Design of a combinatorial logic circuit which makes it possible to...

Design of a combinatorial logic circuit which makes it possible to complement (in complement to 2) binary numbers represented on 4 bits each. This circuit must generate at output the 2's complement of the binary number present at its input. Answer the following questions :

(i) complete the following truth table. A,B,C,D represent the input binary number to be complemented using 2's complement. W,X,Y,Z represent the two's complement outputs of the input binary numbers. The variable D is the least significant bit and A the most significant bit of the binary number.

(ii) Simplify using the Karnaugh diagram the function W in its Sum of Products form.

iii) Show that the function W can be realized with exclusive-OR gates

(iv) Realize the function Z in its Product of Sums form with a decoder which is built with NAND logic gates (see next figure) and external gates connected to the output of the decoder

(v) Realize the function X with a Multiplexer