Digital Logic · Logic minimization and hazards

A 4-bit input ABCD is interpreted as an unsigned binary integer N=8A+4B+2C+D

Problem

A 4-bit input $ABCD$ is interpreted as an unsigned binary integer $N=8A+4B+2C+D$. Two functions are defined from $N$. - $F_{\mathrm{BCD}}$ is an incompletely specified function of a binary-coded-decimal digit: $F_{\mathrm{BCD}}=1$ if $N\in\{1,3,5,7,9\}$, $F_{\mathrm{BCD}}=0$ if $N\in\{0,2,4,6,8\}$, and $N\in\{10,11,12,13,14,15\}$ are don’t-cares. - $F_{\mathrm{full}}$ is the fully specified function that is $1$ if and only if $N\in\{1,3,5,7,9\}$ and is $0$ for every other 4-bit combination, including $10$ through $15$. Use variable order $ABCD$ ($A$ most significant) and a 4-variable Karnaugh map with rows $AB$ and columns $CD$ in Gray-code order $00,01,11,10$. Minimality means fewest product terms, then fewest total literals. (a) Obtain a minimal SOP for $F_{\mathrm{BCD}}$ and state the don’t-care assignments it uses. (b) Obtain a minimal SOP for $F_{\mathrm{full}}$. (c) State whether the two minimal expressions are identical. If they are not, name one minterm index at which they differ.

Hint

Invalid BCD codes may be used as $1$s for $F_{\mathrm{BCD}}$ but are hard zeros for $F_{\mathrm{full}}$.

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