Arithmetic operations in binary

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Arithmetic operations in binary

• Addition / subtraction
• 01011
• 111
• ?
• Method exatly the same as decimal

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Arithmetic operations in binary

• Addition
• X xn xi x0
• Y yn yi y0
• __________________
• di
• di (xi yi ) mod r carry-in

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Addition

• For ( i 0n ) do
• di (xi yi carry-in) mod r
• carry-out (xi yi carry-in) div r
• End for

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Arithmetic operations in binary

• Subtraction
• Can you write an equivalent method (algorithm)
  for subtraction?
• Algorithm a systematic sequence of
  steps/operations that describes how to solve a
  problem

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Arithmetic operations in binary

• Subtraction
• X xn xi x0
• ? Y yn yi y0
• __________________________________________________
  _
• di
• For ( i 0n ) do
• di ( xi ? yi ? bi-1 ) mod r
• borrow-out ( xi ? yi ? bi-1 ) div r
• End for

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Subtraction
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Arithmetic operations in binary

• Multiplication
• For ( i 0n ) do
• di ( xi ? yi ci-1 ) mod r
• ci ( xi ? yi ci-1 ) div r
• End for
• Is this correct?

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Negative numbers (4 traditions)
Signed magnitude Radix complement Diminished
radix complement Excess-b (biased)
e.g. n 4, r 10 7216 --gt 9999 - 7216 1
2784 (10s complement) n 4, r 2 0101 --gt
1111 - 0101 1 1011 (2s complement)
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diminished radix complement is
e.g. n 4, r 10 7216 --gt 9999 - 7216 2783
(9s complement) n 4, r 2 0101 --gt 1111 -
0101 1010 (1s complement)
Note operation is reversible (ignoring
overflow) i.e. complement of complement returns
original pattern (for both radix and diminished
radix forms) for zero and its complement 10s
complement 0000 --gt 9999 1 10000 --gt
0000 9s complement 0000 --gt 9999 --gt 0000
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Arithmetic with complements n 4 r 2
Excess-8

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Signed number representations

• N rn ( rn N ) N
• ( N N ) mod rn 0
• Example (n 4)
• 5 ? 0101
• -5 ? 1011
• 10000 ? 24
• N N mod 24 0
• N 5 r 10 N 32546
• N 105 32546 (67454)10

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Twos complement arithmetic

• ( an-1, an-2, a0 ) in 2-s complement is
• Example (n 4)
• 0101 ? ?0?23 1?22 0?2 1?1 4 1 5
• 1011 ? ?1?23 0?22 1?2 1?1 ?8 2 1
  ?5
• Addition/subtraction (n 5)
• 10 ? 01010
• 3 ? 00011
• 01101 ? 13

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Twos complement arithmetic

• Addition/subtraction (n 5)
• 10 ? 01010
• 7 ? 00111
• Overflow 10001 ? ?15
• 15 ? 01010
• ?13 ? 10011
• Discard 100010 ? 2

When can overflow happen? Only when both operands
have the same sign and the sign bit of the result
is different.
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Cyclic representation (n 4, r 2) avoid
discontinuity between 0111 and 1000
Add x move x positions clockwise Subtract
x move x positions counterclockwise move (16 -
x) positions clockwise (i.e. add radix
complement)
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How to detect discontinuity?
no overflow (in 4-bit register) but, carry into
most significant bit is 1 while carry out is 0
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Same circuitry signed numbers add subtract
(use 2s complement of subtrahend) Intel
architecture OF (overflow flag) detects
out-of-range result
unsigned numbers same protocol but
discontinuity is now between 0000 and
1111 detect with overflow for
addition lack of overflow for
subtraction Intel uses CF (carry flag) to
detect out-of-range result
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Codes 4-bit codes for digits 0
through 9 (BCD 8421 weighted) 2421 and Excess-3
are self-complementing (complement bits to get
9s complement representation) Biquinary has two
ranges 01 and 10 one-bit error produces
invalid codeword
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Number representation inside computer

• Integers represented in 2s complement
• Single precision 32 bits
• Largest positive integer is 231-1
• 2,147,483,647
• Smallest negative integer is -231
• 2,147,483,648

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Number representation inside computer

• Floating point
• Scientific notation
• 0.0043271 0.43271?10-2
• normalized number
• The digit after the decimal point is ? 0
• Normalized notation maximizes the use of
  significant digits.

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Floating point numbers

• Floating point
• N (-1)S ? m ? rE
• S 0 ? positive
• S 1 ? negative
• m ? normalized fraction for radix r 2
• As MSB digit is always 1, no need to explicitly
  store it
• Called hidden bit ? gives one extra bit of
  precision

S E m
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Floating point formats

• IEEE format (-1)S?(1.m)?2E-127
• DEC format (-1)S?(0.m)?2E-128

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Floating point formats

• Different manufacturers used different
  representations

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Mechanical encoder with sequential sector
numbering At boundaries can get a code
different than either adjacent sector
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Gray code
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Gray code algorithm input (binary)
output (Gray code)
e.g. n 3 6 -gt 110 -gt 101 n 4 10 -gt 1010
-gt 1111
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Representing non-numeric data

• Code systematic and preferably standardized way
  of using a set of symbols for representing
  information
• Example BCD (binary coded decimal) for decimal
  s
• It is a weighted code ? the value of a symbol
  is a weighted sum
• Extending number of symbols to include alphabetic
  characters
• EBCDIC (by IBM) Extended BCD Interchange Code
• ASCII American Standard Code for Information
  Interchange

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Codes

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Cyclic codes

• A circular shift of any code word produces a
  valid code word (maybe the same)
• Gray code example of a cyclic code
• Code words for consecutive numbers differ by one
  bit only

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Ascii, ebcdic codes State codes, e.g.
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Consequences of binary versus one-hot coding
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n-cubes of codewords
Hamming distance between x and y is count of
positions where x-bit differs from y-bit Also
equals link count in shortest path from x to y
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Gray code is path that visits each vertex
exactly once
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Error-detecting codes
concept choose code words such
that corruption generates a non-code word to
detect single-bit error, code words must
occupy non-adjacent cube vertices
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Error-correcting codes minimum distance between
code words gt 1
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Write minimum distance as 2c d 1 bits
gt corrects c-bit errors and detects (c
d)-bit errors Example min distance 4 2(1)
1 1
But also, 4 2(0) 3 1
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Why? suppose minimum distance is h
c
c
h - 2c
?
d h - 2c -1
pair of closest nodes maximally distant from
left with entering correction zone 2c d 1
2c (h - 2c - 1) 1 h
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Hamming codes
number bit positions 1, 2, 3, ... n from right to
left bit position is a power of 2 gt check bit
else gt information bit e.g. (n
7) check bits in positions 1, 2, 4 information
bits in positions 3, 5, 6, 7
Create group for each check bit Express check
bit position in binary Group all information
bits whose binary position has a one in same
place e.g. (n 7) check information 1 (001) 3
(011), 5 (101), 7 (111) 2 (010) 3 (011), 6
(110), 7 (111) 4 (100) 5 (101), 6 (110), 7 (111)
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Code information packets to maintain even parity
in groups e.g. (n 7) packet is 1011 gt
positions 7, 6, 5, 3 7 6 5 4 3 2 1 1 0 1 x 1 x x
Consult group memberships to compute check
bits check information 1 3, 5, 7 gt bit 1 is 1
2 3, 6, 7 gt bit 2 is 0 4 5, 6, 7 gt bit 4
is 0 Code word is 1010101
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Note Single bit error corrupts one or more
parity groups gt minimum distance gt 1 Two-bit
error in locations x, y corrupts at least one
parity group gt minimum distance gt 2 Three-bit
error (i.e. 1, 4, 5) goes undetected gt minimum
distance 3 3 2(1) 0 1 2(0) 2 1 gt
can correct 1-bit errors or detect errors of
size 1 or 2.
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Group 8 Group 4 Group 2 Group 1
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

• Pattern generalizes to longer bit strings
• Single bit error corrupts one or more parity
  groups gt minimum distance gt 1

• Consider two-bit error in positions x and y. To
  avoid corrupting all groups -- avoid group 1
  bit 1 (lsb) same for both x and y -- avoid group
  2 bit 2 same for both -- avoid group 4 bit 3
  same for both, etc. -- forces x y
• So true two-bit error corrupts at least one
  parity group gt min distance gt 2
• Three-bit error (two msb and lsb) goes
  undetected gt minimum distance 3
• Conclude process always produces a Hamming code
  with min distance 3

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Traditional to permute check bits to far
right Used in memory protection schemes
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Add traditional parity bit (for entire word) to
obtain code with minimum distance 4
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For minimum distance 4
Number of check bits grows slowly with
information bits
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Two-dimensional codes (product codes)
min distance is product of row and column
distances for simple parity on each (below) min
distance is 4
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Scheme for RAID storage CRC is extension of
Hamming codes each disk is separate row column
is disk block (say 512 characters) rows have CRC
on row disk columns have even parity
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Checksum codes mod-256 sum of info bytes becomes
checksum byte mod-255 sum used in IP
packets m-hot codes (m out of n codes) each
valid codes has m ones in a frame of n bits min
distance is 2 detect all unidirectional
errors bit flips are all 0 to 1 or 1 to 0
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Serial data transmission (simple) transmit clock
and sync with data (3 lines) (complex) recover
clock and/or sync from data line
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Serial line codes
NRZ clock recovery except during long sequences
of zeros or ones NRZ1 nonreturn to zero, invert
on one zero gt no change, one gt change
polarity RZ clock recovery except during long
sequences of zero DC balance Bipolar return to
zero (aka alternate mark inversion send one as
1 or -1) Manchester zero gt 0 to 1 transition,
one gt 1 to 0 transition at interval center