De-Multiplexer

A de-multiplexer (DEMUX) takes a single data input and routes it to one of 2^n output lines, chosen by n select lines. It is the exact inverse of a multiplexer.

1. 1-to-4 De-Multiplexer

   Y0 = D . S1' . S0'
   Y1 = D . S1' . S0
   Y2 = D . S1  . S0'
   Y3 = D . S1  . S0

   Hardware: 2 inverters + 4 three-input AND gates

The unselected outputs are forced to 0 — only one line carries the data at any instant.

2. 1-to-8 De-Multiplexer

   Y(i) = D . (minterm i of S2 S1 S0)

   Y0 = D.S2'S1'S0'      Y4 = D.S2 S1'S0'
   Y1 = D.S2'S1'S0       Y5 = D.S2 S1'S0
   Y2 = D.S2'S1 S0'      Y6 = D.S2 S1 S0'
   Y3 = D.S2'S1 S0       Y7 = D.S2 S1 S0

3. DEMUX and Decoder — the same silicon

A 1-to-2^n de-multiplexer is identical to an n-to-2^n decoder with an enable input, where the DEMUX data line plays the role of the decoder's enable.

   Decoder view :  select lines = the code inputs, D = ENABLE
                   With D = 1, exactly one output goes high -> pure decoding.

   DEMUX view   :  select lines = the address, D = the data being routed.

   Same circuit, two names, chosen by what you connect to D.

This is why manufacturers sell one chip (e.g. 74138) labelled "3-to-8 decoder / 1-to-8 demultiplexer".

4. Cascading

   Build a 1-to-16 DEMUX from two 1-to-8 DEMUXes:

      Data D  -> both chips' data inputs
      S2 S1 S0 -> both chips' select inputs
      S3       -> ENABLE of chip 1 (active when S3 = 0)
                  ENABLE of chip 2 (active when S3 = 1)

   S3 chooses which half is live; S2..S0 choose the line within that half.

5. Implementing Boolean Functions with a DEMUX/Decoder

Because each output is one minterm, a DEMUX plus an OR gate realises any function:

   F(A,B,C) = Σm(1, 3, 5, 7)

   Feed A B C to the select lines, tie D = 1.
   F = Y1 + Y3 + Y5 + Y7    (one 4-input OR gate)

   For the COMPLEMENT, OR the remaining outputs, or use NAND outputs:
   F' = Y0 + Y2 + Y4 + Y6

With active-LOW decoder outputs (the common case) use a NAND gate instead of an OR gate — the result is the same function.

6. MUX vs DEMUX

7. Where MUX + DEMUX Are Used Together

This is time-division multiplexing — four channels share one physical wire because the MUX and DEMUX step through the same select sequence in lock-step. Telephone trunks, serial buses and memory data paths all use this structure.

8. Applications

Next: the decoder, which is this same circuit used for its most important CPU role — address decoding.