Question

How would you build a complete 32-bit adder. Explain and show how your 32 bit
adder is designed and verify it's operation using the simulator Logisim.

Answer 1

Start by constructing a structural element i.e., the truth table for a 1-bit full adder:

Outputs Inputs Comment Row out 0 | 0+0+0=00, 0+0+1=012 0+1+0=01, 0+1+1=10, 1+0+0=01, 1 1+0+1=10, 1+1+0=102

Now, look at all cases (rows) for which the sum output (S) is 1. Do you notice anything special? All rows for which the sum (S) is 1 have an odd number of enabled inputs. This means only one or all of the inputs x,y or Cin can be 1 for S to be 1. This is exactly the XOR function.

Step 0

Make a truth table with input columns x, y and Cin. In one column, give the result of x XOR y. In another column, give the result for (x XOR y) XOR Cin.
You should now see that: S = x XOR y XOR Cin

Step 1 The sum-of-products equation for the carry output (Cout) is: Cout = x'·y·Cin + x·y'·Cin + x·y·Cin + x·y·Cin' , is not a minimal expression. Show step by step how you can reduce the expression for Cout to end up with: Cout = Cin·(x XOR y) + x·y

Step 2 It's now time to implement your 1-bit full adder in Logisim. z Start Logisim. On the department Unix System, type logisim in a shell and press enter. If you work on a laptop or form home, downlad and install Logisim

Open up add.circ in Logisim. Start by double-click on add1 to select the add1 circuit.

Use this circuit to construct your 1-bit full adder. You are free to move the input- and output pins around, but, don't change their orientations or relative positions.
Sum: z Start by adding XOR-gates to produce the sum S from the inputs A, B and Cin. z Use the text tool to add a comment to the circuit giving the equation for the S output.
Cout: z Add the necessary XOR, AND-gates and OR-gates to produce Cout. z Use the text tool to add a comment to the circuit giving the simplified equation for the Co output.

Step 3
Complete the add8 circuit by combining eight 1-bit adders.
Add three splitters to the circuit. Each splitter should have an input bit width of 8 and a fan out of 8. Attach two east-facing splitters to the 8-bit inputs A and B. Attach a west-facing splitter to the 8-bit output S.
Create eight instances of the add1 circuit.
Connect the S outputs of the eight add1 instances to the splitter for the 8-bit S output.
Connect the carry inputs and outputs of the eight add1 instances so that carries will propagate appropriately from the Ci input, through the 1-bit adders, to the Co output.
Connect the A inputs of the eight add1 instances to the splitter for the A input. z Connect the B inputs of the eight add1 instances to the splitter for the B input.
Change the values of the Ci , A, and B inputs and observe the Co and S outputs to verify the correct operation of the circuit

Step 4

Complete the add32 circuit by combining four 8-bit adders.
You will find three splitters in the circuit. Each splitter has an input bit width of 32 and a fan out of 4. Thus, each connection to a splitter represents 8 bits. z Create four instances of the add8 circuit.
Connect the 8-bit S outputs of the four add8 instances to the splitter for the 32-bit S output.
Connect the carry inputs and outputs of the four add8 instances so that carries will propagate appropriately from the Ci input, through the 8-bit adders, to the Co output.
Connect the 8-bit A inputs of the four add8 instances to the splitter for the A input.

Step 5

Within the main circuit, you will find a 32-bit adder connected side-by-side with your add32 circuit. Change the values of the Ci , A, and B inputs and observe the Co and S outputs to verify the correct operation of your add32 circuit.