BCD to HEX Decoder $($ROM$)$:

Inputs: $4-$bit BCD $($Binary$-$Coded Decimal$)$ input lines $(0$ to $9$ in binary$).$

Outputs: $7$ outputs $($a to g$)$ for the $7-$segment display segments.

Address lines: $4$ bits for $16$ possible BCD codes $(0000$ to $1001$ in binary$).$

Connection to $7-$Segment Display:

Connect the $7$ outputs of the ROM to the corresponding segments $($a to g$)$ of the $7-$segment common cathode display.

Decoder Block:

Draw a block labeled "Decoder" that represents the BCD to HEX decoding process.

Connections:

Connect the BCD input lines to the input of the BCD to HEX decoder.

Connect the output lines of the BCD to HEX decoder to the $7-$segment display.

The ROM stores the truth table and provides the appropriate outputs based on the BCD input, thus lighting up the correct segments to display the corresponding HEX code on the $7-$segment display.