Chapter10-ALU.ppt
- 1. 1 Chapter 10-Arithmetic-logic units • An arithmetic-logic unit, or ALU, performs many different arithmetic and logic operations. The ALU is the “heart” of a processor—you could say that everything else in the CPU is there to support the ALU. • Here’s the plan: – We’ll show an arithmetic unit first, by building off ideas from the adder-subtractor circuit. – Then we’ll talk about logic operations a bit, and build a logic unit. – Finally, we put these pieces together using multiplexers. • We use some examples from the textbook, but things are re-labeled and treated a little differently.
- 2. Henry Hexmoor 2 The four-bit adder • The basic four-bit adder always computes S = A + B + CI. • But by changing what goes into the adder inputs A, B and CI, we can change the adder output S. • This is also what we did to build the combined adder-subtractor circuit.
- 3. Henry Hexmoor 3 It’s the adder-subtractor again! • Here the signal Sub and some XOR gates alter the adder inputs. – When Sub = 0, the adder inputs A, B, CI are Y, X, 0, so the adder produces G = X + Y + 0, or just X + Y. – When Sub = 1, the adder inputs are Y’, X and 1, so the adder output is G = X + Y’ + 1, or the two’s complement operation X - Y.
- 4. Henry Hexmoor 4 The multi-talented adder • So we have one adder performing two separate functions. • “Sub” acts like a function select input which determines whether the circuit performs addition or subtraction. • Circuit-wise, all “Sub” does is modify the adder’s inputs A and CI.
- 5. Henry Hexmoor 5 Modifying the adder inputs • By following the same approach, we can use an adder to compute other functions as well. • We just have to figure out which functions we want, and then put the right circuitry into the “Input Logic” box .
- 6. Henry Hexmoor 6 Some more possible functions • We already saw how to set adder inputs A, B and CI to compute either X + Y or X - Y. • How can we produce the increment function G = X + 1? • How about decrement: G = X - 1? • How about transfer: G = X? (This can be useful.) This is almost the same as the increment function! One way: Set A = 0000, B = X, and CI = 1 A = 1111 (-1), B = X, CI = 0 A = 0000, B = X, CI = 0
- 7. Henry Hexmoor 7 The role of CI • The transfer and increment operations have the same A and B inputs, and differ only in the CI input. • In general we can get additional functions (not all of them useful) by using both CI = 0 and CI = 1. • Another example: – Two’s-complement subtraction is obtained by setting A = Y’, B = X, and CI = 1, so G = X + Y’ + 1. – If we keep A = Y’ and B = X, but set CI to 0, we get G = X + Y’. This turns out to be a ones’ complement subtraction operation.
- 8. Henry Hexmoor 8 Table of arithmetic functions • Here are some of the different possible arithmetic operations. • We’ll need some way to specify which function we’re interested in, so we’ve randomly assigned a selection code to each operation. S2 S1 S0 Arithmetic operation 0 0 0 X (transfer) 0 0 1 X + 1 (increment) 0 1 0 X + Y (add) 0 1 1 X + Y + 1 1 0 0 X + Y’ (1C subtraction) 1 0 1 X + Y’ + 1 (2C subtraction) 1 1 0 X – 1 (decrement) 1 1 1 X (transfer)
- 9. Henry Hexmoor 9 Mapping the table to an adder • This second table shows what the adder’s inputs should be for each of our eight desired arithmetic operations. – Adder input CI is always the same as selection code bit S0. – B is always set to X. – A depends only on S2 and S1. • These equations depend on both the desired operations and the assignment of selection codes. Selection code Desired arithmetic operation Required adder inputs S2 S1 S0 G (A + B + CI) A B CI 0 0 0 X (transfer) 0000 X 0 0 0 1 X + 1 (increment) 0000 X 1 0 1 0 X + Y (add) Y X 0 0 1 1 X + Y + 1 Y X 1 1 0 0 X + Y’ (1C subtraction) Y’ X 0 1 0 1 X + Y’ + 1 (2C subtraction) Y’ X 1 1 1 0 X – 1 (decrement) 1111 X 0 1 1 1 X (transfer) 1111 X 1
- 10. Henry Hexmoor 10 Building the input logic • All we need to do is compute the adder input A, given the arithmetic unit input Y and the function select code S (actually just S2 and S1). • Here is an abbreviated truth table: • We want to pick one of these four possible values for A, depending on S2 and S1. S2 S1 A 0 0 0000 0 1 Y 1 0 Y’ 1 1 1111
- 11. Henry Hexmoor 11 Primitive gate-based input logic • We could build this circuit using primitive gates. • If we want to use K-maps for simplification, then we should first expand out the abbreviated truth table. – The Y that appears in the output column (A) is actually an input. – We make that explicit in the table on the right. • Remember A and Y are each 4 bits long! S2 S1 A 0 0 0000 0 1 Y 1 0 Y’ 1 1 1111 S2 S1 Yi Ai 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1
- 12. Henry Hexmoor 12 Primitive gate implementation • From the truth table, we can find an MSP: • Again, we have to repeat this once for each bit Y3-Y0, connecting to the adder inputs A3-A0. • This completes our arithmetic unit. S1 0 0 1 0 S2 1 0 1 1 Yi Ai = S2Yi’ + S1Yi
- 13. Henry Hexmoor 13 Bitwise 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 X and Y, apply the operation on each pair of bits Xi and Yi: • We’ve already seen this informally in two’s-complement arithmetic, when we talked about “complementing” all the bits in a number. 1 0 1 1 AND 1 1 1 0 1 0 1 0 1 0 1 1 OR 1 1 1 0 1 1 1 1 1 0 1 1 XOR 1 1 1 0 0 1 0 1
- 14. Henry Hexmoor 14 • Languages like C, C++ and Java provide bitwise logical operations: & (AND) | (OR) ^ (XOR) ~ (NOT) • These operations treat each integer as a bunch of individual bits: 13 & 25 = 9 because 01101 & 11001 = 01001 • They are not the same as the operators &&, || and !, which treat each integer as a single logical value (0 is false, everything else is true): 13 && 25 = 1 because true && true = true • Bitwise operators are often used in programs to set a bunch of Boolean options, or flags, with one argument. • Easy to represent sets of fixed universe size with bits: – 1: is member, 0 not a member. Unions: OR, Intersections: AND Bitwise operations in programming
- 15. Henry Hexmoor 15 • 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 192.168.10.43 with a “subnet mask” to find the “network address,” or which network the machine is connected to. 192.168. 10. 43 = 11000000.10101000.00001010.00101011 & 255.255.255.224 = 11111111.11111111.11111111.11100000 192.168. 10. 32 = 11000000.10101000.00001010.00100000 • You can use bitwise-OR to generate a “broadcast address,” for sending data to all machines on the local network. 192.168. 10. 43 = 11000000.10101000.00001010.00101011 | 0. 0. 0. 31 = 00000000.00000000.00000000.00011111 192.168. 10. 63 = 11000000.10101000.00001010.00111111 Bitwise operations in networking
- 16. Henry Hexmoor 16 Defining a logic unit • A logic unit supports different logical functions on two multi-bit inputs X and Y, producing an output G. • This abbreviated table shows four possible functions and assigns a selection code S to each. • We’ll just use multiplexers and some primitive gates to implement this. • Again, we need one multiplexer for each bit of X and Y. S1 S0 Output 0 0 Gi = XiYi 0 1 Gi = Xi + Yi 1 0 Gi = Xi Yi 1 1 Gi = Xi’
- 17. Henry Hexmoor 17 Our simple logic unit • Inputs: – X (4 bits) – Y (4 bits) – S (2 bits) • Outputs: – G (4 bits)
- 18. Henry Hexmoor 18 Combining the arithmetic and logic units • Now we have two pieces of the puzzle: – An arithmetic unit that can compute eight functions on 4-bit inputs. – A logic unit that can perform four functions on 4-bit inputs. • We can combine these together into a single circuit, an arithmetic-logic unit (ALU).
- 19. Henry Hexmoor 19 Our ALU function table S3 S2 S1 S0 Operation 0 0 0 0 G = X 0 0 0 1 G = X + 1 0 0 1 0 G = X + Y 0 0 1 1 G = X + Y + 1 0 1 0 0 G = X + Y’ 0 1 0 1 G = X + Y’ + 1 0 1 1 0 G = X – 1 0 1 1 1 G = X 1 x 0 0 G = X and Y 1 x 0 1 G = X or Y 1 x 1 0 G = X Y 1 x 1 1 G = X’ • This table shows a sample function table for an ALU. • All of the arithmetic operations have S3=0, and all of the logical operations have S3=1. • These are the same functions we saw when we built our arithmetic and logic units a few minutes ago. • Since our ALU only has 4 logical operations, we don’t need S2. The operation done by the logic unit depends only on S1 and S0.
- 20. Henry Hexmoor 20 4 4 4 4 4 A complete ALU circuit G is the final ALU output. • When S3 = 0, the final output comes from the arithmetic unit. • When S3 = 1, the output comes from the logic unit. Cout should be ignored when logic operations are performed (when S3=1). The arithmetic and logic units share the select inputs S1 and S0, but only the arithmetic unit uses S2. The / and 4 on a line indicate that it’s actually four lines.
- 21. Henry Hexmoor 21 Comments on the multiplexer • Both the arithmetic unit and the logic unit are “active” and produce outputs. – The mux determines whether the final result comes from the arithmetic or logic unit. – The output of the other one is effectively ignored. • Our hardware scheme may seem like wasted effort, but it’s not really. – “Deactivating” one or the other wouldn’t save that much time. – We have to build hardware for both units anyway, so we might as well run them together. • This is a very common use of multiplexers in logic design.
- 22. Henry Hexmoor 22 The completed ALU 4 4 4 4 • This ALU is a good example of hierarchical design. – With the 12 inputs, the truth table would have had 212 = 4096 lines. That’s an awful lot of paper. – Instead, we were able to use components that we’ve seen before to construct the entire circuit from a couple of easy-to-understand components. • As always, we encapsulate the complete circuit in a “black box” so we can reuse it in fancier circuits.
- 23. Henry Hexmoor 23 ALU summary • We looked at: – Building adders hierarchically, starting with one-bit full adders. – Representations of negative numbers to simplify subtraction. – Using adders to implement a variety of arithmetic functions. – Logic functions applied to multi-bit quantities. – Combining all of these operations into one unit, the ALU. • Where are we now? – We started at the very bottom, with primitive gates, and now we can understand a small but critical part of a CPU. – This all built upon our knowledge of Boolean algebra, Karnaugh maps, multiplexers, circuit analysis and design, and data representations.