ECE232: Hardware Organization and Design Lecture 7: Binary Numbers - PowerPoint PPT Presentation

ECE232: Hardware Organization and Design Lecture 7: Binary Numbers and Adders Adapted from Computer Organization and Design , Patterson & Hennessy, UCB

Computer Organization 5 classic components of any computer  Computer Processor Memory Devices (CPU) Input Control Datapath Output Today we will look at datapaths ( adder, multiplier, … )  ECE232: Adders 2

Unsigned Binary Integers Given an n-bit number  x n-1 x 0        n 1 n 2 1 0  x x 2 x 2 x 2 x 2   n 1 n 2 1 0 Range: 0 to +2 n – 1  Example  • 0000 0000 0000 0000 0000 0000 0000 1011 2 = 0 + … + 1 ×2 3 + 0×2 2 +1×2 1 +1×2 0 = 0 + … + 8 + 0 + 2 + 1 = 11 10 Using 32 bits  • 0 to +4,294,967,295 ECE232: Adders 3

2 ’s -Complement Signed Integers Given an n-bit number  2’s comp. decimal binary x n-1 x 0 1000 -8         n 1 n 2 1 0 1001 -7  x x 2 x 2 x 2 x 2   n 1 n 2 1 0 1010 -6 1011 -5 Bit n-1 is sign bit  1100 -4 1/0 for negative/non-negative numbers • 1101 -3 Range: – 2 n – 1 to +2 n – 1 – 1 1110 -2  1111 -1 Example  0000 0 1111 1111 1111 1111 1111 1111 1111 1100 2  0001 1 = – 1×2 31 + 1×2 30 + … + 1 ×2 2 +0×2 1 +0×2 0 0010 2 = – 2,147,483,648 + 2,147,483,644 = – 4 10 0011 3 Using 32 bits  0100 4 – 2,147,483,648 to +2,147,483,647 • 0101 5 Most-negative: 1000 0000 … 0000 0110 6 • 0111 7 Most-positive: 0111 1111 … 1111 • ECE232: Adders 4

Signed Negation To get – X complement X and add 1  • Complement means 1 → 0,     x x 1111...111 1 0 → 1 2    Example: negate +2 x 1 x  • +2 = 0000 0000 … 0010 2 • –2 = 1111 1111 … 1101 2 + 1 = 1111 1111 … 1110 2 Subtraction: y – x = y + (  x +1)  Representing a number using more bits  Sign Extension • Preserve the numeric value Replicate the sign bit to the left  Examples: 8-bit to 16-bit  • +5: 0000 0101 => 0000 0000 0000 0101 • – 5: 1111 1011 => 1111 1111 1111 1011 ECE232: Adders 5

Overflow in 2’s Comp Add/Subtract (1) Example -  01001 9 11001 -7 1 00010 2 Carry-out discarded - does not indicate overflow In general, if X and Y have opposite signs - no overflow  can occur regardless of whether there is a carry-out or not Examples -  ECE232: Adders 6

Overflow in 2’s Comp Add/Subtract (2) If X and Y have the same sign and result has different sign -  overflow occurs Examples -  10111 -9 10111 -9 1 01110 14 = -18 mod 32 • Carry-out and overflow 01001 9 00111 7 0 10000 -16 = 16 mod 32 • No carry-out but overflow ECE232: Adders 7

Ripple Carry Adder  Addition • most frequent operation • used also for multiplication and division • fast two-operand adder essential  Simple parallel adder • for adding x n -1 , x n -2 ,..., x 0 and y n -1 , y n -2 ,…, y 0 • using n full adders  Full adder • combinational digital circuit with input bits x i , y i and incoming carry bit c i , producing output sum bit s i and outgoing carry bit c i +1 • incoming carry for next FA with input bits x i +1 , y i +1 • s i = x i  y i  c i • c i +1 = x i  y i + c i  ( x i + y i ) ECE232: Adders 8

Full-Adder (FA) Examine the Full Adder table  Cin x y Cin Cout S x Cout 0 0 0 0 0 0 0 1 0 1  0 1 0 0 1 y 0 1 1 1 0 Sum 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 In general, for bit i : c i+1 = x i y i + c i (x i +y i ) Cout = x • y + Cin • (x + y) where c i+1 = Cout, c i = Cin S = x’y’c + x’yc’ + xy’c’ + xyc = x  y  c Half adder has 2 inputs. In principle HA is same as FA, with C in set to 0. ECE232: Adders 9

Parallel Adder: Ripple Carry In a parallel arithmetic unit  All 2n input bits available at the same time • Carry propagates from the FA to the right to FA to the left • Carries ripple through all n FAs before we can claim that the • sum outputs are correct and may be used in further calculations Each FA has a finite delay  ECE232: Adders 10

Example x3,x2,x1,x0=1111 • y3,y2,y1,y0=0001 •  FA - operation time - delay • Assuming equal delays for sum • and carry-out Longest carry propagation • chain when adding two 4-bit numbers In synchronous arithmetic units -  time allowed for adder's operation is worst-case delay - n  FA ECE232: Adders 11

Subtraction using Ripple Carry Adder Suppose you are performing X-Y operation  Complement Y bits • Force C 0 to 1 • add • Example: X = 0101, Y = 0010; Compute X – Y  First step: Complement Y  1101 • Second step: add 0101 + 1101 + 1 = 0011  ECE232: Adders 12

Carry Select Adder Principle: speculative  Carry propagate delay n-bit adder n-bit adder CP(2n) = 2*CP(n) CP(2n) = CP(n) + CP(mux) Compute both, select one n-bit adder n-bit adder n-bit adder 0 1 MUX Carry-select adder Cout ECE232: Adders 13

Summary Implement hardware for performance  Ripple Carry: least hardware, slowest  Carry Select: even faster, even more hardware  Other techniques available, e.g., Carry skip adder  Combination of these techniques – hybrid adders  ECE232: Adders 14