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Computer Architectures - Digital Circuits - Binary arithmetic Circuits

However, there is a problem. It turns out that the normal way of creating such circuits would often use up way too many gates. We must search for different ways.

Binary integer addition

For binary integer addition, we can sacrifice our requirement on circuit depth that we previously had, in order to use fewer gates. The resulting circuit is of a type that we call iterative combinatorial circuit, in that it contains several copies of a simple element. For binary addition, that simple element is called a full adder.

A full adder is a combinatorial circuit (or actually two combinatorial circuits) of three inputs and two outputs. Its function is to add two binary digits plus a carry from the previous position, and give a two-bit result, the normal output and the carry to the next position. Here is the truth table for a full adder:

 

   x y c-in | c-out s
   ------------------
   0 0  0   |    0  0
   0 0  1   |    0  1
   0 1  0   |    0  1
   0 1  1   |    1  0
   1 0  0   |    0  1
   1 0  1   |    1  0
   1 1  0   |    1  0
   1 1  1   |    1  1

Here, we have used variable names x and y for the inputs, c-in for the carry-in, s for the sum output and c-out for the carry-out.

A full adder can be trivially built using our ordinary design methods for combinatorial circuits. Here is the resulting circuit diagram:

The next step is to combine a series of such full adders into a circuit that can add (say) two 8-bit positive numbers. We do this by connecting the carry-out from one full adder to the carry-in of the full adder immediately to its left. The rightmost full adder takes a 0 on its carry-in.

Here, we have used subscript i for the i-th binary position.

As you can see, the depth of this circuit is no longer two, but considerably larger. In fact, the output and carry from position 7 is determined in part by the inputs of position 0. The signal must traverse all the full adders, with a corresponding delay as a result.

There are intermediate solutions between the two extreme ones we have seen so far (i.e. a combinatorial circuit for the entire (say) 32-bit adder, and an iterative combinatorial circuit whose elements are one-bit adders built as ordinary combinatorial circuits). We can for instance build an 8-bit adder as an ordinary two-level combinatorial circuit and build a 32-bit adder from four such 8-bit adders. An 8-bit adder can trivially be build from 65536 (216) and-gates, and a giant 65536-input or-gate.

Another intermediate solution consists of building so-called carry-accelerator circuits. [To be completed...]

Binary subtraction

 

Binary multiplication and division

Binary multiplication is even harder than binary addition. There is no good iterative combinatorial circuit available, so we have to use even heavier artillery. The solution is going to be to use a sequential circuit that computes one addition for every clock pulse. We will discuss this more in a later section since it needs mechanisms we have not discussed yet.

 


 

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