[HDLBits] Truthtable1

For a boolean function of N inputs, there are 2N possible input combinations. Each row of the truth table lists one input combination, so there are always 2N rows. The output column shows what the output should be for each input value.

Row	Inputs			Outputs
number	x3	x2	x1	f
0	0	0	0	0
1	0	0	1	0
2	0	1	0	1
3	0	1	1	1
4	1	0	0	0
5	1	0	1	1
6	1	1	0	0
7	1	1	1	1

The above truth table is for a three-input, one-output function. It has 8 rows for each of the 8 possible input combinations, and one output column. There are four inputs combinations where the output is 1, and four where the output is 0.

Synthesizing a circuit from a truth table

Suppose we want to build the above circuit, but we're limited to using only the set of standard logic gates. How would you build arbitrary logic functions (expressed as a truth table)?

One simple method to create a circuit that implements the truth table's function is to express the function in sum-of-products form. Sum (meaning OR) of products (meaning AND) means using one N-input AND gate per row of the truth table (to detect when the input matches each row), followed by an OR gate that chooses only those rows that result in a '1' output.

For the above example, the output is '1' if the input matches row 2 or row 3 or row 5 or row 7 (This is a 4-input OR gate). The input matches row 2 if x3=0 and x2=1 and x1=0 (This is a 3-input AND gate). Thus, this truth table can be implemented in canonical form by using 4 AND gates that are ORed together.

A Bit of Practice

Create a combinational circuit that implements the above truth table.

module top_module( 
    input x3,
    input x2,
    input x1,  // three inputs
    output f   // one output
);
    //f=(!x1x2!x3)+(!)
    assign f=((~x1)&x2&(!x3))|(x1&x2&(!x3))|(x1&(!x2)&x3)|(x1&x2&x3);
endmodule