Implement the logic circuit in Figure 4.23 using NOR gates only.Figure 4.23 Logic circuit...

Implement the logic circuit in Figure 4.23 using NOR gates only.

Figure 4.23 Logic circuit for Example 4.6.

Example 4.6

Consider the minimum-cost sum-of-products expression

and assume that the inputs x₁ to x₄ are available only in their true form. Then the expression defines a circuit that has four AND gates, one OR gate, two NOT gates, and 18 inputs (wires) to all gates. The fan-in is three for the AND gates and four for the OR gate. The reader should observe that in this case we have included the cost of NOT gates needed to complement x₁ and x₂, rather than assume that both true and complemented versions of all input variables are available, as we had done before.

Factoring x₃ from the first two terms and x₄ from the last two terms, this expression becomes

Now let g(x₁, x₂) = ₁x₂ + x₁₂ and observe that

Then f can be written as

which leads to the circuit shown in Figure 4.23. This circuit requires an additional OR gate and a NOT gate to invert the value of g. But it needs only 15 inputs. Moreover, the largest fan-in has been reduced to two. The cost of this circuit is lower than the cost of its two-level equivalent. The trade-off is an increased propagation delay because the circuit has three more levels of logic.

Figure 4.23 Logic circuit for Example 4.6.

In this example the subfunction g is a function of variables x₁ and x₂. The subfunction is used as an input to the rest of the circuit that completes the realization of the required function f. Let h denote the function of this part of the circuit, which depends on only three inputs: g, x₃, and x₄. Then the decomposed realization of f can be expressed algebraically as

The structure of this decomposition can be described in block-diagram form as shown in Figure 4.24.

Figure 4.24 The structure of decomposition in Example 4.6.