion is called a canonical sum.
Implementing a logical function as a canonical sum is very straight-
forward but may also be very wasteful. It is often possible to simplify the
canonical sum and implement the same function using fewer and sim-
pler logic gates. The canonical sum of the example function in Fig. 6-7
would require 4 three-input gates and 1 four-input gate to implement.

V(WX + YZ ) = V + (WX + YZ ) = V + WX & YZ = V + (W + X )(Y + Z )
Figure 6-6 Applying DeMorgan’s theorem.