Decoders

Start here A hotel has 8 rooms and a 3-digit binary room code. When the code 101 comes in, only room 5’s light should turn on, and every other light stays off. A circuit that does this is called a decoder: it turns a code into “exactly one line on”.

• n inputs → 2ⁿ outputs.
• For each input pattern, exactly one output is 1, and all the rest are 0.
• The output that turns on is the one whose number matches the input.

Names: a decoder with 2 inputs is a 2:4 decoder (or 2-to-4). With 3 inputs it’s a 3:8 decoder. With 4 inputs it’s a 4:16 decoder.

Equations: $D_{0} = \overline{A} \overline{B} , D_{1} = \overline{A} B , D_{2} = A \overline{B} , D_{3} = A B$.

Key idea: each decoder output is a minterm. $D_{k} = m_{k}$.

Lecture example — $A = 1 , B = 0$ Input pattern 10 is the number 2, so only $D_{2} = 1$. The outputs are 0, 0, 1, 0.

• Memory addresses: the processor sends a row number; a decoder switches on that one row of memory.
• Instructions: the processor reads an instruction’s opcode (the part that says which operation); a decoder turns on the one circuit that does that operation.

Since every decoder output is a minterm, you can make any function of n variables:

1. Connect the variables to the decoder inputs.
2. Find the minterms where $F = 1$.
3. OR those outputs together.

Lecture exercise — $Y = A + B + C + D$ Y is 1 in every row except row 0 (0000). So $Y = D_{1} + D_{2} + \ldots + D_{15}$ (an OR of 15 outputs). Smarter: $Y =$ $\operatorname{NOT} D_{0}$. One inverter on $D_{0}$ does the same job.

Trick: if F has many 1s, use the 0s instead: $F = \operatorname{NOT}$(OR of the 0-row outputs). That is a NOR gate.

Inside a decoder, each input goes through a NOT (for the barred letters), then an AND gate. So the longest path is:

delay = NOT delay + AND delay

• 4:16 decoder with $\operatorname{NOT} = 1$ ns and 4-input $\operatorname{AND} = 2$ ns → 1 + 2 = 3 ns.
• 3:8 decoder built from NOT and 2-input ANDs (all 1 ns): a 3-input AND needs two 2-input ANDs in a chain. So 1 + 1 + 1 = 3 ns.

In one line: A decoder has n inputs and 2ⁿ outputs; exactly one output is 1, and each output is a minterm, so decoder + OR gate can build any function. Exam words: decoder, 2:4, 3:8, 4:16, one-hot output, minterm, memory address, opcode, decoder delay.