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

Adapted from Computer Organization and Design, Patterson & Hennessy, UCB

ECE232: Hardware Organization and Design

Lecture 7: Binary Numbers and Adders

ECE232: Adders 2

Computer Organization

• 5 classic components of any computer
• Today we will look at datapaths (adder, multiplier, …)

Processor (CPU) Computer Control Datapath Memory Devices Input Output

ECE232: Adders 3

Unsigned Binary Integers

• Given an n-bit number

1 1 2 n 2 n 1 n 1 n

2 x 2 x 2 x 2 x x     

   



• Range: 0 to +2n – 1
• Example
• 0000 0000 0000 0000 0000 0000 0000 10112

= 0 + … + 1×23 + 0×22 +1×21 +1×20 = 0 + … + 8 + 0 + 2 + 1 = 1110

• Using 32 bits
• 0 to +4,294,967,295

x0 xn-1

ECE232: Adders 4

2’s-Complement Signed Integers

• Given an n-bit number

1 1 2 n 2 n 1 n 1 n

2 x 2 x 2 x 2 x x      

   



• Bit n-1 is sign bit
• 1/0 for negative/non-negative numbers
• Range: –2n – 1 to +2n – 1 – 1
• Example
• 1111 1111 1111 1111 1111 1111 1111 11002

= –1×231 + 1×230 + … + 1×22 +0×21 +0×20 = –2,147,483,648 + 2,147,483,644 = –410

• Using 32 bits
• –2,147,483,648 to +2,147,483,647
• Most-negative:

1000 0000 … 0000

• Most-positive:

0111 1111 … 1111 x0 xn-1

2’s comp. binary decimal

1000

• 8

1001

• 7

1010

• 6

1011

• 5

1100

• 4

1101

• 3

1110

• 2

1111

• 1

0000 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7

ECE232: Adders 5

Signed Negation

• To get –X complement X and add 1
• Complement means 1 → 0,

0 → 1

x 1 x 1 1111...111 x x

2

      

• Example: negate +2
• +2 = 0000 0000 … 00102
• –2 = 1111 1111 … 11012 + 1

= 1111 1111 … 11102

• Subtraction: y – x = y + (x +1)
• Representing a number using more bits
• 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

Sign Extension

ECE232: Adders 6

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 7

• If X and Y have the same sign and result has different sign -
• verflow 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

Overflow in 2’s Comp Add/Subtract (2)

ECE232: Adders 8

Ripple Carry Adder

• Addition
• most frequent operation
• used also for multiplication and division
• fast two-operand adder essential
• Simple parallel adder
• for adding xn-1,xn-2,...,x0 and yn-1,yn-2,…,y0
• using n full adders
• Full adder
• combinational digital circuit with input bits xi,yi and

incoming carry bit ci, producing output sum bit si and

• utgoing carry bit ci+1
• incoming carry for next FA with input bits xi+1,yi+1
• si = xi  yi  ci
• ci+1 = xi  yi + ci  (xi + yi)

ECE232: Adders 9

Full-Adder (FA)

• Examine the Full Adder table

0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1

x y Cin Cout S

Cout = x • y + Cin • (x + y) S = x’y’c + x’yc’ + xy’c’ + xyc = x  y  c In general, for bit i: ci+1 = xi yi + ci (xi+yi) where ci+1 = Cout, ci= Cin

Half adder has 2 inputs. In principle HA is same as FA, with Cin set to 0.

x y

Cin Cout

Sum



ECE232: Adders 10

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 11

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 12

Subtraction using Ripple Carry Adder

• Suppose you are performing X-Y operation
• Complement Y bits
• Force C0 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 13

Carry Select Adder

• Principle: speculative

n-bit adder n-bit adder Carry propagate delay CP(2n) = 2*CP(n) n-bit adder n-bit adder n-bit adder 1 Cout CP(2n) = CP(n) + CP(mux)

Carry-select adder

Compute both, select one

MUX

ECE232: Adders 14

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