Priority encoder; "e" and "f" segments

Priority encoder

A priority encoder is an encoder that gives the binary code of the highest-priority active input when more than one input is active at the same time. It also has a valid output $V$ that shows at least one input is active. Example: in a 4:2 priority encoder with $D_3$ highest, if $D_1$ and $D_2$ are both 1, the output is $10$ (code of $D_2$). The 74148 is an 8:3 priority encoder used for interrupt handling and keyboard encoding.

Segment "e"

Segment e is ON for digits 0, 2, 6, 8 (inputs 10–15 are don't-cares).

$$e = \Sigma m(0,2,6,8) + d(10\text{–}15)$$

Groups: corners (0, 2, 8, 10) give $B'D'$; column 10 (2, 6, 10, 14) gives $CD'$.

$$e = B'D' + CD' = D'(B' + C)$$

 B --[NOT]--+
            +--[OR]--+
 C ---------+        +--[AND]--- e
 D --[NOT]-----------+

Segment "f"

Segment f is ON for digits 0, 4, 5, 6, 8, 9.

$$f = \Sigma m(0,4,5,6,8,9) + d(10\text{–}15)$$

Groups: rows 11 and 10 give $A$; column 00 gives $C'D'$; cells 4, 5, 12, 13 give $BC'$; cells 4, 6, 12, 14 give $BD'$.

$$f = A + C'D' + BC' + BD'$$

 C' -+-[AND]-- C'D' --+
 D' -+                |
 B  -+-[AND]-- BC' ---+
 C' -+                +--[OR 4]--- f
 B  -+-[AND]-- BD' ---+
 D' -+                |
 A  ------------------+

Both circuits are checked against the digit table: e.g. for digit 6 ($0110$): $e = 0 + 1\cdot1 = 1$ and $f = 0 + 0 + 0 + 1 = 1$, both segments lit as required.