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The ALU

The ALU. ALU includes combinational logic. Combinational logic  a change in inputs directly causes a change in output, after a characteristic delay. Different from sequential logic which only changes on the clock .

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The ALU

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  1. The ALU • ALU includes combinational logic. • Combinational logic  a change in inputs directly causes a change in output, after a characteristic delay. • Different from sequential logic which only changes on the clock. • Two major components of combinational logic are – multiplexors & decoders. • 2-input multiplexor (or selector) is implemented with gates below a b a b c c s s gate implementation symbol

  2. c Multiplexors (MUXes) s • Multiplexors can have any number of inputs (in theory) • Multiplexors can apply to buses  multiplied for • many lines. • Example: 1 x 2 multiplexor on 32 bits bus. 0 1 2 3 4 5 6 7 a31 b31 M c31 3 X 8 multiplexor a30 b30 s2 c30 M s1 s0 . . . . . . 32 a b 32 c a0 b0 M c0 32 symbol s

  3. 0 1 2 3 4 5 6 7 0 1 2 DECODER Decoders Inputs Outputs I2 I1 I0 O7 O6 O5 O4 O3 O2 O1 O0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 3 X 8 Decoder • Each combination of the inputs enables exactly one output.

  4. The ALU • The ALU provides the basic logical and arithmetic functions: AND, OR plus addition. • Subtraction in 2's complement  invert +1. • Shift, multiplication and division are usually outside the basic ALU. Logical operations a b 0 1 result MUX select (AND or OR) 1 bit logical unit for AND/OR operations

  5. outputs inputs abCinsumCout 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 Cin a b Cout 1 bit FULL adder (3,2) a b Carry out Carry in Cout Cin sum sum = (a  b  Cin) + (a  b  Cin) + (a  b  Cin) + (a b  Cin) = a + b + Cin Cout =(b  Cin) + (a  Cin) + (a  b) = ((a + b)  Cin) + (a  b) Adder hardware for Cout in 2 layers

  6. a b sum Cout Cin sum a b Cout Full Adder from Half Adders Half adder Full adder from 2 half adders + or gate

  7. 1 Bit Simple ALU 1 bit simple ALU for logical / arithmetic operations Cin select 2 a b 0 1 2 result + Cout

  8. 1 Bit Enhanced ALU Enhanced for subtraction invert Cin select 2 a b 0 1 2 result + 0 1 1's complement Cout 2's complement: use Cin = 1 subtraction: a + b + 1 = a + (b + 1) = a + (-b) = a - b

  9. Cin operation = invert + select 3 Ripple Carry Type Adder Cin ALU0 Cout a0 b0 32 bit ADDER with ripple carry: result 0 • To produce a 32 bit result, we connect 32 single bit units together. • This type of ALU adder is called a ripple adder • Carry bits are generated in sequence. • Bit 31 result and Cout is not correct until it receives Cin from previous unit, which is not correct until it receives Cin from its previous unit, etc. • Total operation time is proportional to word size (here 32). Cin ALU1 Cout a1 b1 result 1 Cin ALU2 Cout a2 b2 result 2 . . . . . . Cin ALU31 Cout a31 b31 result 31 Cout

  10. Carry Lookahead • Ripple arithmetic operations are too slow for high performance • We can calculate all carries in 2-level logic, avoiding the ripple. We know that any logical function can be represented in canonical form (sum of products) but it requires more gates  too expensive. • Carry bit i has two possibilities: either a b and no carry in, or a + b and carry in. But carry in itself is the same combination of the bits previous to those that created it. Hence two level logic has 2n terms for n bits. • Practical adders use carry lookahead. • factors out two basic functions which give us the carries • Generate - does bit i create a carry by itself? • Propagate - does bit i send a carry ahead to the next position? gi = (ai  bi) pi = (ai + bi) ci+1 = gi + pi ci

  11. Illustration of Carry Lookahead for 4 Bit Adder cin = c0 g3 p3 g1 p1 g0 p0 g2 p2 c1 = g0 + (p0c0) c2 = g1 + (p1g0) + (p1 p0c0) c3 = g2 + (p2g1) + (p2p1g0) + (p2p1p0c0) c4 = g3 + (p3g2) + (p3p2g1) + (p3p2p1g0) + (p3p2p1 p0c0) Carry Lookahead Unit a3 b3 a1 b1 a0 b0 a2 b2 cin = c0 ALU3 ALU2 ALU1 ALU0 p0 g0 p3 g3 p2 g2 p1 g1 Cout result 0 result 3 result 2 result 1

  12. Carry Lookahead Concept Generalized to Higher Levels Pi = pi0  pi1 pi2 pi3 Ci+1 = Gi + Pi Ci Gi = gi3 + gi2 pi3 + gi1  pi2 pi3 + gi0 pi1 pi2 pi3 Cin = C0 G3 P3 G2 P2 G1 P1 G0 P0 Carry Lookahead on 4-bit units (provides logarithmic complexity) Cin = C0 a12..15 b12..15 a8..11 b8..11 a4..7 b4..7 a0..3 b0..3 4 4 4 4 4 4 4 4 Cout 4-bit Adder 3 12 - 15 4-bit Adder 2 8 - 11 4-bit Adder 1 4 - 7 4-bit Adder 0 0 - 3 p3x g3x p2x g2x p1x g1x p00,..., p03 g00,..., g03 4 4 4 4 4 4 4 4 result 0..3 result 12..15 result 8..11 result 4..7

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