Digital Logic · Latches, flip-flops, and sequential circuits

A 4-to-2 priority encoder has request inputs I_3,I_2,I_1,I_0, binary outputs A_1,A_0…

Problem

A 4-to-2 priority encoder has request inputs $I_3,I_2,I_1,I_0$, binary outputs $A_1,A_0$, and a valid flag $GS$. Input $I_3$ has the highest priority and $I_0$ the lowest. The contract, for every one of the sixteen input words, is: - $GS=1$ if and only if at least one request is $1$; - if $GS=1$, then $A_1A_0$ is the binary index of the highest-priority request that is $1$; - if $GS=0$, then $A_1A_0=00$. (a) Write sum-of-products expressions for $GS$, $A_1$, and $A_0$ that realize this contract on all sixteen input words. (b) Evaluate the triple $(A_1A_0,GS)$ for each of the request vectors $I_3I_2I_1I_0\in\{0000,0001,0010,0100,1000,1010,0110\}$. (c) A proposed implementation uses $A_1=I_3+I_2$, $A_0=I_3+I_1$, and $GS=I_3+I_2+I_1+I_0$. Identify every vector among the seven listed in (b) for which this proposal violates the contract, and state the incorrect $(A_1A_0,GS)$ it produces on each such vector.

Hint

Priority means a higher request hides every lower request: write each output bit as “the winner’s index has this bit set,” not as a plain OR of all requests that would set it.

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