# Arithmetic-logic units

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1 Arithmetic-logic units An arithmetic-logic unit or ALU performs many different arithmetic and logical operations. The ALU is at the heart of a processor you could say that everything else in the computer is there to support the ALU. Today we ll see how you can build your own ALU. First we explain how several basic operations can be implemented using just an unsigned adder. Then we ll talk about logical operations and how to build a corresponding circuit. Finally, we ll put these two pieces together. We show the same examples as the book (pp ), but things are re-labelled to be clearer in LogicWorks. Some inputs (CI and S) are also treated differently. July 2, Howard Huang

2 It s the adder-subtractor again! An arithmetic unit is a circuit that supports several different arithmetic operations. We ve already seen a simple arithmetic unit that supports two functions addition and subtraction. That circuit has two four-bit data inputs and Y, and a function selection input Sub. The four-bit output G is either + Y or Y, depending on the value of the Sub input. Sub G + Y Y July 2, 23 Arithmetic-logic units 2

3 Hierarchical circuit design This circuit is based on a four-bit unsigned adder, which always outputs the sum of its inputs, S = A + B + CI. To perform addition or subtraction, all we did was vary adder inputs A, B and CI according to the arithmetic unit inputs, Y and Sub. The output G of the arithmetic unit comes right out of the adder. Sub Adder inputs A B CI Adder output S Y + Y Y Y July 2, 23 Arithmetic-logic units 3

4 Some more possible functions We can follow the same approach to implement other functions with our four-bit unsigned adder as well! S = (transfer) S = + (increment) S = (decrement) A B CI S (transfer) + (increment) (decrement) July 2, 23 Arithmetic-logic units 4

5 The role of CI Notice that for the transfer and increment operations, the adder has the same A and B inputs, and only the CI input differs. In general we can always create additional functions by setting CI = instead of CI =, and vice versa. Another example involves subtraction. We already know that two s complement subtraction is performed by setting A = Y, B = and CI =, so the adder outputs + Y +. If we keep A = Y and B =, but set CI to instead, we get the output + Y. This turns out to be a ones complement subtraction. July 2, 23 Arithmetic-logic units 5

6 Table of arithmetic functions Here are the different arithmetic operations we ve seen so far, and how they can be implemented by an unsigned four-bit adder. There are actually just four basic operations, but by setting CI to both and we come up with eight operations. Notice that the transfer function can be implemented in two ways, and appears twice in this table. Arithmetic operation A + B + CI + + Y + Y + + Y + Y + (transfer) (increment) (add) (C subtraction) (2C subtraction) (decrement) (transfer) Required adder inputs A B CI Y Y Y Y July 2, 23 Arithmetic-logic units 6

7 Selection codes We can make a circuit that supports all eight of these functions, using just a single unsigned adder. First, we must assign a three-bit selection code S for each operation, so we can specify which function should be computed. Selection code S2 S S Arithmetic operation A + B + CI + + Y + Y + + Y + Y + (transfer) (increment) (add) (C subtraction) (2C subtraction) (decrement) (transfer) Required adder inputs A B CI Y Y Y Y July 2, 23 Arithmetic-logic units 7

8 Generating adder inputs The circuit for our arithmetic unit just has to generate the correct inputs to the adder (A, B and CI), based on the values of, Y and S. Adder input CI should always be the same as selection code bit S. The adder s input B is always equal to. Adder input A depends only on S2, S and Y. Selection code S2 S S Arithmetic operation A + B + CI + + Y + Y + + Y + Y + (transfer) (increment) (add) (C subtraction) (2C subtraction) (decrement) (transfer) Required adder inputs A B CI Y Y Y Y July 2, 23 Arithmetic-logic units 8

9 Here is a diagram of our arithmetic unit so far. We ve already set the adder inputs B to and CI to S, as explained on the previous page. All that s left is to generate adder input A from the arithmetic unit input Y and the function selection code bits S2 and S. From the table on the last page, we can see that adder input A should be set to, Y, Y or, depending on S2 and S. Building the input logic Y Y July 2, 23 Arithmetic-logic units 9 S2 S A

10 Primitive gate-based input logic We ll build this circuit using primitive gates. If we want to use Karnaugh maps for simplification, then we should first expand the abbreviated truth table, since the Y in the output column is actually an input. Remember that A and Y are each four bits long! We are really describing four functions, A3, A2, A and A, but they are each generated from Y3, Y2, Y and Y in the same way. S2 S Y i A i S2 S A Y Y July 2, 23 Arithmetic-logic units

11 We can find a minimal sum of products from the truth table. Primitive gate implementation S S2 Y i A i = S2 Y i + S Y i Again, we have to repeat this once for each adder input bit A3-A. You can see this repetition in the circuit diagram here. July 2, 23 Arithmetic-logic units

12 Our complete arithmetic unit S2 S S G + + Y + Y + + Y + Y + July 2, 23 Arithmetic-logic units 2

13 Bitwise logical operations Most computers also support logical operations like AND, OR and NOT, but extended to multi-bit words instead of just single bits. To apply a logical operation to two words and Y, apply the operation on each pair of bits i and Y i. AND OR OR We ve already seen this informally in two s complement arithmetic, when we talked about complementing all the bits in a number. July 2, 23 Arithmetic-logic units 3

14 Bitwise operations in programming Languages like C, C++ and Java provide bitwise logical operations. & (AND) (OR) ^ (OR) ~ (NOT) These operations treat each integer as a bunch of individual bits. 3 & 25 = 9 (because & = ) They are not the same as the C operators &&, and!, which treat each integer as a single logical value ( is false, everything else is true). 3 && 25 = (because true && true = true) Bitwise operators are often used in programs to set a bunch of Boolean options, or flags, with one argument. For instance, to initialize a doublebuffered, RGB-mode window with a depth buffer using OpenGL: glutinitdisplaymode(glut_double GLUT_RGBA GLUT_DEPTH); July 2, 23 Arithmetic-logic units 4

15 Bitwise operations in networking IP addresses are actually 32-bit binary numbers, and bitwise operations can be used to find network information. For example, you can bitwise-and an address like with a subnet mask to find the network address, or which network the host is connected to =... & = =... You can use bitwise-or to generate a broadcast address for sending data to all machines on a local network = = =... July 2, 23 Arithmetic-logic units 5

16 Defining a logic unit A logic unit implements different logical functions on two multi-bit inputs and Y, producing an output G. We ll design a simple four-bit logic unit that supports four operations. S S G Y + Y Y Again, we have to assign a selection code S for each possible function. July 2, 23 Arithmetic-logic units 6

17 Implementing bitwise operations Bitwise operations are applied to pairs of corresponding bits from inputs and Y, so we can generate each bit of the output G in the same way. S S G i i Y i i + Y i i Y i i We can implement each output bit G i as shown on the right, using a multiplexer to select the desired primitive operation. The complete logic unit diagram is given on the next page. July 2, 23 Arithmetic-logic units 7

18 Our complete logic unit S S G Y + Y Y July 2, 23 Arithmetic-logic units 8

19 Combining the arithmetic and logic units We can combine our arithmetic and logic units into a single arithmetic-logic unit, or ALU. The ALU will accept two data inputs and Y, and a selection code S. It outputs a result G, as well as a carry out CO (only useful for the arithmetic operations). Since there are twelve total arithmetic and logical functions to choose from, we need a four-bit selection input. We ve added selection bit S3, which chooses between arithmetic (S3=) and logical (S3=) operations. S3 S2 x x x x S S G + + Y + Y + + Y + Y + and Y or Y xor Y July 2, 23 Arithmetic-logic units 9

20 A complete ALU circuit CO should be ignored when logic operations are performed (S3=). G is the ALU output. If S3=, the output comes from the arithmetic unit. If S3=, the output comes from the logic unit. July 2, 23 Arithmetic-logic units 2

21 Comments on the multiplexer Both arithmetic and logic units are active and produce outputs. The multiplexer determines whether the final result G comes from the arithmetic or logic unit. The output of the other one is ignored. In programming, you d use an if-then statement to select one operation or the other. This is useful since programs execute serially, and we want to avoid unnecessary work. Our hardware scheme may seem like wasted effort, but it s not really. Disabling one unit or the other wouldn t save that much time. We have to build the hardware for both units anyway. You can think of this as a form of parallel processing, where several operations are done together. This is a very common use of multiplexers in logic design. July 2, 23 Arithmetic-logic units 2

22 An external view of the ALU S3 S2 S S G + + Y + Y + + Y + Y + x x x x and Y or Y xor Y July 2, 23 Arithmetic-logic units 22

23 Summary In the last few lectures we looked at various arithmetic issues. You can build adders hierarchically, starting with half adders. A good representation of negative numbers simplifies subtraction. Unsigned adders can implement many other arithmetic functions. Logical operations can be applied to multi-bit values. Where are we now? We started at the very bottom with primitive gates, but now we can understand ALUs, a critical part of any processor. This all built upon our knowledge of Boolean algebra, Karnaugh maps, multiplexers, circuit analysis and design, and data representations. July 2, 23 Arithmetic-logic units 23

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