An electronic game uses an array of seven LEDs (light-emitting diodes) to display the resu...

An electronic game uses an array of seven LEDs (light-emitting diodes) to display the results of a random roll of a die. A decoder is to be designed to illuminate the appropriate diodes for the display of each of the six die values. The desired display patterns are shown in Figure.

Figure Patterns for Dice for Problem

(a) Use a 3–to–8-line decoder and OR gates to map the 3-bit combinations on inputs X2, X1, and X0 for values 1 through 6 to the outputs a through g. Input combinations 000 and 111 are don’t-cares.

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(b) Note that for the six die sides, only certain combinations of dots occur. For example, dot pattern A = 5d 6 and dot pattern B = 5a, g 6 can be used for representing input values 1, 2, and 3 as {A}, {B}, and {A, B}. Define four dot patterns A, B, C, and D, sets of which can provide all six output patterns. Design a minimized custom decoder that has inputs X2, X1, and X0 and outputs A, B, C, and D, and compare its gate-input cost to that of the 3-to-8 decoder and OR gates in part (a).