# Building a 2's complementer (need help visualising how to make a state diagram and table)

Just to be fair, this is a previous exam question and was an assignment question. I've already handed in the assignment and now I'm trying to see why I got it wrong -- to study for my final.

I fully understand how to find the 2's complement of a binary number, however, implementing a circuit to find the complement is driving me crazy. I'm given a shift register (that stores my binary number) and a flip flop (using D for simplicity). The process I follow for building sequential circuits is this:

• I look at the question and derive a state diagram.
• From the state diagram, I draw a state table and find the input of my flip flops
• I use maps and stuff to finally build the circuit.

I'm trying to visualize how to actually draw the state diagram for this complementer. I know it depends on the previous value and stuff, but if someone could take the time to explain in plain english how he/she would go about building the state diagram, I would be vey grateful.

• you might want to tag this something like 'sequential logic' – JustJeff Apr 13 '11 at 1:24
• not enough rep to make new tags :( – n0pe Apr 13 '11 at 1:33

That link you gave says it all:

Hint: - 2’s complement of a number can be obtained by keeping the least significant bits as such until the first 1, and then complementing all bits

"example 001010 → 110110" that is to say

   001010 -> flip bits -> 110101 -> add 1 -> 110110
msb     lsb             msb    lsb         msb    lsb

msb    lsb
input   001010
output  110110
<--- time


So there are two states - "I haven't seen a one yet" and "I have seen a one previously". Given my state and my current input, my output and next state are fully determined.

If my current state is "I haven't seen a one yet" and my input is a 0, my output is a 0 and my next state is (still) "I haven't seen a one yet"

If my current state is "I haven't seen a one yet" and my input is a 1, my output is a 1 and my next state is (changes to) "I have seen a one previously"

If my current state is "I have seen a one previously" and my input is a 0, my output is a 1 and my next state is (still) "I have seen a one previously"

If my current state is "I have seen a one previously" and my input is a 1, my output is a 0 and my next state is (still) "I have seen a one previously"

Write that out as a truth table encoding "I haven't seen a one yet" as 0 and "I have seen a one previously" as 1, and you should be home free.

To "wire it up" the current data input bit and the current state go into logic that feeds the input to the D-flip flop and separate logic that feeds the input to the shift register, the D-flip-flop holds the "state variable", and both are clocked by the data clock. And to be complete you need some kind of reset logic - left as an exercise to the reader. The "current state" is the output of the D-flip-flop, and next state is the input to it...

       +----------------------------+
|                            |
|  +---------+     +-----+   |
+--+ Logic A +-----+ DFF +---+
Data-+-|--+         |     |     |
| |  +---------+     | CLK |
| |                  +--+--+
| |                     |
| |                     +--------Clock
| |                  +--+--+
| |  +---------+     | CLK |
| +--+ Logic B +-----+     |
+----+         |     | S R |
+---------+     +-----+


You know the 2's complement of each of the binary values 0000, 0001, 0010, 0011, ... 1111, right?

There are at least 2 ways to get the 2's complement of a number in a SIPO shift register:

• direct combinatorial logic. Take the 4 bits of the value, (simultaneously) invert all the bits and (simultaneously) add one. (I think a bunch of half-adder circuits are adequate here -- you can copy those out of your textbook, right?) There is no flip-flop or other internal state with this approach.

• Serial incrementer. Starting with the least-significant bit, shift one bit at a time into a serial incrementer (a single half-adder and a single D flip flop). The serial incrementer generates (one bit at a time), from the current bit from the shift register and a Carry_in bit, the Carry_out bit and the Sum bit. Every clock cycle, the old Carry_out bit is latched into your D flip flop and becomes the new Carry_in bit, and the next bit comes out of the shift register. Then the half-adder combinational logic combines them to produce another Carry_out bit and Sum bit. (The flip-flop also needs some way to reset the carry bit it holds to "1" when starting over with the least-significant bit of a new number).

When adding 2 bits, the Carry_out is 1 only when both bits are 1; otherwise the Carry_out is zero.

• I get your implementation, but my question uses a shift register and a flip flop along with external gates. We aren't allowed to use half-adders. This is basically what I have to come up with k5rec.blogspot.com/2006/12/digital-electronics-serial-twos.html . But I can't figure out HOW to get that state diagram. – n0pe Apr 13 '11 at 2:00
• The serial incrementer method here only handles 1 bit at a time. Therefore uses a 1 bit half adder which is 2 external gates. – pre_randomize Jan 31 '13 at 23:20

state diagram is just the output to next

for a mod 3 count up counter the state diagram would be

000 -> 001 -> 010 -> back to 000

since it counts from 0 to 2 and reset on 3

• I understand that. I need to make a diagram for a 2's complementer. – n0pe Apr 13 '11 at 1:20