Logic Gates Puzzle 11

There are three input bits A₀, A₁ and A₂, but eight output bits X₀-X₇.

Create a truth table with all input combinations and outputs. Do you see the pattern now?

This digital circuit is a binary decoder, more specific a 3-to-8 line decoder or one-hot binary decoder. For each binary input number, it will set exactly one output to high, while all other outputs stay low.

See these Wikipedia pages for details: Binary Decoder, One-Hot.

It has many uses:

• As multiplexer: E.g. one of eight memory chips is enabled, based on the highest three bits.
• As display: The binary value is displayed by lighting up one LED in a bar or ring.
• As display multiplexing support: Eight 7-segment displays are controlled with just 10 lines from the microcontroller.

About this Specific Design

Because only one output is high at a time, you can create a design like this by creating a set of boolean expressions:

# inputs
a0 = ...
a1 = ...
a2 = ...

# outputs
x0 = ...
x1 = ...
x2 = ...
...
x7 = ...

If you look at the table of input values, you see a simple pattern:

abc  ab bc ca
000  00 00 00  ab = 0, bc = 0
100  10 00 01  bc = 0, a = 1
010  01 10 00  ca = 0, b = 1
110  11 10 01  ab = 1, c = 0
001  00 01 10  ab = 0, c = 1
101  10 01 11  ca = 1, b = 0
011  01 11 10  bc = 1, a = 0
111  11 11 11  ab = 1, bc = 1

So, you have this ab, bc and ca combinations you want to check for 11 or 00. This is easily done with a NOR for 00 and AND for the 11 state.

# inputs
a0 = ...
a1 = ...
a2 = ...

a01on = a0 and a1
a12on = a1 and a2
a02on = a0 and a2

a01off = not (a0 or a1)
a12off = not (a1 or a2)
a02off = not (a0 or a2)

# outputs
x0 = a01off and a12off
x1 = a12off and a0
x2 = a02off and a1
x3 = a01on and not a2
x4 = a01off and a2
x5 = a02on and not a1
x6 = a12on and not a0
x7 = a01on and a12on

Now, if you compare this to the logic circuit, you can see why all the AND gates line up with the outputs, and why there are three additional AND and NOR gates directly at the inputs.

• Rearrange the logic circuit: Use only three NOT and eight 3-input AND gates.
• Reverse it: Create the logic circuit to encode a binary number from eight inputs.